MECHANICS OF THE 21ST CENTURY
Mechanics of the 21st Century Proceedings of the 21st International Congress of Theoret...

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MECHANICS OF THE 21ST CENTURY

Mechanics of the 21st Century Proceedings of the 21st International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 15--21 August 2004

Edited by

WITOLD GUTKOWSKI Polish Academy of Sciences, Warsaw, Poland and

TOMASZ A. KOWALEWSKI Polish Academy of Sciences, Warsaw, Poland

This eBook does not include ancillary media that was packaged with the printed version of the book. A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

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Contents

Committees

viii

Preface

ix

Acknowledgments

xi

Exhibitors

xii

Congress Statistics

xiii

Opening Ceremony

xv

Closing Ceremony

xxiii

Scientiﬁc Program

xxx

Interplay between Air and Water Leen van Wijngaarden

1

Stochastic Dynamics of Engineering Systems Kazimierz Sobczyk

19

Multibody Dynamics: Bridging for Multidisciplinary Applications Jorge A.C. Ambr´sio ´

61

Rapid p Formation of Strongg Gradients and Diﬀusion in the Transport of Scalar and Vector Fields Konrad Bajer

89

Wave-Vortex Interactions in the Atmosphere, and Climate Prediction Onno Bokhove

103

Near-critical Point Hydrodynamics and Microgravity Daniel A. Beysens

117

Flaw Tolerant Nanostructures of Biological Materials Huajian Gao, Baohua Ji, Markus J. Buehler, and Haimin Yao

131

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ICTAM04

Transport and Mixing in the Atmosphere Peter H. Haynes

139

Variational and Multiscale Methods in Turbulence Thomas J. R. Hughes, Victor M. Calo, and Guglielmo Scovazzi

153

Mechanics of Thin Film Structures Henrik Myhre Jensen

165

Nonlinear Dynamics in Ocean Engineering Edwin J. Kreuzer, Wolfgang M. Sichermann

173

A Bridge g between the Micro- and Mesomechanics of Laminates: Fantasy or Reality? Pierre Ladev`eze Turbulence and Large-Eddy Simulations Marcel R. Lesieur Nano Mechanical Analysis of IFM Force Proﬁles on Self-Assembled Monolayers Mingji Wang, Kenneth M. Liechti, Vibha Srinivasan, John M. White, Peter J. Rossky Collisional Granular Flows with and without Gas Interactions in Microgravity Michel Y. Louge and Haitao Xu

187 203

217

229

Probability Phenomena in Perturbed Dynamical Systems Anatoly Neishtadt

241

Mechanics of Rubberlike Solids Ray W. Ogden

263

Elastic Wave Propagation p g Development for Structural Health Monitoring Wieslaw Ostachowicz

275

On the damping of a piezoelectric truss Andre Preumont

287

Strength of Nanostructures Rodney S. Ruoﬀ, ﬀ Nicola M. Pugno

303

Micromechanics of Cells Erich Sackmann, Andreas Reuther, and Doris Heinrich

313

Elastic Interactions of Biological Cells Samuel A. Safran, A. Nicolas, U. S. Schwarz

329

Contents

vii

Electrokinetic Flow Instabilities in Microﬂuidic Systems Hao Lin, Michael H. Oddy and Juan G. Santiago

343

Molecular Mechanics of Cytoskeletal Components M. Atakhorrami, K.M. Addas, M. Buchanan, G.H. Koenderink, F.C. MacKintosh, J.X. Tang, Christoph F. Schmidt

355

Topics in Astrophysical Fluid Dynamics Edward A. Spiegel

365

Miniaturization of Explosive Technology and Microdetonics D. Scott Stewart

379

Foams in Microgravity Denis Weaire and Simon Cox

387

Author Index

395

21st International Congress of Theoretical and Applied Mechanics President and Chairman of Local Organizing Committee

Witold Gutkowski

Co-Chairmen

Michal Kleiber Wlodzimierz Kurnik

Secretary-General

Tomasz A. Kowalewski

Members of Local Organizing Committee Konrad Bajer (Warszawa) Romuald B¸edzinski ´ (Wroclaw) Tadeusz Burczynski ´ (Gliwice) Tadeusz Chmielniak (Gliwice) Krzysztof Dolinski ´ (Warszawa) Stanislaw Drobniak (Cz¸¸estochowa) Jozef ´ Giergiel (Krak´ ´ ow) Stanisaw Kocanda ´ (Warszawa) Jozef ´ Kubik (Bydgoszcz) Zenon Mroz ´ (Warszawa) J´ ´ ozef Niziol (Krak´ ´ ow) Wojciech Nowacki (Warszawa) Janusz Orkisz (Krak´ ´ ow) Andrzej Palczewski(Warszawa)

Wojciech Pietraszkiewicz (Gda´ n ´ sk) Stanislaw Radkowski (Warszawa) Czeslaw Rymarz (Warszawa) Kazimierz Sobczyk (Warszawa) Jaroslaw Stefaniak (Pozna´ n) ´ Jacek Stupnicki (Warszawa) Andrzej Styczek (Warszawa) Gwidon Szefer (Krak´ ´ ow) ´ Eugeniusz Switonski ´ (Gliwice) Andrzej Tylikowski (Warszawa) Zenon Waszczyszyn (Krak´ ´ ow) Zbigniew Wesolowski (Kielce) Wadyslaw Wlosinski ´ (Warszawa) Czesaw Wo´zniak (Cz¸¸estochowa)

Piotr Perzyna (Warszawa)

Henryk Zorski (Warszawa) ˙ Micha Zyczkowski (Krak´ ´ ow)

Henryk Petryk (Warszawa)

Members of the IUTAM Congress Committee Hassan Aref (USA) member of XCCC Ted Belytschko (USA) representative of IACM Martin Bendsoe (Denmark) representative of ISSMO Dimitri Beskos (Greece) David Bogy (USA) Dick van Campen (Netherlands) David Durban (Israel) Fernand Ellyin (Canada) representative of ICM Juri Engelbrecht (Estonia) Norman Fleck (UK) Ben Freund (USA) Graham Gladwell (Canada) Peter Gudmundson (Sweden) Michael Hayes (Ireland) representative of ISIMM Tsutomu Kambe (Japan) Bhushan Karihaloo (UK) representative of ICF Alfred Kluwick (Austria) Valery Kozlov (Russia) Edwin Kreuzer (Germany)

Yu Ku (USA) Gary Leal (USA) representative of ICR Peter Lugner (Austria) representative of IAVSD Fernando Lund (Chile) Keith Moﬀatt (UK) Chairman, member of XCCC Peter Monkewitz (Switzerland) Rene Moreau (France) member of XCCC, representative of HYDROMAG Niels Olhoﬀ (Denmark) member of XCCC Timothy Pedley (UK) Secretary, member of XCCC Bernhard Schreﬂer (Italy) member of XCCC Kazimierz Sobczyk (Poland) Pierre Suquet (France) Ernst Tuck (Australia) Manuel Velarde (Italy) representative of CISM Eiichi Watanabe (Japan) Feng-Gan Zhuang (China)

Preface

The 21st International Congress of Theoretical and Applied Mechanics (ICTAM04) took place August 15 – 21, 2004, in Warsaw, Poland. It was organized by Polish National Committee of IUTAM, Institute of Fundamental Technological Research of the Polish Academy of Sciences (IPPT PAN) and Warsaw University of Technology. The Congress venue was the main building of Warsaw University of Technology. The idea of congresses devoted to mechanics, can be traced back to a conference on problems of ﬂuid mechanics in Innsbruck, 1922. It was organized by four individuals, whose names, are and will, remain very well known to next generations of scientists, C. W. Oseen, T. Levi-Civite, T. von K´ a´rm´an, and L. Prandtl. This conference was so fruitful, that the organizers decided to arrange similar meetings in the future, every four years, and to extend the scope of the future meetings to include solid mechanics. Hence a series of meetings started eighty years ago with the 1st ICTAM held in Delft, Netherlands. From the meetings of the Congress Committee sprang the idea of a more permanent organization to look out for the world interests in the mechanical sciences. Thus, IUTAM, the International Union of Theoretical and Applied Mechanics, was formed on September 26, 1946. In 1947 IUTAM became a member of ICSU, the International Council of Scientiﬁc Unions, itself founded in 1931. The highest authority of IUTAM is the General Assembly, with delegates from the Adhering Organizations, each of which is aﬃliated with a national learned society in a given country.

Scientiﬁc Program Contemporary mechanics poses both, the fundamental problems from the area of pure science, and its strong links with modern technology. It spreads over such areas of our knowledge as oceanography, physical chemistry, biology, medicine, geophysics and astrophysics. Hence, any conclusions deduced in the framework of mechanics, are likely to have

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a value for other ﬁelds. We may easily ﬁnd prove of it in the scientiﬁc program of the ICTAM04. It consists of plenary opening and closing lectures, sectional lectures, mini-symposia, and contributed papers presented in lecture and seminar presentation sessions. These were intended to cover all aspects of mechanics. All contributed papers were peer reviewed. Recommendations had been received from Pre-selection Committees of the National Committees of the nine countries: Canada, France, Germany, Japan, PR China, Poland, Russia, UK and USA. Moreover, recommendations had been received from the Chairs of the Mini-Symposia and of the Pre-nominated Sessions with classiﬁcations of the papers submitted within the topics of their Symposia/Sessions. Finally, the International Papers Committee paid careful attention to the above recommendations. Accordingly, of the 2086 eligible submissions 1574 contributions were invited by the IPC for their presentation. The total number of submitted and accepted papers represents a quite substantial enhancement relative to the previous congresses, providing evidence of vitality of the contemporary mechanics. This volume of Proceedings consists of a book with full texts of invited talks and attached CD-ROM with Extended Summaries of 1225 papers presented during the Congress by authors1 . We have tried to assemble the paper volume with an extensive index of names and papers collected on the CD-ROM. We hope that this volume – pre-ordered at the Congress in record numbers – will be found useful not only as a document of the event but to assess achievements and new paths of research in mechanics of 21st Century. W. GUTKOWSKI T.A. KOWALEWSKI

1 From 1273 papers presented during the Congress we exclude multiple presentations and those given by proxy.

xi

Acknowledgments We would like to express our thanks to our colleagues from the IUTAM Congress Committee, and from the International Papers Committee. Their advice and eﬀorts have helped us to overcome some problems, connected, as usual, with the organization of world-wide meetings. Special gratitude goes to the Chairs responsible for Mini-Symposiums and Pre-nominated Sessions. Their work in organizing sessions, encouraging papers submission in the area of interest, and later reviewing all papers cannot be overestimated. Smilingly, we are deeply indebted to members of the nine National Committees for their valuable contribution to the reviewing procedure. The organization and execution of the Congress was performed by the Institute of Fundamental Technological Research of Polish Academy of Sciences and Warsaw University of Technology. The detailed work of organization was due to many persons from both institutions, personnel of PCO Mazurkas Travel as well as numerous external co-workers. Without their extremely valuable help organization of such meeting would be impossible. Thank you all so much! W. GUTKOWSKI T.A. KOWALEWSKI

Exhibitors Cambridge University Press Dantec Dynamics EDEN Elsevier Era Business Intelligent Laser Applications International Publishing Service Institute of Fundamental Technological Research of Polish Academy of Sciences Kluwer Polish Tourist Organisation Springer Strategic Test Scandinavia Testlab

Congress Statistics A detailed list of ICTAM04 participants is enclosed on the CD-ROM. Here, for brevity, we give the basic statistics of the Congress. The total number of pre-registered to ICTAM04 participants was 2928. The Congress organizers received 2245 abstracts of contributed papers before the ﬁnal deadline of January 16, 2004. Not all abstracts were followed by an extended summary. Hence, the International Papers Committee invited 1574 contributions reviewing 2186 submitted extended summaries. Finally 1235 contributed papers and 38 invited talks were given in Warsaw. Fluid Mechanics was a subject of 550 presentations, Solid Mechanics was selected as a subject of 611 presentations, and 112 contributions aimed to cover problems involving both areas of mechanics and education in mechanics. The total number of participants, accepted papers and presentations represent a quite substantial enhancement relative to the previous congresses (Table 1), providing evidence of vitality of the contemporary mechanics. Table 2 displays brieﬂy country statistics of the participants and presentations. Table 1.

ICTAM04 compared with four previous congresses

ICTAM

Submitted Papers

Grenoble 1988 Haifa 1992 Kyoto 1996 Chicago 2000 Warsaw 2004

1262 1183 1642 1953 2245 (2186)

Presented papers/ presented from organizers country 573 n. a. 420 n. a. 703 192 1126 445 1273 144

Participants/ participants from organizers country 951 340 525 85 936 332 1430 587 1515 194

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15 3 5 3 2 1 3 1 1 2

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

9 8 8 8 7 7 7 7 5 5 4 4 4 3 3 3 2 2 2 2 1 1 1 1 1 1 1 1515

8 8 8 6 5 4 5 4 5 5 4 3 2 3 2 3 1 2 1 1 1 1 1 1 1 1 1273

Invited lectures

Country Portugal Estonia Iran Norway Bulgaria Ireland Latvia South Africa Romania Mexico Belarus Singapore Turkey Lithuania Serbia Slowenia Chile Kazakhstan Libya Saudi Arabia Algeria Armenia Croatia New Zealand Nigeria Quatar Slovakia Total:

Presentations

185 144 153 100 98 72 75 41 38 36 27 21 19 15 13 17 15 11 15 14 14 11 7 11 9 7 10 9

Participants

204 194 170 118 107 89 79 54 43 38 37 30 25 19 19 19 18 18 18 17 14 13 12 12 12 10 10 9

Invited lectures

Country USA Poland France Germany UK Japan Russia China Netherlands Ukraine Israel Denmark Italy Belgium Canada S. Korea Austria Sweden Taiwan Czech Republic Spain Brazil Finland India Switzerland Australia Hungary Greece

Presentations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

ICTAM04 country statistics

Participants

Table 2.

1

38

Opening Ceremony2 Professor W. Gutkowski Distinguished Guests, Esteemed Colleagues, Ladies and Gentlemen, It is my great honor and pleasure to welcome you to the 21st International Congress of Theoretical and Applied Mechanics. I welcome you on behalf of the Local Organizing Committee in Poland, here in Warsaw in this historical and beautiful ediﬁce of the Warsaw University of Technology. The ediﬁce, a witness of many scientiﬁc, cultural and political events. Just sixty years ago, in August heavy ﬁghting was going on here during the Warsaw Uprising. Today you are in the country of growing economy, and for several months a member of European Community. Mister President, Distinguished Guests and Colleagues, Ladies and Gentlemen! It is my great honor and privilege to announce the opening of the 21st International Congress of Theoretical and Applied Mechanics! We are here, scientists from 55 countries, joined by a common passion, a passion for Mechanics. Ladies and Gentlemen, let me now introduce our distinguished guests and organizers: Professor Keith Moﬀatt, President of the International Union of Theoretical and Applied Mechanics and President of the Congress Committee of IUTAM, Professor Werner Schiehlen, Vice-President of IUTAM, Professor Ben Freund, Treasurer of IUTAM, Professor Dick van Campen, Secretary General of IUTAM, Professor Tim Pedley, Secretary General of IUTAM, Professor Micha Kleiber, Minister of Science and Information Technology of Polish Government, Chairman of the State Committee for Scientiﬁc Research, Co-chairmen of the Local Organizing Committee, Professor Janusz Lipkowski, Vice-President of Polish Academy of Sciences, Professor Stanisaw Mankowski, ´ President of the Warsaw University of Technology, Professor Wodzimierz Kurnik, Vice-President of the Warsaw University of Technology, Co-Chairman of the Local Organizing Committee, and Professor Tomasz Kowalewski, Secretary-General of the Congress. Ladies and Gentlemen, to start the Congress I will kindly request Professor Keith Moﬀatt of the Cambridge University, President of the International Union of Theoretical and Applied Mechanics and President of the IUTAM Congress Committee, to address the Congress.

2 The unabridged version of the account of the Opening Ceremony can be found on the accompanying CD-ROM.

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Professor K. Moﬀatt Distinguished Guests, Dear Colleagues, Ladies and Gentlemen, I am honoured to stand before you in this magniﬁcent setting of the Warsaw University of Technology, and to say some words of welcome on behalf of IUTAM, the International Union of Theoretical and Applied Mechanics. This is the 21st Congress in our history, so here we celebrate our ‘coming-of-age’. We also celebrate the 80th anniversary of the ﬁrst Congress of Applied Mechanics (as it was then called), which was held in Delft in 1924. We should pay tribute to the great scientists who had the vision to initiate this series of quadrennial Congresses, particularly J.M. Burgers, Theodore von Karman, Ludwig Prandtl and G.I. Taylor. I note that this year is the centenary of Prandtl’s seminal paper on boundary-layer theory. An IUTAM Symposium on “One Hundred Years of Boundary Layer Research” has been held just last week in G¨ o¨ttingen, in recognition of the crucial role that this branch of mechanics has played in Aerodynamics and many other ﬁelds of application. IUTAM itself grew from the early Congresses, and was formally established in 1948; Poland was one of the earliest members and has been an Adhering Organisation of IUTAM since 1952. The Congress Committee is appointed by the General Assembly of IUTAM, but otherwise retains the autonomy that it enjoyed from the outset. This means that when things go wrong, IUTAM can blame the Congress Committee; of course, when things go right, IUTAM shares in the credit! There were 207 participants at that ﬁrst Congress in Delft; these early pioneers would be gratiﬁed to know that there are over 1500 participants at this 21st Congress, as there were also at the 20th Congress in Chicago four years ago. It is a measure of the continuing vigour of our subject that the Congress attracts such strong and widespread participation. Of course we must be careful not to equate quantity with quality; we do however strive to maintain high quality in the papers selected for presentation whether as lectures or as seminar/poster presentations. Here I pay tribute to the hard work of our International Papers Committee, which, advised by National Committees and by Convenors of Mini-Symposia and Pre-nominated Sessions, had to select the papers from more than 2000 that were submitted; their task was extremely demanding, but the quality and range of the programme before us for this Congress will I hope convince you that they have done an excellent job, in spite of the great diﬃculties. It is highly appropriate that we hold this Congress in Warsaw, a city of grace and beauty, which has emerged from the dreadful traumas of the last century to face the new century with greatly renewed vigour and

Opening Ceremony

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optimism. One cannot visit this land without being deeply conscious of the long years of oppression suﬀered by the Polish people. It is hard now for us to comprehend the tragedy of the Warsaw uprising, which erupted exactly sixty years ago, and which has been commemorated here this month. The long bleak years of the Cold War gave little solace to the Polish people. Yet throughout these years, there remained for our community a ﬂicker of light in the gloom: I refer to the biennial meetings in ﬂuid mechanics that were organised in Poland throughout the 60s and 70s and well into the 80s by our dear colleague Wladek Fiszdon, meetings which enabled scientiﬁc contacts between East and West to be established and nurtured across and despite the Iron Curtain, and which helped to keep alive the universal and apolitical spirit of our subject. Wladek Fiszdon became a member of the General Assembly of IUTAM in 1971; he is now a life-member in recognition of his devoted services. His health unfortunately does not allow him to be with us today, but I am sure you would wish me to convey to him the very good wishes of the whole IUTAM community. IUTAM is an organisation founded on principles of friendly collaboration between scientists irrespective of race, creed or gender. It is a privilege to come to its Congresses, and to renew acquaintance with so many old friends and colleagues. It is also a particular privilege for me to welcome to this Congress younger scientists who may be attending an IUTAM Congress for the ﬁrst time. I remember vividly my own such experience: With the encouragement of my late mentor George Batchelor, I attended the 10th Congress in Stresa, Italy, in 1960. It was an exhilarating experience, which opened my eyes to the great scope and challenges of our vast ﬁeld of endeavour. If this is your ﬁrst IUTAM Congress, I hope that you will ﬁnd it a similarly exhilarating experience, and that you will look forward to succeeding Congresses with ever-keener anticipation. G.I. Taylor once told me that he liked to save his best paper in each four-year period for presentation at ICTAM; we should do well to follow his good example! The time for thanks will come at the end of the Congress on Friday, but I’d like now to say at least a preliminary thank you to the local team who have worked so hard on all the preparations: the Congress President Witold Gutkowski, the Secretary-General of the Congress Tomasz Kowalewski, and the whole Local Organising Committee. They have done a superb job, and I sense that this is going to be a correspondingly superb Congress. Thank you, Witold and Tomasz! Now without further ado, I have much pleasure in conﬁrming the Opening of ICTAM04.

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Welcome to Warsaw, and we wish you a fruitful Congress; or, as it is said here: Witamy w Warszawie i z˙ yczymy udanego Kongresu! Professor W. Gutkowski Thank you very much, Keith, for your kind words about Polish contribution to international mechanics. Let me also express our great appreciation for you, IUTAM Bureau and Executive Committee of the Congress Committee for the support and valuable advice during four years of preparation for the Congress. Let me ask now Professor Micha Kleiber, Minister of Science and Information Technology of Polish government, Chairman of the State Committee for Scientiﬁc Research and Co-chairman of the Local Organizing Committee to address the Congress. Professor M. Kleiber Mr. President of IUTAM, Very Distinguished Guests of the Congress, Colleagues and Friends, On behalf of the government of the Republic of Poland I have the honour and great pleasure to welcome you to the 21st International Congress of Theoretical and Applied Mechanics here in Warsaw. I hope you will forgive me if I start on a rather personal note but I want to emphasize how special for me is this opportunity of addressing you, dear colleagues, in my current double capacity as a co-chairman of the Congress and as a representative of the Polish public administration. Frankly, I have to admit that because of my other duties I have not done much in terms of the Congress organization, and it is my colleagues at the Institute of Fundamental Technological Research of Polish Academy of Science and the Warsaw University of Technology who at the end of the week will deserve to be congratulated for their exceptional engagement and, I am sure, organizational perfection. The Congress is something very special in every respect – it attempts to summarize what we have accomplished in the last 4 years, it facilitates everyone’s research planning for the future, it stimulates interactions with other ﬁelds of science and technology, it strengthens our professional and personal links, it gives us a marvellous feeling of being a group of people coming from so many, sometimes very distant, places all over the world, but a group of people who share similar interests, similar vision of rationality in all human endeavors, similar perception of ways to better understand and improve the world around us. I am one of those who consider the ﬁelds of mechanics as a true research success story of last decades – contrary perhaps to some people

Opening Ceremony

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from outside of our profession, I have no doubts that the progress we have achieved together deﬁnitely places mechanics on one of the very top locations in the hierarchy of modern research disciplines – with tremendous impact on both our perception of the physical world and the means to implement new technologies so much improving the quality of our life. We are very glad to host you, distinguished delegates, in Poland, in the city of Warsaw. You will ﬁnd here, I am sure, traditional hospitality and openness of the people, great public interest in the Congress debates and interesting encounters with the country and its capital so much rooted in the complex history of Europe – country which has only recently managed to deﬁnitely overcome its politically so much unfortunate and undeserved past, country which, as a new member of the European Union, looks forward with unmatched optimism and aspirations to contributing to the progress in science and its positive implications for all of us. I wish all the participants vivid, instructive presentations and discussions and, at the same time, enjoyable stay in Warsaw. Professor W. Gutkowski Thank you, Professor Kleiber. I would like to express my gratitude for your initiative to organize the Congress in cooperation with the Institute of Fundamental Technological Research and the Warsaw University of Technology, in this beautiful ediﬁce. This has been a great idea. Special thanks also for strong ﬁnancial support of the Congress by the Ministry of Science and Information Technology. May I ask Professor Janusz Lipkowski, Vice President of the Polish Academy of Sciences, representing the President of the Polish Academy of Sciences, to the podium. Professor J. Lipkowski Mr. President, Excellences, Distinguished Guests and Participants of the Congress, The Polish Academy of Sciences consists of three parts. First, it is a corporation of elected members, the most distinguished scientists of our country who represent the Polish science. The Academy also organizes the scientiﬁc life in the country through its scientiﬁc committees. There are more than 100 committees in all scientiﬁc disciplines and scientists from all institutions in Poland, universities, institutes, societies etc. serve as members of the committees. One of these, Polish National Committee of the International Union of Theoretical and Applied Mechanics, is coorganizing the present Congress. Last but not least, the Academy has scientiﬁc institutes performing high level research studies.

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About 9% of all scientiﬁc ‘population’ in Poland are employed in the institutes while their output measured by the number of scientiﬁc publications amounts to more than one quarter of all scientiﬁc output in the country. We are proud of our institutes; please feel invited to visit any of them during your stay in Poland, you will be more than welcome. I am pleased to mention that one of the institutes, the Institute of Fundamental Technological Research, is also a coorganizer of the Congress. By the way, it is the mother institution of our Minister of Scientiﬁc Research and Information Technology who served as its director before joining the government of Poland. Ladies and Gentelmen, I made this brief introduction to our Academy of Sciences in the belief that you would become interested in developing scientiﬁc cooperation with our Academicians and institutions. We do count on international joint research which is an obvious choice in the XXIst century. I wish you a very fruitful conference, and a pleasant and enjoyable stay in Poland! Professor W. Gutkowski Thank you very much, Professor Lipkowski. The Polish Academy of Sciences gave us very strong support when we were proposing the organization of the Congress here in Warsaw. Almost all members of the Local Organizing Committee are members of the Committee of Mechanics of the Academy. Let me now request Professor Stanisaw Ma´ nkowski, ´ President of the Warsaw University of Technology, to address the Congress. Professor St. Ma´ n ´ kowski Professor, Minister, Ladies and Gentleman, Distinguished Guests, The Warsaw University of Technology, with its tradition of teaching in the ﬁeld of technical and exact sciences reaching the year 1826, the symbol of which is the Main Building, erected over a 100 years ago and where the Large Hall is an expression of beauty, but also of application of the theoretical and applied mechanics, would like to extend a warm welcome to the participants of this, already the 21st , congress. The Warsaw University of Technology which teaches approximately 32 000 students is the largest technical university in this part of Europe. Teaching and research are inseparable. Therefore I believe that the 21st International Congress of Theoretical and Applied Mechanics is organised in the right place. I wish the authors of the presented scientiﬁc contributions many practical applications, accurate theoretical descriptions and many quota-

Opening Ceremony

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tions. I would like to express my gratitude to the International Committee for this decision, and the Polish National Committee, the Institute of the Fundamental Technological Research of the Polish Academy of Sciences and the Team of the Warsaw University of Technology led by Vice-President Prof. Wlodzimierz Kurnik – for the passion and eﬀort put into the organisation of the congress. Professor W. Gutkowski Thank you very much. It is my great pleasure to say that Professor Mankowski, ´ together with Professor Wodzimierz Kurnik, Vice-President of the University, have undertaken an enormous organizing task to host us in this historical building. Ladies and Gentlemen, I believe you will like the city of Warsaw, and other places in Poland. There are so many interesting historical and cultural places to visit. Palaces, beautiful gardens, museums, concert halls and theaters. Unfortunately, within a couple of days you will be able to see just a few of them. So, we are bringing to you here a small fragment of our cultural heritage – the music of Chopin. For a quarter of an hour, Mr. Marcin Rudzinski, ´ a student of the Fryderyk Chopin Academy of Music, will present us some of the nicest pieces of this famous composer. ... Ladies and Gentlemen, Dear Colleagues, Mechanics is ﬂourishing! No doubt! Increasing number of submissions from ICTAM to ICTAM is showing theoretical and applied importance of mechanics and its multidisciplinary relations with other sciences. In Kyoto there were 1642 submitted papers, in Chicago 1953, and in Warsaw 2186! Thank you for coming, and for contributing with your papers to the development of Mechanics in the 21st century. We can say that all submitted papers present the latest, up-to-date results of our research. Really latest! Suﬃce it to say that on January 5, four days before the submission deadline, only 600 papers reached our server. I needn’t say that no smile could be seen on our faces. Then during the last four days, the number of incoming papers started to grow exponentially with time. Finally, at midnight on the 9th of January the number of submitted paper reached 21 hundred! The authors were then really anxious to send their latest results. Let me mention an e-mail message we received in the late afternoon on Friday. The author of the message was kindly asking if the deadline at midnight should be considered at Greenwich time or at local time! The authors constitute the core and the essence of the Congress. However, its realization strongly depends on sponsoring. Let me then ac-

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knowledge numerous supporting institutions and sponsors who helped us to start the organization of ICTAM04. First of all, I would like to express our thanks to the Polish Universities and Institutes supporting the Congress, and contributing, at the same time, to the exhibition “Mechanics in Poland”. ... Speaking about Polish Mechanics, I would like to turn your attention to a booklet by Professor Zbigniew Olesiak, which you will ﬁnd in your congress bag. The book shows the 19th century roots of our mechanics embedded in our diﬃcult history of that time3 . Let me next express gratitude to numerous scientiﬁc journals for publishing our announcements. This certainly increased the number of people interested in the congress. Let me express appreciation for our friends, organizers of the Chicago ICTAM, for their support, very much needed at the beginning of the Congress preparations. It would be impossible to mention all our sponsors by name. To all of them presented on these two banners, I am sending, on behalf of the Local Organizing Committee our deepest thanks. Ladies and Gentlemen. Before we start our work I will ask Professor Tomasz Kowalewski, Secretary-General of the Congress to give you some latest information. Professor T. Kowalewski I am very touched to see here, in Warsaw, so many old friends and all those numerous new ones, with whom I had pleasure to exchange thousands of emails during the last four years. Thank you very much for coming . . . – began his talk Secretary-General of the Congress. After passing on the latest technical information concerning organizational matters, Prof. Kowalewski expressed his gratitude to numerous young volunteers who were of great help in the last stages of the congress preparation. Professor W. Gutkowski Thank you, Tomasz. In a few minutes Professor Leen van Wijngaarden will present an Opening Lecture. Presiding the session are Professor Hassan Aref from the USA and Michal Kleiber from Poland. I cordially wish you successful presentations and discussions of your papers. Enjoy your stay in Warsaw! Thank you for your attention. 3 The

text of the book can be found on the accompanying CD-ROM.

Closing Ceremony1

Closing Address by Professor Keith Moﬀatt, President of IUTAM Dear Colleagues, Ladies and Gentlemen, We have now come to the Closing Ceremony of this Congress. We shall announce the Bureau Prizes in the course of this ceremony. But my ﬁrst duty is to call on Professor Tim Pedley, Secretary of the Congress Committee, to present his report on ICTAM04. Congress Report by Professor Tim Pedley, Secretary General of the Congress Committee Well, Ladies and Gentlemen, another ICTAM has come to an end. It has been a wonderful Congress and the biggest yet — even slightly bigger than the Millennium Congress in Chicago four years ago. There were 1445 active participants, including students, up from 1430, from 57 diﬀerent countries (up from 54). That last number is a welcome increase in view of IUTAM’s aim of spreading interest and understanding in mechanics more and more widely in the world. ... This has been a resoundingly successful Congress, both scientiﬁcally and organisationally. On behalf of the IUTAM Congress Committee, and of yourselves, I would like to express our profound thanks to the President of the Congress, Professor Witold Gutkowski, and to the SecretaryGeneral, Professor Tomasz Kowalewski, for their immense contribution to ensuring that success. Possibly most impressive, on the organisational front, was the provision of fully synchronised computer facilities in all the lecture rooms, with the uploading of all computerised presentations in advance. ... On behalf of the Congress Committee, I would also like to thank the members of the International Papers Committee (IPC) for their dedicated hard work in evaluating and selecting the papers to be presented at the Congress. These were Professors Peter Monkewitz, Howard Stone, Bernhard Schreﬂer, Kazimierz Sobczyk and Viggo Tvergaard. A lot of 1 The unabridged version of the account of the Closing Ceremony can be found on the accompanying CD-ROM.

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reading was required in February and March of this year, and culminated in an intense four day meeting in Warsaw at the end of March. The IPC were guided by the recommendations of the chairs of the mini-symposia and the pre-nominated sessions, and by the national committees of nine major countries: Canada, China, France, Germany, Japan, Poland, Russia, UK and USA. All those involved in the paper assessment process deserve our warm thanks. ... Professor Keith Moﬀatt Now, Ladies and Gentlemen, I have great pleasure in announcing that the Congress Committee agreed yesterday on the location of the next IUTAM Congress, which will be held in August 2008. As you will know, we have now held 21 Congresses, all in the Northern hemisphere. The next Congress will, for the ﬁrst time, be in the Southern hemisphere. The Australasian bid to hold the Congress in Adelaide, South Australia, has been successful. The Congress President will be Professor Ernie Tuck, and I have pleasure in now inviting him to say a few words. Professor Tuck expressed his gratitude for electing Adelaide for the next congress, inviting all participants to Australia in 2008. Acknowledgments by Professor Witold Gutkowski, President of ICTAM04 and Chairman of the Local Organizing Committee Mister President, Ladies and Gentlemen, Dear Colleagues, The organization of an ICTAM consists, among other things, in cooperation of many bodies: Congress Committee, International Papers Committee, Chairs of Sessions, and National Committees of IUTAM. I would like to express our gratitude to all those, for their very eﬀective cooperation, which facilitated our complex work of the Local Organizing Committee. Sincere thanks! As you already know, there are three institutions standing behind the organization of ICTAM, here in Warsaw. The idea of organizing an ICTAM in Poland has its long history. It originated in the Committee of Mechanics of the Polish Academy of Sciences and at the some time the Polish National Committee of IUTAM. Without the intellectual encouragement of the whole Polish mechanics community, the organization of the congress in our country wouldn’t be possible. This is then right time and right place to express my sincere appreciation to all members of the Committee of Mechanics,

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for emboldening us in the organization of ICTAM04. Thanks to you we could disseminate information about the Congress to a very broad international scientiﬁc community. Sincere thanks! The Institute of Fundamental Technological Research, known as IPPT, has also been, for years, strongly supporting the organization of an ICTAM in Poland. When the Congress Committee of IUTAM invited us to organize the congress, the institute became the center of preparatory works. All scientists and administration were highly supportive. IPPT hosted meetings of IUTAM Bureau and XCCC, as well as the International Papers Committee. I wish to express my sincere thanks to the directors of the institute, Professors Kleiber and Nowacki, for their help and for their patience, when watching the disorder inevitable in such situations and paying sharply growing telephone bills. Thank you, Professor Kleiber, thank you, Professor Nowacki, thank you, all friends from IPPT. The idea of organizing the Congress in Warsaw found many supporters at the largest university in Poland, the Warsaw University of Technology. The university has oﬀered its beautiful ediﬁce we are now in as a site of the congress. The organization of ICTAM here was strongly encouraged by Professor Stanisaw Mankowski, ´ President of the university and the host of the Welcome Reception. Thanks to the team led by Professor Wlodzimierz Kurnik, Vice President of the University and Co-chairman of the Local Organizing Committee, together with Professor Stanisaw Radkowski, we had this beautiful building prepared for our gathering. (. . . ) Professors Ma´ n ´kowski, Kurnik and all friends from the university — many thanks! Such a great event as our congress couldn’t be organized without the help of a professional bureau organizing large conferences. We asked Mazurkas Travel to: take care of your money, make hotel reservation, serve lunches and coﬀees, prepare poster stands, transportation, organization of excursions and many other things for good comfort of participants. Let me then express my thanks to Mazurkas Travel and its director Mrs. Barbara Zygmunt. The dissemination of information started with the ﬁrst announcement and then continued with the second. The nice booklet of the second announcement is the work by Professor Krzysztof Doli´ n ´ski, member of the Local Organizing Committee. ... It is impossible, to mention by names all young volunteers, mostly PhD students. It has been a real pleasure to see all those strongly motivated young girls and boys working hard and trying to do their

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best. They were led by two extremely active and eﬃcient members ´ eczkowska and Dr. Kamil of the Congress Bureau — Mrs. Izabella Sl¸ Kulesza. ´ e¸czkowska, Dr. Kulesza and all young Professor K. Doli´ n ´ski, Mrs. Sl¸ volunteers, your contribution to the successful Congress cannot be overestimated. Thank you! Last but not least, I am acknowledging the heart, soul, brain and calculator of the Congress organization — Professor Tomasz Kowalewski, Secretary General of ICTAM 2004! You have been meeting him from the beginning, at the congress page, that is since September 2000. Tomasz has loaded for you all update information. For the ﬁrst time in the history of IUTAM Congresses, participants could see the program and read an abstract and extended summary of an arbitrary paper, still being home, before reaching the congress site. This was Tomasz’s idea and his realization. The perfect preparation of the meeting of the International Paper Committee helped much to take right decisions on accepting papers for presentations. Again, this was Tomasz’s idea and realization with the help of several PhD students. Suﬃce it to say that each of 2000 papers could be found within seconds by title, author’s name or country. And with all that, thousands, thousands of e-mails sent and replied to. It is impossible to enumerate all the works he has done, but I should add at least two others. He kept our ﬁnances under strict control and he infected the minds of all young people who worked with him with his enthusiasm for organization of ICTAM 2004. ... There are not enough words to express my deepest thanks to you and I hope all participants will join me in conveying you our sincere thanks for your great contribution to the success of the Congress. Thank you, Tomasz. In order to commemorate your great contribution to organization of the Congress, let me hand you this 19th century statuette of a Bacchanate, a woman, as we know enjoying good wines. The statuette is inscribed in Polish: To Professor Tomasz Kowalewski for his outstanding contribution to organization of ICTAM04. Organizing Committee. IUTAM Bureau Prizes — Professor Keith Moﬀatt (certiﬁcates), Professor Witold Gutkowski (statuettes) The members of the Bureau have this week been attending many lectures and seminar/poster presentations given by young researchers. The word ’young’ is of course a relative term: I regard everyone younger than myself as young! But to be eligible for a Bureau prize, you must be really

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young, and that means under 35 years of age. The Bureau met today over lunch to agree on the winners, and I have to say ﬁrst that the competition was a stiﬀ one — we may well say of Olympic standard; and three ’gold medallists’ have emerged from this competition. The prizes consist of a certiﬁcate, a cheque for $500, and, as a special gesture from the local organisers, a beautiful owl, symbol of wisdom, cast in bronze. The three prizes are awarded as follows: Ingo Kaiser, of the Institute of Robotics and Mechatronics, Wessling, Germany, for his lecture on “The running behaviour of an elastic wheelset”; Ingo gave a beautiful demonstration of the eﬀect of elastic deformation on the stability wheels running on a track; you should be warned of this if you plan to return home by rail after the Congress! Taisuke Sugii, of the University of Tokyo, Japan, for his seminar/poster presentation entitled “Molecular dynamics study of permeation process of small molecules through a lipid bilayer”; biomechanics is a ﬁeld of tremendous potential and rapidly growing importance; this paper provides an excellent example of the application of mechanical principles at the biomolecular level.

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Pantxica Otheguy, of LadHyX, Ecole Polytechnique, Palaiseau, France, for her lecture on “Instability of corotating vertical vortices in a stratiﬁed ﬂuid: why strongly stratiﬁed turbulence is not similar to 2D turbulence”; Pantxica gave a beautiful interpretation of the ‘zig-zag instability’ to which such vortices are subject, illustrated by a video of an experiment showing the nonlinear outcome of this instability. We congratulate these three young scientists on the excellence of their presentations. Ladies and Gentlemen, This week the General Assembly of IUTAM met twice, and as part of its business elected its new Bureau to serve our Union for the next four years from 1st November 2004. The President will be Professor Ben Freund from Brown University, Providence, Rhode Island, USA, who has served IUTAM so successfully as Treasurer for the past eight years. Professor Dick van Campen will continue as Secretary-General for a further four-year term. Dick has been extremely successful in implementing recommendations of a previous Appraisal Committee, both as regards setting up Working Parties in nine diﬀerent priority areas, and in developing an excellent website for IUTAM, at which information about all our activities can be easily accessed. We are most grateful to him for his willingness to continue to carry this heavy responsibility. Professor Juri Engelbrecht, from Estonia, who has served on the Bureau for eight years, has accepted the important position of Treasurer. I myself continue, according to IUTAM rules, as Vice President for the next four years. Four other members of the Bureau have been elected by the General Assembly. These are: Professors Tsutomu Kambe (Japan), Alfred Kluwick (Austria), Niels Olhoﬀ (Denmark) and Zhemin Zheng (China). I wish to thank the retiring members of the Bureau, Carlo Cercignani, Roddam Narasimha, Jean Salencon ¸ and Werner Schiehlen (retiring VicePresident) who have served with great conscientiousness and wisdom. We have also elected four new members-at-large of the General Assembly, in recognition of their distinction in mechanics and their service to our Union: these are Professors Andy Acrivos (USA), Sol Bodner (Israel), Werner Schiehlen (Germany) and Franz Ziegler (Austria). So you see that we have done our best to ensure that IUTAM will continue to be well governed! Now I just have to add some words of thanks myself: ﬁrst to my old friend and colleague, Tim Pedley, who will continue as Secretary of the Congress Committee for a further four years. This is a task of great

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responsibility, which Tim has carried out with ﬂair and eﬃciency. I ask you to join me in thanking him. Second, I wish to thank Witold Gutkowski, who has been such a gracious host to us here in Warsaw. The smooth running of the Congress owes much to his eﬀorts over the last four years. Finally, I wish to echo Professor Gutkowski’s remarks concerning Tomasz Kowalewski, Secretary-General of the Congress. Tomasz has been indefatigable in his control of every detail of this immense organisation, in which he has shown amazing skill, patience, good humour and eﬃciency, and we all owe both Tomasz and Witold a resounding vote of thanks. And even more ﬁnally, I wish to thank all of you, Invited Sectional Lecturers, Convenors of Mini-Symposia, Chairmen of Pre-nominated Sessions, contributors of lectures and seminar/poster presentations, and all of you who by your presence have made this Congress such a brilliant success. I wish you all a safe journey home, and will look forward to seeing you again down under in Adelaide in August 2008, if not before!

Keith Moﬀatt

Scientiﬁc Program On the following pages the contents of the scientiﬁc program of the Congress are listed. The program consists of plenary opening and closing lectures, eighteen sectional lectures, six minisymposia, and sixty prenominated sessions devoted to all aspects of mechanics. OL CL SL

Opening Lecture Closing Lecture Sectional Lectures

Mini Symposia MS1 MS2 MS3 MS4 MS5 MS6

Smart materials and structures Tissue, cellular and molecular biomechanics Mechanics of thin ﬁlms and nanostructures Microﬂuids Microgravity ﬂow phenomena Atmosphere and ocean dynamics

Pre-Nominated Sessions on Fluid Mechanics FM1 FM2 FM3 FM4 FM5 FM6 FM7 FM8 FM9 FM10 FM11 FM12 FM13 FM14 FM15 FM16 FM17 FM18 FM19 FM20 FM21 FM22 FM23 FM24 FM25 FM26

Biological ﬂuid dynamics Boundary layers Combustion and ﬂames Complex and smart ﬂuids Compressible ﬂow Computational ﬂuid dynamics Convective phenomena Drops and bubbles Environmental ﬂuid dynamics Experimental methods in ﬂuid mechanics Flow control Flow in porous media Flow instability and transition Flow in thin ﬁlms Fluid mechanics of materials processing Fluid mechanics of suspensions Granular ﬂows Low-Reynolds-number ﬂow Magnetohydrodynamics Multiphase ﬂows Solidiﬁcation and crystal growth Stirring and mixing Topological ﬂuid mechanics Turbulence Vortex dynamics Waves

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Pre-Nominated Sessions on Solid Mechanics SM1 SM2 SM3 SM4 SM5 SM6 SM7 SM8 SM9 SM10 SM11 SM12 SM13 SM14 SM15 SM16 SM17 SM18 SM19 SM20 SM21 SM22 SM23 SM24 SM25 SM26 SM27

Computational solid mechanics Contact and friction mechanics Control of structures Damage mechanics Dynamic plasticity of structures Elasticity Experimental methods in solid mechanics Fatigue Fracture and crack mechanics Functionally graded materials Impact and wave propagation Material instabilities Mechanics of composites Mechanics of phase tranformations (jointly with IACM) Mechanics of porous materials Mechatronics Multibody dynamics Plasticity and viscoplasticity Plates and shells Rock mechanics and geomechanics Solid mechanics in manufacturing Stability of structures Stochastic micromechanics Structural optimization Structural vibrations Vehicle dynamics Viscoelasticity and creep

Topics Involving Both Fluid Mechanics and Solid Mechanics FSM1 FSM2 FSM3 FSM4 FSM5 FSM6 FSM7

Acoustics Chaos in ﬂuid and solid mechanics Continuum mechanics Fluid-structure interaction Mechanics of foams and cellular materials Multiscale phenomena in mechanics Education in mechanics

The description of the session consists of the session ID, the descriptive name of the session (e.g. ‘Smart materials and structures’) and the list of responsible co-chairs in parentheses. The list of the contributions follows. It is sorted by the unique lecture ID. The Mini Symposia (MS1. . . MS6), apart from the Lecture Presentations and Seminar Presentations contain Introductory Lectures. The latter are distinguished by the boldface letters of the author names.

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OL – Opening Lecture 10498 Leen van Wijngaarden (Netherlands): Interplay between Air and Water

CL – Closing Lecture 10697 Kazimierz Sobczyk (Poland): Stochastic Dynamics of Engineering Systems

SL – Sectional Lectures 10042 Konrad Bajer (Poland): Rapid Formation of Strong Gradients and Diﬀusion in the Transport of Scalar and Vector Fields 10143 Raymond W. Ogden (UK): Mechanics of Rubberlike Solids 10158 Edward Spiegel (USA): Problems in Astrophysical Fluid Dynamics 10495 Harry L. Swinney (USA): Scaling in Quasi-2D Turbulence Experiments in a Rotating Flow 10508 Pierre J. Ladev´eze (France): A Bridge Between the Micro- and Mesomechanics of Laminates: Fantasy or Reality? 10512 Roland Keunings (Belgium): Non-Newtonian Fluid Mechanics Using Molecular Theory 10529 D.A. Saville (USA): Electrokinetics & Electrohydrodynamics in Microﬂuids 10544 Edwin Kreuzer (Germany): Nonlinear Dynamics in Ocean Engineering 10551 Anatoly Neishtadt (Russia): Probability Phenomena in Perturbed Dynamical Systems 10731 Marcel Lesieur (France): Turbulence and Large-Eddy Simulations 10772 Huajian Gao (Germany): Nanoscale Mechanics of Biological Materials 10880 Daniel A. Beysens (France): Near-Critical Point Hydrodynamics and Microgravity 10930 Andre Preumont (Belgium): Some Issues in Active Vibration Control of Smart Structures 11040 Jorge Ambr´ ´ osio (Portugal): Multibody Dynamics: Bridging for Multidisciplinary Applications 11327 Erich Sackmann (Germany): Microviscoelasticity of Cells: Cells as Viscoplastic Bodies 12160 John F. Brady (USA): Suspensions: From Micromechanics to Macroscopic Behavior 12781 Peter B. Rhines (USA): Ocean Circulation and its Inﬂuence on Climate 13003 Thomas J.R. Hughes (USA): Variational and Multiscale Methods in Turbulence

MS1 – Smart materials and structures (J. Holnicki-Szulc, Poland & N. Sottos, USA) 10680 Wieslaw M. Ostachowicz (Poland): Elastic Wave Propagation Development for Structural Health Monitoring 10722 Richard D. James (USA): A Way to Search for Smart Materials with Unprecedented Physical Properties 13010 Scott R. White (USA): Autonomic Healing of Polymers and Composites 10136 Mieczyslaw S. Kuczma (Poland): Composite Plates with Active Fibres 10784 Hans Irschik (Austria): Transient Eigenstrains Without Incremental Displacements in a Hyperelastic Body 10821 Hisaaki Tobushi (Japan): Shape Fixity and Shape Recovery of Shape Memory Polymer and their Applications 10985 Ji-Hwan Kim (S. Korea): Vibration Control of Stiﬀened Plates with Integrated Piezoelectrics 11685 Nancy R. Sottos (USA): Stress Eﬀects on Ferroelectric Thin Film Patterning, Properties and Performance 11731 Kaushik Bhattacharya (USA): A Novel Approach to the Application of Ferroelectric Thin Films to Micro-actuation

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11752 Sven Lentzen (Germany): A Geometrically Non-Linear Finite Shell Element with Piezoelectric Layers 11857 Cun-Fa Gao (Japan): Thermal-Induced Fracture of Electroded Piezoelectric Composites 12011 Rivka Gilat (Israel): Thermal Buckling of Active Composite Plates with Shape Memory Alloy Fibers 12016 Daining Fang (China): Study of Non-Linear Magnetomechanical Constitutive Relations of Ferromagnetic Materials 12018 Fumihiro Ashida (Japan): Optimum Control of Thermoelastic Deformation in a Smart Composite Disk 12027 Catherine L. Brinson (USA): SMA Hybrid Composites: Self-healing, Self-Stiﬀening and Shape Control Simulations 12147 Sadagopan Narayanan (India): Active Control of FGM Plates Using Distributed Piezoelectric Sensors and Actuators 12637 Michal Landa (Czech Republic): Ultrasonic Characterization of Phase Transformation in NiTi Wire During Thermomechanical Loading 12786 Mitsunori Denda (USA): Upper and Lower Bounds of Electric Induction Intensity Factors for Multiple Piezoelectric Cracks by the BEM 13007 Andrew Smyth (USA): Direct Identiﬁcation of the State Equation in Complex Nonlinear Systems 13011 Alan Jones (USA): Self-Healing Polymer Composites for Extended Fatigue Life 13015 Piotr Pawlowski (Poland): The Concept of Multifoldig and Its Experimental Validation

MS2 – Tissue, cellular and molecular biomechanics (D. Barth´ ´es-Biesel, France & A. Hoger, USA) 10709 Christoph F. Schmidt (Netherlands): Molecular Mechanics of Cytoskeletal Components 10933 Susan S. Margulies (USA): Tissue Mechanics 12811 Samuel A. Safran (Israel): Elastic Interactions of Biological Cells 10627 Erik van der Giessen (Netherlands): Micromechanics of Cytoskeletal Actin Networks 10670 Tobias Olsson (Sweden): Residual Stress Fields in Soft Tissues 10689 Katarzyna A. Rejniak (USA): From Individual Cells To Complex Tissues – an Immersed Boundary Approach 10838 Kazimierz Piech´ or (Poland): Travelling Waves in a Model of Skin Pattern Formation 10988 Norman A. Fleck (UK): Mechanics of Deep Penetration of Soft Solids 11102 Andrejs Cebers (Latvia): Mechanics of Elastic and Viscous Magnetic Filaments 11148 Samuel Sideman (Israel): Intracellular Control Mechanisms of Cardiac Contraction & Energetics 11581 Yoel Forterre (France): Mechanics of Venus’ ﬂytrap Closure 11808 Jaroslaw Piekarski (Poland): Approximations of Stiﬀness Tensor of Bone – Determining and Accuracy 11866 Thomas R. Powers (USA): Theory of Polymorphism in Bacterial Flagella 11961 Dominique Barth´ ´es-Biesel (France): Hydrodynamic Interaction Between Two Bioartiﬁcal Capsules in Shear Flow 12031 Christoph Hartmann (Germany): Stress and Strain in a Yeast Cell under High Hydrostatic Pressure 12059 Alexander V. Kondrachuk (Ukraine): Models of Hair Cell Bundle Functioning 12064 Edoardo Mazza (Switzerland): Measuring the Mechanical Properties of Soft Biological Tissues 12100 Stanisaw Jemiolo (Poland): Anisotropic Hyperelastic and Pseudo-Hyperelastic Materials and Applications to Soft Tissue Modelling 12192 Taisuke Sugii (Japan): Molecular Dynamics Study of Permeation Process of Small Molecules Through A Lipid Bilayer

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12315 Tomasz Lekszycki (Poland): Modeling of Periodic Load Eﬀects in Bone Tissue Adaptation 12453 Ken-ichi Tsubota (Japan): A Particle Method Computer Simulation of Blood Flow 13049 Luigi Gambarotta (Italy): Wrinkling and Buckling of Isotropic Biological Tissues

MS3 – Mechanics of thin ﬁlms and nanostructures (H.M. Jensen, Denmark & Z. Suo, USA) 11070 Henrik M. Jensen (Denmark): Mechanics of Thin Film Structures 11594 Rodney S. Ruoﬀ (USA): Mechanics of Nanostructures 12132 Kenneth M. Liechti (USA): A Hybrid Molecular/Continuum Analysis of IFM Experiments on a Self-assembled Monolayer 10105 Min Zhou (USA): Thermomechanical Continuum Representation of Atomistic Deformation at Arbitrary Time and Size Scales 10699 Rui Huang (USA): Ratcheting-induced Wrinkling of an Elastic Film on a Metal Layer Under Cyclic Temperatures 11122 Akihiro Nakatani (Japan): Atomistic Study of Size Eﬀect in Torsion Tests of Nanowire 11279 Wei Yang (China): Microstructural and Atomistic Simulation for Deformation of Nano-grained Metals 11444 Kinga Nalepka (Poland): Energy-Based Approach to Limit States in Nanostructures. Calculation of the Critical Values of Energy from Firt Principles 11474 Pradeep Sharma (USA): Size-dependent Elastic State of Embedded Nano-inclusions & Quantum Dots 11599 Mikhail N. Perelmuter (Russia): Fracture Criterious for Bridged Crack: from Macro to Nanoscale 12181 Fenghuan Sha (China): Investigation of Wave Propagation in Multiwall 12251 Malgorzata Chwa (Poland): Homogenisation Models of Carbon Nanocomposites Mechanical Properties 12314 Xi-Qiao Feng (China): Micro- and Nano- mechanics of Carbon Nanotubes Composites 12384 Jurg Dual (Switzerland): Characterization of MEMS Materials 12386 Daniel S. Balint (UK): An Analytical Model of Oxide Rumpling as the Mechanism Leading to Failure in Thermal Barrier Coatings 12432 Raymond Parnes (Israel): Instabilities of Composite Materials Reinforced by NanoFibres: a Re-examination of Elastic Buckling 12486 Xuejun J. Zheng (China): Interfacial Adhesion of PZT Ferroelectric Thin Films Determined by Nano-Indentation Method 12627 Simon P.A. Bill (UK): A Cellular Automaton for Modelling Evolution of Heteroepitaxial Systems 12705 Taher Saif (USA): Mechanical Behaviour of Nano Grained Metals 12778 Kyung-Suk Kim (USA): Nano-scale Planar Field Projection of Atomistic Decohesion of Crystalline Solids

MS4 – Microﬂuids (R.J. Adrian, USA & J. Santiago, USA) 12960 Patrick Tabeling (France): Slip, Patterns, and other small Things in Microﬂuidic Systems 12961 J. Santiago (USA): Electrokinetic Flow Instabilities in Microﬂuidic Systems 12976 D. Scott Stewart (USA): Miniaturization of Explosive Technology and Microdetonics 10723 Carlo Cercignani (Italy): Plane Poiseuille Flow in a Rareﬁed Gas with General Boundary Conditions 10971 Anna Kucaba-Pi¸etal (Poland): Water Flows in Copper and Quartz Nanochannels 11437 Nicolas G. Hadjiconstantinou (USA): A Second-Order Slip Model for Early-TransitionRegime Flows 11604 Kenneth S. Breuer (USA): Direct Measurement and Simulation of Apparent Slip Velocities in Sub-Micron-Scale Flows

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11607 Eiichiro Yamaguchi (USA): Theoretical and Experimental Study of Microchannel Blockage Phenomena 11777 Todd M. Squires (USA): Induced-Charge Electro-Osmosis: Theory and Microﬂuidic Applications 12047 Laure Menetrier (France): Using Microﬂuidics to Investigate Reaction-diﬀusion Phenomena in Simple Flows 12130 Guillaume Degr´e (France): Magnetic Particles Aggregation in the Presence of a Hydrodynamic Shear 12536 Marie Caroline Jullien (France): Chaotic Mixing and Resonances in a Microﬂuidic System 12673 Pierre Joseph (France): An Accurate Velocity Proﬁle Measurement System for Microﬂuidics: A Direct Measurement of the Slip Length 12966 Piotr Garstecki (USA): Tunable Microﬂuidic Bubble Generator

MS5 – Microgravity ﬂow phenomena (J.I.D. Alexander, USA & P. Neitzel, USA) 10598 Denis Weaire (Ireland): Foams, Films and Surfaces in Microgravity 10714 Michel Y. Louge (USA): Collisional Granular Flows with and without Gas Interactions in Microgravity 11133 George M. Homsy (USA): Microgravity and Microscale Fluid Mechanics 10958 Dmytro V. Yevdokymov (Ukraine): Hydrodynamic Eﬀect of Slow Phase Transitions in Microgravity 10980 M.C. Charrier Mojtabi (France): Heat Transfer Due to High Frequency Vibration: a New Approach for Achieving Thermally Optimum Geometry Under Microgravity Conditions 11757 Vladislav V. Pukhnachov (Russia): Mathematical Models of Microconvection for Isothermally Incompressible and Weakly Compresible Liquids 11843 Michael K. Ermakov (Russia): Onset of Oscillations in High-Prandtl Thermocapillary Liquid Bridges: Linear-stability Analysis vs. Experiment 12402 Tatyana P. Lyubimova (Russia): Spherical Two-phase Interface in a Near-critical Fluid. Gradient Approach 12447 Iwan Alexander (USA): Capillary Pressure of a Liquid Between Uniform Spheres Arranged in a Square-packed Layer 12651 G. Paul Neitzel (USA): Recent Advances in Permanent Noncoalescence and Nonwetting

MS6 – Atmosphere and ocean dynamics (M.E. McIntyre, UK & J. Sommeria, France) 10716 Onno Bokhove (Netherlands): Wave Vortex Interactions in the Atmosphere and Oceans; with Applications to Climate 10803 Olivier Talagrand (France): Assimilation of Observations into Numerical Models 10987 Peter Haynes (UK): Transport and Mixing in the Atmosphere 10261 Peter J. Thomas (UK): Modelling Oceanographic Coastal Currents in Small-scale and Large-scale Laboratory Experiments 10513 Pierre Carlotti (France): Near-surface Turbulence in a Neutral Planetary Boundary Layer: Comparison of LES with the CASES’99 Experiment Observations 10634 Gordon Swaters (Canada): Meridional Flow of Source Driven Grounded Abyssal Flow in a Wind Driven Basin with Topography 10977 Michael E. McIntyre (UK): Remote Recoil and Wave Capture: Wave–vortex Interactions in Atmosphere-Ocean Models 11152 Boris Galperin (USA): Anisotropic Large-Scale Turbulence on Giant Planets and in the Ocean 11336 Jean Noel Reinaud (UK): Strong Vortex Insteractions in Quasi-Geostrophic Flows 11499 Ross Griﬃths (Australia): Turbulent Horizontal Convection and the Global Thermohaline Circulation of the Oceans

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11547 Joel Sommeria (France): Instability of Gravity Driven Coastal Current in a Turntable Experiment 11585 Marius Ungarish (Israel): Intrusive Gravity Currents in a Stratiﬁed Ambient – Novel Theoretical Results and Insights 11732 Shinya Shimokawa (Japan): Irreversible Transition to a State with Higher Entropy Production in Oceanic General Circulation 11829 Jan-Bert Fl´ or (France): Interactions of Planar Waves with a Baroclinic Vortex 11942 Semion Sukoriansky (Israel): A New Spectral Closure Model of Turbulent Flows with Stable Stratiﬁcation and Its Application to Atmospheric SBLS 12035 Dieter Etling (Germany): Roll Vortices in the Atmospheric Boundary Layer 12104 Jacques Vanneste (UK): Spontaneous Generation of Inertia-Gravity Waves by Balanced Motion 12140 Jun-Ichi Yano (France): The Energy Cycle of the Tropical Madden-Julian Oscillations seen through Wavelets 12157 Oliver Buhler (USA): Wave Capture and Wave-vortex Duality 12169 Keke Zhang (UK): A New Theory For Convection In Rapidly Rotating Spherical Systems 12194 Takeshi Miyazaki (Japan): Vortex-based Models of Quasigeostrophic Turbulence 12210 Jonas Nycander (Sweden): Generation of Internal Waves in the Deep Ocean by Barotropic Tides 12294 Pantxika Otheguy (France): Instability of Corotating Vertical Vortices in a Stratiﬁeld Fluid: Why Strongly Stratiﬁeld Turbulence is not Similar to 2D Turbulence 12408 Yuliya Kanarska (Ukraine): Laboratory and Numerical Modelling of Exchange Flows 12463 Alastair D. Jenkins (Norway): Wave-Mean Flow Interaction in Coupled AtmosphereIce-Ocean Systems 12494 Peter L. Read (UK): Multiple Jet Formation in a Convectively Driven Flow on a Betaplane 12601 Pascale Bouruet-Aubertot (France): Intermittency in Stratiﬁed Turbulence Produced by Breaking Internal Gravity Waves 12722 Fabrice Veron (USA): Measurements of the Inﬂuence of Ocean Surface Kinematics on Air-sea Heat Fluxes 12776 Francisco J. Beron-Vera (USA): Linear Waves and Baroclinic Instability in an Inhomogeneous-Density Layered Primitive-Equation Ocean Model 13017 Silvia Ferrarese (Italy): Simulation of Sea Surface Temperature Trends Under Severe Wind Forcing With a Full Atmosphere-Ocean Coupled Model

FM1 – Biological ﬂuid dynamics (M. Gharib, USA & F. van de Vosse, Netherlands) 10132 Takeshi Sugimoto (Japan): Mechanics of the Bounding Flight Revisited 10741 Thomas Podgorski (France): Deformation of Vesicles Flowing Through a Capillary 11013 Charles N. Baroud (France): How to Breathe in a Liquid-Filled Lung: Symmetry of Airway Reopening 11187 Tony W.H. Sheu (Taiwan): Computational Exploration of Liver Acinus Microstructure 11441 Tomonobu Goto (Japan): Bacterium Swimming Motion Close to a Wall 11851 Maheshwaran K. Kolandavel (UK): A CFD Study of the Eﬀects of Physiological Vessel Wall Motion on Oxygen Transport in Coronary Arteries 11900 Peter Vennemann (Netherlands): In Vivo PIV Measurement in the Embryonic Chicken Heart 11908 David M. Lewis (UK): A Model of Plankton Dynamics Coupled with a LES of the Surface Mixed Layer 11943 Adrian S. Carabineanu (Romania): Self-Propulsion of an Oscillatory Wing 12063 Oleksiy S. Galaktionov (Netherlands): Bioirrigation in Marine Sediments: Ecological Conclusions from Numerical Modelling

Scientiﬁc Program

xxxvii

12124 Cyrus K. Aidun (USA): Direct Numerical Simulation of Red Blood Cell Flow and Aggregation 12295 Andrew L. Hazel (UK): Three-Dimensional Airway Reopening – Finite-ReynoldsNumber Eﬀects 12400 Philippe Marmottant (Netherlands): Cell Permeabilisation and Transport Focused Around Oscillating Microbubbles 12409 Maciej K. Ginalski (Poland): Computational Model of Selected Transport Processes in an Infant Incubator 12423 Manouk Abkarian (USA): Red Blood Cell Dynamics, Deformation and Rheology via Microﬂuidic Experiments 12522 Masanori Nakamura (Japan): Flow in an Integrated Model of Heart and Aorta 12655 Sang-Joon Lee (S. Korea): In Vivo Visualization of the Water Reﬁlling Process in Xylem Vessels Using Synchrotron X-Ray Micro-Imaging 12784 Mory Gharib (USA): On the Issue of Optimal Trans-Mitral Flow

FM2 – Boundary layers (P.W. Duck, UK & A. Kluwick, Austria) 10103 James P. Denier (Australia): The Development (and Suppression) of Very Short-Scale Instabilities in Buoyant Boundary Layers 10275 Yury S. Kachanov (Russia): 3D Distributed Boundary-Layer Receptivity to NonStationary Free-Stream Vortices in Presence of Surface Roughness 10948 Matthias H. Buschmann (Germany): Extending the Generalized Logarithmic Law to the Wall 11017 Ihor Nesteruk (Ukraine): Sub- and Supersonic Shapes without Separation and Cavitation 11083 Bernhard Scheichl (Austria): Non-Unique Quasi-Equilibrium Turbulent Boundary Layers 11084 Stefan Braun (Austria): Near Critical Unsteady Three-Dimensional Triple Deck Flows 11210 Frank T. Smith (UK): Long Layers Exhibiting Local Jumps, in Industrial and Biomedical Applications 11339 Victor V. Kozlov (Russia): Secondary Instability of Stationary Vortex Packets in a Swept Wing Boundary Layer 11397 Anatoly I. Ruban (UK): Discontinuous Solutions of the Boundary-Layer Equations 11780 Vladimir B. Zametaev (Russia): New Numerical Method for Complex Interacting Flows 11887 Peter W. Carpenter (UK): Why do Dolphins Have Cutaneous Ridges? 11984 Patrick D. Weidman (USA): Two-Fluid Jets and Wakes 12030 Herbert Steinrueck (Austria): The Trailing Edge Problem for Mixed Convection Flow Past a Horizontal Plate 12062 Tomas Vit (Czech Republic): Experimental and Theoretical Study of Heated Coanda Jet 12241 Matthew R. Turner (UK): A Combined Numerical and Asymptotic Approach to Boundary Layer Receptivity Problems 12320 Owen R. Tutty (UK): Flow Along a Long Thin Cylinder 12631 Narayanan Vinod (India): Aspects of the Laminar-Turbulent Transition in Axisymmetric Boundary Layers

FM3 – Combustion and ﬂames (N. Peters, Germany & P. Wola´ n ´ ski, Poland) 10367 Ay Su (Taiwan): Enhancement of the Impinging Diﬀusion Flame by Splash Plate 10747 Oluwole Daniel Makinde (South Africa): Exothermic Explosions in a Slab: a Case Study of Series Summation Technique 10918 David Lo Jacono (Switzerland): A Nearly 1-D Non-Premixed Flame Near Extinction. Cell Formation and the Eﬀect of the Direction of Bulk Flow 11031 Aidarkhan Kaltayev (Kazakhstan): Simulation of Flame Propagation in a Tube with Obstacles

xxxviii

ICTAM04

11888 Tim Broeckhoven (Belgium): Large Eddy Simulation of Piloted and Bluﬀ – Body Diﬀusion Flame 11914 Artur Tyliszczak (Poland): Inﬂuence of the Subgrid Models on Combustion Modelling 12199 Yeshayahou Levy (Israel): Chemical Aspects of the Flameless Oxidation Applied for GasTurbine Combustor 12348 Pedro J. Coelho (Portugal): Experimental and Numerical Investigation of a Flameless Oxidation Combustor 12745 Piotr Wola´ n ´ ski (Poland): Detonations of Hexane Vapor/Droplets-Air Mixtures 12843 Arkadiusz Kobiera (Poland): Simulation of Ram Accelerator with PETN Layer 12911 Zbigniew A. Walenta (Poland): Simple Model of a Detonating Gas for use with the Direct Monte-Carlo Simulation Technique

FM4 – Complex and smart ﬂuids (B. Khusid, USA & A. Yarin, Israel) 10140 Konstantin G. Kornev (USA): Capillary Microﬂuidics for Viscoelastic Fluids 10753 Boris Khusid (USA): Field-Induced Dielectrophoresis and Phase Separation in Suspention 10932 Alexander L. Yarin (Israel): Electrospinning of Nanoﬁbers from Polymer Solutions 11165 Ping Sheng (China): The Giant Electrorheological Eﬀect in Suspensions of Nanoparticles 11184 Sawomir Blo´ nski ´ (Poland): Electrospinning of Liquid Jets 11209 Kevin D. Dorfman (France): Modeling DNA Separations in Self-Assembled Magnetic Arrays: Comparison of Theory and Experiment 11603 Ilker Bayer (USA): Contact Angle Dynamics of Droplets Impacting on Flat Substrates 11648 Semyon P. Levitsky (Israel): Dissipation Features at Nonlinear Pulsations of Bubbles in Viscoelastic Fluids 11954 Aleksey N. Rozhkov (Russia): Break up of Polymer Solution Drop Impacting a Small Target 11982 Markus Zahn (USA): Ferrohydrodynamic Hele-Shaw Cell Flows and Instabilities with Simultaneous DC Axial and In-Plane Rotating Magnetic Fields 12254 Daniel A. Weiss (Germany): Spray Impact on Solid Walls of Non-Newtonian Fluids, Including Yield Stress and Thixotropic Behavior 12319 Eyal Zussman (Israel): Nanowires Assembly Using Microﬂuidic: an Experimental Investigation 12437 Antonio Castellanos (Spain): Particle Manipulation in Microﬂuidics: the Role of Dielectrophoresis, Electrohydrodynamics and AC Electrokinetics 12894 Paul C. Duineveld (Netherlands): Non-Newtonian Eﬀects of Ink-Jet Printed Droplets 12910 Agnieszka Sowicka (Poland): Conditions for Creating Thin Liquid Layers at the Contact Surface of Two Other Liquids

FM5 – Compressible ﬂow (H.G. Hornung, USA) 10531 Mikhail S. Ivanov (Russia): Hysteresis-Related Phenomena in Shock Wave Reﬂection 10789 Alfred Kluwick (Austria): Nonclassical Dynamics of Laminar Dense Gas Boundary Layers 10945 Helmut E. Sobieczky (Germany): Analytical Models for Shocks in Compressible Flow 10984 Holger Babinsky (UK): LDA Investigation of a Transonic Bump Flow 11049 Kazuyoshi Takayama (Japan): Unsteady Drag Force Measurements of Shock Loaded Bodies Suspended in a Vertical Shock Tube 11072 Beric W. Skews (South Africa): Shock Wave Reﬂection in a Non-Circular Inlet 11560 Susumu Hasegawa (Japan): Numerical Optimization of 2D Scramjet Inlets 11730 John K. Hunter (USA): The Mach Reﬂection of Weak Shocks 11864 Susumu Kobayashi (Japan): Eﬀect of Surface Roughness on Mach Reﬂection 12066 Vladimir V. Serebryakov (Ukraine): On the Theory for Subsonic, Transonic and Supersonic Flows in Water with Supercavitation 12219 Vaclav Dvorak (Czech Republic): Interaction of Supersonic Flows in an Ejector

Scientiﬁc Program

xxxix

12997 Ryszard Szwaba (Poland): Shock Wave-Boundary Layer Interaction Control by Streamwise Vortices 13016 Joel Delville (France): Correlation of Nearﬁeld Pressure with Mixing Layer Velocity in a Supersonic Jet

FM6 – Computational ﬂuid dynamics (L. Kleiser, Switzerland & W. Schroeder, Germany) 10163 Randolph C.K. Leung (China): Non-Reﬂecting Boundary Condition for Direct Aeroacoustic Computation 10857 Jaime Klapp (Mexico): Treesph Simulations of Choked Flow Systems Using Smoothed Particle Hydrodynamics 10921 Piotr Boro´ n ´ ski (France): Poloidal-Toroidal Decomposition in Cylindrical von Karman Flow 11348 Manuel Garcia-Villalba (Germany): On Inﬂow Boundary Conditions for Large Eddy Simulation of Turbulent Swirling Jets 11456 Rubens Campregher (Brazil): Numerical Simulation of the Flow over a BackwardFacing Step in a Beowulf-Class Cluster 11485 Alexey N. Kudryavtsev (Russia): Development and Paractical Application of WENO Schemes for Compressible Fluid Flow Computations 11569 Jae-Woo Lee (S. Korea): Numerical Analysis and Design Optimization of Lateral Jet Controlled Missile 11578 Tadeusz Chmielniak (Poland): Numerical Prediction of Energy Dissipation in Condensing 11602 Erik R. Dick (Belgium): A Pressure-Correction Method for All Mach Numbers 11702 Jan Vimmr (Czech Republic): Numerical Computation of Compressible Viscous Flow Through a Male Rotor-Housing Gap of Screw Compressors 11763 Piotr G. Kowalczyk (Poland): Fast Numerical Method for the Boltzmann Equation on Nonumiform Grids 11937 Lorena A. Barba (UK): Computation of Viscous Vortices with Fully Meshless Method 12076 Sawomir Kubacki (Poland): Dirichlet/Dirichlet and Neumann/Neumann Parallel Non-Overlapping Domain Decomposition Method 12337 Milan Schuster (Czech Republic): CFD Methods in Industrial Applications Vehicle External Aerodynamics and Aerodynamic Interaction of Moving Vehicles 12510 Guillaume Dufour (France): Numerical Error Evaluation for Tip Clearance Flow Calculations in Centrifugal Compressor 12565 Andrzej Styczek (Poland): Simulation of a Viscous Flow Past a Three Dimensional Obstacle Using Vortex Particles 12583 Petros Koumoutsakos (Switzerland): Multiscale Simulations Using Particles 12618 Jerzy Majewski (Poland): Investigation of WENO Scheme for 3D Unstructured Grids 13030 Fedderik van der Bos (Netherlands): Commutator – Errors in Large-Eddy Simulation of Turbulent Flow 13051 Patrick Bontoux (France): Three-Dimensional Rayleigh-Benard Instability in a Supercritical Fluid by Direct Numerical Simulation

FM7 – Convective phenomena (G. de Vahl Davis, Australia & K. Zhang, UK) 10306 Alexander V. Getling (Russia): Cellular Compressible Magnetoconvection: a Mechanism for Magnetic-Field Ampliﬁcation and Structuring 10438 Martin P. King (Italy): Scaling Laws for Thermal Convections 10538 Aleksandra Aleksandrovna Bozhko (Russia): On Features of Magnetic Convection in Ferroﬂuid 10914 Katarzyna Boro´ n ´ ska (France): Multiplicity of Patterns in Cylindrical Convection 11053 Igor Rogachevskii (Israel): Large-Scale Semi-Organized Structures in Geophysical Turbulent Convection

xl

ICTAM04

11160 Tomasz Michalek (Poland): Natural Convection for Anomalous Density Variation of Water — Numerical Benchmark 11251 C.M. Rindt (Netherlands): 3D Flow Transition Behind a Heated Cylinder 11440 Jonathan M. Aurnou (USA): Experimental Studies of Planetary Core Convection and Dynamo Processes 11673 Antonio Cenedese (Italy): Penetrative Convection in Stratiﬁed Fluids: Velocity Measurements by Image Analysis 11905 Heiko Meironke (Germany): Experimental and Numerical Studies of Convection Flow in a Cylindrical-Conical Fermenting Tank 11972 Vanda Bouch´e (Italy): Sea Convective Motions Driven by Random Buoyancy Inputs 12126 Michael Le Bars (UK): Thermochemical Convection in Two Superimposed Miscible Viscous Fluids 12168 Xinhao Liao (China): Nonlinear Convective Patterns in Spherical Rayleigh-Benard Systems 12173 Elzbieta ˙ Fornalik (Poland): Visualization of the Flow Structure and Temperature Field in the Region of Mixed Convection 12220 Anne Sergent (France): Large Eddy Simulation of Rayleigh-Benard Convection in an Inﬁnite Fluid Layer 12267 Aleksander Alekseevich Kozlov (Russia): The Inﬂuence of Translational Vibration of Circular Polarization on Fluid Convection Stability and Flow Patterns 12344 Dmitry V. Lyubimov (Russia): Thermal Buoyancy Convection in Systems with Deformable Interfaces 12730 Alexander A. Smirnovskii (Russia): Convective Phenomena in Rotating Annuli Heated on Periphery 12850 Avshalom Manela (Israel): On the Rayleigh-Benard Problem in the Continuum Limit 13005 Vasiliy A. Novosiadliy (Russia): The Inﬂuence of Vibration on the Onset of Marangoni Convection in Horizontal Fluid Layer 13022 Hiroyuki Ozoe (Japan): Air Convection in a Cubic Enclosure with Laterally Shifted Electric Coil without a Gravity Field.

FM8 – Drops and bubbles (J. Eggers, UK & A. Prosperetti, USA) 10580 Yuriy A. Semenov (Ukraine): Method for Solving Nonlinear Problems on Unsteady Free-Boundary Flows 11026 Miguel F. Moctezuma Sanchez (Mexico): Bubble Wall Interaction and Bubble Pairs Motion Using Potential Flow Theory 11250 Olga M. Lavrenteva (Israel): Locomotion of a Viscous Drop, Induced by the Internal Secretion: Boundary Eﬀects 11289 Gary L. Leal (USA): Theoretical Studies of Flow-Induced Coalescence 11299 Teruo Kumagai (Japan): Occurrence of Micro-Bubbles During the Coalescence of Two Bubbles 11317 Nicolas Bremond (Netherlands): Atomization of an Undulating Liquid Sheet 11330 Eric Lauga (USA): Evaporation-Driven Assembly of Colloidal Particles 11411 Eligiusz Wajnryb (Poland): High-frequency Linear Viscosity of Emulsions Composed of Two Viscoelastic Fluids 11462 Christian Wagner (Germany): Molecule Conﬁgurations in a Droplet Detachment Process of a Semdilute Xanthan Solutions 11614 Christophe Josserand (France): Spreading and Retraction of Impacting Drops 11669 Slavka S. Tcholakova (Bulgaria): Main Factors Controlling the Emulsiﬁcation Process under Turbulent Conditions. Experiment and Data Interpretation 11670 Teresa Parra (Spain): Water Mist Behavior as Flame Supressant 11723 Stefan Zaleski (France): Numerical Simulation of Liquid-Gas Interfaces with Applications to Atomization 11855 Ernest O. Tuck (Australia): Viscous Extensional Flow and Drop Break-Oﬀ Under Gravity

Scientiﬁc Program

xli

11911 Jacques J. Magnaudet (France): Evolution of a Pair of Spherical Bubbles Rising Side by Side at Moderate Reynolds Number 11979 Patrick Le Qu´ ´er´e (France): On the Numerical Simulation of Two Phase Liquid-Vapor Phenomena 11999 Manish Arora (Netherlands): Cavitation Inception on Micro-particles: a Self Propelled Particle Accelerator 12106 Peter D.M. Spelt (UK): Level-Set Simulations of Shear Flow with Inertia Pas a Droplet Adhering to a Wall with Moving Contact Lines 12163 Marianne Francois (USA): Modelling Surface Tension Using a Ghost Fluid Technique within a Volume of Fluid Formulation 12190 Andrei S. Topolnikov (Russia): Dynamics of Bubble Supercompression in Organic Liquids 12311 Ryszard Pohorecki (Poland): Hydrodynamics of Gas Bubbling through Organic Liquids 12329 Michael A. Rother (USA): Surfactant Eﬀects on Buoyancy-Driven Coalescence of Spherical Drops 12388 Laurent Duchemin (UK): Static Shapes of Levitating Viscous Drops 12429 Wendy W. Zhang (USA): A Long-Wavelength Model of Viscous Entrainment 12547 Ulderico P. Bulgarelli (Italy): Entrainment of Air Bubbles During Strong VorticityFree-Surface Interaction 12548 Marco A. Fontelos (Spain): Spreading of Charged Microdroplets 12596 C. Wang (Singapore): Multiple Bubbles Dynamics Using Level Set Indirect Boundary Element Method 12741 Andrew J. Griggs (USA): Low-Reynolds-Number Motion of a Drop Beween Two Parallel Plane Walls 12873 Andrzej Zachara (Poland): Thermodynamic Parameters of Vapour Bubble Growth by Image Analysis 13025 Salima Rafai (France): Singular Droplets

FM9 – Environmental ﬂuid dynamics (H.E. Huppert, UK & R. Narasimha, India) 10366 Owen M. Phillips (USA): The Growth and Structure of Double-Diﬀusive Cells Adjacent to a Side-Wall in a Salt-Stratiﬁed Enviroment 10467 Zbynek Janour (Czech Republic): Flow and Dispersion in the Atmospheric Boundary Layer Investigation by Physical Modelling 10576 Krzysztof Dekajo (Poland): Experiments on Up-slope to Down-slope Transition in an Inclined Box Filled with Water 10733 Mohammad J. Kazemzadeh-Parsi (Iran): Analysis of Double-Free Surface Flow through Gates Using Element-Free Galerkin Method 11164 Hitoshi Miyamoto (Japan): Free Surface Behavior in Turbulent Open-Channel Flows 11182 Gregory F. Lane-Serﬀ (UK): Integral and Laboratory Modelling of Sedimentation from Turbulent Buoyant Jets 11294 Falin Chen (Taiwan): Modiﬁed Shallow Water Equations for Inviscid Gravity Currents 11346 Ramon Fernandez-Feria (Spain): Dam-Break Flow for Arbitrary Slope of the Bottom 11433 Oksana E. Poloukhina (Russia): Extended Nonlinear Theory for Topographic Rossby Waves 11703 Andrew A. Osiptsov (Russia): The Propagation of Viscous Gravity Currents over a Rigid Conic Surface 12072 James G.A. Croll (UK): An Alternative Model for “Pingo” Formation in Permafrost Regions 12099 Tarmo Soomere (Estonia): Fast Ferry Traﬃc as a New Forcing Factor of Enviromental Processes in Non-Tidal Sea Areas 12266 Jaroslaw Ciechanowski (Poland): Dynamics of Separation Zone behind the 2D Hill in Oscillating Incident Wind. 12270 Arne Moe (Norway): Eﬀects of Curvature in Avalanche Deﬂecting Dams

xlii

ICTAM04

12396 Jim N. McElwaine (UK): Lobe and Cleft Formation at the Head of a Gravity Current 12410 Sabine Decamp (France): Experimental and Numerical Simulation of Dense Water Overﬂows on a Continental Slope 12592 Szymon P. Malinowski (Poland): High Resolution Modelling of Atmospheric Flow over Southern Poland 12626 Alfred J. W¨ u ¨ est (Switzerland): Formation and Rapid Expansion of Double Diﬀusive Layering in Lake Nyos 12738 John E. Holeman (USA): Joint Urban 2003 Surface Energy Budget Measurements and Analysis 12980 A.S. Vasudeva Murthy (India): Nocturnal Temperature Inversions Under Calm Clear Conditions: an Analytical Study

FM10 – Experimental methods in ﬂuid mechanics (A. Leder, Germany & J. Westerweel, Netherlands) 10445 Grazia Lamanna (Germany): On the Evaporation of a Monodisperse Droplet Stream at High-Pressure 11088 Jerry Westerweel (Netherlands): HPIV using Polarization Multiplexing Holography in Bacteriorhodopsin (bR) 11455 Valery Chernoray (Sweden): Time-resolved Wall Shear Stress Measurements using MEMS 11909 Raymond P.H.M. Bergmann (Netherlands): Void Collapse and Jet Formation: The Impact of a Disk on a Water Surface. 11916 Andrzej S. Witkowski (Poland): Comprehensive Experimental and Computational Investigations of the Unsteady Flow in an Axial Flow Low Speed Compressor Stage 12070 Tov Elperin (Israel): Experimental Detection of the New Phenomenon of Turbulent Thermal Diﬀusion 12213 Nikita A. Fomin (Belarus): 3D Vortices Structure Monitoring in Turbulent Flows by Digital Speckle Photography 12719 Anna Matvienko (Canada): Thermal-Wave Resonator Cavity: Modelling and Applications for Water Mixtures 12855 Piotr M. Korczyk (Poland): Particle Image Velocimetry (PIV) for Cloud Droplets – Laboratory Investigations 13048 Albert Baars (Germany): Optical Diagnosis Systems for Measuring Thermoﬂuiddynamicals Phenomena in Liquid Biosystems Under Ultra High Pressure

FM11 – Flow control (J.B. Freund, USA & M. Gad-el-Hak, USA) 10128 Mohamed Gad-el-Hak (USA): Liquids: The Holy Grail of Microﬂuidics Modeling 10872 Ramesh K. Agarwal (USA): Active Control of Shock/Boundary Layer Interaction in Transonic Flow Over Airfoils 11043 Rong F. Huang (Taiwan): Manipulating a Vee-Shaped Bluﬀ Body Wake Using a Fluidic Oscillator 11394 Sedat F. Tardu (France): Response on the Near Wall Turbulence to Localized Unsteady Blowing Periodical and Dissymetric in Time 11636 Haecheon Choi (S. Korea): Passive Control of Turbulent Flow behind a Model Vehicle for Drag Reduction Using Wake Disrupter 11918 Andrzej Szumowski (Poland): Control of Internal Supersonic Flow Separation 12013 Junwoo Lim (USA): A Singular Value Analysis of Boundary Layer Control 12068 Tom Weier (Germany): Separation Control by Stationary and Time Periodic Lorentz Forces 12156 Victor F. Kopiev (Russia): On the Possibility and Prospects of Turbulent Flow Noise Control 12244 Herv´ ´ e Illy (France): Control of Flow Oscillations over a Cavity by Means of a Spanwise Cylinder

Scientiﬁc Program

xliii

12258 Zdenek Travnicek (Czech Republic): Synthetic Jet Actuation at the Resonance Frequency 12305 Alan Guegan (France): Optimal Energy Growth and Optimal Control of the Swept Attachment – Line Boundary Layer 12387 Tim Colonius (USA): Feedback Control of Vortex Shedding in a Separated Diﬀuser 12491 Seichiro Izawa (Japan): Reduction of Aerodynamic Noise Induced by Flow over a Cavity 12602 Philippe Konieczny (France): Control of Turbulent Streaks by Active Wall Movement 12624 Mark Pastoor (Germany): Model-Based Control of Shear Flows Using Low-Dimensional Galerkin- and Vortex Models 12736 Marek Morzy´ n ´ ski (Poland): Numerical Analysis of the Wake Control behind a Circular Cylinder with Oscillatory Rotation

FM12 – Flow in porous media (Abder Kader Mojtabi, France & V. Nikolaevskiy, Russia) 10199 Vasiliy Govorukhin (Russia): Numerical Investigation of Convective Regimes in a Planar Filtrational Convection Problem 10234 Michel Quintard (France): Dissolution in Porous Media: Upscaling, Instabilities and Heterogeneity Eﬀects 10548 Guzel T. Bulgakova (Russia): Instability and Dynamic Chaos of Non-equilibrium Filtration of Gaseous Liquid 11109 Victor N. Nikolaevskiy (Russia): Plastic Mass Flow of Sand Under Action of Pore Pressure Gradient 11115 Dmitry Nikolaevich Mikhaylov (Russia): P-Waves Behavior at Transition from Liquid to Gas-Saturated Porous Media 11248 Wlodzimierz Bielski (Poland): Nonstationary Flow of Stokesian Fluid through Elastic Skeleton with Hierarchical Structure 11343 Franck Plouraboue (France): Conﬁned Air-liquid Drainage: Local Analysis and Invasion Percolation Model 11487 Thomas Loimer (Austria): A Joule-Thomson Process of a Wetting Fluid Near Saturation 11994 Piotr Szymczak (Poland): Microscopic Simulations of the Dissolution of Rock Fractures 12376 Marie-Christine N´ ´eel (France): Fractional Model for Solute Spreading in Randomly Heterogeneous Porous Media 12528 Liana Kovaleva (Russia): Oscillatory Modes of Adsorption in the Flow of Multicomponent Systems 12611 Mieczyslaw Cieszko (Poland): Extended Description of Pore Space Structure and Fluid Flow through Anisotropic Porous Materials 12617 Robert P. Behringer (USA): Onset of Convection for a Miscible Fluid in a Porous Medium

FM13 – Flow instability and transition (P. Huerre, France & P.A. Monkewitz, Switzerland) 10185 Aomar Ait Aider (France): Instabilities in a Taylor-Dean Open Flow 10188 Fran¸¸cois Lusseyran (France): Shear Layer Instability and Frequency Modes Inside an Open Cavity 10313 Bruno Eckhardt (Germany): Travelling Waves and Transition to Turbulence in Pipe Flow 10487 Joseph T.C. Liu (USA): Nonlinear Mechanics of Wavy Instability of Steady Longitudinal Vortices and Drag Rise in Boundary Layer Flow 10489 J.M. Floryan (Canada): Stability of Flow in a Rough Channel 10525 St´ ´ephane Leblanc (France): Stability of Lagrangian Ideal Flows 10640 Laurette Tuckerman (France): Instability Thresholds of Flow Between Exactly CounterRotating Disks

xliv

ICTAM04

10916 Peter W. Duck (UK): Transient Growth in Developing Plane and Hagen Poiseuille Flow 11186 Denis Martinand (UK): Three-dimensional Global Modes in Spatially Varying RayleighBenard-Poiseuille Convection 11396 Laurent Lacaze (France): Elliptical Instability in a Rotating Spheroid 11461 Jacob Cohen (Israel): The Instability of a Localized Vortex Disturbance in Uniform Shear Flow 11647 Maher Lagha (France): Subcritical Transition to Turbulence in Plane Couette Flow 11995 Eckart Meiburg (USA): Three-Dimensional Vortex Breakdown in Swirling Jets and Wakes 12150 Jonathan J. Healey (UK): A New Convective Instability with Growth Normal to a Boundary Layer 12233 Guillemette G. Caulliez (France): By-pass Laminar-Turbulent Transition of the WindDriven Free Surface Flow 12280 Denis S. Goldobin (Russia): Inﬂuence of Swirl Vibrations on Flow in Long Cylinder 12291 Masato Nagata (Japan): Nonlinear Stability of Rotating Channel Flow 12372 Uwe Ehrenstein (France): On Instability Mechanisms in a Separating Boundary-layer Flow 12381 Leandro Franco de Souza (Brazil): Gortler Vortex Secondary Stability: Varicose Mode 12425 Fran¸¸cois Gallaire (France): Spiral Vortex Breakdown as a Global Mode 12431 Dwight Barkley (UK): Computational Study of Turbulent-Laminar Bands in Couette Flow 12483 Arnaud Antkowiak (France): A Generic Mechanism for By-Pass Transition in Vortices 12508 Thierry Feraille (France): Global Stability of the Flow Induced by Wall Injection 12509 Christophe Millet (France): Acoustic Field Generated by Instability Waves in the Transonic Regime 12542 Ilmars Grants (Latvia): Nonlinear Transition of a Flow Driven by a Rotating Magnetic Field 12711 Jean-Marc Chomaz (France): Fully Nonlinear Global Modes and Transition to Turbulence in Open Flows 12766 Cherif Nouar (France): Stability of Plane Poiseuille Flow and Energy Growth in the Case of a Bingham Fluid 13019 Jens Norkaer Sørensen (Denmark): Instability of the Far Wake Behind a Wind Turbine

FM14 – Flow in thin ﬁlms (N. Aksel, Germany & V. Shkadov, Russia) 10220 Seraﬁm Kalliadasis (UK): Dynamics of a Reactive Falling Film at Large Peclet Numbers 10543 Alexander Oron (Israel): Long-Wave Marangoni Instability in Binary-Liquid Films with Soret Eﬀect 10557 Vasilis Bontozoglou (Greece): Solitary Waves on Liquid Film Flowing Along a Periodic Wall 10642 Nuri Aksel (Germany): Eﬀect of Bottom Undulations on the Stability of Film Flow Down Inclined Planes 10928 Andreas Wierschem (Germany): Hydraulic Jumps and Resonance in Gravity-Driven Flows of Liquid in Inclined Wavy Channels: Transition and Hysteresis 11089 Jens G. Eggers (UK): Hydrodynamic Theory of De-Wetting 11413 Norbert Alleborn (Germany): Linear Response of a Viscous Liquid Sheet 11477 Le Han Tan (Australia): Experimental and Numerical Study of Marangoni-Natural Convection 11509 Dirkjan B. van Dam (Netherlands): Layer Thickness Distribution of Thin-Film InkJet Printed Structures. 11739 Takao Yoshinaga (Japan): Ampliﬁcation of Nonlinear Disturbances on a Falling Liquid Sheet

Scientiﬁc Program

xlv

12155 Gregory P. Chini (USA): Thin Film Flows Near Isolated Humps and Interior Corners 12293 Tatiana Gambaryan-Roisman (Germany): Gravity- and Shear- Driven Thin Films Flow on Heated Hicrostructured Walls 12656 Jaroslav Tihon (Czech Republic): Hydrodynamics of the Solitary Waves Travelling Down a Liquid Film 12858 John Tsamopoulos (Greece): Transient Displacement of Viscoelastic Liquids by Air

FM15 – Fluid mechanics of materials processing (F. Dupret, Belgium & R. Moreau, France) 10363 Jeﬀrey J. Derby (USA): Analysis of Flow-Induced, Step-Bunching Instabilities During the Growth of Crystal from Liquid Solutions 10860 Ren´ ´ e J. Moreau (France): Relevance of Alfven Waves in Process Metallurgy under a High Magnetic Field 11386 Mohammed El Ganaoui (France): Eﬀect of Thermal Boundary Modulation in a Restricted Fluid Domain of a 3D Vertical Bridgman Apparatus 11893 Viatcheslav V. Kolmychkov (Russia): 3D Computer Simulation of Time-Depended Solutal Convection 11903 Geoﬀrey M. Evans (Australia): Liquid and Gas Jets Impinging on a Moving Wetted Surface 11929 Vadim I. Polezhaev (Russia): Convective Instabilities in Czochralski Model 12349 Andreas Cramer (Germany): New Possibilities for Velocity Measurements and Model Experiments in Liquid Metal Processing 12591 Fran¸¸cois Dupret (Belgium): Dynamic Simulation of the Entire Crystal Growth Process: Multi-Scale Analysis of Melt Flow Transients 12636 Othman Bouizi (France): Sensitive Regions and Optimal Perturbations in the Floating Zone Using the Adjoint System

FM16 – Fluid mechanics of suspension (R. Bonnecaze, USA & E. Guazzelli, France) 10248 Fran¸cois Feuillebois (France): Eﬀective Viscosity of an Inhomogeneous Dilute Suspension Flowing Along a Wall 10361 Andreas Acrivos (USA): Velocity Fluctuations in Non-Brownian Suspensions Undergoing Simple Shear Flows 10603 Vishwajeet Mehandia (India): The Collective Dynamics of Self-Propelled Particles 10993 Krzysztof Sadlej (Poland): Microstructure of a Dilute Sedimenting Suspension 11236 B.U. Felderhof (Germany): Sedimentation of Dilute Suspensions 11409 Maria L. Ekiel-Je˙z˙ ewska (Poland): Relaxation Time for Sedimenting Spheres of a Suspension with Periodic Boundary Conditions 11418 Anthony Ladd (USA): Numerical Simulations of Particle Suspensions in a Rotating Flow 11431 Gerhard Naegele (Germany): A Mode-Mode Coupling Scheme of Colloidal Electrolyte Friction 11473 Helen J. Wilson (UK): The Eﬀect of Diﬀerent Particle Contacts on Suspension Rheology 11480 Evgeny S. Asmolov (Russia): Evolution of Suspension Sedimenting in a Container Bounded by Horizontal Walls 11588 Elisabeth Guazzelli (France): Spreading Fronts and Fluctuations in Sedimentation: Part I Experiments 11725 Alan L. Graham (USA): Constant Force and Constant Velosity Momentum Tracers in Concentrated Suspensions 11749 Cyril Cassar (France): Flow of a Concentrated Suspension Down a Rough Plane 11850 Jeﬀrey F. Morris (USA): Inertial Migration of Rigid Spherical Particles in Poiseuille Flow 11884 Gabriel Seiden (Israel): Segregation of Suspended Particles in a Rotating Fluid-Filled Horizontal Cylinder – Experiment and Theory

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11991 Roger T. Bonnecaze (USA): Migration of Buoyant Mono- and Bi- Disperse Suspensions in Low Reynolds Number Pressure-Driven Pipe Flow 12005 David Saintillan (USA): Dynamic Simulations of the Instability of Sedimenting Fibers 12056 Howard Stone (USA): Mobility of Membrane-Trapped Particles: Protrusion into the Surrounding Fluid 12345 E. John Hinch (UK): Spreding Fronts and Fluctuations in Sedimentation: Part II Computer Simulations 12452 Ileana C. Carpen (USA): Single Particle Motion in Colloidal Dispersions 12574 Michel Cloitre (France): Slip and Flow in Pastes

FM17 – Granular ﬂows (R. Behringer, USA & I. Goldhirsch, Israel) 10253 Detlef Lohse (Netherlands): Impact 10371 Joe D. Goddard (USA): Maximum-Entropy Estimates and Virtual Thermomechanics for Granular Assemblies 10524 Herbert E. Huppert (UK): Granular Column Collapse 10959 J. Rajchenbach (France): Gravity Flow of a Densely-Packed Granular Material 11169 Irena Sielamowicz (Poland): Particle Image Velocimetry Analysis of Granular Material Flows 11751 Maxime Nicolas (France): Pore Pressure Relaxation During Granular Compaction 11775 Pierre Jop (France): Granular Flows on a Heap 11876 Osamu Sano (Japan): Collapse, Growth and Merging of Cavity Regions in a Granular Material Due to Viscous Flow 11883 Akiko Ugawa (Japan): Undulations and Ripples of a Thin Granular Layer Due to Vertical Vibration 12073 Christine Hrenya (USA): Species Segregation Driven by a Granular Temperature Gradient 12308 Isaac Goldhirsch (Israel): Kinetics of Weakly Frictional Granular Gases 12338 John R. de Bruyn (Canada): Morphology and Scaling of Impact Craters in Granular Media 12399 Lou Kondic (USA): Extended Granular Temperature 12445 J.C. Tsai (USA): Evolution of Internal Structure of Sheared Dense Granular Flows: Crystallization and History-Dependent Final States 12790 Radoslaw L. Michalowski (USA): Arching in Granular Media 12883 Renaud L. Delannay (France): Transverse Motion, Segregation and Rotations in 2D Granular Flows 13029 Sylvain Courrech du Pont (UK): Velocity Proﬁles During Granular Avalanches

FM18 – Low-Reynolds-number ﬂow (R.H. Davis, USA & C. Pozrikidis, USA) 10297 Ryszard Staroszczyk (UK): Radially Symmetric Polar Ice Sheet Flow with Evolving Anisotropic Fabric 10565 Robert B. Jones (UK): Hydrodynamic Interaction of a Spherical Particle in Poiseuille Flow Between Planar Walls 10766 Mark G. Blyth (UK): Two-Layer Stagnation Point Flows 10983 Lisa A. Mondy (USA): Free Surface Deformation in Suspensions Near a Rotating Rod 11300 Bogdan Cichocki (Poland): Particles Located on a Planar Free-Surface-Hydrodynamic Interactions in Quasi-Two-Dimensional System 11460 Alexander Prokunin (Russia): Microcavitation and Detachment of a Stokes Particle in Near-Wall Slow Motion 11915 Emin Fuad Kent (Turkey): Flow Visualization Experiments of Cellular Stokes Flows Induced by Rotation of a Cylinder Variously Positioned Inside Channels 11922 Izabella Pie´ n ´ kowska (Poland): Many-Sphere Hydrodynamic Interactions: Weak Convective Inertia Eﬀects 12164 Michal Branicki (UK): Viscous Eddy Structures in an Oscillating Cylinder with Sharp Corners

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xlvii

12859 George K. Karapetsas (Greece): Transient Squeeze Flow of Viscoplastic Liquids 12981 Devanayagam Palaniappan (Qatar): Slow Rotation of a Double Sphere in a Viscous Fluid

FM19 – Magnetohydrodynamics (P.A. Davidson, UK & A. Thess, Germany) 10025 Antoine Sellier (France): Migration and Interaction of two Conducting Particles Freely Immersed in a Liquid Metal 10235 Vladimir Shtern (USA): Bifurcation of Conical Magnetic Field 10931 Ulrich M¨ u ¨ller (Germany): Complementary Experiments at the Karlsruhe Dynamo Test Facility 11120 Jurij B. Kolesnikov (Germany): Liqiud Metal Flow Under Inhomogeneous Magnetic Field 11124 Peter A. Davidson (UK): Small-Scale Motion in the Core of the Earth 11429 Hubert Baty (France): Magnetohydrodynamic Instabilities of Astrophysical Jets 11458 Yuji Hattori (Japan): Magnetohydrodynamic Motion of Toroidal Magnetic Eddies 11621 Jacques L´ ´eorat (France): Fluid Dynamos and Precession Driving 11637 Jungwoo Kim (S. Korea): Large Eddy Simulation of Magnetic Damping of Jet 11681 Daniel P. Lathrop (USA): Observations of the Magnetorotational Instability in Spherical Couette Flow 11809 Steinar Borve (Norway): Simulating the Orszag-Tang vortex using RSPH 11906 Uwe Krieger (Germany): Homogenisation of Electrically Heated Glass Melts by Lorentz Forces 12090 Andre D. Thess (Germany): A Model for Liquid Metal Current Limiters 12107 Krzysztof A. Mizerski (Poland): The Magnetohydrodynamic Couette Flow in a Plane and Spherical Geometries with Singular Hartmann Boundary Layers 12120 Avalos-Zu˜ n ˜iga (France): Mean Electromotive Force for a Ring of Helical Vortices 12125 Andreas Tilgner (Germany): Numerical Simulations of Dynamo Experiments 12290 Vaclav Kocourek (Germany): Stability of Liquid Metal Drops Aﬀected by HighFrequency Magnetic Fields 12330 Nathanael ¨ Schaeﬀer (France): Quasi-Geostrophic Dynamos 12363 Serge Barral (Poland): Model of Gas Flow Inside a Plasma Thruster 12587 Jorg ¨ Stiller (Germany): Numerical Study of the Flow in a Finite Cylinder Driven by a Rotating Magnetic Field 12605 Hartmut Brauer (Germany): Interface Reconstruction in Cylindrical Two-Compartment-Systems Using Magnetic Field Tomography 12613 Frank Stefani (Germany): Contactless Inductive Flow Tomography: Theory and Experiment 12979 Jonathan A. Mestel (UK): Dynamo Action in Steady Helical Pipe Flow

FM20 – Multiphase ﬂows (S. Balachandar, USA & J. Magnaudet, France) 10416 Konstantin Volkov (UK): Large-Eddy Simulation of Particle Dispersion in the Duct with Fluid Injection 10690 Maria Joseﬁna Ferreira (Portugal): Sings of Flooding Instability in Inclined Liquid Films at High Pressure and Mass Transfer in High Density Gas Slugs 10843 Daniel Joseph (USA): Viscous and Viscelastic Potential Flow 11189 Urbano J. S´ ´ anchez Dominguez (Spain): Separation and Sorting of Heavy Particles Suspended in a Fluid by Settling in a Periodic Vorticity Field 11655 Elena Trostinetsky (Israel): Gas-Liquid Interfractial Distribution in Inclined Downward Pipe Flow 11745 Stanislaw Anweiler (Poland): Videogrametry in Fluidized Bed Reactors 12092 Thomas Seon (France): Gravity Induced Mixing of Miscible Fluids in Vertical and Inclined Tubes 12139 Frederic Risso (France): Oscillatory Motion of Freely-Moving Light Bodies: from Cylinders to Disks

xlviii

ICTAM04

Touvia Miloh I. (Israel): Non-Uniform Flow Hydrodynamics of Deformable Shapes Veronique Roig (France): Mean Motion Induced in a Liquid by Rising Bubbles Jacobus J. Derksen (USA): Plane Couette Flow of Dense Liquid-Particle Suspensions Daniel Zaj¸ac (Poland): Image Processing Method in Estimation of Bubble Column’s Work 13012 Cristian Marchioli (Italy): Statistics and Preferential Distribution of Micro-Particles in Turbulent Boundary Layer: Implications for Resuspension Mechanisms

12342 12383 12397 12886

FM21 – Solidiﬁcation and crystal growth (M. Glicksman, USA & M.G. Worster, UK) 11058 Chuan F. Chen (USA): Experimental Observations of Hydrate Formation in a Convection Tank 11208 Chih-Ang Chung (Taiwan): Morphological Stability of Directional Solidiﬁcation under Temperature Modulations 11270 Peter Guba (UK): Nonlinear Oscillatory Convection in Mushy Layers 11320 Pascale Aussillous (UK): Magnetic Resonance Imaging of Structure and Convection in Solidifying Mushy Layers 11328 Andrew Thompson (USA): Solidiﬁcation and Compositional Convection of a Ternary Alloy 11331 Michael A. Gonik (Russia): AHP Setup for Low Laminar Melt Flow Study in Crystal Growth 11362 Liliana Braescu (Romania): Optimization of the Growth Conditions of a Nd:YVO4 Cylindrical Bar 11364 Jerzy Banaszek (Poland): Front Tracking Technique on a Fixed Grid in Modelling of Binary Mixture Soldiﬁcation with Natural Convection 11926 George G. Tsypkin (Russia): Salt Precipitation in Geothermal Reservoirs 12276 Jacqueline Ashmore (UK): Boundary-Layer Analysis of Chimney Structures in Mushy Layers 12278 Irina Fayzrakhmanova (Russia): Travelling Magnetic Field Inﬂuence on Crystal Growth by Bridgman Method 12312 Gustav Amberg (Sweden): A Semi-Sharp Phase Field Method for Quantitative Phase Change Simulations 12365 Sven Eckert (Germany): Directional Solidiﬁcation of Pb-Sn Alloys Aﬀected by a Rotating Magnetic Field 12543 Marc Georgelin (France): Cell Shapes in Directional Solidiﬁcation: a Global Study 12743 Leszek Czechowski (Poland): Convection Driven by Tidal Heating: Numerical Model and Parameterized Theory 12768 Daniel M. Anderson (USA): Ternary Alloy Convection in Mushy Layers 13008 Chang Kyun Choi (S. Korea): Convective Instabilities During Solidiﬁcation of a Mushy Layer

FM22 – Stirring and mixing (H. Aref, USA & E. Villermaux, France) 10130 Frank C.G.A. Nicolleau (UK): Development of the Fractal Dimension of Material Elements in Homogeneous Isotropic Turbulence Using Kinematic Simulation 10497 Stephen M. Cox (Australia): Chaotic Advection in a Mixer with Changing Geometry 10532 Patrice Meunier (France): Enhanced Mixing by Vortices 10896 Jorg ¨ Schumacher (Germany): Geometric Features of High-Schmidt Number Scalar Mixing 11190 Alain Pocheau (France): Front Propagation in Laminar Cellular Flows: an Experimental Study 11293 Stephen Wiggins (UK): On the Design of 3D Micromixers Having the Bernoulli Property 12048 Alain Pumir (France): Intermittent Distribution of Heavy Inertial Particles in Turbulent Flows

Scientiﬁc Program

xlix

12110 Alexandre Gourjii (Ukraine): Chaotic Stirring of Passive Fluid by a Vortex Pair in Circular Domain 12158 Mark A. Stremler (USA): Chaotic Advection and Mixing in Pulsed Source-Sink Systems 12209 Shenqyang Shy (Taiwan): On Dissipative Structures of Stirring-Grids Turbulence 12257 Tatyana S. Krasnopolskaya (Ukraine): Evaluation of Transport Properties by Exchange Matrix Method 12326 Emmanuel Villermaux (France): Mixing Is an Aggregation Process 12415 Jerzy Baldyga (Poland): Stirring and Mixing Eﬀects in Agglomerative Precipitation 12439 Dmitri L. Vainchtein (USA): Resonances and Mixing in Stokes Flows 12442 Philip Boyland (USA): Mixing in Multiconnected Planar Domains 12545 Richard J. Keane (UK): Eulerian Measures for Lagrangian Stirring in a Thermally Driven Flow 12610 Marek Jaszczur (Poland): An Analysis of Mixing Process in a Static Mixer 12633 Dennis van der Woude (Netherlands): Stirring by Blinking Rotlets in a Bounded Stokes Flow 12731 Alain Bergeon (France): Weak Inertia and Mixing Between Rough Surfaces 12761 Frederic Bottausci (USA): Active Shear Superpositon Micromixer

FM23 – Topological ﬂuid mechanics (P.L. Boyland, USA & K. Ohkitani, Japan) 10247 Morten Brons (Denmark): Streamline Topology of the Nearwake of a Circular Cylinder at Low Reynolds Numbers 10938 Koji Ohkitani (Japan): Eulerian-Lagrangian Analysis of Navier-Stokes Turbulence 11060 Matthew D. Finn (UK): Topological Chaos in Simple Mixers 11166 Tsutomu Kambe (Japan): Gauge Principle for Ideal Fluids and Variational Principle 11660 Yoshi Kimura (Japan): Particle Transport by a Vortex Soliton 11677 Robert W. Ghrist (USA): Generic Hydrodynamic Instability 12484 Mitsuaki Funakoshi (Japan): Relation Between Mixing Eﬃciency and Geometrical Property of Stable Manifolds 12868 Dmytro I. Cherniy (Ukraine): Topological Aspects of the Tornado Problem

FM24 – Turbulence (Olivier Metais, France & R.D. Moser, USA) 10149 Julian Andrzej Domaradzki (USA): Large Eddy Simulations of Decaying Rotating Turbulence 10455 Maxim S. Loginov (Germany): Large-Eddy Simulation of Shock-Wave / TurbulentBoundary-Layer Interaction 10506 Sebastien Poncet (France): Experimental Study of Rotor-Stator Flows with Centripetal Fluxes 10564 Marta Waclawczyk (Poland): PDF Computation of Turbulent Flows with a New Near-Wall Model 10937 Naoya Takahashi (Japan): Interaction Between a Columnar Vortex and External Turbulence 11116 Rainer Friedrich (Germany): Turbulence Scalings in Supersonic Channel Flow 11151 Enrico Pasero (Italy): On the Scale Similarity in Large Eddy Simulation 11161 Tomasz Lipniacki (Poland): Two Scale Approach to Anisotropic Turbulence in Hel II 11256 Stefan Hickel (Germany): Optimization of an Implicit Subgrid-Scale Model for LES 11303 Agnes Maurel (France): Study of the Turbulent Energy Spectrum Build Up in an Experimental Vortex Burst 11454 Bernd R. Noack (Germany): Empirical Galerkin Models for Incompressible Flow — Pressure-Term and ’Subgrid’ Turbulence Representations 11488 Yu-L. Liu (China): Orthonormal Wavelet Analysis of CGT in Fully Developed Asymmetric Turbulent Channel Flow 11802 Xiangyu Hu (Germany): The Cellular Structure and Its Tracks of a H2 /O2 /Ar Detonation Waves

l

ICTAM04

11814 Frederic Moisy (France): Energy Spectrum in Rotating Turbulence 12004 Akira Rinoshika (Japan): Three-Dimensional Turbulent Structures of Diﬀerent Scales 12151 Tomomasa Tatsumi (Japan): Inertial Similarity of Velocity Distributions in Homogeneous Isotropic Turbulence 12430 Robert M. Kerr (UK): A New Mixed Nonlinear LES Models for Boundary Layers 12564 Thomas Indinger (Germany): 3D-Measurements in an Adverse-Pressure-Gradient Turbulent Boundary Layer over Smooth and Ribbed Surfaces 12652 Song Fu (China): POD Analysis of Coherent Structures in Turbulent Flows 12675 Vladimir I. Borodulin (Russia): Resonant Interactions of 3D Instability Waves in an Airfoil Boundary Layer for Harmonic and Broadband Perturbations 12740 Pierre Comte (France): Compressibility Eﬀects and Sound Propagation in Turbulent Channel Flow 12965 Hassan M. Nagib (USA): Impact of Pressure-Gradient Conditions on High Reynolds Number Turbulent Boundary Layers

FM25 – Vortex dynamics (G.J.F.van Heijst, Netherlands & E. Krause, Germany) 10683 11391 11467 11615 11676 11683 11741 11964 12032 12041 12051 12080 12108 12161 12196 12275 12690 12748 12821 12836

Miguel A. Herrada (Spain): New Means of Vortex Breakdown Control Paul Billant (France): Instabilities of a Vortex Pair in a Stratiﬁed and Rotating Fluid Eugene Benilov (Ireland): Stability of Oceanic Vortices: a Solution to the Problem Yasuhide Fukumoto (Japan): Curvature Instability of a Vortex Ring Fernando L. Ponta (USA): Numerical Experiments on Vortex Shedding From an Oscillating Cylinder Denis Blackmore (USA): Bifurcation of Motions of Three Vortices and Applications Hung-Cheng Chen (Taiwan): Strong Cyclonic Vortices over Topography on a BetaPlane Gert Jan F. van Heijst (Netherlands): Spontaneous Sign Reversals in Self-Organized States of Forced Two-Dimensional Turbulence on a Bounded Square Domain Felix B. Kaplanski (Estonia): A Model for the Formation of ’Optimal’ Vortex Rings with Taking into Account Viscosity Katsuya Ishii (Japan): Numerical Simulation of Vortical Flows Using a Highly Accurate Finite Diﬀerence Scheme Michael D. Patterson (UK): The Development of an Axisymmetric Gravity Current Ewa Tuliszka-Sznitko (Poland): Numerical Investigation of the Laminar-Turbulent Transition of the Flow in a Rotor-Stator Cavity Ramiro Godoy-Diana (France): Viscous Vertical Length Scale Selection in Stratiﬁed Fluids S. Balachandar (USA): On Local Vortex Identiﬁcation Vyacheslav V. Meleshko (Ukraine): The Modelling of The Dynamics of Hairpin Vortex Packets in Wall Turbulence Marcin Kurowski (Poland): Coherent Structure of Point Vortices Inﬂuenced by Uniform Straining Flow Wolfgang Schr¨ ¨ oder (Germany): Strong Shock-Vortex Interaction a Numerical Study Oscar U. Velasco Fuentes (Mexico): Isolated Vortices over Seamounts: Laboratory Experiments and Numerical Simulations Klaus W. Hoyer (Switzerland): Three Dimensional Velocity Field of Vortices Impinging on a Wall Obtained by Scanning Particle Tracking Velocimetry Pawel Regucki (Poland): Study of the Vortex Rings Interaction by 3d Vorticity Particle-In-Cell Method

FM26 – Waves (W.K. Melville, USA & V.I. Shrira, UK) 10245 Lev Shemer (Israel): Unidirectional Steep Waves in Wave Tanks 10746 Tetsu Hara (USA): Wave Breaking and Equilibrium Surface Wave Spectra 10762 T.R. Akylas (USA): Propagation and Interactions of Nonlinear Internal Gravity Wave Beams

Scientiﬁc Program

li

10861 Vladimir E. Zakharov (USA): Weak-Turbulent Theory of Wind-Driven Sea 11126 Hu Huang (China): Shallow-Water Theory for Wave-Current-Bottom Interactions 11134 Gennady El (UK): Unsteady Undular Bore Transition in Fully Nonlinear Dispersive Wave Dynamics 11144 Anatoli Ivanovich Dobrolyubov (Belarus): The Theory of Travelling Deformation Waves and Its Applications in Biomechanics, Engineering, and Geophysics 11179 William R. Phillips (USA): The Spacing of Langmuir Circulation in Strong Wavy Shear 11199 Thomas Peacock (USA): Experiments on Rotating and Reﬂecting Internal Wave Beams 11230 Paul A. Hwang (USA): Spatio-Temporal Measurements of Capillary-Gravity Waves 11290 G´ ´ erard Iooss (France): Standing Gravity Waves in Deep Water 11345 Emilian I. R´ ´ ar´ au (UK): Nonlinear Three-Dimensional Free Surface Flows in Finite and Inﬁnite Depth 11417 Dorian Fructus (Norway): Dynamics of Crescent Wave Patterns in a Channel 11468 Colm Howlin (Ireland): Evolution of Packets of Surface Gravity Waves over Smooth Topography 11548 Walter Craig (Canada): Three Dimensional Gravity Water Waves 11591 Vasyl P. Lukomsky (Ukraine): Sharpening and Breaking of Subharmonic Gravity Waves on Deep Water 11680 Xin Zhang (USA): Short Wind Waves and Surface Wind Drift 11746 Takeshi Kataoka (Japan): Transverse Instability of Surface Solitary Waves 11803 Jan Erik Weber (Norway): A Lagrangian Approach to Wave-Induced Oceanic Mass Transport 12087 Victor I. Shrira (UK): Eﬀect of Horizontal Component of the Coriolis Force on Propagation of Near-Inertial Waves in the Ocean 12089 Fred´ ´ ´ eric Dias (France): Generalized Internal Solitary Waves and Fronts 12268 Igor A. Brovchenko (Ukraine): Intermittent Mixing by Multiscale Breaking of Wind Waves: Implications for Oil Dispersion 12424 Chantal Staquet (France): Focusing of an Inertia-Gravity Wave Packet with a Baroclinic Shear Flow 12715 Jean-Marc Vanden-Broeck (UK): Steep Capillary Waves in Electriﬁed Fluid Sheets 12813 Georgy I. Burde (Israel): Bi- Directional Water Waves and Integrable High Order KDV Equations

SM1 – Computational solid mechanics (T. Belytschko, USA & P. Wriggers, Germany) 10270 Chung-Yue Wang (Taiwan): Elastic-Plastic Large Deformation Analysis of 2D Frame Structure 10274 Terumi Touhei (Japan): Multiscale Analysis of Scattered Elastic Waves Based on the Lippmann-Schwinger Equation ¨ (Germany): Model Updating a Multicriteria Optimization 10423 Hans H. Muller-Slany Process in Mechanics 10559 Yuan Lin (China): Experiment and Quasicontinuum Simulation of Nanoindentation of Single Crystal Copper 10665 Vasilios G. Mokos (Greece): A BEM Solution to Transverse Shear Loading of Beams 10728 Anatoli Stulov (Estonia): Mechanical Features of Piano Hammer Felt 10829 Larry D. Libersky (USA): A Dual Particle Computational Method for Continua 11154 Jaroslaw Knap (USA): Mesh Optimization for the Quasicontinuum Method: A Generalization of VALE 11193 Jiann-Tsair Chang (Taiwan): Derivation of the Higher-Order Stiﬀness Matrix of a Space Frame Element for Geometric Nonlinear Analysis of Structrues 11202 Pavlo A. Steblyanko (Ukraine): The Method of Solving of Non-Stationary Coupled Problems of the Theory Thermal-Plasticity for the Rotation Shells

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11211 Julia Mergheim (Germany): A New Approach for the FE Modelling of Cohesive Cracks 11291 Stanislaw A. Lukasiewicz (Canada): Eﬀective Solution for Finite Element Problems with Nonlinear Constraints 11298 Etienne L.G. Pecquet (Belgium): Lingopti Project: Semi-Continuous Casting Process of Copper-Nickel Alloys 11334 George Mejak (Slovenia): Two Scale Finite Element Method 11389 Grzegorz W. Zboi´ n ´ ski (Poland): Problems of Application of Hierarchical Modelling, Displacement FEM and a Posteriori Residual Error Estimation to Static and Dynamic Adaptive Analysis of Complex Structures 11482 Florian Kovacs (Hungary): Statics And Kinematics of Symmetric Swelling Viruses 11580 Jixin Yang (China): A Numerical Approach for Large-Scale Computation CEM 11720 Zhenhan Yao (China): Some Investigations on FM Bem in Solid Mechanics 11769 Anthony Nouy (France): Radial-Type Approximation Technique for a Space-Time Multiscale Computational Strategy 11797 Julien R´ ´ethor´e (France): An Energy Conserving Scheme for Time Dependent Problems Using the Extended Finite Element Method 11817 Seyoung Im (S. Korea): Development of a Novel ’Crack’ Finite Element for Propagation Simulation 12054 Alexandre V. Vakhrouchev (Russia): Modelling of Static and Dynamic Processes of Nanoparticles Interaction 12101 Andras Lengyel (Hungary): Singularities of the Four-Sided Antiprism Ring 12115 Ercan Guerses (Germany): Analysis of Evolving Deformation Microstructures in Instable Inelastic Solids Based on Energy Relaxation Methods 12131 Rene L.J.M. Ubachs (Netherlands): Microstructural Behaviour of Solder Joints 12167 Sergey N. Medyanik (USA): Molecular Mechanics Simulations of Carbon Nanostructures Using Multi-Scale Boundary Conditions 12177 Hirohisa Noguchi (Japan): Multiscale Buckling Analyses of Corrugated Fiberboard 12237 Pierre Feissel (France): Modiﬁed Error in Constitutive Relation and Its Application to Dynamic Tests with Corrupted Boundary Conditions 12310 Juha A. M¨ ¨ akipelto (Finland): Geometry Based Rational Enrichment Functions for Triangular Plane Elasticity Element 12340 Marek S. Kara´s (Poland): Solving of Indirect Problems Using Treﬀtz Method 12351 Alexey V. Borisov (Russia): Tensor Invariants and Mechanisms of Transition to Chaos in Nonholonomic Dynamical Systems 12359 Mathieu Cloirec (France): Analysis of a Structural Detail Using a Two-Scale Approach 12441 Robert B. Haber (USA): Adaptive Discontinuous Galerkin Method for Elastodynamics on Unstructured Spacetime Grids 12455 Marino Arroyo (USA): Continuum Mechanics and Carbon Nanotubes 12458 Katerina D. Papoulia (USA): Toward Convergence in Initially Rigid Cohesive Fracture Models 12534 Stefan Loehnert (Germany): Computational Homogenisation of Microheterogeneous Materials Including Decohesion at Finite Strains 12562 Antoni John (Poland): The Load Cases in Numerical Model of Pelvic Bone with Artiﬁcial Acetabulum 12579 Andrzej Siemaszko (Poland): Shakedown Safety Criterion in Reliability Analysis 12584 Eiris F.I. Boerner (Germany): A New Finite Element Formulation Based on the Theory of a Cosserat Point 12607 Ilson P. Pasqualino (Brazil): Arc-Length Method for Explicit Dynamic Relaxation 12723 Ekaterina Viatkina (Netherlands): Modelling of Non-Uniform Deformation of Metals with Dislocation Cell Structure 12727 Frederic Grondin (France): The Numerical Homogenization of the Concrete Behavior 12737 A. Amine Benzerga (USA): Discrete Dislocation Calculations of the Stored Energy of Cold Work

Scientiﬁc Program

liii

12844 Huu Nam Tran (Czech Republic): Deformation Analysis of Inﬂated Cylindrical Membrane of Composite Material with Rubber Matrix Reinforced by Cords 12845 Dimitri E. Beskos (Greece): Dynamic Analysis of Gradient Elastic Solids by BEM 12901 Kian-Meng Lim (Singapore): Variable-Order Singular Boundary Element for Calculation of Three-Dimensional Stress Intensity Factors 12924 Ellen Kuhl (Germany): Application of the Material Force Method to Structural Optimization

SM2 – Contact and friction mechanics (A. Klarbring, Sweden & G. Szefer, Poland) 10211 Mykhaylo G. Pantelyat (Ukraine): Thermocontact Interaction of Bodies of Revolution During Induction Heating 10342 Victor M. Musalimov (Russia): Dynamic Characteristics and Monitoring of Rubbing Surfaces Quality 10459 Herman N.V. Parland (Finland): Contact Mechanical Analysis of Elastic Multibody Structures 10686 L.E. Anderson (Sweden): Existence and Uniquness of Steady State Solutions in Thermoelastic Contact With Frictional Heating 10718 Hamid Reza Irannejad (Iran): FE Analysis of Bond for Smooth FRP Rods Embedded in Concrete 10999 Sergey A. Chizhik (Belarus): Modelling of Contact of Structured Materials Based on Data from Scanning Probe Microscopy 11008 Graham J. Weir (new zealand): A Universal Property of Geometrical Hardening 11074 Istvan Paczelt (Hungary): Contact Optimization Problems Associated with the Wear Process 11135 Feodor M. Borodich (UK): Molecular Adhesive Contact for Indenters of Non-Ideal Shapes 11196 Irina G. Goryacheva (Russia): Adhesive Component of the Rolling Friction Force 11215 Leon M. Keer (USA): Fundamental Relations for Frictional and Adhesive Nanoindentation Tests 11365 Denis Elaguine (Sweden): Hertz Contact at Finite Friction and Arbitrary Proﬁles 11452 Jozef ´ Joachim Telega (Poland): Frictional Contact with Wear Diﬀusion 11481 Jan Awrejcewicz (Poland): On the Contact Thermoelastic Problem with Frictional Heating, Wear and Auto-Vibrations 11611 Ganna Shyshkanova (Ukraine): Three-Dimensional Problem of the Contact by Doubly Connected Domain Taking into Account Roughness and Friction 11613 Alfred Zmitrowicz (Poland): Evolutions of Friction Anisotropy and Heterogeneity 11697 Victor A. Shevchuk (Ukraine): Inverse Problems of Thermoelasticity for Frictionally Interacting Layers 11778 Alexey A. Kireenkov (Russia): Multidimensional Model of Combined Dry Friction 11933 Yuriy Letser (Ukraine): Numerical Modeling of Contact Fracture of Elasto-Plastic Cracked Bodies 11987 Demirkan Coker (USA): Crack-Like and Pulse-Like Modes of Frictional Sliding along an Interface Under Dynamic Shear Loading 12082 Leila Abdou (France): Experimental and Numerical Study of the Brick-Mortar Interface 12086 Markus Lindner (Germany): Experimental and Analytical Investigation of Rubber Friction 12382 Marius Cocou (France): A Dynamic Unilateral Contact Problem for a Cracked Body 12412 Larissa Gorbatikh (USA): A Simple Model to Account for the Locking Eﬀect Between Two Rough Surfaces under Cyclic Loading 12553 Stanislaw Stupkiewicz (Poland): Boundary Layers Induced by Contact of Rough Bodies 12559 Alexandr A. Olshevsky (Russia): The Accounting of Surface Roughness in Contact of Arbitrary Shaped Bodies Using FEM

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ICTAM04

12563 Thibaut Putelat (UK): Frictional Sliding of a Multislip System 12646 Antonio Pinto da Costa (Portugal): Second-Order Cone Complementarity Formulation for Quasi-Static Incremental Frictional Contact Problem in Three-Dimensional Space 12773 Yves Gonthier (Canada): A Novel Contact Model Based on Volumetric Information

SM3 – Control of structures (F. Chernousko, Russia & S. Pellegrino, UK) 10098 10457 10458 10879 11036 11163 11219 11243 11265 11287 12756 13004

Anupam S. Ahlawat (India): Coupled Optimal Design of Building with TMD Bartlomiej Blachowski (Poland): Optimal Vibration Control of Guyed Masts Pawel Holobut (Poland): Time-Optimal Control of Hydraulic Manipulators Kazuo Tanizawa (Japan): Surface Accuracy of Inﬂatable Reﬂector Covered with Stretched Cable Daniela G. Marinova (Bulgaria): H-inf Control for Smart Multistory Building Structures Firdaus E. Udwadia (USA): Exact Tracking Control for Nonlinear Structural and Mechanical Systems Nikolai N. Bolotnik (Russia): Pre-Acting Impact Isolation Systems Felix L. Chernousko (Russia): Control of Multibody Systems Moving along a Plane Agnessa S. Kovaleva (Russia): Control of Random Dynamics of a Rigid Rocking Block Jong-Dar Yau (Taiwan): Suppression of Train-Induced Vibrations of Continuous Truss Bridge by Hybrid TMDs Walerian Szyszkowski (Canada): Optimization of Active Control of Structural Vibration by the Beam Analogy Andre Fenili (Brazil): Control of a Nonlinear Slewing Flexible Beam

SM4 – Damage mechanics (M. Chrzanowski, Poland & P. Steinmann, Germany) 10011 Noel ¨ Challamel (France): Stability and Creep Damage of Quasi-Brittle Materials 10456 Akrum Abdul-Latif (France): Modeling of the Damage Evolution at the Granular Scale in Polycrystals under Complex Cyclic Loadings 10865 Robert Svendsen (Germany): Continuum Thermodynamic and Variational Modeling and Simulation of Ductile Failure at Large Deformation with Application to Engineering Structures 10960 Kari J. Santaoja (Finland): Material Models for Hookean Materials with Voids or Cracks 10961 Chi L. Chow (USA): Localized Necking Criterion Based on Acoustic Tensor for Materials with Anisotropic Damage 10974 Ren´ ´ e Billardon (France): An Elasto-Viscoplastic Model Coupled to Damage and Grain Growth to Take Account of Material Variability 11228 Andrzej Litewka (Portugal): Damage and Failure of Brittle Solids 11406 Sabine Ricci (Germany): Numerical Analysis of Nonlocal Anisotropic Continuum Damage 11414 Ekkehard Ramm (Germany): Discrete Models and Their Application in Damagemechanics 11415 Ron H.J. Peerlings (Netherlands): A Nonlocal Plasticity – Damage Formulation Based on the Micromechanics of Defect Growth 11665 Marcin Chrzanowski (Poland): Propagation of Cracks in Terms of Continuum Damage Mechanics 11779 Larisa V. Stepanova (Russia): An Asymptotic Analysis of Mode I Crack in Creeping Damaged Solids 11796 Jacek J. Skrzypek (Poland): Damage Acquired Anisotropy in Elastic-Plastic Materials 11885 Jean-Louis Chaboche (France): A CDM Approach of Ductile Damage with Plastic Volume Changes

Scientiﬁc Program

lv

11886 Issam Doghri (Belgium): Micromechanical Modelling of the Deformation and Damage of Inelastic Brittle Three-Phase Composites: Application to Fiber-Reinforced Concrete 12021 Fran¸cois Hild (France): Damage Field Identiﬁcation using Full-Field Displacement Measurements 12036 Jian-Ying Wu (China): A New Energy-Based Elastoplastic Damage Model for Concrete 12088 Henning Schuette (Germany): Lifetime Prediction with a Damage Model Based on Mixed-Mode Microcrack Propagation 12116 Serdar Goektepe (Germany): A Micromechanically Based Network Model for Rubbery Polymers Incorporating Mullins-Type Stress Softening 12277 Gilles Lubineau (France): Computational Micro-Meso Modeling for Laminates Under Thermomechanical Fatigue and an Oxidizing Atmosphere 12298 Ilaria Monetto (Italy): A Non-Associative Anisotropic Damage Model for Brittle Materials 12333 Andrzej Stachurski (Poland): Robust Identiﬁcation of an Augmented Gurson Model for Elasto-Plastic Porous Media 12335 Artur W. Ganczarski (Poland): Low Cycle Fatigue Based on Unilateral Damage Evolution 12341 Thierry J. Massart (Belgium): Coupled Meso-Macro Simulation of Masonry Cracking 12749 George Chatzigeorgiou (Greece): Coupling Between Progressive Damage and Permeability of Concrete 12913 Vincent P. Godard (France): Anisotropic Damage Model for Concrete Including Unilateral Eﬀects

SM5 – Dynamic plasticity of structures (N. Jones, UK & T. Wierzbicki, USA) 11022 Vikram S. Deshpande (UK): Blast Resistance of Clamped Sandwich Beams 11051 Tongxi Yu (China): Collision Between Two Deformable Structures 11107 D. Karagiozova (Bulgaria): Counterintuitive Response of Long Circular Tubes to Axial Impact 12043 Rami Masri (Israel): Self Similar Dynamic Expansion of a Spherical Cavity in Elastoplastic Media 12949 Piotr Perzyna (Poland): Numerical Investigation of Dynamic Shear Bands in Inelastic Solids as a Problem of Mesomechanics 12982 Narinder K. Gupta (India): On Non-Axisymmetric Collapse of Thin Tubes

SM6 – Elasticity (R. Kienzler, Germany & L. Wheeler, USA) 10442 Katarzyna Kowalczyk-Gajewska (Poland): On Invariants of the Elasticity Tensor for Orthotropic Materials 10972 Tsolo P. Ivanov (Bulgaria): Motion and Stability of an Elastic Heavy Top 11220 David M. Haughton (UK): Stability of Compressible Elastic Blocks 11523 Reuven Segev (Israel): Generalized Stress Concentration Factors 11526 Lenser A. Aghalovyan (Armenia): On Asymptotic Method of Static and Dynamic Boundary Problems Solution 11901 Charles Ruimy (France): Axisymmetric Force Solution for a Semi-Inﬁnite Cubic Solid 11960 Tungyang Chen (Taiwan): An Exactly Solvable Microgeometry in Torsion 11998 Xiaojing Zheng (China): A New Nonlinear Constitutive Relation for Magnetostrictive Materials 12039 Gaza H. Maluleke (South Africa): Nonlinear Radial Oscillations of Anisotropic ThinWalled Cylindrical Tubes 12122 Marcelo Epstein (Canada): Nonlocal Eshelby Entities: a One-Dimensional Example 12189 Iwona M. Jasiuk (USA): Analysis of Trabecular Bone as a Hierarchical Material 12289 Valery P. Matveyenko (Russia): Investigation of Couple-Stress Eﬀects in Elastic Bodies Under Deformation

lvi

ICTAM04

12324 Serge N. Gavrilov (Russia): New Analytical Approach for Investigation of NonStationary Dynamics of Media with Moving Inhomogeneities 12354 Gearoid P. Mac Sithigh (USA): Agmon’s Condition for Incompressible Elasticity: a Variational Formulation 12434 Elena F. Grekova (Russia): Modelling of Complex Elastic Crystals by Means of Micromorphic Gyrocontinua 12575 Vyacheslav V. Lyakh (Ukraine): Truncated Elastic Wedge under Torsional Load 12685 Valeriy A. Buryachenko (USA): Homogenization of Triply Periodic Elastic Media with Random Imperfections 12987 Roberta Sburlati (Italy): On the Impact Law in Elastic Plate-Like Bodies

SM7 – Experimental methods in solid mechanics (I. Emri, Slovenia & J.L. Freire, Brazil) 10492 Keyu Li (USA): An Optical Strain Rosette/Ring – Core Method Applied on Laser Weld 10691 Masashi Sato (Japan): Estimation of Principal Axes of Inertia 11556 Michael N. Osipov (Russia): The Whole Field Non-Destructive Optical Slicing Method in Three-Dimensional Photoelasticity 11617 Sylwester Samborski (Poland): Porous Ceramics – Experimental Research and Modelling 11764 Yilong Bai (China): Critical Sensitivity in Rock Experiments 12272 Vsevolod V. Shchennikov (Russia): Phase Transitions and Mechanical Properties of Ternary Chalcogenides 12551 Saed Mousavi (Sweden): SHPB Technique for Identiﬁcation of Complex Modulus Under Condition of Non-Uniform Stress 12666 Michal A. Miskiewicz (Poland): In-Situ Observation of Fatigue Crack Growth in Carbon Steel 12799 Tadeusz Uhl (Poland): New Solutions in Experimental Modal Analysis of Mechanical Structures

SM8 – Fatigue (J. Dominguez, Spain & K. Reifsnider, USA) 10698 Jaime Dominguez (Spain): Inﬂuence of Contact Conditions on Fretting Fatigue Under Spherical Contact 11371 Krystyna Majorkowska-Knap (Poland): Fatigue Investigations into a Composite Glider Structure 11686 Yi Sun (China): A Microscopic Mechanics Model 11737 Byeongchoon Goo (S. Korea): Fatigue Life Prediction Considering Residual Stress Relaxation 11920 Pavlo Maruschak (Ukraine): Eﬀects of Frequency Temperature and Loading Waveform on Fatigue Crack Growth Rate in Steel 15Kh13MF 11927 Steﬀen Brinckmann (Netherlands): Stress Concentrations Caused by Dislocations at the Free Surface 12034 Ki-Seok Kim (S. Korea): Probabilistic Analysis of Fatigue Crack Growth using Moment Method 12217 Ren´ ´ e C. Alderliesten (Netherlands): Energy Release Rate Approach for Delamination in a Fatigue Crack Conﬁguration in Glare 12228 Marion Risbet (France): Modelling Fatigue Crack Growth with Time-Derivative Equations 12391 Youshi Hong (China): Characteristics of Very-High-Cycle Fatigue for a High Carbon Low Alloy Steel 12914 Dorota I. Koca´ n ´ da (Poland): Modelling of Short Fatigue Crack Growth in a Metal in HCF Range

Scientiﬁc Program

lvii

SM9 – Fracture and crack mechanics (D. Gross, Germany & A. Needleman, USA) 10250 10334 10338 10419 10429 10579 10646 10676 10824 10864 10897 10910 11045 11195 11321 11425 11484 11568 11626 11856 11910 11934 11980 12044 12171 12174 12249 12378 12393 12492 12798

Michael P. Wnuk (USA): A Fractal Cohesive Crack Model Daniel Kujawski (USA): Inﬂuence of Stress State on Crack-Tip Driving Force Mokhtar Adda-Bedia (France): Branching Instability of Brittle Fracture Vidya Sagar Remalli (India): Size Eﬀect in Tensile Fracture of Concrete – A Study Based on Lattice Model Applied to CT-Specimen Octavian Pop (France): Numerical and Experimental Study of the Plastic Zone in the Vicinity of the Crack Tip by the Optical Caustics Method Vera E. Petrova (Russia): Thermoelastic Problems for a Bimaterial with Defects/Singularities Parissa Hosseini-Tehrani (Iran): Dynamic Crack Analysis Under Thermal Shock Huijian Li (China): Experimental Investigation on Concrete Shear Crack Extension Igor Guz (UK): Eﬀect of Inter- and Intralaminar Damage on the Compressive Fracture of Hyperelastic Materials Liviu Marsavina (Romania): Experimental and Numerical Crack Growth in a Special Geometry Nikita F. Morozov (Russia): Elastodynamics Problems in Domains with Edges Alan Needleman (USA): 3D Microstructural Eﬀects on Plane Strain Ductile Crack Growth Andreas Ricoeur (Germany): Weight Functions for Cracks in Piezoelectrics Oleksandr V. Menshykov (Ukraine): Elastodynamic Contact Problem for Two PennyShaped Cracks Jos´ ´e Dominguez (Spain): Numerical Approach for Dynamic Fracture in Piezoelectric Solids Yasuhide Shindo (Japan): Finite Element Analysis of Fracture and Polarization Switching Behavior in Modiﬁed Small Punch Testing of Piezoelectric Ceramics Axel M¨ u ¨ller (Germany): On Crack Assessment at Bimaterial Interfaces Hai-Tao Wang (China): A Quasi-Spherical Coordinate System and Its Application to the Determination of Vertex-Type Singularities Asher A. Rubinstein (USA): Failure Model of Protective Coatings Alla V. Balueva (USA): Modeling of Environment Assisted Delamination I. Quasistatic Case Ewa M. Turska (Poland): The Inﬂuence of Remote Stresses on the Near Crack Tip Stress Field Yu Shouwen (China): The Elasto-Plastic Thin Film/Substrate Via Buckle-Driven Delamination and Crack Growth Andrzej Kaczy´ n ´ ski (Poland): On 3-D Thermoelastic Problems of Interfacial Cracks in a Periodic Stratiﬁed Space Karsten Kolk (Germany): Automatic 3D Crack Growth Simulation Based on Boundary Elements Irene Arias (USA): Massively Parallel Simulations of Dynamic Fracture and Fragmentation of Brittle Solids Ping Wang (China): the Shield Eﬀect of Phase Transformation Stress Field at Crack Tip Volodymyr V. Loboda (Ukraine): Contact Zone Approach to the Analysis of Interface Cracks in Thermomechanically Loaded Piezoelectric Bimaterials Krishnaswamy Ravi-Chandar (USA): Interaction of Propagting Cracks and Shear Waves Jean-Baptiste M. Leblond (France): Disorder of the Front of a Tensile Tunnel-Crack Propagating in Some Inhomogeneous Medium Yichun Zhou (China): Creep Deformation in Thermal Barrier Coatings Due to the Eﬀect of Thermal Growth Oxidation and Temperature Gradient Abdulhamid Al-Abduljabbar (Saudi Arabia): Numerical Analysis of Strain Hardening and Pressure Sensitivity Eﬀects on J-Integral

lviii

ICTAM04

SM10 – Functionally graded materials (R. Batra, USA & G.H. Paulino, USA) 10045 Andrzej Tylikowski (Poland): Dynamic Stability of Functionally Graded Plate Under In-Plane Compression 10135 Lizhi Sun (USA): Micromechanics-Based Elastic Model for Functionally Graded Materials with Particle Interactions 10364 Jiann-Quo Tarn (Taiwan): A State Space Formalism for Piezothermoelasticity of Functionally Graded Materials 10501 Weichen Shi (China): Conservation Laws of Functionally Graded Materials in Elastodynamics 10836 Chuanzeng Zhang (Germany): Transient Dynamic Crack Analysis in FGMs Under Impact Loading 11177 Dhirendra V. Kubair (India): Asymptotic Ananlysis of a Stationary Crack in a Ductile Functionally Graded Material 11551 Linzhi Wu (China): The Plane Crack Problem in a Functionally Graded Orthotropic Strip 11572 Bing-Zheng Gai (China): Frictional Slip Between a Gradient Non-Homogeneous Layer and a Half-Space in Anti-Plane Elastic Wawe Field 11592 Juri Engelbrecht (Estonia): Wave Propagation in Functionally Graded Materials 11671 Eduard Rohan (Czech Republic): Adaptive Modelling of Microscopic Heterogeneous Medium Undergoing Large Deformation 12093 Joel ¨ A. Pouget (France): Actuator and Sensor Modelling for Laminated Piezoelectric Plates 12273 Arthur H. England (UK): Complex-Variable Methods Applied to Functionally-Graded Elastic Plate Problems 12461 Takemasa Seto (Japan): Study of Two-Dimensional Elasticity on FGM 12465 Yoshihiro Ootao (Japan): Three-Dimensional Transient Thermoelastic Analysis of Orthotropic Functionally Graded Rectangular Plate 12594 R.C. Batra (USA): Adiabatic Shear Bands in Functionally Graded Materials 12665 Emilio C.N. Silva (Brazil): Topology Optimization Applied to the Design of Functionally Graded Material (FGM) Structures 12968 Sathyanaraya Hanagud (USA): First Principles-Based Equations of State for Functionally Graded Materials 12978 Minoru Taya (USA): Design of FGM Bimorph Piezo-Actuators 13018 Lavinia S.A. Borges (Brazil): Thermoelastic Analysis of Functionally Graded Materials Submitted to Shocks

SM11 – Impact and wave propagation (A. Norris, USA & K. Wilma´ n ´ ski, Germany) 10072 Bettina Albers (Germany): Surface Waves on an Impermeable Boundary of a Poroelastic Medium 10114 Alexander M. Samsonov (Russia): Bulk Solitons do not Decay in Elastic Wave Guides 10344 Toshiaki Hata (Japan): Stress-Focusing Eﬀect in a Spherical Zirconia Inclusion with Dynamically Transforming Strains 10422 Toshiro Maeda (Japan): Simultaneous Simulation of Dispersion Curves and H/V Spectra 10426 Nikolai N. Myagkov (Russia): Nonlinear Waves in Shock-Loaded Solids 10612 Krzysztof Wilma´ n ´ ski (Germany): Critical Time for Acoustic Waves in Weakly Nonlinear Poroelastic Materials 10891 Semra Ahmetolan (Turkey): Rayleigh-Like Surface Waves on a Nonlinear Layered Elastic Half Space 11085 Istvan A. Veres (Switzerland): Non-Destructive Testing of Wood by Wave Propagation 11099 Ji Wang (China): A Two-Dimensional Analysis of Surface Acoustic Waves in Finite Anisotropic Solids with Electrodes 11242 Karima R. Khusnutdinova (UK): Nonlinear Wawe Processes in a Bi-Layer

Scientiﬁc Program

lix

11271 Gabriel E. Chao (Netherlands): Shock-Induced Surface Waves in Porous Reservoirs 11360 Janusz Klepaczko (France): Trapping of Plastic Waves by Adiabatic Deformation 11377 Kateryna V. Terletska (Ukraine): Modeling of Solitary Impulses in a Composite Material Using Wavelet Analysis 11401 Igor Selezov (Ukraine): Some Degenerated and Extended Wave Models of Elasto- and Hydrodynamics with Finite Velocity Disturbance Propagation 11596 Angelo Morro (Italy): Uniqueness Results for the Reﬂection-Transmission Problem 11620 Andres Braunbrueck (Estonia): Wave Interaction Resonances in Inhomogeneous Elastic Materials 11682 Martin Ostoja-Starzewski (Canada): Acceleration Wavefronts in Random Media 11812 Philippe S. Boulanger (Belgium): Inhomogeneous Circularly Polarized Waves in Orthorhombic Crystals 11813 Andrus Salupere (Estonia): Hidden and Driven Solitons in Microstructured Media 11853 Witold Kosi´ n ´ ski (Poland): Thermo-Mechanical Wave Propagation in Materials with Internal State Variables 11989 Sergey K. Kanaun (Mexico): Self-Consistent Methods in the Problem of Elastic Wave Propagation Through Matrix Composite Matrials 12203 Anton G. Pegushin (Russia): Waves of Deformation Propagation in Nonlinear Viscously Elastic Porous Material 12313 Bernhard Pichler (Austria): Elastoplasticity of Gravel Protecting Rockfall-Endangered Steel Pipelines 12418 Toshihiko Sugiura (Japan): Flaw Identiﬁcation by Angle Beam Electromagnetic Acoustic Transducers 12481 Anton M. Krivtsov (Russia): Impact Fracture of Rock Materials Due to Percussive Drilling Action 12514 Anil C. Wijeyewickrema (Japan): Dispersion and Stability Analysis of Waves in PreStressed Imperfectly Bonded Layered Composites 12568 Olari Ilison (Estonia): On the Propagation of Solitary Waves in Microstructured Solids 12763 Chokri Zammali (France): Level-Sets and Mixed Approaches for Dynamic Contact Problems 12772 Mahmoud I. Hussein (USA): Analysis and Design of Dispersive Materials and Structures

SM12 – Material instabilities (D. Bigoni, Italy & H. Petryk, Poland) 10265 Ahmed Benallal (France): Material Instabilites in Thermo-Mechanical Processes 10314 Peter B. Beda (Hungary): Dynamical Systems Theory in Material Instabilities 10867 Eugene I. Ryzhak (Russia): An Idea and Theory of Hypothetical Device for Investigating the Localization Phenomena 10996 Henryk Petryk (Poland): Incremental Energy Minimization in Material Instability Problems 11254 Davide Bigoni (Italy): Dynamics of Perturbations and Shear Band Instabilities 11277 Paul Steinmann (Germany): On Convexity Conditions in Spatial and Material Settings of Hyperelasticity 11432 Yoshihiro Tomita (Japan): Computational Characterization of Micro- to Macroscopic Mechanical Behavior of Carbon Black-Filled Rubber 11705 Jose Merodio (Spain): Material Instabilities of Fiber-Reinforced Nonlinearly Elastic Solids 12025 Walter J. Drugan (USA): Ultrastiﬀ Elastic Composites via Negative Stiﬀness Inclusions, and Material Stability Implications 12223 Pavel V. Tkachev (Russia): Stability of Ideal Inﬁnite Crystal Under Finite Uniform Deformation 12419 Igor Dobovsek (Slovenia): Explosive Instabilities in a Class of Hyperelastic Materials with Higher-Order Gradients

lx

ICTAM04

12682 Joao A. Martins (Portugal): On the Concept of “Dynamic (In)Stablility of QuasiStatic Paths” 12751 Yves M. Leroy (France): Strain Localization at the Brittle-Ductile Transition of the Earth’s Continental Crust.

SM13 – Mechanics of composites (S. Adali, South Africa & N.A. Fleck, UK) 10189 Mohammad Reza Khoshravan (Iran): Numerical Evaluation of Mixed Mode Delamination in a U.D. Glass/Epoxy Composite in 2D and 3D States 10244 Ryszard Pyrz (Denmark): Interfacial Properties of Nanowire-Polymer Composites 10347 Jozef ´ Ignaczak (Poland): Plane Harmonic Waves in a Microperiodic Layered Thermoelastic Solid Revisited 10412 Tong-Earn Tay (Singapore): Damage Progression by the Element-Failure Method (EFM) and Strain Invariant Failure Theory (SIFT) 10483 Hossein M Shodja (Iran): Eﬀective Properties of Solids Containing Randomly Distributed Multi-Phase Spherical Particles 10555 Lidiya Nazarenko (Ukraine): Porous Anisotropic Composites under Microfructures 10667 Petri J. Kere (Finland): Reissner-Mindlin-Von Karman Type Plate Modle for Postbuckling Analysis of Laminated Composite Structures 10745 Kanmi Aderogba (Nigeria): Three-Dimensional Transmission in Plane Layered Elastic Composites 11014 Y. Jack Weitsman (USA): Aspects of the Mechanical Response of Randomly Reinforced, Chopped Fiber Strand, Polymeric Composites 11080 Javier LLorca (Spain): Computational Modeling of Deformation and Damage in Particle-Reinforced Composites 11162 Martin G. Andrews (USA): Elastic Interaction of Multiple Delaminations in Laminated Structures 11176 Federico J. Sabina (Mexico): Overall Properties of Periodic Biocomposites 11180 Christophe Bouvet (France): Damage Tolerance of Composite Structures with Thermal Shield 11387 Mike J. Cliﬀord (UK): Can it Be Made? Predicting the Formability of Textile Composite Components 11426 Fumio Narita (Japan): Electroelstic Fields Concentrations and Polarization Switching by Circular Electrodes in Piezoelectric Disk Composites 11457 Aleksander Muc (Poland): Fuzzy Set Approach to Modelling Composite Mechanical Properties 11507 Amna Rekik (France): Evaluation of Linearization Procedures Sustaining Nonlinear Homogenisation Theories 11610 Tomo Takeda (Japan): Three-Dimensional Thermoelastic Analalysis of Plain Weave Glass/Epoxy Composities with Cracks at Cryogenic Temperatures 11747 Andras Szekrenyes (Hungary): Advanced Beam Model for Fiber-Bridging in Unidirectional Composite Double-Cantilever Beam Specimens 11782 John R. Willis (UK): Interfacial Jump Conditions in Strain-Gradient Plasticity and Relations of Hall-Petch Type 11794 Brian Nyvang Legarth (Denmark): A Study of Particle Debonding with Anisotropy 11805 Emmanuelle Chabert (France): Nonlinear Aﬃne Extension of the Three-Phase Sphere Model 11895 Barbara Gambin (Poland): H-Convergence and Multilayering in Piezocomposites 12148 Jorn S. Hansen (Canada): A Homogenization Based Laminated Beam Theory 12214 Ulrik Borg (Denmark): Compressive Strength of Fiber Composite with Porosity 12242 Zhong Ling (China): Thermal Residual Stress in Al2O3/SiCnano Ceramic Composites Measured by Nanoindenter 12321 Heinz E. Pettermann (Austria): Composites with Planar Random Fiber Arrangements 12489 Jan Schjødt-Thomsen (Denmark): Inclusion Dispersion: Eﬀects on Stress and Eﬀective Properties

Scientiﬁc Program

lxi

12496 Shiguo Long (China): Thermal Fatigue of MMC Induced by Laser Heating 12531 Lingadahally S. Ramachandra (India): Thermo-Mechanical Stability and Vibration Analysis of Composite Shells 12560 Aleksander Muc (Poland): New Trends in Optimal Design of Composite Materials 12576 Robert Boehm (Germany): An Anisotropic Damage Model for the Prediction of the Degradation Behaviour of Novel Textile Reinforced Composites 12588 Vladyslav Danishevskyy (Ukraine): Asymptotic Study of Imperfect Interfacial Bonding in Periodic Composite Materials 12667 Antoni A. Gaka (Poland): T-Inclusion Regions for the Eﬀective Transport Coeﬃcients of Two–Phase Media 12680 Akke S.J. Suiker (Netherlands): Crack Tunneling in Laminates 12732 Ryszard Wojnar (Poland): Macroscopic Relations for Nonlinear Thermodiﬀusion in Heterogeneous Elastic Medium 12818 Arwen Smits (Belgium): Study of the Usability of Various Cruciform Geometries for Biaxial Testing of Fiber Reinforced Composites 12970 Marek Leﬁk (Poland): Incremental Eﬀective Constitutive Law for Composite Material in the Form of Artiﬁcial Neural Network 13009 Ali Daneshmehr (Iran): Analysis of Thick Laminated Panel With Piezoelectric Sensors Based on Three-Dimensional Theory of Elasticity

SM14 – Mechanics of phase transformations (F.D. Fischer, Austria & A. Molinari, France) 10046 Wojciech K. Nowacki (Poland): Temperature and Strain Rate Eﬀects on TRIP Sheet Steel. Measurement of Temperature by Infrared Thermograph 10088 Thomas Antretter (Austria): A Numerical Approach to Martensitic Phase Transformations 10434 Tatsuo Inoue (Japan): Macro-, Meso- and Micro-Scopic Metallo-Thermo-Mechanics 10648 Sergio R. Turteltaub (Netherlands): Multiscale Modeling of Steels assisted by Transformation-Induced Plasticity 10721 Isaac V. Chenchiah (Germany): The Nature of Stress and Strain Fields in Shape Memory Polycrystals 11308 Elzbieta ˙ Alicja Pieczyska (Poland): Shape Memory Alloy Under Strain and Stress Controlled Conditions – Thermomechanical Aspects of Martensite and Reverse Transformations 11325 Valery I. Levitas (USA): High Pressure Mechanochemistry: Conceptual Multiscale Theory and Interpretation of Experiments 11393 Arkadi Berezovski (Estonia): Stress-Induced Martensitic Phase Transition Front Propagation 11395 Shangping Chen (Netherlands): Modeling Martensite Transformation in the ElastoPlastic Material at Finite Strain 11625 Sabine M. Schl¨ ¨ ogl (Germany): Modeling of the Microstructural Evolution in Cr-Mo Steels During Tempering and Hydrogen Exposure 11837 Alexander B. Freidin (Russia): Equilibrium and Stability of Two-Phase Deformations within the Framework of Phase Transition Zones 11852 Cristian Faciu (Romania): On Modeling the Longitudial Impact of Two Shape Memory Bars 12077 Thorsten Bartel (Germany): A Micromechanical Model for Single-Crystal ShapeMemory-Alloys 12121 Claus Oberste-Brandenburg (Germany): Simulation of Discontinuity Movement by Boundary Elements 12225 Christian Lexcellent (France): Determination of Phase Transformation Yield Surface of Anisotropic Shape Memory Alloys 12421 Fabrice Barbe (France): Numerical Determination of Diﬀusional Transformation Induced Plasticity from Computations of Random Microstructures

lxii

ICTAM04

12747 Salem Meftah (France): Numerical Analyses of the Interaction Classical Plasticity – TRIP 12881 Qingping Sun (China): Nucleation and Motion of Phase Boundary in Shape Memory Alloy Microtubes

SM15 – Mechanics of porous materials (W. Ehlers, Germany & J.M. Huyghe, Netherlands) 10198 Tim Ricken (Germany): Biodegradation in Porous Landﬁll Bodies 10790 Michio Kurashige (Japan): Mandel and Cryer Problems For Fluid-Saturated Foams With Negative Poisson’s Ratio 10792 Wolfgang Ehlers (Germany): Localization and Stability of Unsaturated Soil 11119 Martin Schanz (Germany): Convolution Quadrature Based Boundary Element Method for Quasi-Static Poroelasticity 11194 Bernd Markert (Germany): Theory and Numerics of Multicomponent Mixture Models for Soft Biological Tissues 11463 Jorg ¨ Hohe (Germany): Probabilistic Homogenization of Hyperelastic Solid Foams 11941 Luc Dormieux (France): Coupling Between Permeability and Damage: a Micromechanical Approach 12114 Stefano Dal Pont (France): Thermo-Hydro-Chemical-Mechanical Analysis of Concrete at High Temperatures 12358 Jacek Tejchman (Poland): FE-Investigations on Shear Localizations in Granular Bodies within Hypoplasticity 12518 Ragnar Larsson (Sweden): Modelling of Composites Processing Using a Two-Phase Porous Media Theory 12566 Michal Pakula (Poland): Wave Propagation in High Porosity Bones – a Cellular Model 12567 Mariusz Kaczmarek (Poland): Soft Porous Media Model of Magnetic Fluid 12577 J´ ´ ozef Kubik (Poland): Mechanics of Saturated High Porosity / Soft Materials 12614 Daniela M. Bauer (France): A Three Layer Porous Media Model of Cutaneous Circulation with Application to Mechanical Skin Irritation 12713 Jo¨ ¨el Sarout (France): Identiﬁcation of Some Chemoporoelastic Parameters of a Reactive Shale from Experimental Data 12771 Michael W. Crochet (USA): Mesoscale Predictions for the Thermomechanics of Granular Energetic Composites 12903 Alan C.F. Cocks (UK): The Structure of Constitutive Laws for Powder Metallurgical Components 12917 Csaba I. Sinka (UK): Experimental Characterisation and Numerical Modelling of Density Distribution in Tablets 13006 Jacek Banaszak (Poland): Stresses and Fractures in Capillary – Porous Materials Under Drying

SM16 – Mechatronics (W.O. Schiehlen, Germany & M. Tomizuka, USA) 10510 Ya-Pu Zhao (China): Stability Analyses of Electrostatic Torsional RF MEMS Switch 10553 Dominique de Blaise (France): Improvement of Positioning Accuracy of Delta Parallel Robot 10809 Friedrich G. Pfeiﬀer (Germany): Dynamics and Control of a Hydraulically Driven Boring Plant 10900 Dalius Mazeika (Lithuania): Investigation of Powerful and High Precision Piezoelectric Actuator for Two-Dimensional Positioning 11282 Wim Symens (Belgium): Gain-Scheduling Control of Machine Tools with Varying Structural Flexibilities 12417 Horst Schulte (Germany): A Systematic Load Identiﬁcation Procedure for Parallel Robot Manipulators

Scientiﬁc Program

lxiii

12546 Alfredas Busilas (Lithuania): Development of Positioning of Mechanisms with Piezoelectric Engines 12712 Li-Sheng Wang (Taiwan): Hierarchical Tracking Control of Wheeled Mobile Robot 12758 Matthias Weber (Germany): Rapid Prototyping of Model Based Control Algorithms for Diesel-Engines with Turbocharger

SM17 – Multibody dynamics (M. Geradin, Italy & F. Pfeiﬀer, Germany) 10026 Wojciech Blajer (Poland): A Geometrical Framework for Modeling and Simulation of Nonholonomic Mechanical Systems 10533 Gakhip Ualiyev (Kazakhstan): Research of Movement of the Mechanism Suﬃcient with Elastic Part 10624 Katica Stevan´ ´ ovi´ ´ c Hedrih (Serbia): Homoclinic Orbits Layering in the Coupled Rotor Nonlinear Dynamics and Chaotic Clock Models 10647 Pavel Polach (Czech Republic): Design of the Rear Carriage Stabilizer of a Low-Floor Articulated Trolleybus 10779 Yuriy N. Kononov (Ukraine): Stabilization by Rotating Rigid Bodies for Unstable Rotation of a Rigid Body with Cavities Containing a Fluid 10893 Oleg N. Dmitrochenko (Russia): Simulation of Constrained Rigid and Elastic Bodies Without Constraint Equations 11020 Harry J. Dankowicz (USA): Analysis of Grazing Bifurcations in Impact Microactuators 11223 Nils Guse (Germany): Low Energy Control of Periodic Motions in Manufacturing 11352 Amalia Pielorz (Poland): Selected Problems of Discrete-Continuous Mechanical Systems with Local Nonlinearities 11361 Gilles Saussine (France): Modeling Ballast Behavior Using a Three-Dimensional Polyhedral Discrete Element Method 11375 Arun K. Banerjee (USA): Eﬃcient Generalized Speeds in a Recursive Formulation of Flexible Multibody Dynamics 11640 Dmitry Yu. Pogorelov (Russia): On Approximate Jacobian Matrices in Simulation of Stiﬀ Multibody Systems 11674 Jean Lerbet (France): Intrinsic Formulation of Dynamics of Curvilinear Systems 11688 Dmitry G. Agapov (Russia): Simulation of Track Ballast 11700 Kazuo Tanizawa (Japan): Conﬁguration Control and Dynamic Analysis of Redundant Link-Type Manipulators 12029 Johannes Gerstmayr (Austria): The Absolute Coordinate Formulation with Reduced Strain and Stiﬀening 12071 Juan Valverde (Spain): Stability Analysis of a Tethered System 12133 Beate Muth (Germany): Simulation of Contacting Spatial Polyhedral Particles 12134 Robert Seifried (Germany): Plastic Deformation by Impacts in Multibody Systems 12208 Olivier Bruls (Belgium): A Systematic Model Reduction Method for the Control of Flexible Multibody Systems 12464 Kohichi Miura (Japan): Expression on the Deﬂection of a Flexible Thin Rod and Its Measurement 12537 Naoki Sugano (Japan): Dynamic Analysis and Vibration Control of The Planar Beams Moving Along the Axial Direction 12539 Guy Bessonnet (France): Generating Optimal Motions of Constrained Multibody Systems 12569 Sine Leergaard Pedersen (Denmark): Contact Problems in Roller Chain Drive Systems. 12582 Katsuhisa Fujita (Japan): Motion and Vibration Control of the Lift Mechanism of a Ladder Truck

lxiv

ICTAM04

SM18 – Plasticity and viscoplasticity (E. Van der Giessen, Netherlands & P. Perzyna, Poland) 10064 Ryszard B. P¸echerski (Poland): Metal Forming Processes Conditioned by Cyclic Loading. A New Challenge for the Theory of Plasticity 10383 Milan V. Micunovic (Serbia): Quasi Rate-Independent Viscoplastic FCC-Polycrystals 10393 George Z. Voyiadjis (USA): Physically Based Thermomechanical Modeling of Metals over a Wide Range of Strain Rates and Temperatures 10652 Norimasa Chiba (Japan): Plastic Properties Identiﬁcation With Plural Sharp Indenters 10734 Alexis Rusinek (France): Advanced Thermo-Visco-Plastic Constitutive Relations for Direct Applications in Numerical Analyses 10797 Giulio Maier (Italy): Calibration of Anisotropic Elastic-Plastic Models for Thin Layers and Foils in Microtechnologies: Two Novel Techniques 10955 John D. Clayton (USA): Simulation of Dynamic Polycrystalline Thermoelastoviscoplasticity and Fracture 11005 Ricardo A. Lebensohn (USA): Assessing Diﬀerent Self-Consistent Approximations by Comparison with Full-Field Simulations in Viscoplastic Polycrystals 11110 Andr´ ´ e Dragon (France): Three-Dimensional Modelling of Thermo-Elasto / Viscoplastic Solids Containing Adiabatic Shear Bands 11185 Wiera Oliferuk (Poland): Energy Storage Rate in Non-Homogeneous Deformation 11295 Kazuwo Imai (Japan): Dynamic Behavior of Many-Dislocation Systems in Silicon 11309 Alain L. Molinari (France): The Eshelby Problem for Elastic-Viscoplastic Materials 11344 J.A.W. van Dommelen (Netherlands): Multiscale Modeling of the Structure-Property Relationship for Semicrystalline Materials 11350 Mamoru Mizuno (Japan): Modeling of Viscoplastic Constitutive Equation for Polymers by Taking into Account Strain Recovery 11410 Peter Gudmundson (Sweden): Thickness Dependent Yield Strength of Thin Films 11498 Stephane Andr´ ´e Berbenni (USA): Yield Surfaces Using an Extension of the Regularized Schmid Law to Polycrystalline Materials 11790 Christian F. Niordson (Denmark): Size-Eﬀects in Void Growth 11820 Amit Acharya (USA): On the Accounting of Dislocation Internal Stress in Continuum Plasticity 11930 Nicolaie Dan Cristescu (USA): Steady-Flow of a Non-Homogeneous Bingham Material Over a Wedge 11953 Anguel I. Baltov (Bulgaria): Modelling of Elastic-Plastic or Viscoplastic Materials Sensitive to the Type of Processes – Diﬀerent Approaches 11963 Marc G. Geers (Netherlands): Strain Gradient Crystal plasticity Incorporating Grain Boundary Eﬀects 11975 Wiktor L. Gambin (Poland): Metal Forming and Texture Development Modelling 12058 Dieter Weichert (Germany): Limit and Shakedown Analysis with Decohesive Eﬀects 12339 Fahmi Za¨ri ¨ (France): A Constitutive Law for Glassy Polymers and Blends 12469 Cedric Doudard (France): Development and Identiﬁcation of a Probabilistic TwoScale Model for High Cycle Fatigue Prediction 12549 Thomas B¨ ¨ ohlke (Germany): Modeling the Crystallographic Texture Evolution Based on the Maximum Entropy Method 12634 Absamad El Adb (France): An Elastoplastic Model for Prediction of Permanent Deformations of Unbound Granular Pavement Layers 12697 Maxime Sauzay (France): Intragranular Kinematic Hardening Modelling and Validation 13038 Oliver Pierard (Belgium): Mean Field Homogenization of Elasto-(Visco) Plastic Composites: Formulation for Time-Dependent and Independent Behaviors

Scientiﬁc Program

lxv

SM19 – Plates and shells (H. Mang, Austria & E. Ramm, Germany) 10071 Gennadiy Lvov (Ukraine): The Stress Analysis of the Multilayered Plates and Shells with Defects of the Structure 10233 Parthasarathi Mandal (UK): Some New Thoughts on the Buckling of Thin Cylindrical Shells 10287 Wojciech Pietraszkiewicz (Poland): Continuity Conditions in Elastic Shells with Phase Transformation 10770 Kenzo Sato (Japan): Analytical Solution of Bending of a Clamped Elliptical Plate Under Lateral Load and In-Plane Force 11021 Herbert A. Mang (Austria): Sensitivity Analysis Concerning the Initial Postbuckling Behavior of Elastic Structures 11238 Jan Sladek (Slovakia): Meshless LBIE Formulations for Viscoelastic Thin Plates 11815 Maryna A. Varyanychko (Ukraine): Eﬀect of a ’Static’ Resonance in Elastic ThinWalled Cylinders 11890 Izabela Lubowiecka (Poland): Energy-Conserving Scheme for Nonlinear Dynamics of Shells – Numerical Examples 12224 Takeshi Sakiyama (Japan): Mindlin Cylindrical Panels with Twist and Double Curvature 12328 Zdzislaw Nowak (Poland): Application of the Return Mapping Algorithm to Perzyna Viscoplasticity for Plane Stress 12346 Eelco L. Jansen (Netherlands): A Perturbation Method for Nonlinear Vibrations of Structures 12350 Lidija V. Kurpa (Ukraine): Nonlinear Vibrations of Shallow Shells and Thin Plates of Arbitrary Shape

SM20 – Rock mechanics and geomechanics (Z. Mr´ ´ oz, Poland & I. Vardoulakis, Greece) 10231 Zdzislaw A. Wi¸eckowski (Poland): The Material Point Method in Soil Mechanics Problems 10374 Leopold Kruszka (Poland): Dynamic Behaviours of Soils and Rocks in a Wide Pressure Range 10866 Niels P. Kruyt (Netherlands): Micromechanical Study of Macroscopic Friction and Dissipation in Idealised Granular Materials: The Eﬀect of Interparticle Friction 11584 Marcin Ma´ ´zdziarz (Poland): Inﬂuence of Contact Phenomena on Structure-Subsoil Interaction: Finite Elements Method Analysis 11667 Nathalie Boukpeti (Belgium): Modeling of Static Liquefaction and Evolving Failure Modes 11845 Joseph F. Labuz (USA): Development of Shear Banding in Sandstone 12364 Bojan Guzina (USA): On the Use of Second-Order Topological Information for Subsurface Imaging by Elastic Waves 12462 Yuji Kishino (Japan): Incremental Nonlinearity in Constitutive Relation for Granular Media 12779 Martin J. Schmidt (USA): A High-Pressure Hish Strain Rate Elastic-Viscoplastic Model for Cementitious Materials 12848 Jan Maciejewski (Poland): The Inﬂuence of Teeth on the Earth-Working Processes

SM21 – Solid mechanics in manufacturing (B. Heimann, Germany & T. Inoue, Japan) This session has been cancelled

SM22 – Stability of structures (Z. Gaspar, Hungary & S. Kyriakides, USA) 10378 J. Blachut (UK): Shallow Spherical Caps Under External Pressure 10692 Nobutada Ohno (Japan): Elastoplastic Microscopic Bifurcation and Post-Bifurcation Behavior of Periodic Cellular Solids 10915 Zsolt Gaspar (Hungary): Statical Models to Illustrate Special Instabilities

lxvi

ICTAM04

11097 Sergio Pellegrino (UK): Wrinkles in Square Membranes 11172 Ioannis G. Raftoyiannis (Greece): Nonlinear Dynamic Stability of Multi-Suspended Roof Systems 11174 Alain Combescure (France): Coupling of Axisymmetric and 3D Shell Models for Non Linear Elastoplastic Buckling Prediction of Mainly Axisymmetric Shells 11600 Theodoro A. Netto (Brazil): Dynamic Arrest of Propagating Buckles in Oﬀshore Pipelines 11653 Suresh Shrivastava (Canada): Bifurcation Buckling of Sandwich Plates and Shells in Plastic Range 11816 Hans Troger (Austria): On the Stability of the Sky-Hook 11939 Alexander P. Seyranian (Russia): Stability of Parametrically Excited Structures: New Results 11962 Philippe Le Grognec (France): A Uniﬁed Treatment for the Elastoplastic Bifurcation of Structural Elements 12084 Hans Obrecht (Germany): Buckling and Imperfection – Sensitivity of Axially Compressed Cylindrical Shells with Compliant Cores 12297 Esben Byskov (Denmark): Stability of Shear-Flexible Frames 12420 Djenane C. Pamplona (Brazil): Instabilities of Initially Stressed Hyperelastic Cylindrical Membrane and Shell Under Internal Pressure 12558 Dinar R. Camotim (Portugal): Generalised Beam Theory Formulation to Analyse the Post-Buckling Behaviour of Orthotropic Laminated Plate Thin-Walled Members 12669 Simon D. Guest (UK): The Stiﬀness of Prestressed Frameworks: A Unifying Approach 12688 Giles W. Hunt (UK): Twist Buckling and the Foldable Cylinder: An Exercise in Origami 12931 Szymon Imieowski (Poland): On Stability of Systems Subject to Generalized Follower Force 12934 Jaroslaw Weronko (Poland): Dynamic Instability of a High-Speed Flexible Shaft with a Massive Disc and Follower Load. 12967 Ciprian D. Coman (UK): Secondary Bifurcations and Localisation of Buckle Patterns

SM23 – Stochastic micromechanics (Y. Brechet, France & Y. Shibutani, Japan) 10242 Andrejs Krasnikovs (Latvia): Creep Rupture and Fiber Breaks Accumulation in Unidirectional Composite 12380 Jan Zeman (Czech Republic): Homogenization of Plain Weave Composites with Imperfect Microstructure 12571 Yoji Shibutani (Japan): Collective Prismatic Dislocation Loops Mechanism

SM24 – Structural optimization (K.K. Choi, USA & J. Herskowits, Brazil) 10048 Yoon-Young Kim (S. Korea): Shape Sensitivity Analysis for Fixed-Grid Analysis Based on Oblique Boundary Curve Approximation 10052 Tomasz Lewi´ n ´ ski (Poland): Perturbation of the Compliance Functional Due to the Apperance of a Small Cavity in an Elastic Body 10433 Niels Olhoﬀ (Denmark): Topology Optimization of Vibrating Structures with Hydrodynamic Surface Pressure Loading 10594 Jos´ ´e N. Herskovits (Brazil): A Technique for Nonsmooth Optimization Based on the Interior Point Feasible Directions Algorithm 10621 Timo J. Saksala (Finland): Nash Equilibrium in Bicriteria Structural Optimization 10669 Pauli Pedersen (Denmark): On Shape Optimization for Eigenvalue Problems 11015 Valeri Markine (Netherlands): Shape Optimisation of Railway Wheel Proﬁle 11137 Tadeusz S. Burczy´ n ´ ski (Poland): Shape Optimization of Thermomechanical Structures in the Presence of Convection and Radiation Using Parallel Evolutionary Computation 11316 Jeong Sam Han (Germany): Eﬃcient Optimization of Transient Dynamic Problems for a Micro Accelerometer Using Model Order Reduction

Scientiﬁc Program

lxvii

11349 Krzysztof Dems (Poland): Damage Identiﬁcation in Structures by Means of Thermographic Methods ˙ (Poland): Topological Optimization for Contact Problems 11486 Antoni Zochowski 11532 Yuanxian Gu (China): Coupled Sensitivity and Design Optimization for ThermoStructural Systems 11623 George I.N. Rozvany (Hungary): New Classes of Analytically Derived Optimal Topologies and Their Numerical Conﬁrmation 11634 Pawel Smas (Poland): Optimal Structures for Buckling Forces and Buckling Displacements 11744 Søren Halkjaer (Denmark): Optimization of Beam Properties with Respect to Maximum Band-Gap 11760 Slawomir Czarnecki (Poland): Optimal Layout of Two Materials within the Core Layer of a Sandwich Plate. Relaxed Formulation and Its Computiational Algorithm 11826 Su-Young Chang (S. Korea): Material Cloud Method for Topology Optimization 11858 Doo-Ho Lee (S. Korea): Optimal Design of Unconstrained Damping Layer on Beams 11936 Niels L. Pedersen (Denmark): On Separation of Eigenfrequencies in Two-Material Structures 12074 Gregor Kotucha (Germany): Density Gradient Based Regularization of Topology Optimization Problems ˙ 12075 Michal Zyczkowski (Poland): New Results of Structural Optimization for Post-Buckling Behaviour 12079 Thomas Buhl (Denmark): Compliant Mechanism Design for Adaptive Trailing Edge Flaps 12098 Erik Lund (Denmark): Structural Optimization of Composite Shell Structures Using a Discrete Constitutive Parameterization 12185 Gil Ho Yoon (S. Korea): Why Parameterizing Element Connectivity for Topology Optimization? 12200 Sami Holopainen (Finland): Topology Optimization of the Geometrically Nonlinear Structures Made of Rubber-Like Material 12222 Jakob S. Jensen (Denmark): Optimal Design of Lossy Bandgap Structures 12287 Lijuan Li (China): Non-Gap Design Method and Test for Post-Tensioned Prestressed R.C. Structure 12302 Atsushi Kawamoto (Denmark): Design of Articulated Mechanisms with a Degree of Freedom Constraint Using Global Optimization 12413 Michal Nowak (Poland): Simulation of Trabecular Bone Adaptation – Creating the Optimal Structure 12573 Sandor Kaliszky (Hungary): Optimal Design of Elasto-Plastic Structures Subjected to Normal Loads and Earthquake 12628 Andrzej Garstecki (Poland): Optimal Force Action and Reaction in Structural Design and Identiﬁcation 12679 Piotr Kowalczyk (Poland): DSA for Elastic-Plastic Finite Rotation Shells under Dynamic Loads 12789 Florin Bobaru (USA): Optimization of Functionally Graded Materials with Temperature Dependent Properties. A Meshfree Solution 12870 Vassili Toropov (UK): Optimum Blank Design for Deep Drawing Using Interaction of High and Low Fidelity Simulation 12916 Henrik T. Møller (Denmark): Computational Tricks for Eﬃcient Design Sensitivity Analysis 13013 Gengdong D. Cheng (China): The Concurrent Design of Materials and Structures for Cellular Materials on Eﬃciency of Heat Dissipation

SM25 – Structural vibrations (I. Blekhman, Russia & K. Popp, Germany) 10166 Jianbing Chen (China): Extreme Value Distribution and Dynamic Reliability of Stochastic Structures 10408 Alexander Vakakis (Greece): Experimental Study of Nonlinear Energy Pumping

lxviii

ICTAM04

10470 Marian Wiercigroch (UK): Nonlinear Vibrations of Jeﬀcott Rotor with Preloaded Snubber Ring 10656 Victor Z. Gristchak (Ukraine): Non-Linear Stochastic Vibration Problems for the Plates with Time-Dependent Parameters 10750 Debasish Roy (India): A Multi-Step Transversal Linearization Method in Nonlinear Dynamics 10863 Stephen H. Crandall (USA): Equivalent Stochastic Linearization as an Alternative to Solving the Fokker-Planck Equation 11055 Iliya I. Blekhman (Russia): Nonlinear Eﬀects, Observed in the Process of the Liquid Flowing Out of the Vibrating Vessels: Theory, Experiment and Applications 11066 Akihiko Higashi (Japan): Propagation Analysis of Flexural Waves by Wavelet Transform 11079 Oleg Gendelman (Israel): Bifurcations of Damped Nonlinear Normal Modes: Linear Oscillator with Strongly Nonlinead Attachment 11082 Gamal Mohamed Ashawesh (Libya): Eﬀect of Root Flexibility on the Aeroelastic Analysis of a Composite Wing Box 11246 Herve Riou (France): Reanalysis of an SEA High – Frequency Vibration Calculation Based on the VTCR 11261 Utz von Wagner (Germany): Active Control of Disk Brake Sqeal 11273 Marcin Luczak (Poland): Experimental and Theoretical Modal Analysis of Three Support Rotor Test Rig Using LMS CADA-X and ABAQUS 11338 Jacob P. Meijaard (UK): Stability of a Rotor with Periodically Varying Angular Velocity 11384 Tibor Tarnai (Hungary): Paradoxical Behaviour of Vibrating Systems Challenging Rayleigh’s Theorem 11494 Carlos E.N. Mazzilli (Brazil): Imperfection Sensitivity of Circular Arch’s Non-Linear Modes 11536 Alexei A. Mailybaev (Russia): Optimal Shapes of Parametrically Excited Beams 11679 Alexander J. Fidlin (Germany): Non Trivial Eﬀect of Strong High-Frequency Excitation on a Nonlinear Controlled System 11699 Igor Zeidis (Germany): An Aproach to Worm-Like Motion 11712 Alla D. Firsova (Russia): Dynamics of a Rotor Rolling Along a Circular Surface 11716 Yuri Leonidovich Menshikov (Ukraine): The New Statement of Problem of Unbalance Identiﬁcation 11719 Ekaterina V. Shishkina (Russia): Vibrorheology: Main Results, New Problems 11759 Vladimir Zeman (Czech Republic): Nonlinear Vibrations of Gear Drives 11783 Rob H.B. Fey (Netherlands): Passive Vibration Control of a Piecewise Linear Beam System 11834 Ye Ping Xiong (UK): A power Flow Mode Theory Based on Inherent Characteristics of Damping Distributions in Systems and Its Applications 11867 T.H. Young (Taiwan): Stability of a Spinning Disk Under a Stationary Oscillating Unit 11904 Seyed Saleh Hosseini Yazdi (Iran): High Revolving Speed Spindles Deﬁnition Due to Transient Vibration Conditions 12026 Fadi Dohnal (Austria): Suppressing Self-Excited Vibrations in a Coupled Pendulum System 12322 Ingo Kaiser (Germany): The Running Behaviour of an Elastic Wheelset 12404 Andrew N Norris (USA): Thermoelastic Relaxation in Thin Plates with Applications to MEMS and NEMS Oscillators 12498 Jacek Cie´ ´slik (Poland): Estimation of the Vibration Energy Characteristics for Joints of Constructional Elements 12511 Marek S. Kozien (Poland): Sound Radiation by the White Noise Excited Viscoelastic Shallow Shells 12535 Thomas Marc Richard (Belgium): Self-Excited Stick-Slip Oscillations of Drag Bits

Scientiﬁc Program

lxix

12593 Yoshikazu Sugiura (Japan): Vibration Characteristics of the Main Tower, the Byaon Temple 12632 Viktorija E. Volkova (Ukraine): Application of Extended Phase Space to Investigation of Forced Biharmonic Oscillations 12659 Robert Jankowski (Poland): Non-Linear Modelling of Earthquake Induced Pounding of Buildings 12694 Jon J. Thomsen (Denmark): Discontinuous Transformations and Averaging for VibroImpact Analysis 12759 Pedro M. Ribeiro (Portugal): Experimental Analysis of Modal Interactions in the Non-Linear Vibrations of a Plate 12770 Arkadiusz M¸ezyk ˙ (Poland): Optimum Selection of Design Features of Electromechanical Drive Systems Incorporating a Control Unit 12785 Christian Seidel (Germany): Mode Switching of Rain-Wind Induced Vibrations 12814 Pankaj Wahi (India): Regenerative Tool Chatter Near a Codimension-2 Hopf Point Radkowski (Poland): Characteristics of Vibroacoustic Signals in Diagnosing 12977 Stanisaw Early Stages of Defects 12984 Tadeusz Majewski (Poland): Entering the Excitation into a Mechanical System with Dynamic Eliminators of Vibration 12989 Gayane Manucharyan (Ukraine): Frictional Auto-Oscillations under the Action of Almost Periodic and Periodic Excitations

SM26 – Vehicle dynamics (S. Iwnicki, UK & R. Sharp, UK) 10701 Robin S. Sharp (UK): Optimal Path Following Road Vehicle Steering Control 10909 Simon Iwnicki (UK): Simulation and Testing of a Wheelset with Induction Motor Driven Independent Wheels 11589 Vladislav Yazykov (Russia): Railway Vehicle Simulation Using Non-Elliptical WheelRail Contact Model 11672 Georg Rill (Germany): A Modelling Technique for Fast Computer Simulations of Conﬁgurable Vehicle Systems 11795 Per-Anders J¨ onsson (Sweden): Experimental and Theoretical Aanalysis of Freight Wagon Link Suspension 11832 Zbigniew Lozia (Poland): Mathematical Models and Simulation of Stick-Slip Processes in a Car Steering System

SM27 – Viscoelasticity and creep (F. Cocks, UK & N. Ohno, Japan) 10018 Boris P. Maslov (Ukraine): Nonlinear Overall Viscoelastic Properties of the Random Multicomponent Media 10906 Renata S. Engel (USA): Sintering Simulation of Stainless Steel Powder Compacts 11201 Alan R.S. Ponter (UK): Characterisation of the Cyclic Behaviour of Elastic-PlasticCreeping Bodies 11368 Holm Altenbach (Germany): A Creep Continuum Damage Theory for Beams, Plates and Shells 11506 Jinghong Fan (USA): Multiscale Modeling Schemes Spanning a Large Range of Scales 11827 Ji-Hyun Cho (S. Korea): Constitutive Modeling of Rubber Components Under Small Vibration Superimposed on Large Static Deformation Considering Strain-Dependent Properties 11957 Zbigniew L. Kowalewski (Poland): An Inﬂuence of Cold Work on Creep of Engineering Materials 11981 Roman Lackner (Austria): Multi-Scale Model for Low-Temperature Creep of Asphalt 12450 Fusahito Yoshida (Japan): A Model of Cyclic Viscoplasticity with Special Reference to Yield-Point Phenomena

lxx

ICTAM04

FSM1 – Acoustics (T. Geers, USA & N. Peake, UK) 10294 Nobumasa Sugimoto (Japan): Dissipative Eﬀects on Propagation of the Acoustic Solitary Waves 10907 Jonathan B. Freund (USA): An Empirical ’lower bound’ on Free-Shear-Flow Noise 11249 Jeremy Astley (UK): Special Short Wave Finite Elements for Flow Acoustics 11337 Sergey V. Sorokin (Denmark): Wave Propagation in and Sound Emission from a Sandwich Plate Under Heavy Fluid Loading 11373 David S. Burnett (Italy): 3-D Structural Acoustics Modeling with HP-Adaptive Finite Elements 11508 Dmitry V. Churochkin (Russia): The Low-Temperature Acoustical and Thermal Properties of Materials Due to the Dynamics of Linear Topological Defects 11561 Takao Suzuki (USA): Shock Leakage Through a Vortex-Laden Mixing Layer Causing Jet Screech 11896 Natasha V. Movchan (UK): Transmission of Elastic Waves and Localised Modes in Composite Structures 11973 Murthy N. Guddati (USA): Arbitrarily Wide-Angle Wave Equations and their Applications to Unbounded Domain Modeling and Subsurface Imaging 12149 Tatiana Andreeva (USA): Ultrasonic Travel-Time Technique for Diagnostic of GridGenerated Turbulence 12180 Jeﬀ D. Eldredge (USA): The Acoustics of Two-Dimensional Leapfrogging Vortex Interactions 12477 Edward J. Kerschen (USA): A Theoretical Model for Resonances in Flow Past a Cavity 12497 Paul W. Hammerton (UK): Structure of Sonic Booms in a Medium with Multiple Relaxation Modes 12516 Alexander Alexeev (Germany): Gas Oscillations in a Closed Tube at Resonance 12527 Iain D.J. Dupere (UK): The Eﬀect of Viscosity on the Propagation of Acoustic Waves Through Fine Cylindrical Meshes 12533 Lars V. Hansen (Denmark): Modelling of Hydrophone Based on a DFB Fiber Laser 12555 J´ ´ ozef Lewandowski (Poland): Numerical Analysis of the Texture and Acousto-Elastic Properties of Prestressed Polycrytalline Aggregate 12586 Rossano Stefanelli (Switzerland): Measurements and Calculations Related to Curve Squealing in the Railway System 12643 Fernando Lund (Chile): Acoustic Wave Propagation Through a Random Array of Dislocations 12691 George Biros (USA): Distributed Parameter Control of a 2D Acoustic Helmholtz Problem on a Halfspace 12717 Loukas F. Kallivokas (USA): Frequency- and Directionality- Continuation Schemes for Scatterer Shape Detection in Acoustics

FSM2 – Chaos in ﬂuid and solid mechanics (I. Mezic, USA & G. Rega, Italy) 10252 Gabor Stepan (Hungary): Nonlinear Dynamics of High-Speed Milling 10499 John S. Hogan (UK): The Eﬀect of Smoothing on Bifurcation and Chaos Computations in Non-Smooth Mechanics 11183 Remco I. Leine (Switzerland): A Set-Valued Force Law For Spatial Coulomb-Contensou Friction 11241 Ugo Galvanetto (UK): Chaotic Attractors with Long Regular Sequences 11302 Ekaterina E. Pavlovskaia (UK): Reduction of Multidimensional Flow to Low Dimensional Map for Piecewise Smooth System Experiencing Chaos 11363 P. Piiroinen (UK): Numerical Detection and Continuation of Sliding Bifurcations in a Dry-Friction Oscillator 11416 Elzbieta ˙ Tyrkiel (Poland): On Generating Chaotic Dynamics in Nonlinear Vibrating Systems 11535 Ken Kiyono (Japan): Low-Dimensional Chaotic Dynamics in Dripping Faucets

Scientiﬁc Program

lxxi

12014 Kohei Yamasue (Japan): Inﬂuence of Remaining Chaos on Convergence of Solutions in Time Delayed Feedback Controlled Duﬃng System 12129 Giuseppe Rega (Italy): A Dynamical Systems Analysis of the Overturning of Rigid Blocks 12468 Igor Mezic (USA): On the Nonlinear Dynamics of Multicomponent Dynamical Systems 12482 Theodoros Karakasidis (Greece): Short-Time Dynamical Behavior of Fluids at the Atomic Scale 12674 Li-Qun Chen (China): Nonlinear Dynamics of Axially Moving Viscoelastic Strings Based on Translating Eigenfunctions n ´ ski (Poland): Nonlinear Oscillators with Time Delays 12927 Zbigniew Peradzy´ 12972 Radoslaw Iwankiewicz (South Africa): Non-Linear Oscillator Under Random RenewalDriven Trains of Impulses

FSM3 – Continuum mechanics (K.R. Rajagopal, USA & G. Saccomandi, Italy) 11096 11140 11227 11247 11347 11638 11642 11704 11818 11840 11950 12392 12406

James Casey (USA): Pseudo-Rigid Bodies Viewed as Globally Constrained Continua Lev Steinberg (USA): Constitutive Equations of Mesoelastic Deformation Luis A. Dorfmann (Austria): Nonlinear Response of Magnetoelastic Solids Bohuslav Striz (Czech Republic): Application of Continuum Mechanics in the Textile Fabrics Gerard A. Maugin (France): Generalized Continuum Mechanics: Three Paths Victor M. Tigoiu (Romania): Viscoelastic Fluid Flows in a Falling Cylinder Viscometer and the Evaluation of Shear Viscosity Stanisaw Tokarzewski (Poland): Fundamental Inequalities for the Bounds on the Eﬀective Transport Coeﬃcients of Two–Phase Media Glenn B. Sinclair (USA): On the Source of Singularities in Mechanics Michael A. Hayes (Ireland): Extended Polar Decompositions for Finite Plane Strain Wlodzimierz Doma´ nski ´ (Poland): Nonlinear Waves in Elastic Solids Michel Destrade (France): Explicit Secular Equations for Surface and Interface Waves in Anisotropic Solids Andreas Menzel (Germany): Views on Material Forces in Multiplicative Elastoplasticity Jan J. Sawianowski (Poland): Aﬃne Symmetry in Mechanics of Discrete and Continuous Systems

FSM4 – Fluid-structure interaction (J. Grue, Norway & M.P. Paidoussis, Canada) 10520 Elena G. Gavrilova (Bulgaria): Coupled Frequancies of a Fluid-Structure Interaction Cylindrical System 10826 Matej Vesenjak (Slovenia): Fluid Structure Interaction in Multiphase Mixing Vessel 10911 Tatiana Khabakhpasheva (Russia): Piston Impact Onto the Boundary of Two-Layer Fluid 11078 Michael P. Pa¨ ¨ıdoussis (Canada): Nonlinear Dynamics of Pinned-Pinned Cylinders in Axial Flow 11668 Ming-Jyh Chern (Taiwan): Interaction of Oscillating Flow with a Pair of Side-By-Side Square Cylinders 11736 Anthony D. Lucey (Australia): The Hydroelastic Destabilisation of Finite Compliant Panels 11758 Hiroshi Kagemoto (Japan): Water-Surface Dynamics Among a Periodic Array of Floating Bodies Subject to Regular Incident Waves 11801 Charlotte Py (France): The Mixing Layer Instability of Wind Over a Flexible Crop Canopy 11833 Jing Tang Xing (UK): An Updated Arbitrary-Lagrangian-Eulerian Description in Continuum Mechanics and Its Application to Nonlinear Fluid-Structure Interaction Dynamics

lxxii

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11912 Wolfgang M. Sichermann (Germany): Slender Body Theory Approach to Nonlinear Ship Motions 11986 Andrzej Herczy´ n ´ ski (USA): Inverse Magnus Force in Free Molecular Flow 12521 Yu Zhou (China): Eﬀect of an Oscillating Cylinder on a Neighbouring Cylinder Wake 12557 John R. Chaplin (UK): Disturbed-Laminar Flow Over an Oscillating Cylinder 12956 John Grue (Norway): PIV Experiments on Vortex Induced Vibrating Cylinders at High Reynolds Numbers 13050 Ahcene Bouabdallah (Algeria): Inﬂuence of the Circular Cylinder Cross-Section Variation on the Near Wake Behaviour

FSM5 – Mechanics of foams and cellular materials (S. Hilgenfeldt, Netherlands & D.L. Weaire, Ireland) 10597 Sascha Hilgenfeldt (Netherlands): Bubble Shapes in Foams: The Importance of Being Isotropic 11095 Benjamin Dollet (France): Mechanics of Bidimensional Liquid Foams 11240 Martin E. Glicksman (USA): Average N-hedra as Descriptors of 3D Network Cells 11292 H.X. Zhu (UK): Modelling the Round-oﬀ and the Tensile/Compressive Failure Behaviour of Plant and Vegetable Tissues 11605 Stelios Kyriakides (USA): On the Crushing Response of Open Cell Foams 11663 Alfonso H.W. Ngan (China): A Statistical Mechanics Theory of Random Honeycomb and Open-Cell Foam Structures 12052 Isabelle Cantat (France): Dissipation in 2D Foam Flow 12212 Arnaud Saint-Jalmes (France): Surfactant and Protein Foams: Diﬀerences in Drainage and Rheology 12373 Adrian D. Staicu (Netherlands): Determining Stress During Finger Propagation in 2D Foams 12395 Jing Tian (UK): Thermal Flow through Brazed Woven Screens 12435 James E. Coons (USA): Drainage of Emulsion and Foam Films in Scheludko-Exerowa Cells 12578 Vincent Labiausse (France): Shear-Induced Normal Stress Diﬀerences in Aqueous Foams 12851 Stephen J. Neethling (UK): The Dispersion of Particles within Foams 12986 Pacelli L. Zitha (Netherlands): Iinvestigation of Foam Development in Porous Media Using X-Ray Computed Tomography 13033 Fabian Lipperman (Israel): Nucleation of Cracks in Two-Dimensional Periodic Cellular Material

FSM6 – Multiscale phenomena in mechanics (A. Carpinteri, Italy & C. Miehe, Germany) 10399 Justyna Czerwi´ n ´ ska (Germany): Simulations of Micro- and Nano- Channel Flows by a Dissipative Particle Dynamics Method 11224 Alberto Carpinteri (Italy): Multi-Scaling Approach in the Mechanics of Disordered Materials 11237 Wieslaw Larecki (Poland): Grad-Type Expansion About Nonequilibrium States for the Relaxion-Time Approximation of the Boltzmann-Peierls Equation 11340 Heike Emmerich (Germany): Two-Scale Simulations of Epitaxial Surfaces 11471 Pilar Ariza (Spain): A Geometrical Theory of Discrete Dislocations in Lattices, with Applications to Dislocation Dynamics and Crystal Plasticity 11476 Krishna Garikipati (USA): Stress-Defect Interactions at Molecular / Continuum Scales 11806 Bhushan L. Karihaloo (UK): Deterministic Size Eﬀect in the Strength of Cracked Quasi-Brittle Structures 11830 Varvara Kouznetsova (Netherlands): Multi-Scale Second-Order Computational Homogenization of Heterogeneous Materials 11862 Yutaka Shimomura (Japan): Jumping of a Spinning Spheroid 12040 Luca Placidi (Germany): Characteristics of Orientation and Grain-Size Distributions

Scientiﬁc Program

lxxiii

12078 Antoine Gloria (France): Numerical Homogenization of a Locally Hyperelastic Constitutive Law 12112 Joachim Dettmar (Germany): Multiscale Analyses of Granular Media at Finite Strains Based on Micro-Macro Transitions with Diﬀerent Boundary Constraints 12113 Christian Miehe (Germany): Exploitation of Incremental Energy Minimization Principles in Computational Multiscale Analyses of Inelastic Solids 12118 Martin Becker (Germany): Non-Convex Homogenization of Inelastic Composites with Interaction of Material and Structural Instabilities on Diﬀerent Scales 12334 Luciano Colombo (Italy): Physical Modeling of Fracture Mechanics in Complex Materials 12336 Frederic Legoll (France): Analysis of a Variational Method Coupling Discrete and Continuum Mechanics

FSM7 – Education in mechanics (R. Engel, USA & B. Karihaloo, UK) 10003 Hassan Aref (USA): Toys and Games in Mechanics Education 10016 Keith Moﬀatt (UK): African Institute for Mathematical Sciences: a Capacity Building Initiative in which IUTAM Has an Active Involvement 10226 Carl T. Herakovich (USA): On Mechanics/Engineering Science Education 11724 Aleksandr Kositsyn (Ukraine): Mechanics – a New Internet Tutor 11765 Yilong Bai (China): Teaching Mechanics as an Engineering Science in China 11869 Vasily Yaremchuk (Russia): Education and Tutorial on Fluid Mechanics on the Basis of Computer Laboratory 12259 Vitauts Tamuzs (Latvia): Education in Mechanics in Latvia Higher Schools 12488 Yasuaki Nohguchi (Japan): Simulator, Nohguchi Bottle, of Soil Liquifaction for Education 12603 Anders Bostr¨ om (Sweden): Mechanics Education in Sweden 12642 Donovan L. Evans (USA): Rigid Body Dynamics: Student Misconceptions and Their Diagnosis 12752 Kamal B. Rojiani (USA): Web-Based Instructional Units for Teaching Mechanics

Professor Leen van Wijngaarden delivers the ICTAM04 Opening Lecture

INTERPLAY BETWEEN AIR AND WATER Leen van Wijngaarden J.M.Burgers Centre for Fluid Dynamics University of Twente, Enschede, The Netherlands. [email protected]

Abstract

In the Prologue I recall, among others, the period of the Cold War in which, thanks to Polish colleagues, scientiﬁc contacts between East and West were maintained . After that, several aspects of the ﬂow of mixtures of air and water will be discussed and illustrated by examples. Finally I will give some comments on the diﬀerences and similarities between fundamental and applied science and scientists.

Keywords: Multiphase ﬂow, bubbly ﬂows

1.

Prologue

It is a great honour to be invited to deliver the Opening Lecture at ICTAM 2004, especially now that it is here in Warsaw, a city of great signiﬁcance for Mechanics. It reminds me of the Cold War when East was East and West was West. They could nevertheless meet here in Poland, where Wladek Fiszdon organized once in two years a “Symposium on Advanced Problems and Methods in Fluid Mechanics”. Participation was on invitation and those invited travelled to Warsaw and stayed there one night. The next day they were transported by bus to some place found by magician Wladek where there was food and accommodation, modest but suﬃcient. One could meet in this way with famous Russian scientists as Barenblatt, Zel‘dovich, Ladyshenskaya and others. The ﬂuid dynamics community is greatly indebted to Wladek Fiszdon for organizing these Symposia. Unfortunately, his health condition does not allow him to be here today. From this place I would like to thank him for all he did for Fluid Mechanics in this way. The ﬁrst time that I was invited to participate in such an event was in 1969 in Kazimierz (not named after my friend and colleague Kazimierz Sobczyk who will present the Closing Lecture next Friday). George Bat1 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 1–16. © 2005 Springer. Printed in the Netherlands.

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chelor was a key ﬁgure in these Symposia. He had great authority (he was a Foreign Member of the Polish Academy of Sciences), Wadek Fiszdon asked his advice whom to invite and he was always very relaxed and willing to lecture on everything that he was working on. I remember very well that he gave a lecture on the sedimentation problem on which he was working at the time and what was to become the subject of his celebrated paper “Sedimentation in a dilute dispersion of spheres” [1]. This concerns the velocity with which a cloud of heavy particles sedimentates in a ﬂuid. The, until that time unsolved, diﬃculty in this and similar problems is that the velocity which a small particle induces in its vicinity falls oﬀ very slowly, as the reciprocal distance from its center. The calculation of the average sedimentation speed results, because of this in not uniformly convergent integrals, with which J.M. Burgers struggled already in the 1930‘s. George found a way, a renormalization, to overcome this diﬃculty. His renormalization technique has found wide application in other areas. His presentation in Kazimierz induced me to think about the analogous problem where a cloud of bubbles rises under buoyancy.

2.

Air and Water

The ﬂow around a bubble is, to a good approximation, a potential ﬂow. The velocity which one bubble induces in another falls oﬀ as (distance from centre to centre) −3 . In contrast with the falling particle inertia eﬀects are here dominant, the Reynolds number is large. This (distance)−3 behaviour is faster than that with the falling particle but not fast enough to overcome diﬃculties with not uniformly convergent integrals. When a bubble is accelerated, the surrounding liquid exerts a reaction force on the bubble, which is proportional to the acceleration. The multiplying factor has dimension of mass and is called virtual or added mass, because in calculations it can be treated as a virtual mass of the bubble which is itself of course almost massless. It appears that this mass depends on the presence of nearby bubbles in a manner which gives rise to convergence problems. Consider N bubbles in a conﬁguCN ). When there is always one ration CN with probability density P (C bubble in the point r 0 , such a conﬁguration is indicated with CN −1 | r 0 and the corresponding probability density with P (C CN |r0 ). In the course of the calculation one needs to know the average velocity in the centre of a bubble in the presence of all the others, and with respect to the volume velocity U 0 of the suspension, u − U0 = 1/N !

{u(r0 , CN ) − U0 }P (C CN |r0 )dC CN .

(1)

Interplay between Air and Water

3

For low concentration by volume one considers, just as in the case of dilute gases, the interaction between two particles only or, in this case, bubbles. Then Eq. (1) becomes (2) u − U0 = {u(r0 + r, r0 ) − U0 }P (r0 + r|r0 )d3 r. The quantity in curly brackets in Eq. (2) behaves at large distance r as r−3 and therefore the integral does not converge. The essence of Batchelor‘s renormalisation technique starts in this case with noting that if in Eq. (1) we take just P (C CN ), that means when we consider the average velocity in a point whether in ﬂuid or in gas, then the result is zero, CN )dC CN . (3) 0 = 1/N ! {u(r0 , CN ) − U0 }P (C When reduced to the interaction of two bubbles also this integral does not converge. The only diﬀerence with the right-hand side of Eq. (1) is that there is in the latter always a bubble in r 0 and in Eq. (3) – sometimes. However we know the exact result Eq. (3). Now we subtract Eq. (3) from Eq. (1). Since in the absence of long-range order in the suspension we have at a large distance from r 0 P (C CN |r0 ) = P (C CN ), the diﬀerence of the two integrals converges when the conﬁguration is reduced to two bubbles and this overcomes the problem because we are left with the calculation of the integral {u(r0 , r0 + r) − U0 }{P (r0 + r|r0 ) − P (r0 + r)}d3 r, which is now convergent. Although this problem could be solved, [2], the general problem to understand the dynamics of air-water mixtures is today far from being solved. Particle – liquid ﬂow can either show random conﬁgurations or ﬁxed conﬁgurations. With air and water many more topologies are possible. I mentioned as a ﬁrst example the bubbly suspension, a common device in the chemical industry where it serves as a reactor column. Some more examples are: Niagara Falls (American and Canadian). Air is ﬁrst entrained by river water falling down and mixes with this during its fall. The air leaves eventually together with only a little bit of water, forming with it a spray, or mistﬂow. There are in fact

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Figure 1. The Niagara Falls, in the foreground the horseshoe-shaped Fall on the Canadian side, and the American Fall in the background.

Interplay between Air and Water

5

two falls, one on the Canadian side, with the shape of a horseshoe, and one on the American side. Although the picture is not very good, you can clearly see that with the Canadian fall water droplets are in the upward ﬂow entrained with the air above the original level from where they originated. They derive some energy from the air. This does not happen at the American fall. The reason seems to me that there is, in the restricted space available for the downﬂow of the Canadian one, a pressure built up pushing a strong upward air ﬂow entraining droplets. Breaking wave with trapped air. In the case of a breaking wave air is trapped in the overturning wave. This entrained air plays an important role in the dynamics of the wave, and of its impact on walls. A striking eﬀect of the trapped air is a tremendous change in the compressibility of the mixture. Even an air concentration of a few percent dramatically alters the sound velocity which is directly related to the compressibility. This can be made clear as follows. Denoting the velocity of sound of the mixture with cm , we have from thermodynamics (cm )2 = (dp/dρ)s ,

(4)

Figure 2. A breaking wave at Coogee Beach, Sydney, Australia. Photograph taken by D.H. Peregrine, University of Bristol, and reproduced here with his permission.

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where p, ρ and s denote pressure, density and entropy respectively. The density is made up from that of air, with volume fraction α, and that of water, with volume fraction 1 − α. Referring to air and water with subscripts a and w respectively, we have therefore ρ = α ρa + (1 − α) ρw .

(5)

We neglect a possible velocity diﬀerence between air and water. Then the mass of air in a unit mass of the mixture is constant, whence (6) ρa α/{ρa α + ρw (1 − α)} = constant. We assume further that the pressure in water and air is the same (later we shall see when this ceases to be realistic). Then it follows from Eq. (4)–Eq. (6) that for α not too close to zero or to unity (cm )2 = γp/ρw (1 − α)α,

(7)

γ being the ratio of speciﬁc heats of the air. In Fig. 3 graphs of the complete expression are shown for various ambient pressures. Even with a volume concentration α of air of one percent, the velocity of sound is in an air-water mixture only 100 m/s, far below the sound speed in either air or water. (Air was, of course, also involved in the generation of the wave. That is an old problem in hydrodynamics.

Figure 3. The sound velocity, c, in a mixture of air and water. The air concentration by volume α is indicated along the horizontal axis, the sound velocity along the vertical axis.

Interplay between Air and Water

7

In the last 50, or so, years much has been clariﬁed but it is not completely solved.) The air trapped in water has also a profound eﬀect on the radiation of sound when the ﬂow is turbulent. The late Sir James Lighthill has shown in one of his ﬁnest contributions to ﬂuid mechanics, [3], that turbulence produces, ineﬃciently, quadrupole sound. The presence of air gives a new, by far dominating, monopole contribution, which leads as shown by Crighton and Ffowcs Williams [4], for not too low void fraction, to a sound emission larger by a factor (cw /cm )4 which can be for air and water of the order 106 or an intensity increase of 60 dB. Think about this when you hear these waves speak! Cavitation. Another two-phase situation is encountered in cavitation, for example at a hydrofoil, see Fig. 4. Due to the low pressure in the ﬂow along the hydrofoil, a propeller blade, microscopic nuclei become unstable and grow to macroscopic size. In a region of higher pressure these bubbles, ﬁlled with air and vapour, collapse again and may cause at the ﬁnal stage of the collapse considerable damage to the blade.

Figure 4. Cavitation on a ship‘s model propeller turning in a water tunnel. There is cavitation on the blade but also in the tip vortex. Courtesy of the Maritime Research Institute of the Netherlands (MARIN) at Wageningen.

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In all these cases there is interaction between the gaseous phase and the liquid phase and the title of my lecture refers to this. In a further part of the lecture I shall in a symbolic way imagine the air which is above as the domain of fundamental research and the water, down below,as applied science, which will give opportunity to share some thoughts about science and engineering with you. In the spray above the Niagara waterfall we have a lot of air and some water, in the case of a bubbly suspension we have a lot of water and some air. At both ends there are unsolved problems forming a challenge for ﬂuid mechanics.

3.

Bubbly Flow

Compared with the Niagara fall, the rising suspension looks quiet and peaceful. But lo and behold what happens when we increase, with bubbles of about 1mm radius, the void fraction to about 25 %. A violent transition to slug ﬂow occurs. (During the presentation a video of the transition to slug ﬂow was shown). Both phenomena, the homogeneous rise at low concentration and the transition to slug ﬂow are ill understood. Let us start with the former. The interest in bubbles has always been great. Bubbles smaller than about 0.8 mm radius rise in a straight line. For example bubbles rising in champagne or beer. The application in champagne is due to Dom P´ ´erignon who was a contemporary of Sir Isaac Newton. Their simultaneous occupations are described by Michel Onfray [5] as “while the ﬁrst (P´ ´erignon) prepares beverages with rising bubbles, the second (Newton) derives formulas based on falling fruits“ (my translation from the French text). This quotation from the chapter ´ des bulles” of Onfray‘s book illustrates on a day like “Une petite th´orie this very appropriately the unity of mechanics. Bubbles with a radius above 0.8 mm do not rise linearly in water but perform spiralling or zigzag motion, in contrast to falling particles which fall in a straight line. This was already known to Leonardo da Vinci, who made a sketch of what he saw, Fig. 5, and is therefore called nowadays Leonardo‘s Paradox, see e.g. Ohl, Tijink and Prosperetti [6]. Recently, see e.g. de Vries et al. [7], it has been observed that these spiralling bubbles have a wake trailing behind them consisting of two vorticity bearing threads, see Fig. 6. The relevant bifurcation has been also described numerically, Maugin & Magnaudet [8], but the underlying physical mechanism is not yet understood. The ﬂow around a bubble rising in clean water is well described by potential ﬂow supplemented with thin boundary layers. These, of the

Interplay between Air and Water

9

Figure 5. Sketch by Leonardo da Vinci of a spiralling bubble (Courtesy of Mus´ ´ee du Louvre, Paris). The “ Paradox”, for further details see [6], is in the fact that a falling particle has a straight trajectory but a rising bubble – a spiralling path.

Figure 6. Two mutually perpendicular projections of a bubble spiralling in hyperclean water. The eﬀective bubble radius is about 1 mm. Clearly seen is the double –threaded wake behind the rising bubble. Courtesy of Christian Veldhuis.

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same thickness as with rigid bodies, have now only to overcome the diﬀerence in shear stress between the outer ﬂow and the zero stress at the interface. Just as in the study of dilute gases, it is natural to start with looking at binary interactions. Numerical work and analytical work with this model unfortunately predict clustering of bubbles and not the homogeneously rising bubbles as observed in experiments. This can be made plausible as follows: Imagine two bubbles with their line of centres making, at time t = 0, an angle of more than 36 degrees with the vertical direction which is parallel to gravity. The hydrodynamic forces are such that after a time of order a/V , where a is the bubble radius and V the terminal velocity, the line of centres is horizontal. This is therefore a statistically highly probable situation. But in this conﬁguration there is only an attractive force between the bubbles. They bounce for a while but even when this is a purely elastic collision, the motion gets exhausted by viscous friction, which explains the clustering in the numerical simulation. How in reality the lack of repulsive forces, the basic reason for the clustering, is overcome is another unsolved problem of two-phase ﬂow. I have made recently [9]) the suggestion that the above mentioned twothreaded wakes could play a role here. To continue with unsolved problems in two-phase ﬂow I recall the transition to slug ﬂow. Also this awaits an explanation. There is an analogeous phenomenon in ﬂuid beds. There the homogeneous bed becomes unstable, big gas bubbles are formed, as a result of the instability of concentration waves. It has been thought for some time that transition to slug ﬂow is similarly due to instability of such waves. Evidence, Lammers & Biesheuvel [10] shows this not to be the case. The instability of concentration waves (or void fraction waves) occurs but before slugs appear, there is an intermediate ﬂow regime characterized by a pointed transverse velocity and concentration distribution. You might have the impression by now that in two-phase ﬂow there are mainly riddles and unsolved problems. This is certainly not the case. I will illustrate this with two examples. The ﬁrst is about expansion waves in a two-phase ﬂow. We have seen that the bubbly suspension has a low velocity of sound. So, we can play at low velocities the whole organ of compressible gases. For example the theory and experiments of waves of ﬁnite amplitude. There is, however, an important diﬀerence. If pressure changes become very rapid, the relation Eq. (7) for the speed of sound is no longer valid. My compatriot Marcel Minnaert measured in the 1930‘s the frequency of volume oscillations of small air bubbles in water in an ingenious way, described in [11]. He determined the frequency of the audible popping sound of the bubbles formed in his

Interplay between Air and Water

11

apparatus by means of tuning forks. He also derived a formula for this frequency, f , say, f = 1/(2πa){3γ(p − pv )/ρl }1/2 .

(8)

This is for a bubble with radius of 1mm about 3 kHz. When now in a bubbly suspension pressure changes are not far from this Minnaert frequency, the bubbles do no longer passively follow the pressure changes but pressure diﬀerences between the two phases develop due to the inertia of the liquid. As a result the medium becomes dispersive which expresses itself in various ways. One of these is that the velocity of propagation of a wave of ﬁnite amplitude depends not only on the amplitude, as is the case for “normal” compressible ﬂuids, but also of the frequency, or wavelength. With weak nonlinearity and weak dispersion the famous Korteweg-de Vries equation is valid for the pressure in the wave. Some time ago we did [12] the following experiment: At the entrance of a semi-inﬁnite bubbly ﬂow, a time-dependent pressure was established in the form of a rectangular triangle, a shock wave followed by a rarefaction. For this special initial proﬁle the KdV equation can be solved exactly with help of the so-called inverse scattering theory. The evolution in the mixture of this initial proﬁle is into a train of solitons according to this theory. The nice thing about this is that the associated mathematical equation, a Schrodinger ¨ equation, has for this particular proﬁle an exact solution in the form of an Airy function and that the number of evolving solitons is equal to the zeros of this Airy function in a certain interval. In the experiment that we did, the shape of the evolving waves was not quite that of solitons (they were still evolving) but the number agreed exactly with the predictions, see Fig. 7. Another example is directly connected to Minnaert‘s early ﬁnding. Much later it was discovered that also the sound of rain on a water surface is due to bubbles but in a special way. The falling drop forms a crater in the water which is ﬁlled with air. As the crater closes again the air escapes but sometimes a small air bubble is trapped. This produces noise while oscillating in its Minnaert frequency. Experiments, see Oguz ˇ & Prosperetti [13], with drops of various speeds and sizes show that a bubble is trapped only in a narrow portion of the speed/size plane (see Fig. 8). In nature the speed of the raindrop depends on its size and hence the intersection of this line with the above mentioned area gives the size of the raindrops producing bubbles and accompanying sound. This explains the rather narrow frequency spectrum of rain noise.

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Figure 7. From an initial pressure proﬁle shaped in terms of time as a rectangular triangle by a shock wave followed by an expansion wave, (left-hand side of picture) develops, after the wave has travelled a long distance in the bubbly ﬂow, a series of solitons (right-hand side of picture).

Figure 8. The shaded area bounds that portion out of the diameter/impact plane of falling drops, that marks the occurrence of a trapped air bubble. The broken line represents the relation between diameter and impact velocity of raindrops. Hence the intersection of this line with the shaded area gives the range of raindrops which produce air bubbles and thereby sound. The picture is from [13] and reproduced here with permission of Ann. Rev. Fluid Mech.

Interplay between Air and Water

4.

13

Fundamental and Applied, The Scientist and the Engineer

I will now use the freedom of the lecturer on an occasion as this one to compare in a symbolic sense the interplay just described with the interplay between fundamental and applied science. Fundamental science high up in the air, the applications down below where the water is. Sometimes the interplay is not obvious. Famous are the words spoken by the German mathematician David Hilbert when he was asked to comment on the relation between pure and applied science. He said “Meine Herren, sie haben gar nichts mit einander zu tun” (they have nothing to do with each other). This points at a lack of appreciation. In my case I have been lucky to see a lot of both in my professional life. Both have their peculiarities of which I shall give two examples. First the “∼” and the “=”. What do I mean by this? Suppose you are designing a device which includes ﬂow of water, of kinematic viscosity ν = 10−6 m2 /s, with a velocity U =0.10 m/s in a pipe with diameter D=1 cm. You want to know for the operation of your device, what entry length l is needed for the ﬂow to become fully developed. You turn for advice to a theoretical physicist. He takes his copy of Landau & Lifshitz, Fluid Mechanics [14], from the shelf in his oﬃce and ﬁnds on page 150 l/D ∼ Re. (9) This means that the dimensionless entrance length l is a multiple of order unity times the Reynolds number U D/ν. Landau & Lifshitz point out that the thickness δ of the laminar boundary layer along the pipe wall grows, with x running along the pipe axis, as δ ∼ (νx/U )1/2 , where again ∼ means that the boundary layer thickness is a multiple of order unity times the shown quantity. Putting now the thickness equal to the diameter gives Eq. (9) for the entrance length l. In this particular case Re is 1000. You feel uneasy over it and you ask the physicist, do I really need thousand diameters, which is 10 m? The physicist does not listen anymore. Your problem is now an engineering problem and he does not care. So, you turn to an engineer, for example R.S. Brodkey, who tells you in his book Brodkey 1967 [15] on page 129 that exact calculation gives l/D = 0.06 Re. (10) This is, to your great relief, only 60 diameters or 60 cm. As another example I mention granular materials. This is nowadays a popular subject in physics. It has been, however, widely studied in

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civil engineering and in mechanical engineering since long ago. In the second part of the last century A.W. Jenike (1914-2003) has dominated research in the ﬁeld of bulk solids. The research in this ﬁeld has had results. During the last ICTAM of the last century, Chicago 2000, in one invited lecture, Roux & Radja¨¨ı [16], we read “ The quasi-static behaviour of granular materials is already a mature ﬁeld in which a number of elastico-plastic models reproduce very accurately the available experimental tests. They allow us to design civil engineering structures with conﬁdence”. The authors of these lines come from civil engineering. At the same ICTAM there was another invited lecturer, Goldhirsch [17], a physicist, from whose lecture I quote: “Granular matter is often described as “unpredictable”, “irreproducible” or “erratic”. These and other adjectives used to characterize granular matter are a clear sign that much is still lacking in our understanding of these “materials” (my italics). In my opinion the physicists could have in this case more appreciation for the work done and results obtained by engineers.

5.

Epilogue

The great experience in ICTAM is that both fundamental and applied scientists can listen to each other and talk to each other during and outside the many sessions. And in spite of diﬀerences of approach, illustrated in the previous Section with some examples, there are many aspects in their work that they share. Whereas consultant ﬁrms apply high per-hour rates for every service that they deliver, we all are referees and editors for journals, sit in committees, do work for funding organisations, you name it, without payment or at most a modest compensation for subsistence costs. Why do we do that? There are immaterial rewards in the form of prizes and other signs of recognition. But above all it is out of a sense of duty to the scientiﬁc community. The British writer and philosopher Iris Murdoch [18] points out that our sense of duty stems from the fact that we are not perfect beings, “A totally good being would not experience the call of duty, might be said to lack or not need the concept since all acts and decisions would emerge from virtuous insight and its orderly process”. But just this sense of duty saves us, according to an Editorial in Science [19], from becoming victims of human frailty. I quote from this article entitled “The Roots of Scientiﬁc Integrity”: “The system of rewards and punishments tends to make honest, vigorous, conscientious hardworking scholars out of people who have human tendencies of slothfulness and no more rectitude than the law requires”.

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With this quotation Mr Chairman, intended to make us all feel good, I come to the end of my presentation. I wish you all an enjoyable and rewarding ICTAM 2004 and I thank you for your attention. I thank my colleagues of the Physics of Fluids Group of the University of Twente for their helpful comments, Michel Versluis for teaching me Power Point and Raymond Bergmann, Peter Eshuis, and Christian Veldhuis for their expert help in preparing the Power Point version of this lecture.

References [1] G.K. Batchelor, Sedimentation in a dilute dispersion of spheres, J.Fluid Mech., Vol.52, pp.245-268, 1972. [2] L. van Wijngaarden, Hydrodynamic interaction between gas bubbles and liquid, J.Fluid Mech., Vol.77, pp.27-44, 1976. [3] M.J. Lighthill, On sound generated aerodynamically 1. General theory, Proc.R.Soc.London, Vol.A211, pp.564-587, 1952. [4] D.G. Crighton and J.E. Ffowcs Williams, Sound generation by turbulent twophase ﬂow, J. Fluid. Mech., Vol.36, pp.585-603, 1969. [5] M. Onfray, Gourmande La Raison, Grasset et Fasquelle, 1995. [6] C.D. Ohl, A. Tijink, A. Prosperetti, The added mass of an expanding bubble, J.Fluid Mech., Vol.482, pp.271-291, 2003. [7] A.W.G. de Vries, A. Biesheuvel, L. van Wijngaarden, Notes on the path and wake of a gas bubble rising in pure water, Intn‘l J.Multiphase Flow, Vol.28, pp.1823-1834, 2002. [8] G. Mougin, J. Magnaudet, Path instability of a rising bubble, Phys.Rev. Lett., Vol.88, 014502- 1-014502-3, 2002. [9] L.van Wijngaarden, Bubble velocities induced by trailing vortices behind neighbours, Submitted to J.Fluid Mech., 2004. [10] J.H Lammers, A. Biesheuvel,, Concentration waves and the instability of bubbly ﬂows, J.Fluid Mech., Vol.328, pp.67-93, 1996. [11] M. Minnaert, On musical air bubbles and the sound of running water, Phil. Mag., Vol.16, pp.235-245, 1933. [12] L.van Wijngaarden, Evolving Solitons in Bubbly Flows, Acta Applicanda Mathematicae, Vol.39, pp.507-516, 1995. [13] A. Prosperetti, H.N. Oˇ ˇ guz, The impact of drops on liquid surfaces and the underwater noise of rain, Annu. Rev. Fluid Mech. Vol.25, pp.486-537, 1993. [14] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, 1959. [15] R.S. Brodkey, The Phenomena of Fluid Motion, Addison-Wesley, 1967. [16] S. Roux, R. Radja¨, ¨ Statistical Approach to the mechanical behaviour of granular media, In Mechanics for a New Millenium [Eds.] H. Aref, and J.W. Phillpps, pp.181-197, Kluwer Academic Publishers, 2001.

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[17] I. Goldhirsch, Kinetic and continuum descriptions of granular ﬂows, Mechanics for a New Millenium, Eds. H. Aref and J.W. Phillips, pp.345-359, Kluwer Academic Publishers, 2001. [18] I. Murdoch, Metaphysics as a guide to morals, Penguin Books, 1993. [19] Editorial Essay “The roots of scientiﬁc integrity”, Science, Vol.139, pp.3161, 1963.

Professor Sobczyk delivers the ICTAM04 Closing Lecture

STOCHASTIC DYNAMICS OF ENGINEERING SYSTEMS Origins, challenges and results Kazimierz Sobczyk Institute of Fundamental Technological Research Polish Academy of Sciences ´ ¸etokrzyska 21, 00-049 Warszawa Swi¸ [email protected]

Abstract

This lecture presents a concise exposition of the basic features of contemporary stochastic dynamics of physical/engineering systems with emphasis on its methodological principles, applicatory power and recent challenges.

Keywords: Stochastic systems, random vibration, stochastic degradation models, random loads, reliability assessment, failure models, information dynamics, noise-induced phenomena

1.

Introduction

Historical Origins: Hundred Years from the Beginnings – W. Gibbs (1903), A. Einstein, M. Smoluchowski (1905/06), P. Langevin (1908) Stochastic dynamics is today a greatly advanced ﬁeld of science investigating real dynamical systems with use of stochastic process theory. It develops the models and methods for investigation of various dynamical systems subjected to parametric and external random excitations. The genesis of stochastic dynamics is connected with problems in physics. Although the ﬁrst probabilistic/statistical concepts were introduced to physics already in the 19-th century (kinetic theories of gases; the Maxwell distribution of velocities of the molecules of a gas, Boltzman H-theorem), the ﬁrst years of the 20-th century brought systematic formulations of statistical mechanics/dynamics, including the stochastic description of the phenomenon of the Brownian motion.

19 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 19–60. © 2005 Springer. Printed in the Netherlands.

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One should mention ﬁrst the Gibbsian construction of statistical mechanics (described by Gibbs [1] in 1903) where the problem of time evolution of a large number of material particles (described by Hamiltonian diﬀerential equations) is treated probabilistically. Because of diﬃculties in the exact determination of the dynamical variables and the fact that the systems considered in statistical physics include a very large number of particles, a statistical description of the motion of the system was introduced. The basic role is played by the so-called phase probability density f (x, t), where x denotes a point in the phase space (of generalized coordinates and velocities): x = (x1 , x2 , . . . x6N ), and N denotes the number of particles. Since the motion of the system is governed by the deterministic Hamiltonian equations, the principle of conservation of probability leads to the well-known Liouville equation for f (x, t) ∂f ∂ + {x˙ k f (x, t) } = 0, ∂t ∂xk 6N

f (x, t0 ) = f0 (x0 ),

(1)

k=1

where f0 (x0 ) denotes the probability density of the initial state of the system. The initial value problem described by Eq. (1) can be regarded as the ﬁrst connection between probability and diﬀerential equations. Another phenomenon belonging to physics, which inﬂuenced tremendously the development of probabilistic thinking in natural sciences, is the Brownian motion – an extremely irregular movement of a small particle suspended in a ﬂuid. This phenomenon discovered experimentally by R. Brown in 1827 is one of the most interesting examples of random physical processes. During many years after the Brown’s discovery, various experiments were performed attempting to measure the properties of the Brownian particles (e.g. experiments of F. Exner and R.A. Zsigmondy). Although some qualitative hypotheses were formulated, as well as some results of quantitative nature were provided (e.g. dependence of the particle displacement on its size or the temperature of the medium), none of these ﬁndings was able to shed brighter light on the true nature of the phenomenon of Brownian motion. As S.G. Brush [2] writes in his historical work: “three quarters of a century of experimentation produced almost no useful results, simply because no theorist had told the experimentalists what quantity should be measured!”. The ﬁrst fruitful and breakthrough results came from mathematical model of the Brownian motion by Einstein [3,4] and Smoluchowski [5,6] – (1905/06). These great physicists, proposed - independently and via diﬀerent approaches – the theoretical description and explanation of the Brownian motion. Einstein’s reasoning was inspired by the ideas of the

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Figure 1. Illustration of the trajectory of the Brownian particle on a plane. (cf. www.math.yale.edu)

diﬀusion theory, whereas Smoluchowski’s arguments were based on combinatorics and the mean-free-path approximation of the kinetic theory. It is worth noting that Einstein and Smoluchowski pointed clearly that the basic measurable characteristic of the phenomenon of Brownian motion should be – not, as their predecessors believed, the velocity of particles but – the mean square of their displacements per second. Einstein and Smoluchowski obtained for the mean square displacement x2 in time interval t the same formula (with a slight discrepancy in the numerical factor – due to various approximations used; in his later papers Smoluchowski accepted the Einsten’s numerical factor). This formula is as follows: kT 1 , (2) x2 = t N 3πηr where k is the Boltzmann constant, N is the Avogadro number (a number of point molecules in unit volume), T is the absolute temperature, r is the radius of the spherical particle and η is the viscosity coeﬃcient of the medium. Equation (2) shows that the mean square of the displacement of the Brownian particle grows linearly in time – the result which (as we know today) has had very profound implications in the mathematical theory

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of Brownian motion and stochastic dynamics in general (e.g. the Wiener process/model of Brownian motion). The exact experimental conﬁrmation of the Einstein and Smoluchowski theory was provided by Perrin [7] (using the Brownian motion experiments Perrin determined also the Avogadro number N ). In this context, the value of experimental work of T. Svedberg should also be underlined. Although the scientists mentioned above worked in diﬀerent places, there was among them quite a strong, natural interaction by correspondence. For example, as Pais [8] writes (pp.101) “Six letters between Einstein and Smoluchowski have survived. All show cordiality and great ` natural respect”; cf. also Sredniawa [9]. Nearly in the same time (exactly, in 1908) Langevin [10] formulated a “phenomenological” description of the erratic motion of a “heavy Brownian particle” of mass m immersed in a liquid, using the Newtonian equation for the particle. The interaction of the surrounding ﬂuid with the Brownian particle gives rise to two distinct forces: a dissipative force (due to dynamic friction in the course of a motion of a particle in viscous ﬂuid), and a ﬂuctuation force (arising from the molecular collisions). So, Langevin wrote down a diﬀerential equation of the motion of the particle m

dv = f (t), dt

f (t) = Fr (t) + Firr (t),

(3)

where v is the component of velocity of particle along the x axis and f (t) – the total force caused by a surrounding medium consists of two parts: Fr (t) = −βv(t) being a regular component, and Firr (t) representing irregular or random force acting on the particle by collisions. Denoting: β/m = α,

Firr (t)/m = ξ(t)

one obtains

dv + αv = ξ(t). (4) dt If one assumes that the particle is spherical with radius r and the liquid has a viscosity parameter η then the constant α = 6πrη/m. Symbol ξ(t) represents the unknown force due to the molecular impacts; this force is random in nature and can only by described probabilistically (the Langevin force). Langevin assumed that its mean value (over the ensemble) should be zero and the correlation of each two ﬂuctuating forces at diﬀerent times should be negligibly small when time diﬀerence t2 − t1 is meaningful. The above hypotheses concerning the features of molecular collisions are usually formalized in physics as follows: ξ(t) = 0,

ξ(t1 )ξ(t2 ) = Dδ(t2 − t1 ),

D > 0,

(5)

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23

where δ(t2 −t1 ) is the Dirac function, D is a constant and · denotes averaging over the ensamble of collision process. Therefore, ξ(t) in Eq. (4) is a very peculiar process; today, in stochastic theory it is called a “white noise”. Such a process does not exist in the conventional sense. For this reason, during a long time period it was not clear what should be the appropriate rigorous interpretation of the Eq. (4). Equation (4) is commonly known as the Langevin equation. It can be regarded historically as the ﬁrst stochastic diﬀerential equation. Although the original Langevin equation is linear (and in a scalar form), nowadays (especially in physics) more general, nonlinear equations (also – in vectorial form) with white noise additional excitation are often termed the Langevin equations; various forms of this equation serve as models of real systems in contemporary stochastic dynamics. It is worth noting, that the value for the mean square of the Brownian particle displacement x2 obtained from the Langevin equation (4) agrees exactly with the Einstein-Smoluchowski formula (2) – cf. Lindsay [11].

Origins of Stochastic Dynamics of Real Engineering Systems Uncertainty and random ﬂuctuations are a very common feature of a variety of real dynamical engineering systems. Most of engineering systems (control systems, mechanical, structural, etc.) are subjected to complicated external and internal (time-varying) inﬂuences. These complex excitations and the associated responses can, most rationally, be described in terms of stochastic processes. Among the examples are: structural response due to earthquake, wind load, sea waves etc., random vibration of road vehicles (response to random road roughness), response/and reliability of aerospace structures to random turbulent ﬁeld, response (and stability) of suspension bridges. In the case of such systems as above, the ultimate objective of stochastic dynamics is to provide a new tools for the reliability estimation. In other situations, the qualitative characterization and eﬀects of random excitation are of interest (e.g. stochastic bifurcations, stochastic resonance, eﬀect of random noise on deterministic chaos). Stochastic dynamics of engineering systems emerged nearly exactly ﬁfty years ago. First – in the context of automatic control theory (cf. Booton [12], Kazakov [13]) and, a little later, in the analysis of dynamics of aerospace, mechanical and structural systems. The primary reasons for stochastic analysis was the need to assure a satisfactory performance of engineering systems in the presence of real random noises/excitations.

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For a proper assessment of reliability of a wide class of systems the stochastic analysis of their response turned out to be necessary. In the context of engineering systems of mechanical nature, the pioneering work belongs to the following great scientists (alphabetically): Ariaratnam [14] (1960), Bolotin [15] (1959), Caughey [16] (1959), Crandall [17] (1958), Kozin [18] (1961), Lin [19] (1963), Lyon [20] (1956), Shinozuka [21] (1964). The books by Bolotin [22], Crandall and Mark [23], Lin [24], Robson [25] give an excellent exposition of the early eﬀorts and results. Today, stochastic dynamics of physical/engineering systems has a very extensive literature dealing both with mathematical bases as well as with speciﬁc applications. This literature includes also a number of books-monographs. The books of Bolotin [26], Lin and Cai [27], Roberts and Spanos [28], Sobczyk [29], and Soong and Grigoriu [30] provide an adequate presentation of the existing results.

General Unifying Scheme of Stochastic Dynamics A general methodical scheme of stochastic dynamics of mechanical systems can be illustrated as it is shown in the Fig. 2. The random excitation acting on the system is described by a random/stochastic process (in general vectorial) X(t, γ), where time t belongs to a prescribed time interval [t0 , ∞] or [t0 , T ], whereas γ symbolizes randomness, γ ∈ Γ where Γ is a space of elementary events. More exactly, we have the basic probability space (Γ, F, P ) where F is a family of subsets of Γ (σ–algebra) on elements of which the probability P is deﬁned; 0 ≤ P (A) ≤ 1, A ∈ F . A random variable X(γ) is a measurable function which maps Γ into Rn . A stochastic process X(t, γ) is a function which for each t gives a random variable. This function

Figure 2.

General scheme of stochastic dynamics.

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25

is characterized partially by its mean X(t, γ) = mX (t), by its correlation function KX (t1 , t2 ) = X(t1 , γ)X(t2 , γ) and by other – more complicated – averages. A complete characterization of X(t, γ) is given by probability distributions for various subsets of t-values. Having the probability distribution of X(t, γ) we can calculate the probability of various events associated with X(t, γ), e.g. the probability that values of X(t, γ) belong to a given set D. A stochastic process can be, for example, stationary or non-stationary, Gaussian or non-Gaussian, Markovian etc. For a systematic presentation, a reader is referred to the books on stochastic processes (e.g. see the references in Sobczyk [29]). In stochastic dynamics problems the excitation process X(t, γ) is assumed to be given; most often it must be inferred from the empirical data on real processes. A dynamical system transforms X(t, γ) into another process Y (t, γ); this response process is unknown and should be characterized via the mathematical/stochastic analysis.

Major Challenges The following problems constitute the major tasks of stochastic dynamics of engineering systems. System modeling and characterization of real random excitations. This is a problem of formulation of the governing equations adequately to the speciﬁc system under consideration and selection of the appropriate stochastic processes characterizing external and/or parametric excitations. Although the system modeling follows, to a great extent, the basic principles of model building in deterministic theory, here in stochastic dynamics there are some speciﬁc factors which should be taken into account. For example, in modeling of the system dynamics (under random excitations) with simultaneous degradation taking place in it, the coupled responsedegradation model has to be consistent with the nature of randomness; also, the initial and boundary conditions (which are posed for random functions) should be deﬁned according to stochastic nature of the problem. The characterization of random loads acting on speciﬁc engineering systems is an involved problem itself (we will discuss it in the next section). Characterization of the response; eﬀective solution methods. This is a problem of solving the adequate systems of stochastic diﬀerential equations (mostly nonlinear) with speciﬁed real random excitations. This means that we are looking for a stochastic process, which satisﬁes (in the appropriate sense) a given system

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of equations along with the eﬀective method of quantifying the probabilistic characteristics of this process (moments, probability distribution, etc.). The most notable methods will be discussed in Sec. 5. Failures of stochastic dynamical systems; reliability assessment. The ultimate purpose of stochastic analysis of engineering systems is characterization of safety or reliability of systems of practical importance. Having obtained the probabilistic characteristics of the response (e.g. displacement, stress, etc.) from the dynamic analysis, we can use them to assess the conditions of a system performance for various failure mechanisms. In most cases the reliability of systems in question can be deﬁned and quantiﬁed in terms of some random variables associated with the response process. However, for a wide class of problems (where the Markov process theory can be used) the reliability can be assessed more directly via differential equations for the reliability function (derivable from the governing stochastic diﬀerential equations). Section 6 expounds this problem. Qualitative phenomena / eﬀects. In addition to the problems indicated above and primarily important for engineering practice, there exists a class of interesting questions which are qualitative in nature and are associated with the basic dynamical features of a system in the presence of random noise. Is a random excitation acting on / in a system just an annoying factor - which makes our life more diﬃcult or – maybe – it can generate some new and interesting physical phenomena? It turns out that random perturbations, when combined with nonlinearity, can induce multifarious speciﬁc “noise-induced” phenomena and eﬀects; they have a potential to change some internal features of system dynamics (stability, bifurcations, resonances, etc.). The last part of this lecture will shed some light on these problems.

2.

General Mathematical Model of Stochastic Dynamics

Stochastic Diﬀerential Systems. Basic Interpretations A general model for a wide class of physical and engineering systems subjected to time-varying random disturbances can be represented in

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the vectorial form as follows: dY/dt = F(Y, t) + G(Y, t)X(t, γ),

Y(t0 ) = Y0 ,

t ∈ [t0 , T ],

(6)

where Y(t, γ) = [Y Y1 (t, γ), . . . , Yn (t, γ))] is an unknown vectorial response process, X(t, γ) = [X1 (t, γ), . . . , Xn (t, γ))] is m-dimensional stochastic process (characterizing random excitations acting on the system); F(y, t) = [F F1 (y, t), . . . , Fn (y, t)] is a given vector-valued function of indicated variables describing the regular (deterministic) component of the motion, G(y, t) = [Gij (y, t)]j=1,...,m i=1,...,n is a given matrix-valued function characterizing the state-dependent intensity of the random excitation X(t, γ), Y0 is an initial state of the system (deterministic or random). If G in Eq. (6) does not depend on Y, the model Eq. (6) describes dynamics with an external random excitation. It is seen that the classical Langevin equation (4) is a special case of Eq. (6) when F = −αv,

GX = X/m = ξ.

It is clear that functions F(y, t) and G(y, t) – taking on speciﬁc mathematical forms in modeling real systems – should belong to the class of functions which satisfy the appropriate conditions assuring the existence and uniqueness of a solution of Eq. (6). If the stochastic process X(t, γ) is suﬃciently regular (e.g. continuous and diﬀerentiable) then system Eq. (6) can be called a regular stochastic diﬀerential system. The majority of problems for such systems can be analyzed by use of the methods which are analogous to those in deterministic theory of ordinary diﬀerential equations; in spite of this fact, such stochastic equations give rise to serious solution problems (cf. Sobczyk [29]). If the stochastic process X(t, γ) is very irregular (e.g. white noise, Brownian motion process, jump process) then the system Eq. (6) requires more sophisticated probabilistic analysis. Let us assume that X(t, γ) = ξ(t, γ) where ξ(t, γ)is a vectorial white noise i.e. ξ(t, γ) = [ξ1 (t, γ), . . . , ξm (t, γ)]. In this case the stochastic system Eq. (6) is commonly represented in the form of the Langevin-type equation dY/dt = F(Y, t) + G(Y, t)ξ(t, γ),

Y(t0 ) = Y0 ,

t ∈ [t0 , T ].

(7)

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As we have already mentioned, Eq. (7) – as it stands – should be regarded as a “pre-equation” which needs an appropriate interpretation. The generally accepted interpretation is associated with the equation dY/dt = F(Y, t) + G(Y, t)dW(t, γ),

Y(t0 ) = Y0 ,

t ∈ [t0 , T ], (8)

or, equivalently, with the following stochastic integral equation: t t Y(t) = Y0 + F Y(s), s ds + G Y(s), s dW(s), t0

(9)

t0

where W(t, γ) = [W W1 (t, γ), . . . , Wm (t, γ)] is the m-dimensional Wiener (or Brownian motion) process. Because of the peculiar properties of the Wiener process (e.g. trajectories of W(t, γ) are continuous but nowhere diﬀerentiable and have unbounded variation on each ﬁnite interval), the second integral in Eq. (9) has to be deﬁned in a special way. Two basic deﬁnitions are associated with the names of Ito ˆ and Stratonovich (cf. Arnold [31], Sobczyk [29]). Depending on the choice of the definition of the integral with respect to dW(t), we obtain two diﬀerent interpretations of Langevin-type Eq. (7) via Eq. (8); those are the Itˆ o and Stratonovich interpretations. When G(y, t) occurring in Eq. (7) depends explicitly on y, the Ito ˆ and Stratonovich interpretations lead to diﬀerent solution processes Y(t). There exists, however, simple relationship between the Itoˆ solution and Stratonovich solution. Namely, the Stratonovich solution coincides with the Itoˆ solution of Eq. (8) if the components of the drift term F(y, t) in Eq. (8) are replaced by the following ones 1 ∂Gik (Y, t) + Gjk (Y, t) , 2 ∂Y Yj n

Fi (Y, t) =

Fi∗ (Y, t)

m

(10)

j=1 i=1

where Fi∗ (Y, t), i = 1, . . . , n are the drift components in the Stratonovich equation. The diﬀusion term is the same in both interpretations. In what follows we will adopt the Itoˆ interpretation of the nonlinear Langevin “pre-equation” (7).

Main Theorem; Relation to F-P-K Equation Let us concentrate our attention on the Itˆ oˆ stochastic diﬀerential model Eq. (8). Suppose the following conditions are satisﬁed: (a) the vector-valued function F(y, t) and the (n × m)–matrix valued function G(y, t) are deﬁned and continuous for t ∈ [t0 , T ], y ∈ Rn ,

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(b) the functions F (y, t) and G(y, t) satisfy the Lipschitz condition (with respect to y), (c) the functions F(y, t) and G(y, t) grow (with respect to y) at most linearly, (d) the random variable Y(t0 ) = Y0 is independent of W(t) − W(t0 ) for each t > t0 . Then Eq. (8) has on t ∈ [t0 , T ] a unique solution satisfying the initial condition, almost all realizations of the solution process Y(t) are continuous, and the solution Y(t) is a Markov diﬀusion process on [t0 , T ] with the following drift vector A(y, t) and diﬀusion matrix B(y, t): A(y, t) = F(y, t),

B(y, t) = G(y, t)GT (y, t).

(11)

This means that the transition probability density p(y, t|y0 , t) of process Y(t) satisﬁes the following Fokker-Planck-Kolmogorov (F-P-K) equation n n 1 ∂ ∂p ∂ + [F Fi (y, t)p(y, t)] − [bij (y, t)p(y, t)] = 0, (12) ∂t ∂yi 2 ∂yi ∂yyj i=1

i,j=1

with the appropriate initial and boundary conditions. In Eq. (12) bij (y, t) = {G(y, t)GT (y, t)}ij =

n

Gir (y, t)Gjr (y, t).

(13)

r=1

The theorem stated above shows that a wide class of diﬀusion Markov processes can be constructed – via the stochastic Eq. (8) – on the basis of increments of the Wiener process. It also indicates that a wide class of real dynamical systems modeled by the Langevin-type equation (7) can be characterized by solving the partial diﬀerential Eq. (12). The methods for obtaining solutions of the F-P-K equation (12) have been a subject of a great research eﬀort (cf. Soize [32], Langtangen [33], Spencer and Bergman [34]). Although the progress is signiﬁcant, the effective characterization of the transition probability density for systems of higher dimension (e.g. n > 5) still constitutes a serious problem. The situation is simpler if one is interested in stationary solutions of stochastic systems (8), i.e. when t → ∞, and the F-P-K equation becomes “time-independent”.

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3.

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Engineering Stochastic Vibratory Systems: Response and Degradation Models

Basic Model A general model for dynamics of mechanical/structural systems with random excitation (both – external and/or parametric) can be formulated in the following form ¨ ˙ ˙ MY(t) + CY(t) + R[Y(t), Y(t), X1 (t, γ)] = X2 (t, γ),

(14)

where M and C represent the constant mass and damping matrices, respectively, Y(t) is an unknown response vector process, R characterizes the nonlinear restoring force, and X1 (t, γ), X2 (t, γ) are the random processes characterizing parametric and external excitations, respectively. When the original system is of a continuous type (e.g. beam, plate, shell), the Eq. (14) is a spatially discretized version (e.g. via Galerkin of ﬁnite-element methods) of the original equations and it describes the system response (as a function of time) in ﬁxed spatial points. The stochastic system of Eq. (14), which can be easily represented in the form of a system of the ﬁrst order equations, characterizes (when appropriately speciﬁed) adequately a variety of real systems of engineering practice, e.g. complicated multibody vibrating vehicle systems (cf. Schiehlen [35]), structural/ mechanical vibrating components in bridges, oﬀshore structures as well as aerospace systems (cf. examples in Roberts and Spanos [28], Lin and Cai [27]). In modeling of real physical / engineering problems, the stochastic processes X1 (t, γ) and X2 (t, γ) describing the parametric and external exci-

Figure 3. Exemplary multibody vehicle system with random excitation (cf. W. Schiehlen [35]).

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tations may have quite diverse probabilistic properties. From the point of view of analytical treatment and computational eﬃciency it is most convenient if these random processes can be assumed to be random white noises (i.e. uncorrelated for diﬀerent instants of time); as we know from Sec. 2, in such a case one can make use of the advantages of the theory of Itoˆ stochastic equations (and Markov process theory). It turns out, however, that these theories can also be used for a wide class of random excitations with ﬁnite correlation time; especially for excitations which can be represented as a response of dynamical systems to a white noise excitation ξ(t, γ). An arbitrary Gaussian and stationary random process with rational spectral density can be obtained as an output of a linear system (ﬁlter) with white noise as the input. Therefore, in order to represent the model Eq. (14), with random excitations X1 (t, γ), X2 (t, γ) being real (or, “colored”) random processes, we extend the state space of the system by deﬁning the extended state ˙ The vibratory sysvector [Y1 , Y2 , X1 , X2 ], where Y1 = Y, Y2 = Y. tem governed by Eq. (14) can in this way be represented as the following system of ﬁrst-order Itˆˆo stochastic equations: dY1 (t) = Y2 (t)dt, dY2 (t) = −M−1 [CY2 + R(Y1 , Y2 , X1 (t)) − X2 ]dt, dX1 (t) = −A1 X1 (t)dt + φ1 (t)dW1 (t, γ), dX2 (t) = −A2 X2 (t)dt + φ2 (t)dW2 (t, γ),

(15)

where X1 (t, γ) and X2 (t, γ) are the outputs of the ﬁlter driven by white noises ξ1 (t, γ) and ξ1 (t, γ), respectively; A1 and A2 are the ﬁlters operators (matrices) associated with real excitations X1 (t, γ) and X2 (t, γ), whereas φ1 (t, γ) and φ2 (t, γ) are the time-dependent intensities of white noises ξ1 (t, γ) and ξ2 (t, γ), respectively; W1 (t, γ) and W2 (t, γ) are the Wiener (or Brownian motions) processes in the Itˆ oˆ representation of “pre-equations” of Langevin – type with white noises ξ1 (t, γ) and ξ2 (t, γ), respectively. Therefore, adding the additional ﬁlter equations (for real excitations X1 (t, γ) and X2 (t, γ)), to the original system Eq. (14) enables us to use the Markov process theory, including the analytical and numerical methods developed for Itoˆ stochastic equations (cf. Sobczyk [29]). Of course, the ﬁlter can also be governed by higher order diﬀerential equations. Another possibility of treating system Eq. (14) with real random noises gives the Khasminskii averaging method (cf. Lin and Cai [27], Sobczyk [29], Soong and Grigoriu [30]).

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Stochastic Dynamics of Degrading Systems In fact, stochastic governing equations for many mechanical / structural systems should be represented in a more general form which accounts for inelastic behaviour and simultaneous degradating processes taking place in the system. Above all, these are elastic-plastic vibratory systems (under severe random loadings) in which the restoring force has a hereditary nature (cf. Casciati [36], Irshik and Zigler [37], Wen [38]. Therefore, instead of model (14) we construct the coupled system of equations for the response problem in which the restoring force, in general, depends on the time history of the response. The model is as follows: ¨ ˙ ˙ MY(t) + CY(t) + R[Y(t), Y(t), Z(t), X1 (t, γ)] = X2 (t, γ),

(16)

where Z(t) characterizes various speciﬁc processes responsible for degradation phenomena. Process Z(t) is governed by its own equation (coupled with Y(t)). In general, it has a form ˙ ˙ Z(t) = H[Z(t), Y(t), Y(t)],

Z(t0 ) = Z0 ,

(17)

where function H[z, y, y] ˙ should be constructed for speciﬁc situations. Its mathematical form is inferred from the elaboration of empirical data, or it is derived from the analysis of the physics of the process. It seems that a need for the coupled models dynamics Eqs. (16,17) arose for the ﬁrst time in the analysis of structural response to earthquake. Indeed, structures under strong earthquake excitation become inelastic with restoring forces being nonlinear and depending on the time history of the response. In this situation process Z(t) describes a hysteretic loop and is most often represented by model (17) in which Z(t) = Z(t) is a scalar process, and function H depends only on (z, y) ˙ and has the form (the Bouc-Wen [39] model) n−1 − δ y|z| ˙ n, H[z, y] ˙ = αy˙ − β|y||z||z| ˙

(18)

where α, β, δ, n characterize the amplitude and shape of the hysteretic loop. In the situation considered, a degradation D(t) taking place in the system has been deﬁned in terms of the total hysteretic energy dissipation characterizing the cumulative eﬀect of severe response and expressed by the state variables (z, y). ˙ Therefore, the coupled responsedegradation problem for randomly excited dynamic hysteretic systems is governed by Eqs. (16,17). These equations have been the subject of detailed analysis for many speciﬁc situations under various hypotheses concerning random external excitation (the restoring force R does not depend on X1 (t, γ); cf Wen [39]).

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There is a class of practically important problems governed generally by the coupled Eqs. (16,17) in which Z(t) characterizes directly the degradation of the system. A general model can be formulated in the following form ¨ ˙ ˙ MY(t) + CY(t) + R[Y(t), Y(t), D(t)] = X(t, γ),

(19)

˙ ˙ Q[D(t), D(t), Y(t), Y(t)] = 0,

(20)

where Q[˙] symbolizes a relationship between degradation and response processes. In Eq. (20) dependence on the response can be regarded in some relaxed sense, i.e. degradation rate may depend not on the actual values of Y(t) but – on some functionals of Y(t). In the fatigue degradation problem (more exactly – in the analysis of response of vibrating system with the stiﬀness degradation due to fatigue accumulation), it is natural to quantify the degradation D(t) = D(t) by a “normalized” crack size and adopt as an evolution Eq. (20) one of the “kinetic” equations for fatigue crack growth. These equations, however, ˙ contain the stress intensity factor range, so the degradation rate D(t) depends on the quantity related to Ymax − Ymin . In this situation, Eq. (20) has the form ˙ D(t) = H[D(t), Ymax − Ymin ].

(21)

Another version of an equation for D(t) in the coupled response– degradation problem is obtained if the functional relationship (20) does ˙ not include D(t), and the degradation D(t) depends on some functionals ˙ deﬁned on the response process [Y(t), Y(t)], i.e. (20) takes the form (F denotes here the appropriate functional) ˙ D(t) = F Y(t), Y(t) . (22) Important examples include randomly vibrating systems in which a degradation process depends on the time length which the response Y(t) spends above some critical level y∗ (or, D(t) depends on the number of crossings of the level y∗ by the trajectories of the process Y(t) within a given interval [0, T ]). This is the case of an elastic-plastic oscillatory system with D(t) interpreted as an accumulated plastic deformation generated by the “excursion” of the response process Y(t) into plastic domain (in this situation y∗ = y ∗ may be regarded as the yield limit of the material component in question, cf. Grossmayer [40]. This is also a situation of randomly vibrating plate with fatigue – induced stiﬀness degradation; in this case D(t) is interpreted as accumulated fatigue damage due to exceeding the fatigue limit by the response process.

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The analysis of the stochastic response-degradation problem for elastic - plastic vibratory systems and for the system with fatigue-induced degradation can also be analyzed by the more explicit cumulative model for degradation D(t). We mean the situation in which relationship (20) is represented as follows (in scalar form):

N (t)

D(t) = D0 +

∆i (Y, γ),

(23)

i=1

where ∆i = ∆i (Y, γ) are random variables characterizing the elementary degradations taking place in the system; the magnitude of ∆i depends on the characteristics of the process Y (t) above a ﬁxed (critical) level y ∗ . Process N (t) is a stochastic counting process characterizing the number of degrading events in the internal [t0 , t]. In the case of elastic-plastic oscillator (cf. Grossmayer [40]) ∆i (Y, γ) are the yielding increments taking place in a single yielding duration τY which is related to the time interval which the response process spends above the yield level during a single excursion or during a single clump of excursions. In the case of fatigue ∆i , i = 1, 2, . . . , N (t) can be regarded as the magnitudes of elementary (e.g. within one cycle) crack increments (cf. Sobczyk and Trebicki ¸ [41,42] as well as Sobczyk and Spencer [43]).

4.

Characterization of Real Random Dynamic Loads

General Remarks It is clear that a key factor aﬀecting the system behaviour is the excitation (load) to which a system is exposed. So, the appropriate (adequate to the physical / mechanical situation) modelling of real random loads is a crucial task within the whole methodical eﬀort of applied stochastic dynamics. This is a problem of statistical inference from the empirical data (and from the basic physical mechanisms of the excitation in question) about the most informative features of a random excitation under consideration. Depending on the type of engineering structure and its operational task, we meet various kinds of random load processes. For example: gust wind loads (acting on tall slender, tower-shaped structures such as TV masts, chimneys, some bridges, etc. as well as on various aerospace structures) induced by atmospheric turbulence; sea wave loads (acting on oﬀshore platforms, ships, storage tanks etc.) dominated by gravity forces; earthquake excitations (acting on all structural systems) caused by tectonic phenomena and complex interaction of seismic waves; ground-

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induced excitations (acting on road vehicles) generated by a roughness of real road surfaces; traﬃc loads (acting on long-span bridges) caused by moving vehicles. The character and intensity of the load ﬂuctuations in the examples given above depend on the shape of a structure and its orientation with respect to the load direction. Of course, in the case of turbulent wind its probabilistic characteristics depend additionally on meteorological conditions, the geographical position, the height over the Earth’s surface, etc. So, the statistical inference on the real random excitations acting on engineering systems has to make use of various hypotheses and simpliﬁcations. Let us consider here two important types of random excitation: the sea wave excitation acting on the steel oﬀshore platform and the earthquake excitation.

Characterization of Sea Wave Excitation This type of excitation is generated by wind and sea waves. Waves, in turn, occur as a result of complicated interaction between wind and water. This leads to a loading process that is often described by a series of continuously varying sea states. The nature of oﬀshore loading and the complex interactions likely in the seawater environment make establishment of standard load characteristics (e.g. spectra) for oﬀshore structures much more diﬃcult than for aircraft structures. Although sea motion (or sea states) can be partially characterized by some parameters (e.g., the wave height hs the mean wave period Ts , the wave direction θ), an underlying quantity in stochastic theory is the sea elevation η(x, y, t), which is regarded as a random function of position and time. Probabilistic properties of η(x, y, t) are derived partially from the measurements and partially from hydrodynamic wave theory. In almost all studies in ocean engineering, it is assumed that the sea wave process is a stationary stochastic process. Under such a hypothesis, the process η(x, y, t), for ﬁxed (x, y), is characterized by the spectral density gη (ω). Various forms of the spectral density of sea surface elevation η(t), for ﬁxed (x, y), have been proposed in the literature. The most popular form used in practice is the Pierson-Moskowitz spectrum gη (ω) = Aa2g ω −5 exp[−B(ag /ωv0 )4 ],

ω > 0,

(24)

where A and B are dimensionless constants taken to be A = 8.1 × 10−3 , B = 0.74; v0 is the mean wind velocity at a height of 19.5 m above the still sea surface; and ag is the acceleration of gravity. Using a linearized wave theory, one obtains a relationship between the ﬂuid particle velocity u(x, y, z, t) and the surface elevation η(x, y, t). This relationship allows

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to determine the statistics of particle velocity; e.g. the spectral density of the horizontal component of u and acceleration u˙ (cf. Madsen [44] et al.) If the characteristic dimension of a structure is small compared with the wave length, then the load consists of two basic components: a drag force proportional to the square of the normal component of the incident particle velocity and an inertia or mass force associated with the normal component of the particle acceleration. These forces are combined in the Morison formula for the force per unit length of a ﬁxed cylinder: P = kd u|u| + km u˙ ≡ X(t, γ),

(25)

where u is the incident particle velocity normal to the cylinder, and kd and km are given in terms of the drag and mass coeﬃcients. The total Morison force on a ﬁxed vertical cylinder is obtained by an integration of Eq. (25) with respect to z over the interval [−d, 0]. It should be noticed that even when the sea surface elevation is Gaussian, the non-linearity of the Morison formula yields a force P(t) which, in general, is a non-Gaussian process. The departure from the Gaussian distribution (at a given cross-section) depends on the coeﬃcients kd and km . The non-Gaussian character of the forces acting on oﬀshore structures causes additional problems with their proper characterization. The spectral density, in this situation, provides only a partial characterization of the process. Higher order statistics should be estimated from the data. The local extremes of a random wave force P(t) were investigated by Grigoriu [45]. The application to the response analysis of tension-leg platform can be found e.g. in Spanos and Agarwal [46].

Description of Earthquake Excitation An earthquake action, i.e. a complicated ground motion caused by tectonic phenomena, is a result of complex interaction of seismic waves propagating from the source through inhomogeneous layered media. Multiple scattering of waves at randomly distributed inhomogeneities makes the surface displacement ﬁelds highly unpredictable (cf. Sobczyk [47]). Various stochastic models for a strong ground motion have been proposed. Modelling started from uncorrelated impulses (Housner G.W.) and white-noise representations (Bycroft G.N.) and has been developed to account for non-uniform spectra (Kanai K., Tajimi H.) as well as for the temporal non-stationarity of a random seismic action (Bolotin V.V., Amin M., Ang A.H.S.). These investigations lead to a commonly accepted model for (horizontal) ground acceleration having the form of

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37

a non-stationary modulated stochastic process ¨ γ) = A(t)X1 (t, γ), X(t,

(26)

where A(t) is a deterministic envelope function imposed on stationary process X1 (t, γ). More general model has the form ¨ γ) = Ak (t)Xk (t, γ)I(t), (27) X(t, k

where Xk (t, γ) are stationary random processes, Ak (t) – deterministic functions (envelopes) and I(t) is the Heaviside function. One of the methods of obtaining a tractable model for the reliability predictions consists in treating the system transmitting the motion from the source to the ground surface, as a suitable ﬁlter characterized by a frequency transfer function. This transfer function characterizes approximately the averaged eﬀects of wave propagation through the earth strata. The required frequency transfer function is approximated on the basis of the analytical theory of wave propagation and system identiﬁcation techniques. A speciﬁc common formula for the spectral density of the ground displacement was identiﬁed in: Kanai [48], Ruiz and Penzien [49]. Recently, Suzuki and Minai [50] elaborated the model: ¨ γ) = a1 (t)V (t, γ) + a2 (t)ζ2 (t, γ), X(t,

(28)

where V (t, γ) is the output of a time-dependent linear ﬁlter driven by Gaussian white noise ζ1 (t, γ); ζ2 (t, γ) is another white noise independent of ζ1 (t, γ). There is also possible another way of characterizing the earthquake excitation acting on the structures. It consists in representation of the earthquake process as a series of random impulses. The original idea is attributed to Housner (1947), but the model which we have in mind has the general form N (t) Zk (γ)s(t, τk ), (29) X(t, γ) = k=1

where function s(t, τk ) describes the shape of a pulse at random time instant τk , Zk (γ) is a random amplitude of the k-th pulse and N (t) is a stochastic counting process characterizing the number of impulses in the interval [t0 , t]. Lin [24] indicated the conditions under which there may be valid reasons for modelling an earthquake excitation by uncorrelated or correlated random impulses. The response analysis of

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many speciﬁc systems subjected to excitation in the form (29) has been performed by many authors (cf. book of Iwankiewicz [51] and references therein). The processes of the form (29) can also be used for modelling vehicular traﬃc ﬂow and the corresponding loading of highway bridges.

5.

Characterization of Response: Eﬀective Solution Methods

Existing Approaches The most notable approaches used in engineering stochastic dynamics are as follows. Perturbation method. If the nonlinearity in the system is weak, a small parameter ε 1 is introduced into the governing equations, and the solution process Y(t) is looked for in the form of a series expansion with respect to this small parameter. Introducing this expansion into the equations of dynamics and equating terms of the same order in ε, we obtain a recurrent sequence of linear diﬀerential equations for the successive terms Y0 , Y1 , Y2 , . . . of the expansion. These equations, especially when the excitation is Gaussian, can serve to obtain two ﬁrst statistical moments of the solution. Statistical linearization. A nonlinear stochastic system is replaced by “equivalent” linear equations whose coeﬃcients are determined from the condition of minimum of the diﬀerence between nonlinear and the “equivalent” linear part of the system equations (e.g. mean-square criterion); to evaluate the coeﬃcients of the “equivalent” linear equations one has to assume a speciﬁc form of the probability distribution of the unknown solution. Most often the Gaussian approximation is used (for systems with Gaussian external excitation and very weak nonlinearity). The procedure has been used mostly to compute the second order statistical moments of the stationary response; cf. Spanos [52], Socha and Soong [53]. Equivalent nonlinear systems. A given nonlinear system (with random excitation) which is too complicated for eﬀective quantitative analysis is replaced “equivalently” by another nonlinear system which is simpler to handle mathematically and computationally; e.g. by a nonlinear system

Stochastic Dynamics of Engineering Systems

39

for which a stationary solution exists and its probability density is available (cf. Caughey [54], Cai and Lin [55]). Equations for moments and closure hypotheses. From a given nonlinear stochastic system the equations for the statistical moments of the response are derived, which are diﬀerential for a nonstationary solution process and algebraic for a stationary solution. Because of nonlinearity of the original equations, these moment equations constitute an inﬁnite hierarchy of equations, so the appropriate closure assumptions have been proposed and used to obtain a ﬁnite system of equations (these are various hypothetical relationships between higher- and lower-order moments); cf. Crandall [56], Wu and Lin [57], Sobczyk and Tr¸e¸bicki [58]. Once the equations were “closed” they can be solved and, therefore, give the approximate moments of the response. Stochastic averaging method. This method is associated with the question: can the system with real random excitations be treated by the use of the Itˆ oˆ stochastic diﬀerential equations (without extension of the state space of an unknown process)? A positive answer to this question is due to Stratonovich and Khasminskii. Stratonovich [59] noticed that for a wide class of excitation processes X(t) acting on nonlinear systems, the stochastic eﬀects become truly important for the time intervals of order 1/ε or 1/ε2 . He also enunciated heuristically a theorem assuring that on time intervals of length of order 1/ε2 , process Yε (t) approaches a Markov diﬀusion process. A rigorous mathematical formulation and proof of this theorem was provided by Khasminskii [60] along with explicit formulae for the drift vector and diﬀusion matrix of a limiting diﬀusion process. This theorem constitutes a ground for the eﬃcient method known in stochastic dynamics as the stochastic averaging method; its numerous applications to practical problems can be found in the books cited above. Numerical schemes for stochastic diﬀerential systems. In the last decades various numerical schemes for approximation of the solutions of stochastic diﬀerential and integral equations have been elaborated. Although they can be viewed as being an extension of the corresponding schemes of deterministic numerical analysis (e.g. stochastic Euter scheme, stochastic Runge-Kutta scheme, etc.), in the stochastic case one meets speciﬁc and complex problems. In contrast to the deterministic situation, in the

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case of stochastic Ito ˆ equation diﬀerent schemes can converge to diﬀerent solutions for the same noise sample and initial conditions. In addition we can consider various types of approximation; the most common are: (i) mean-square approximation, (ii) pathwise (or, sample function) approximation, (iii) approximation of moments [f (Y Yt )]. Additional diﬃculties occur in multidimensional case; in this case the solution process can not be, in general, expressed as a continuous functional of the Wiener process alone; the detailed presentation can be found in the book by Kloeden and Platen [61], cf. also Chapter 5 of the book by Sobczyk [29]. Numerical methods for the Fokker-Planck-Kolmogorov equation. Since a wide class of physical engineering stochastic systems can be analysed via the diﬀusion Markov processes (generated by the Itoˆ stochastic diﬀerential equations), the numerical solutions of the appropriate F-P-K equations are of a great interest. The F-P-K equation(12) associated with the basic stochastic model Eq. (8) is a partial diﬀerential equation of parabolic type and, as such, is – in principle – accessible to the existing numerical methods. However, one should keep in mind that the F-P-K equation for the transition probability density p(y, t|y0 , t0 ) is for practical problems an equation in multi-dimensional space with variable coeﬃcients and with speciﬁc conditions associated with the probabilistic nature of unknown function (initial, boundary conditions and global normalization condition for the probability density). So, the applicability of the numerical methods (including FEM) is still limited to systems of lower dimensions, that is to the cases when unknown probability density depends on at most four spatial variables. Extension to higher dimensions, as concluded by the authors of paper by Spencer and Bergman [34] “ . . . while posing no logical problems . . . is beyond the capability of the current computer hardware”. An interested reader is referred to the book by Soize [32], and the paper of Spencer and Bergman [34]. An approach to approximate solving the Fokker-Planck-Kolmogorov equation which has attracted some interest has its roots in R. Feyman’s work on a space-time approach to quantum mechanics and is known as the path integration method; it consists in using the ChapmanSmoluchowski equation for discretized time variable (cf. Wehner and Wolfer [62], Naess and Johsen [63]). But, like in the previous techniques, its application to higher-dimensional systems seems to be diﬃcult. Each of the above approaches to quantitative characterization of the solution (or response) of stochastic systems has its own methodical draw-

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41

backs and restrictions. As far as numerical methods are concerned, the computational eﬃciency (especially, for real multi-dimensional systems) is still not satisfactory. Undoubtedly, elaboration of the eﬃcient computational methods for multidimensional stochastic systems constitutes today one of the greatest challenges within the stochastic dynamics research. In what follows we will report brieﬂy on the approach elaborated recently for evaluation of the probability distribution of the solution of stochastic equations and based on the moment equations and the informational entropy of the system.

Maximum Entropy Method for Stochastic Systems The maximum entropy principle (MEP), originating in statistical physics, states that of all the probability distributions that satisfy the appropriate moment constraints (given information) one should choose the distribution having the largest informational Shannon entropy. Since the entropy characterizes a global uncertainty of a random quantity in question, the principle of maximum entropy means that maximum entropy distribution is maximally noncommittal with regard to missing information. The above idea has been widely used in statistics and in variety of other applications. In its classical formulation MEP deals with random variable with unknown probability density, the partial information about which is given by a ﬁnite number of moments. It has been tempting to try to adopt this principle to determining the unknown probability distribution of the solution of a stochastic system. The ﬁrst attempt toward such a goal has been made by Sobczyk and Tr¸ebicki [64] where the general scheme of the method and some illustrative examples have been presented. In the papers by Sobczyk and Tr¸ebicki [65,66], Tr¸e¸bicki and Sobczyk [67], the idea has been extended to more complicated situations. Let the system of interest be governed by the following stochastic Itˆ o equation for the vector process Y(t) = [Y Y1 (t), . . . , Yn (t)]: dY(t) = F[Y(t)]dt + G[Y(t)]dW(t, γ),

(30)

where W(t, γ) = [W W1 (t, γ), . . . , Wm (t, γ)] is the m-dimensional Wiener process. Under known conditions speciﬁed in Sec. 2 the solution of Eq. (30) is a diﬀusion Markov process with the drift vector A(y) and diﬀusion matrix B(y) deﬁned in Sec. 2 (see Eq. (11)). The equations for moments are derived easily by use of the Itˆ oˆ formula k1 (or, Itˆ oˆ diﬀerentiation rule) to the function hk = Y1 , . . . , Ynkn of the solution and taking the average. The symbol k denotes here the multiindex, i.e. k = (k1 , . . . , kn ); we will denote: |k| = k1 + . . . + kn and

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|k| = 1, 2 . . . , K. The moments of process Y(t) at time t are deﬁned as usual by Y1k1 . . . Ynkn f = hk (Y(t))f , (31) mk = Y where f denotes the mean value of the quantity indicated, i.e., f is the integral of the product function hk (y) = y1k1 . . . ynkn with respect to the true probability density f (y; t) = f (y1 , . . . , yn ; t) of the solution process. The general form of the moment equations is ∂hk 1 ∂ 2 hk dmk (t) Fi Gil Gjl = + . (32) dt ∂Y Yi f 2 ∂Y Yi ∂Y Yj f i

i,j

l

The initial conditions mk (t0 ) are speciﬁed from the given probability density f (y; t0 ) of the initial condition Y0 (γ). According to the spirit of the maximum entropy principle, the approximate probability density p(y; t) of the stochastic process Y(t) governed by the general system Eq. (30) is determined as a result of maximization of the information entropy functional H[p [ ] = − p(y; t)lnp n (y; t)dy (33) under constraints Eq. (32) and the normalization condition p(y; t)dy = 1.

(34)

The integration in Eqs. (33,34) is extended over the range of the possible values Y(t) for each t. Let us notice that constraints Eqs. (32,34) in the maximum entropy scheme can be represented as p(y; t) − 1 = 0, (35) dmk (t)/dt = Qk (y)p , where Qk (y) =

i

Fi

∂hk (y) 1 ∂ 2 hk (y) + Gil Gjl . ∂yi 2 ∂yi ∂yyj l

(36)

i,j

It has been shown (cf. Tr¸e¸bicki and Sobczyk [67]) that the probability density which maximizes the entropy functional Eq. (33) under constraints Eq. (35) has the form K λk (t)Qk (y) , p(y; t) = C(t)exp − |k|=1

(37)

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43

where |k| = k1 + . . . + kn = 1, 2, . . . K < n, ki = 0, 1, . . . , n and C(t) is the normalizing factor equal to e−λ0 (t)−1 . Functions λk (t) being the unknown Lagrange multipliers are determined by substituting density Eq. (37) into constraints Eq. (35). This means that all moments mr , for r > |k| occurring in the set of moment Eq. (35), which is not closed, are calculated with the use of probability density Eq. (37); this is just the maximum entropy closure. In the stationary case when the probability density of the solution process Y(t) does not depend on t, multipliers are constant, i.e., λk (t) = λk and the moment equations are algebraic (cf. Sobczyk and Tr¸ebicki [65]). Hence, instead of Eq. (37) we have K p(y) = C exp − λk Qk (y) . (38) |k|=1

In such a way the problem of determining a probability distribution of the solution of a general stochastic system is reduced to the solution of a system of deterministic equations (diﬀerential or algebraic) for the Lagrange multipliers; the approximate probability density itself is represented in analytical form. A reader interested in details of the method is referred to the papers cited at the beginning of this subsection.

Empirical Characterization of Random Response, Optimal Experiment Design, Remarks Although the theoretical analysis dominates the research in stochastic dynamics, in many situations the empirical information acquired from measurements on randomly vibrating structures is necessary. In addition, the nature of the state (response) variables often does not allow much ﬂexibility as to which states can be measured. The problem consists in estimation of the probabilistic characteristics of random ﬁelds (e.g. displacements, stresses of randomly vibrating beams, plates, shells) on the basis of a statistical sample obtained in a ﬁnite number of points. It is clear that the informational content of the data depends on the number of measurement points and their locations. The experiments should be designed optimally. In distributed parameter systems an important optimal experiment design variable is the spatial location of the measurement sensors. In the paper by Papadimitriou, Haralampidis and Sobczyk [68] a general method was given for optimising the number and positions of sensors on randomly vibrating structures for the purpose of the response prediction at the unmeasured locations. The dynamics of the structure is governed by a linear partial diﬀerential equation subjected to space-time

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random excitation with given mean and correlation function. The response characteristics, when obtained from the random vibration analysis, are used in the kriging method to obtain the response predictions (and the corresponding mean-square errors) at unmeasured locations. The optimal sensor locations are chosen to minimize the total averaged (over all prediction points) mean-square error of the response prediction in unmeasured points. The dependence of the optimal sensor locations on the type of response variable (displacement, strain), the characteristics of the random excitation and number of sensors have been determined and illustrated via numerical calculations. The interested reader will ﬁnd the details of the analysis in the paper cited above, along with the appropriate references.

Information Dynamics, Remarks Analysis of many problems of stochastic dynamics leads naturally to the concepts and tools of the information theory. This is not only due to the fact that the basic notions of mathematical information theory are based on the probability theory, but also because the apparatus of information theory is applicable to any probabilistic system of inference (in which we seek information). When the language and tools of information theory (e.g. Shannon entropy, mutual information between random events and processes, information ﬂow) are used in system dynamics, we come to the notion of information dynamics. The premises, challenges and results of this emerging ﬁeld are presented in Sobczyk [69].

Eﬀects of Spatial Randomness, Remarks In the present lecture I have restricted my attention to dynamical systems in which randomness is time-dependent. The systems material parameters remain deterministic and constant. It is clear however that in reality, an engineering system can never fulﬁll strictly such ideal requirements; manufacturing processes of structural/mechanical components introduce some imperfections and inhomogeneity into the material structure. Therefore, in many situations there is a need for taking into account the spatial randomness of the system properties (e.g. random bending stiﬀness of beams, plates, etc., or span length in an N-span beam). This randomness, when characterized by random variables, is usually called a random disorder (cf. Lin and Cai [27] – Sec. 9). In many situations one should assume that the material property randomly varies in space; this leads to the governing partial diﬀerential equations with spatial randomness in coeﬃcients, and subsequently – to

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random eigenvalue problems (cf. early papers by Shinozuka and Astill [70], Sobczyk [71]) and stochastic ﬁnite elements (Shinozuka and Deodatis [72], Ghanem and Spanos [73]). In a sense, the analysis of wave propagation in stochastic media belongs to this category (cf. the book by Sobczyk [47] and references therein) and it constitutes today a fairly advanced ﬁeld.

6.

Failures of Stochastic Dynamical Systems – Reliability Assessment

General Formulation Next important step in stochastic analysis of dynamical engineering systems – strongly connected with the response characterization – is the evaluation of the system performance and its reliability. The character of systems being a subject of stochastic dynamics (complexity of interaction between the system constituents and external excitation, material property degradation due to dynamics, etc.) requires much more sophisticated formulations and methods than the traditional safety assessment. Not only the probability concepts have to play a key role in the analysis but also (and above all) the fact that system performance changes in time. So, today there is a common agreement between the researchers that failures (of various modes) of dynamical systems should be deﬁned as outcrossings or exits of the appropriate stochastic processes (or, random ﬁelds) out of an acceptable (safety) domain. Let us assume that the system states in each time instant are characterized by a random response process Y(t, γ), i.e. for each t ∈ [t0 , ∞] – by a random vector Y belonging to the state space of the system. In order to describe the system performance and its reliability it is convenient to deﬁne quality states of the system and the quality space. A quality is ˆ For each state yt in the state space there characterized by a vector Y. exists a corresponding quality state y ˆt in the quality space. A set of system states and consequently, a set of corresponding quality parameters admissible from the point of view of quality (or reliability) requirements deﬁnes in the quality space a set Dr which can be interpreted as a safety or reliability domain. A boundary of Dr corresponds to the limit (or critical) states. Reliability function R(t) of a system under consideration is deﬁned as the probability of its admissible performance during the time interval [0, t], i.e.

ˆ ) ∈ Dr ; R(t) = P {Y(τ

τ ∈ [0, t]}.

(39)

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Figure 4.

Illustration of the system reliability problem.

The complement of R(t) to unity deﬁnes the probability of failure, Pf (t) = 1 − R(t)

(40)

i.e. the probability that at least one outcrossing of set Dr by process ˆ Y(t), in the direction normal to the boundary of Dr , will occur within the time interval [0, t]. The time of satisfactory system performance TDr (i.e. the time duraˆ t ∈ Dr , is called a life-time; it is a random variable. tion within which Y Reliability is related to TDr by the formula R(t) = P {T TDr > t}.

(41)

Analysis of the reliability problems depends crucially on the speciﬁc failure mechanisms. Engineering systems (e.g. machines, structures, etc.) subjected to random dynamic load may fail due to various failure modes that can occur during the designed lifetime. They depend on the material properties, system characteristics and excitations. The basic failure modes are: (1) the motion of a system becomes unstable, (2) the system response (or, the appropriate function of it) at a critical location exceeds, for the ﬁrst time, the prescribed safety boundary, (3) the accumulated damage (due to Y(t) – ﬂuctuations) exceeds the ﬁxed critical limit (e.g. fatigue failures). It is clear that evaluation of the reliability function R(t) is a basic problem in the reliability assessment of stochastic dynamical systems. Solving such a problem in practical situations (i.e. obtaining exact values of the probability Eq. (39)) meets serious diﬃculties. So, various approximations and bounds on reliability and failure probability have been proposed (e.g. cf. Bolotin [74], Lin and Cai [27] – Chapter 8). In general, the boundary of safety domain Dr can be random. Therefore, the problem consists in evaluation of the probability of passing of the ˆ process Y(t) outside the safety domain Dr with random boundary; this introduces additional diﬃculties.

Stochastic Dynamics of Engineering Systems

47

Stochastic Stability/Instability As far as stability or instability is concerned, it has always been of a great concern in the analysis of dynamical systems. In the stochastic case, when the system response is generally an n-dimensional stochastic vector process Y – measured from a referenced solution (regarded as the trivial solution) – stability is deﬁned in terms of boundedness and convergence (for t → ∞) of the norm ||Y(t)||. But, these properties can be characterized in diﬀerent probabilistic meanings. So, various stochastic stability notions have been introduced and a variety of stochastic stability criteria obtained. Nowadays the existing literature concerning stability of stochastic dynamical systems is very extensive; it is concerned both with a beautiful mathematical analysis and with speciﬁc applications. The essential advances are concerned with elaboration of the stochastic Lapunov function method and with the analysis of the asymptotic behaviour (when t → ∞) of stochastic systems, including stability, by using the Lapunov exponents; cf. for example: Khasminskii [75], Arnold and Wihstutz [76], Ariaratnam and Xie [77], Wedig [78], Bucher and Lin [79], Tylikowski [80,81]. It is worth noticing that the system stability/instability can be characterized via the basic formula (39). Indeed, the system stability can be deﬁned in terms of probability that the response process Y does not leave a spherical domain, say Dst of radius ε > 0 centered at the equilibrium point y0 , where Dst is a set of points y such that ||y − y0 || < ε.

First–excursion Failures: Stochastic Diﬀusion Markov Systems As we have already stated in Section 7, the transition probability oˆ stochastic dynamical density p(y, t; y0 , t0 ) of the solution process of Itˆ system (let us assume here that its drift and diﬀusion coeﬃcients do not depend explicitly on time) satisﬁes the Fokker-Planck-Kolmogorov Eq. (12) with respect to y and t. It turns out that p(y, t; y0 , t0 ) satisﬁes also – with respect to the “backward” variables y0 , t0 - the backward Kolmogorov equation. Using this equation it can be shown that the reliability function R(t|y0 , t0 ) satisﬁes the equation n n ∂R 1 ∂2R ∂ R+ mi (y0 ) 0 + bij (y0 ) 0 0 = 0. ∂t0 2 ∂yi ∂yi ∂yyj i=1

(42)

i,j=1

This equation is supplemented by the appropriate initial and boundary conditions.

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Equation (42) along with the imposed conditions constitutes a general mathematical basis for characterization of the reliability of a wide class of stochastic dynamical systems. As one could expect, however, closed form solutions are known only for simple cases (cf. the book by Lin and Cai [27]). To make the reliability problem more tractable, instead of evaluation of the (conditional) reliability function we restrict ourselves to the calculation of statistical moments of the probability distribution of the (ﬁrst) passage time TD of the response process Y(t) across the boundary of the safety domain Dr . For example, the mean value of the ﬁrst passage time satisﬁes the equation (known as the Pontryagin equation) 1 d2 dT dT b(y0 ) + m(y0 ) = −1. 2 dy0 dy0

(43)

The boundary conditions for this equation at the ends of the safety interval considered, say [α, β] are: T = 0 for y0 = α, y0 = β, since the ﬁrst passage time is zero when Y (t) starts on the boundary of the safety set. The moments of the ﬁrst passage time have been evaluated for various practical situations. In general, the Galerkin ﬁnite element method can be used (cf. Spencer [82]). However, numerical diﬃculties are usually encountered when we are dealing with higher-dimensional random response processes.

Cumulative Failures: Fatigue in Randomly Vibrating Systems An important deterioration or failure mechanism in structural and mechanical components subjected to time-dependent (deterministic or stochastic) loading is fatigue. According to fracture mechanics, the fatigue is due to nucleation and growth of cracks. In engineering, a measurable characteristic of fatigue is usually the size of a dominant crack, and ultimate failure occurs when this crack reaches the critical size. To capture the basic features of random fatigue crack growth, various stochastic models have been proposed in the literature (cf. Sobczyk and Spencer [43] and references therein). A model which takes into account the empirical information and randomness inherent in the fatigue crack growth consists in randomization of the empirical crack growth equation (e.g. Paris-Erdogan equation) by introducing to the equation an appropriate stochastic process X(t, γ). The resulting equation has the general form dA(t) = F [A(t), ∆S, constants]X(t, γ), (44) dt

Stochastic Dynamics of Engineering Systems

49

where A(t) is the crack size at time t, ∆S characterizes the stress range, and F is the empirical (nonlinear) √ function of the indicated variables. Assuming that X(t, γ) = mx + 2Dξ(t, γ) and using the empirical Paris equation, the following probability density of fatigue life-time TF was derived in Sobczyk [83]

α 1 (α − βt)2 √ , (45) fTF (t) = exp − 2 t 2πt3/2 which is the inverse Gaussian distribution (α, β are constant parameters including Paris constants, A0 , mx , D and ∆S). Such distribution has been earlier hypothesized as a possible lifetime model. The inverse Gaussian distribution for a fatigue life-time has also been derived by Ditlevsen [84] by an alternate approach based on an incremental version of Paris-Erdogan equation. Other approaches to the characterization of the fatigue life-time can be found in some recent publications (cf. Doli´ n ´ski and Colombi [85]).

7.

Qualitative Phenomena: Noise – Induced Eﬀects. Examples

Introductory Remarks In the previous four sections I focused my attention on the quantitative aspects of stochastic dynamics, which are of interest in applied (engineering) problems. One could therefore get an impression that stochastic dynamics is primarily concerned with numerical eﬀects of random noises and it does not deal with the phenomena which might be generated in the system solely by random excitations and which have a power to change the system dynamics qualitatively. Such a view would not be correct. The questions which naturally come to mind are, for example: (i) Does a random excitation (acting on the system) have any inﬂuence on the most essential, internal features of a system dynamics? (ii) Is a random noise just an annoying factor we have to live with or is there any interesting physics induced that is not present when the random ﬂuctuations are absent? All macroscopic systems of interest in physics, biology, engineering, chemistry, etc. are subjected to irregular perturbations, internal and external, which – when combined with nonlinearity – can display a rich variety of speciﬁc “noise-induced” phenomena and eﬀects. For example, stochastic instability, bifurcations, the also called – noise-induced transitions of the equilibrium states of the system, etc. It has to be kept in

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mind that in the presence of random (internal/external) excitations the state of the system is no longer characterized by a simple number (or vector) but by a probability distribution. In order to shed some light on the phenomena and eﬀects which random excitations may generate in the dynamical system we will discuss a few “simple” examples.

Stabilization and Destabilization by Random Noise Since some time it has been observed by engineers (ﬁrst, in radioelectronic systems) that noise can aﬀect a system in two opposite ways. It can destabilize a system as well as stabilize it (cf. the survey article of Roberts and Spanos [86] and the paper by Bucher and Lin [79]). As it is well known, a linear oscillatory system (cf. Arnold [87] et al.) Y¨ (t) + 2β Y˙ (t) + Y (t) = 0

(46)

is stable for β > 0. Let us perturb the constant stiﬀness term by a random noise ξ(t), where ξ(t) is a stationary random process with a given spectral density g(ω) and with intensity σ > 0. So, we have the damped linear oscillator with random restoring force Y¨ (t) + 2β Y˙ (t) + [1 + σξ(t)]Y (t) = 0.

(47)

The Lapunov exponent λ = λ(β, σ), a counterpart of the real parts of the eigenvalues, is the indicator of stability (λ < 0) or instability (λ > 0). For small noise (σ → 0) and underdamped case (β 2 < 1): πg(2 1 − β 2 ) 2 σ + O(σ 3 ). λ = −β + 4(1 − β 2 )

(48)

For overdamped case (β 2 > 1): 2 −1 2 λ = −β + β − 1 − 4[(β − 1)]

∞

e−2τ

√

β 2 −1

K(τ )dτ.

(49)

0

Therefore, in the underdamped case, a small noise destabilizes the motion since a positive term is added to λ = −β. However, in the overdamped case, a small noise stabilizes the motion since a positive term is subtracted from the value λ = −β + (β 2 − 1)1/2 . For a more general analysis of the stabilization by noise cf. Arnold [88] and the references therein, and the book by Khasminskii [75].

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Noise-Induced Bifurcations Stochastic bifurcation analysis deals with the qualitative changes in parameterised families of stochastic systems ˙ Y(t) = F[Y(t), X(t), λ],

(50)

where λ is a parameter. It seems that the bifurcation problems for stochastic equations were ﬁrst studied by physicists (cf. Horsthemke and Lefever [89] and references therein). In these studies the qualitative change of stationary solutions of the Fokker-Planck-Kolmogorov equation is used as an indicator of bifurcation. More explicitly, the extremes of the stationary density pst (y; λ) are regarded as indicators of qualitative changes in the system dynamics; these changes are called the phase transitions. The number and positions of the extrema of pst in the stochastic case and the extrema of the potential (of the system considered) in the deterministic case are the characteristic features of the steady-state behaviour of the system. Let us write down the deterministic system corresponding to Eq. (50) in the form (for n = 1) Y˙ (t) = Fλ (Y (t)).

(51)

It is convenient to represent this equation as d Vλ (Y ), Y˙ (t) = − dY

Vλ (Y ) = −

Y

Fλ (z)dz,

(52)

0

where Vλ (y) is called the potential of Eq. (50). The stable steady states of the system Eq. (51) correspond to the minima of Vλ (y) and the unstable steady states to the maxima. The maxima of pst (y) are the states in whose neighbourhood the system spends relatively much time, and they are most likely to be observed in an experiment; the minima of pst (y) correspond to the valleys of a potential (stable steady states). The minima of pst (y) are the states that the system leaves rather quickly (the unstable steady states). Depending on the value of the bifurcation parameter λ, density pst (y) may exhibit one-peak to two-peak or crater-like density. The analysis of the extrema of stationary probability density in multidimensional cases is, of course, much more involved. The analysis of a nonlinear Duﬃng-Van der Pol oscillators along this line was presented by Wiesenfeld and Knobloch [90]. Let us take the deterministic Duﬃng-Van der Pol oscillator Y¨ = αY (t) + β Y˙ (t) − Y 3 (t) − Y 2 (t)Y˙ (t),

α, β ∈ R1 .

(53)

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For this system in a deterministic case, trajectories do not explode in ﬁnite time, and there exists a (parameter-dependent) attracting compact set. When β is ﬁxed and is less than zero, whereas α varies, the pitchfork bifurcation occurs for α = 0. When α < 0 is ﬁxed and β varies, the Hopf bifurcation can be observed at β = 0 (cf. the bifurcation diagrams in Schenk-Hoppe´ [91]). Let the noisy version of Eq. (53) have the form Y¨ (t) = [α + σξ(t)]Y (t) + β Y˙ (t) − Y 3 (t) − Y 2 (t)Y˙ (t).

(54)

The analysis of a stationary F-P-K equation corresponding to Eq. (54) shows that for β suﬃciently negative, the Dirac delta shape of probability density is the only possible. Increasing β, we observe the birth of a bellshaped density and when β increases further (crossing β = 0) this density undergoes a “P-bifurcation” (at β = βp ) and it becomes crater-like. The above eﬀects of noise on “P-bifurcation” are inferred from the stationary probability density pst (y) which characterizes a system for long times and therefore it does not carry information on the transient states. However, recently a serious eﬀort has also been made to understand the bifurcations of stochastic systems on the dynamic level. Extensive simulations of stochastic dynamic systems, their random attractors and invariant measures provided interesting results. For example, it has been found (cf. Arnold et al. [92]) that random noise splits deterministic multiple eigenvalues. For β 2 < −4α (where α is ﬁxed and negative) and σ = 0, the deterministic linear system has two complex-conjugate eigenvalues 0.5β ± iωd what amounts to just one Lapunov exponent λ1 (β) = β/2 with multiplicity 2; for σ = 0, however, a linearized system has two diﬀerent simple Lapunov exponents.

Chaotic Dynamics Subjected to Random Noise An important inner property of deterministic nonlinear systems is chaos. It causes unpredictability of the long-term behaviour of the system. As one may expect, there has to exist the inﬂuence of external random noise on various characteristics of a chaotic dynamics. The eﬀects of noise on chaos, and – more generally – the interplay between chaos and externally introduced randomness has been a topic of research in nonlinear dynamics and statistical physics (cf. Grasman and Roerdink [93], Kapitaniak [94], Ying Cheng Lai [95] et al.). Among various reported results are the following: in the common route to chaos (the period-doubling bifurcations) the random noise tends to smooth out the transition and induces chaos in the parameter regime where there is no chaos otherwise,

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some authors found (cf. Grasman and Roedrik [93]) that noise may regularize chaotic dynamics causing a decrease of the Lapunov exponent λ (“noise-induced order”), for multidimensional systems (in chaotic regime) represented in the form of a system of ﬁrst-order diﬀerential equations, diﬀerent Lapunov exponents λi , (i = 1, 2, . . . , n) may react diﬀerently to the changes of the noise intensity. To obtain a measure of the averaged eﬀect of noise, some authors compute a “global” exponent deﬁned as

λg = m +

m i=1

λi , |λm+1 |

(55)

where λ1 ≥ λ2 ≥ . . . and m is the largest integer such that λ1 + . . . + λm > 0. For the Van der Pol oscillator (represented by a system of three diﬀerential equations of the ﬁrst order) λg decreases as the noise intensity σ increases. Although the maximum exponent λ1 slightly increases with σ, the system is “regularized” by noise in a global sense.

Stochastic Resonance. Remarks Another phenomenon which makes stochastic dynamics fascinating is stochastic resonance. It occurs as a result of interplay of nonlinearity, periodicity and randomness. Intuition suggests that when noise is added to a signal prior to or during transmission through a system/communication channel, the received signal will be more corrupted (deteriorated) than if the uncorrupted signal had been transmitted. The amount of corruption is usually characterized by the so-called signal – to noise ratio (SNR) of the output. For linear systems, the output SNR decreases monotonically with increasing noise intensity. The peculiarity of stochastic resonance lies in the fact that (for a large class of nonlinear systems) there occurs an increase in the SNR up to a maximum, with added random noise. This phenomenon has attracted much attention in the recent years. Besides numerous theoretical studies also numerical simulations and experimental work are today in the progress (cf. Benzi et al. [96], McNamara and Wiesenfeld [97], and references therein).

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Korteweg de Vries Solitons in Randomly Varying Medium. Remarks Phenomenon which occurs in spatially extended material media (ﬂuids, solids) and results due to the interplay between nonlinearity and dispersion of the medium is known as solitary waves or solitons. One of the best known equations describing solitons is the Korteweg de Vries equation ∂u ∂ 3 u ∂u +u + 3 = 0, ∂s ∂ξ ∂ξ

(56)

where u = u(s, ξ) characterizes the medium disturbance as a function of time and one-dimensional spatial coordinate. A characteristic feature of this equation is the existence of the so-called stationary waves (solitons) which do not change their form during propagation (usually, nonlinearity distorts the wave proﬁle). For example, waves in shallow water, ion-acoustic waves in plasma are governed by the above equation. If, however, the medium is perturbed by a random inhomogeneity (e.g. random roughness of the bottom of the water channel, random impurity of plasma density), the KdV solitary waves are attenuated and the amount of attenuation depends on the magnitude of randomness of the medium (cf. Sobczyk [99] and references therein).

8.

Closing

Stochastic dynamics is still at the stage of its development. Nowadays, the methodology presented in this lecture extends its models and methods to new ﬁelds of human endeavours. For example, a great intellectual eﬀort is concentrated today on stochastic dynamics of economic (and ﬁnancial) systems as well as on the atmospheric processes (weather and climate forecasting). Are there any other, fundamental, expectations concerning probabilistic / stochastic methods in science (and . . . in everyday life)? David Munford, the past president of the International Mathematical Union, in his article “The dawning of the age of stochasticity”, published in the distinguished volume [100] writes: “My overall conclusion is that I believe stochastic methods will transform pure and applied mathematics in the beginning of the third millennium. Probability and statistics will come to be viewed as the natural tools to use in mathematical and scientiﬁc modelling”.

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Acknowledgments I wish to express my sincere thanks to the Congress Committee of IUTAM for selecting stochastic dynamics for a very prestigious, plenary presentation at the XXI-st IUTAM Congress. I was very pleased and honoured to deliver this exceptional lecture. Also I owe a great debt of gratitude to many colleagues of my Institute who have contributed, by their friendly advices, to the clarity of my lecture. In particular, I wish to express my appreciation to dr. Jerzy Trebicki ¸ for his painstaking work on the visualization of my presentation, as well as – for bringing the typescript to its ﬁnal form.

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[56] S.H. Crandall, Non-Gaussian Closure for random vibration of nonlinear oscillators, Int. J. Nonlinear Mech., Vol.15, 303-313, 1980. [57] W.F. Wu, Y.K. Lin, Cumulant-neglect closure for nonlinear oscillators under parametric and external excitations, Int. J. Nonlinear Mech., Vol.19, 349-362, 1984. [58] K. Sobczyk, J. Tr¸¸ebicki, Maximum entropy closure for nonlinear stochastic systems, in: Vibration of Nonlinear, Random, and Time-Varying Systems, Proc. of 1995 Design Eng. Conf., DE-Vol.84-1, ASME, 1995. [59] R.L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, N. York, 1963 (translation from Russian). [60] R.Z. Khasminskii, A limit theorem for the solution of diﬀerential equations with random right-hand side (in Russian), Tieoria Vieroyatn. Prim., Vol.11, 3, 1966. [61] P.E. Kloeden, E. Platen, The Numerical Solutions of Stochastic Diﬀerential Equations, Springer, Berlin, 1992. [62] M.F. Wehner, W. G. Wolfer, Numerical evaluation of path integral solutions to F-P-K equations, Phys. Rev. Vol.A27, 2663-70, 1983. [63] A. Naess, J. M. Johsen, Response statistics of nonlinear, compliant oﬀshore structures by the path integral solution method, Probab. Eng. Mech., Vol.8, 91-106, 1993. [64] K. Sobczyk, J. Tr¸¸ebicki, Maximum entropy principle in stochastic dynamics, Probab. Eng. Mech., Vol.5, 3, 102-110, 1990. [65] K. Sobczyk, J. Tr¸¸ebicki, Maximum entropy principle and nonlinear stochastic oscillators, Physica A, Vol.193, 448-468, 1993. [66] K. Sobczyk, J. Tr¸¸ebicki, Approximate probability distributions for stochastic systems: maximum entropy method, Comput. Methods Appl. Mech. Eng., Vol.168, 91-111, 1999. [67] J. Tr¸¸ebicki, K. Sobczyk, Maximum entropy principle and nonstationary distributions of stochastic systems, Probab. Eng. Mech., Vol.11 (3), 169-178, 1996. [68] C. Papadimitriou, Y. Haralampidis, K. Sobczyk, Optimal experiment design in stochastic structural dynamics, Probab. Eng. Mech., 2004. [69] K. Sobczyk, Information dynamics: Premises, challenges and results, Mech. Systems and Signal Processing, Vol.15(3), 475-498, 2001. [70] M. Shinozuka, C. J. Astill, Random eigenvalue problems in structural analysis, AIAA Journal, Vol.10, 4, 456-462, 1972. [71] K. Sobczyk, Free vibrations of elastic plate with random properties – the eigenvalue problem, J. Sound and Vibration, Vol.21, 4, 1972. [72] M. Shinozuka, G. Deodatis, Response variability of stochastic ﬁnite element systems, ASCE Jour. Eng. Mech., Vol.114, 39, 499-519, 1988. [73] R.G. Ghanem, P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, Berlin, 1991. [74] V.V. Bolotin, Prediction of Service Life of Machines and Structures (in Russian: Mashinostroienje, Moskov, 1984; English edition: ASME Press, N. York, 1989).

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[75] R.Z. Khasminskii, Stability of Diﬀerential Equations with Random Perturbation of Parameters (in Russian) Nauka, Moskov, 1969; English transl. Stochastic Stability of Diﬀerential Equations, Sijthoﬀ and Noordhoﬀ, ﬀ Alphen, 1980. [76] L. Arnold, V. Wihstutz, Lapunov exponents: a survey, in: L. Arnold, V. Wihstutz (Eds.): Lapunov exponents, Lecture Notes in Math. 1186, Springer, 1-26, Berlin, 1986. [77] S.T. Ariaratnam, W.C. Xie, Lapunov exponents and stochastic stability of coupled linear systems under real noise excitation, ASME J. Appl. Mech., Vol.59, 3, 664-673, 1992. [78] W. Wedig, Stability of nonlinear stochastic systems, in: C. Dafermos, G. Ladas, G. Papanicolau (Eds.) Lecture Notes in Pure and Appl. Math., Dekker, N. York, 1988. [79] C.G. Bucher, Y. K. Lin, Eﬀect of spanwise correlation of turbulence ﬁeld on the stability of long-span bridges, J. of Fluids and Structures, Vol.2, 437-451, 1988. [80] A. Tylikowski, Dynamic stability of nonlinear antisymetrically laminated crossply rectangular plates, J. Appl. Mech., ASME, Vol.56, 375-381, 1989. [81] A. Tylikowski, Stabilization of parametric vibrations of a nonlinear continuous system, Meccanica, Vol.38, 6, 659-668, 2003. [82] B.F. Spencer, Reliability of randomly excited hysteretic structures, Lecture Notes in Engineering (C. A. Brebbia , S. H. Orszag, eds.) Springer, N. York, 1986. [83] K. Sobczyk, Modelling of fatigue crack growth, Eng. Fracture Mech., Vol.24, 609-623, 1986. [84] O. Ditlevsen, Random fatigue crack growth – a ﬁrst passage problem, Eng. Fracture Mechanics, Vol.23,2, 467-477, 1986. [85] K. Doli´ n ´ ski, P. Colombi, Fatigue life time under stochastic loading with random overloading pulse trains, Comp. Meth. Appl. Mech. Eng., 168, 1999. [86] J. B. Roberts, P. D. Spanos, Stochastic averaging: An approximate method of solving random vibration problems, Int. J. Nonlinear Mechanics, Vol.21, 111-134, 1986. [87] L. Arnold, G. Papanicolau, V. Wihstutz, Asymptotic analysis of the Lapunov exponent and rotation number of the random oscillator and applications, SIAM J. Appl. Math., Vol.46, 3, 427-450, 1986. [88] L. Arnold, Stabilization by noise, ZAMM, M Vol.70, 7, 235-246, 1990. [89] W. Horsthemke, R. Lefever, Noise-Induced Transitions, Springer, Berlin, 1984. [90] K.A. Wiesenfeld, E. Knobloch, Eﬀect of noise on the dynamics of a nonlinear oscillator, Phys. Rev. Vol.A26, 5, 2946-2953, 1982. [91] K.R. Schenk-Hopp´ ´e, Bifurcation scenarios of the noisy Duﬃng-Van der Pol oscillator, Nonlinear Dynamics, Vol.11, 255-274, 1996. [92] L. Arnold, N. Sri Namachchivaya, K. R. Schenk-Hopp´ ´e, Toward an understanding of stochastic Hopf bifurcation: A case study, Int. J. Bifurc. and Chaos, Vol.6, 11, 1947-1975. [93] J. Grasman, J. B.T.M. Roerdink, Stochastic and chaotic relaxation oscillations, J. Statist. Physics, Vol.54, 3/4, 949-970, 1989.

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[94] T. Kapitaniak, Chaos in Systems with Noise, World Scientiﬁc, Singapore, 1988. [95] Y-Cheng Lai, Z. Liu, L. Billings, I.B. Schwartz, Noise-induced unstable variability and transition to chaos in random dynamical systems, Phys. Rev., Vol.E 67, 026210, 2003. [96] R. Benzi, A. Sutera, A. Vulpiani, The mechanism of stochastic resonance, Journ. of Physics A, Vol.141, L453-L457, 1981. [97] B. McNamara. K. Wiesenfeld, Theory of stochastic resonance, Phys. Rev. Vol.A39, 4854-4869, 1989. [98] L. Schimansky-Geier et al., Noise induced order: Stochastic resonance, Int. J. Bifurc. and Chaos, Vol.8, 5, 869-879, 1998. [99] K. Sobczyk, Korteweg-de Vries solitons in a randomly varying medium, Intern. Journ. Nonlin. Mech., Vol.27, 1, 1-8, 1992. [100] D. Munford, The Dawning of the Age of Stochasticity, in: V. Arnold, M. Atiyah, P. Lax, B. Mazur (Eds.) Mathematics: Frontiers and Perspectives, Amer. Math. Soc., 2000.

MULTIBODY DYNAMICS: BRIDGING FOR MULTIDISCIPLINARY APPLICATIONS Jorge A.C. Ambr´ o´sio IDMEC Instituto Superior Tecnico, ´ Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Abstract

Simple or complex systems characterized by large relative motions between their components ﬁnd in the multibody dynamics formalisms the most general and eﬃcient computational tools for their analysis. Initially restricted to the treatment of rigid bodies, the multibody methods are now widely used to describe the system components deformations, regardless of their linear or nonlinear nature. The ease of including in the multibody models diﬀerent descriptions of the contact problems, control paradigms or equations of equilibrium of other disciplines is demonstrated here to show the suitability of these approaches to be used in multidisciplinary applications

Keywords: Flexible multibody dynamics, contact, biomechanics, vehicle dynamics, railway dynamics, crashworthiness.

1.

Introduction

The design requirements of advanced mechanical and structural systems and the real-time simulation of complex systems exploit the ease of use of the powerful computational resources available today to create virtual prototyping environments. These advanced simulation facilities play a fundamental role in the study of systems that undergo large rigid body motion while their components experience material or geometric nonlinear deformations, such as vehicles, deployable structures, space satellites, machines operating at high speeds or robot manipulators. Some examples of engineering and biological systems for which the large overall motion is of fundamental importance are exempliﬁed in Fig. 1. If on the one hand the nonlinear ﬁnite element method is the most powerful and versatile procedure to describe the ﬂexibility of the system components, on the other hand the multibody dynamic formulations are the basis for the most eﬃcient computational techniques that 61 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 61–88. © 2005 Springer. Printed in the Netherlands.

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Figure 1. Natural biological and artiﬁcial engineering systems for which multibody dynamics provides irreplaceable modeling methodologies.

deal with large overall motion. Therefore, it is no surprise that many of the most recent formulations on ﬂexible multibody dynamics and on ﬁnite element methods with large rotations share some common features. In multibody dynamics methods, the body-ﬁxed coordinate frames are generally adopted to position each one of the system components and to allow for the speciﬁcation of the kinematic constraints that represent the restrictions on the relative motion between the bodies. Several formalisms are published suggesting the use of diﬀerent sets of coordinates, such as Cartesian [1], natural [2] and relative coordinates [3]. Depending on the type of applications, each of these types of coordinates has advantages and disadvantages. Due to the ease of the computational implementation, their physical meaning and the widespread knowledge of their features, all the formalisms presented in this work are based on the use of Cartesian coordinates. The methodological structure of the equations of motion of the multibody system obtained allows the incorporation of the equilibrium equations of a large number of disciplines and their solution in a combined form. The description of the structural deformations exhibited by the system components by using linear [5] or non-linear ﬁnite elements [6] in the framework of multibody dynamics is an example of the integration of the equations of equilibrium of diﬀerent specialties. Of particular importance for the applications pursued with the methodologies proposed is the treatment of contact and impact, which is introduced in the multibody systems equations by using either unilateral constraints

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[7] or a continuous contact force model [8]. The availability of the state variables in the multibody formulation allows for the use of diﬀerent control paradigms in the framework of vehicle dynamics, biomechanics or robotics and their integration with the multibody equations [9]. The coupling between the ﬂuid and structural dynamics equations allows for the development of applications, where the ﬂuid-structure interaction is analyzed, especially for cases with large absolute or relative rotations in the system components, are of importance [10, 11]. The research carried at IDMEC provides the examples oﬀered in this work. Application cases involving the modeling of realistic mechanisms, passive safety of road and rail vehicles, impact and human locomotion biomechanics, automotive and railway dynamics are used to demonstrate the developments reviewed here.

2.

Rigid Multibody Dynamics

A multibody system is deﬁned as a collection of rigid and/or ﬂexible bodies constrained by kinematic joints and eventually acted upon by a set of internal and/or external forces. The position and orientation of each body i in the space is described by a position vector ri and a set of rotational coordinates pi , which are organized in a vector as [1]: qi = [rT , pT ]Ti .

(1)

According to this deﬁnition, a multibody system with nb bodies is described by a set of coordinates in the form: q = [qT1 , qT2 , . . . , qTnb ]T .

(2)

The dependencies among system coordinates, which result from the existence of mechanical joints interconnecting several bodies, are deﬁned through the introduction of kinematic relationships written as [1]: Φ (q, t) = 0,

(3)

where t is the time variable, which is used only for the driving constraints. The second time-derivative of Eq. (3) with respect to time yields: ¨ (q, q, Φ ˙ q ¨, t) = 0 ≡ D¨ q = γ, (4) where D is the Jacobian matrix of the constraints, q ¨ is the acceleration vector and γ is the vector that depends on the velocities and time. The system kinematic constraints are added to the equations of motion using the Lagrange multipliers technique [1]. Denoting by λ the

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vector of the unknown Lagrange multipliers, the equations of motion for a mechanical system are written as

q ¨ f M DT = (5) λ γ D 0 where M is the global mass matrix, containing the mass and moments of inertia of all bodies, and f is the force vector, containing all forces and moments applied to the system bodies plus the gyroscopic forces. The Lagrange multipliers, associated with the kinematic constraints, are physically related with the reaction forces generated between the bodies interconnected by kinematic joints, given by [1] f (c) = −ΦTq λ,

(6)

The usual procedures to handle the integration of sets of diﬀerentialalgebraic equations must still be applied in this case in order to eliminate the constraint drift or to maintain it under control [1, 2].

Forward Dynamics The computational strategy used to solve the forward dynamics of the system, represented by Eq. (5), is outlined in Fig. 2. The solution procedure starts by the determination of the initial positions and velocities of the system components. Next, the system inertia, the Jacobian matrices, the forces and the right-hand-side of the kinematic acceleration constraint equations vectors, are calculated and assembled in the equations of motion. Equation (5) is then solved to ﬁnd the system accelerations, and in the process the Lagrange multipliers. By integrating the current velocities and the system accelerations, at time t, the new positions and velocities for time t + ∆t are calculated by using a variable

Figure 2.

Solution of the forward dynamic analysis of a multibody system.

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65

order, variable time-step integration procedure [1, 2, 4]. The forward dynamics simulation proceeds until the previously set ﬁnal time is reached. The procedure outlined in Fig. 2 is used in general purpose multibody dynamics codes, such as DAP-3D [1]. Throughout this work it is demonstrated that all engineering applications foreseen here are implemented, either by developing speciﬁc kinematic constraints or by implementing force models in Eq. (5).

Application Example of a Roller Coaster. When a body travels along a guide, not only its path has to be followed, but also its spatial orientation has to be prescribed, according to spatial characteristics of the curve. The formulation adopted to implement these kinematic constraints, using the moving Frenet frame associated with the track centerline based on the work by Pombo and Ambrosio ´ [12], is outlined next. Prescribed Motion Constraint. The objective here is to deﬁne the constraint equations that enforce that a point of a rigid body follows the reference path [12]. Consider a point R, located on a rigid body i, that has to follow the speciﬁed path depicted in Fig. 3. The path is deﬁned by a parametric curve g(L), which is controlled by a global parameter L that represents the length travelled along the curve until the current location of point R. The kinematic constraint is Φ(pmc,3) = 0 ≡ rR i − g(L) = 0,

(7)

R where rR i = ri +Ai s i represents the coordinates of point R with respect to the global coordinate system (x, y, z), ri is the vector that deﬁnes the location of the body-ﬁxed coordinate system (ξ, η, ζ)i , Ai is the transformation matrix from the body i ﬁxed coordinates to the global reference frame, and s R i represents the coordinates of point R with re-

x Figure 3.

Local frame alignment constraint.

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spect to the body-ﬁxed reference frame. For notational purposes (·) means that (·) is expressed in body-ﬁxed coordinates. The second part of the constraint ensures that the spatial orientation of body i remains unchanged with respect to the moving frame of the reference path represented in Fig. 3. Consider that (uξ , uη , uζ )i represent the unit vectors associated with the axis of the body-ﬁxed coordinate system (ξ, η, ζ)i . Let the Frenet frame of the general parametric curve g(L) be deﬁned by the principal unit vectors (t, n, b)L . At the initial time of analysis, the relative orientation between the body vectors (uξ , uη , uζ )i and the local frame (t, n, b)L leads to ⎫ ⎧ ⎫ ⎧ T ⎨n · uξ ⎬ ⎨a⎬ Φ(lf ac,3) = 0 ≡ bT · uξ − b = 0. (8) ⎭ ⎩ ⎭ ⎩ T c n · uζ This kinematic constraint ensures that the alignment remains constant throughout the analysis. The transformation matrix from the body i ﬁxed coordinates to the global coordinate system is: Ai = [uξ

uη

uζ ]i

(9)

deﬁning the following unit vectors as: u1 = {1

0

0}T ;

u2 = {0

1

0}T ;

u3 = {0 0 1}T . (10)

Equation (8) is now rewritten in a more usable form as: ⎫ ⎧ ⎫ ⎧ T ⎨n Ai u1 ⎬ ⎨a⎬ Φ(lf ac,3) = 0 ≡ bT Ai u1 − b = 0, ⎭ ⎩ ⎭ ⎩ T c n Ai u3

(11)

which constitutes the second part of the path following constraint.

Roller-Coaster Dynamics. Let the roaller-coaster rail be deﬁned with the spatial geometry described in Fig. 4. The path-following constraint is used to enforce the vehicles to follow the rail for the prescribed geometry. The roller coaster vehicle consists of a train with three cars that are interconnected by linking bars, represented in Fig. 5. The multibody model of the vehicle is assembled using eleven rigid bodies, corresponding to 3 car bodies, 6 wheelsets and 2 connection bars. The complete vehicle model only has 1 d.o.f., which is the longitudinal motion of the cars. The motion of the vehicle is guided by the dynamics described by Eq. (5). A view of the motion of the roller coaster is displayed in Fig. 6 and the details of the analysis are found in reference [13].

Multibody Dynamics: Bridging for Multidisciplinary Applications

Figure 4.

67

View of the roller coaster as used in the simulation.

z 3

6

10

7

11 x

1

Figure 5.

Figure 6.

2

4

5

8

9

Multibody model of the roller coaster vehicle.

Snap shots of the roller coaster motion as observed from the second car.

Note that the study of these vehicles only requires the use of the pathfollowing constraint. The contact forces are not explicitly used but they can be calculated from the Lagrange multipliers associated to the path constraint.

Inverse Dynamics In many applications all external forces are known and the motion of the system is also known. Therefore, the only unknowns are the internal forces. Let the ﬁrst row of Eq. (5) be re-written as M¨ q + DT λ = g

(12)

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which is q. DT λ = g − M¨

(13)

Equation (13) emphasizes that the only unknowns of the system are the Lagrange multipliers. The reaction forces at the joints are given by: g(c) = −DT λ.

(14)

The solution of the equations of motion in inverse dynamics can be used to solve for the internal forces of the human body, i.e., muscle and anatomical joint reaction forces, that develop for known motions.

Application to Biomechanics: Gait Analysis For biomechanical applications in gait a three-dimensional model, presented in Fig. 7, is used [14]. It has a kinematic structure made of thirtythree rigid bodies, interconnected by revolute and universal joints, in such a way that sixteen anatomical segments are represented. 16

3 v20

111

12 3

4

2

5

2

7

v12

14

9

5

17

10

v11

7 122

13

v17

v17 29

v199

28

23

v13 21

v18

22

v211 v22

v11

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v4

15

8

27

26

30

v13

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v16

v15

v15

20

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1

v16 24

v14

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32 25

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15

Figure 7. The biomechanical model, its kinematic structure and a detail of the ankle joint.

Joint Moments-of-Force: A Determinate Problem. To drive the biomechanical model in the inverse dynamic analysis, joint actuators such as the one represented in Fig. 8 for the knee joint, are speciﬁed. The actuators are the kinematic constraints in which the angle between two adjacent segments is a known function of time. These additional equations are added to the system kinematic equation so that the number of

Multibody Dynamics: Bridging for Multidisciplinary Applications

69

O m1 m1

Im1 =

O m2 m2

Im22 = I m3

Figure 8.

O m3

m3

Joint actuator associated with the knee joint and muscle actuator.

non-redundant constraint equations becomes equal to the number of coordinates. Equation (13) is solved to obtain the Lagrange multipliers associated with the joint actuators, representing the net moments-of-force of the muscles that cross those joints. The inverse dynamics problem, as stated here, is totally determined.

Muscle Forces: A Redundant Problem. The solution of the inverse dynamics problem with muscle actuators introduces indeterminacy in the biomechanical system, since it involves more unknowns than equations of motion. By using optimization techniques to ﬁnd the muscle forces that minimize a prescribed objective function, a solution for the problem is obtained. The optimization problem is stated as: minimize F0 (ui ) ⎧ ⎪ ⎨ fj (ui ) = 0, subject to fj (ui ) 0, ⎪ ⎩ lower ui uupper ui i

j = 1, ..., nec , j = (nec + 1) , . . . , ntc , i = 1, . . . , nsv

(15)

where ui are the state variables bounded respectively by ulower and uupper , F0 (ui ) is the objective or cost function to minimize and fi (ui ) are constraint equations that restrain the state variables. The minimization of the cost functions simulate the physiological criteria adopted by the central nervous system when deciding which muscles to recruit and what level of activation to obtain the adequate motion. Several cost functions have been proposed for the study of the redundant problem in biomechanics [15]. The minimization of the sum of the cube of the muscle stresses [16] is often used in applications involving human

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Figure 9.

Figure 10.

Lower extremity muscle apparatus.

Muscle forces for the hamstrings and triceps surae.

Multibody Dynamics: Bridging for Multidisciplinary Applications

locomotion

m m 3 Fl Fl˙ m σ ¯ a F0 = F0m2 m=1 n ma

71

(16)

¯ is the speciﬁc where nma are the number of muscle actuators and σ muscle strength with a constant value of 31.39 N/cm2 [17]. The human locomotion apparatus, represented in Fig. 9, is modeled having the muscles with the physiological data described in Yamaguchi [17]. The state variables associated with muscle actuators represent muscle activations that can only assume values between 0 and 1. To illustrate the type of results obtained for the muscle forces in a case of normal cadence gait of a 50%ile male, the muscle forces for the hamstrings and triceps surae are presented in Fig. 10.

3.

Contact and Impact

Let a triangular patch, where point k of the body shown in Fig. 11 will impact, be deﬁned by points i, j and l. The normal to the outside surface of the contact patch is deﬁned as n = rij × rjl . The position of the point k with respect to point i of the surface is rik = rk − ri

(17)

which is decomposed in a tangential and a normal component, given by (18) rtik = rik − rTik n n; rnik = rTik n n. The necessary conditions for contact are that node k penetrates the ‘front’ surface of the patch, but not through its ‘back’ surface, with which a thickness e is associated. These conditions are written as 0 rTik n e.

k

riik

rikn = ( ik

rlli

l k*

i

Figure 11.

rikt = rik − ( ik

rij

(19)

)n n

)n

r jjl j

Contact detection between a ﬁnite element node and a surface.

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The remaining necessary conditions for contact result from the need for the node to be inside of the triangular patch. These three extra conditions are t T ˜ rik rij n 0;

T T t t ˜ rik rjl n 0 and ˜ rik rki n 0.

(20)

Equations (19) and (20) are necessary conditions for contact. However, depending on the contact force model actually used, they may not be suﬃcient to ensure eﬀective contact.

Unilateral Constraints If contact between a node and a surface is detected, a kinematic constraint is imposed. For ﬂexible bodies let us assume a fully plastic nodal contact, i.e., the normal components of the node k velocity and acceleration, with respect to the surface, are null during contact (−) (−)T (−) (−)T ¨k = q ¨k − q ¨k n n (21) q˙ k = q˙ k − q˙ k n n; q (−)

(−)

where q˙ k and q ¨k represent the nodal velocity and acceleration immediately before impact respectively . The kinematic constraint implied by Eqs. (21) is removed when the normal reaction force between the node and the surface becomes opposite to the surface normal, i.e., fkn = −ffkT n > 0.

(22)

It should be noted that the contact force is related to the Lagrange multiplier associated by the kinematic constraint deﬁned by Eqs. (22). Therefore, the change of sign of the force is in fact the change of sign of the multiplier. This contact model is not suitable to be used directly with rigid body contact. The sudden change of the rigid body velocity and acceleration would imply that the velocity and acceleration equations resulting from the kinematic constraints would not be fulﬁlled. Other forms of this contact model can be found in the work by Pfeiﬀer and Glocker [7].

Continuous Contact Force Model An alternative description of contact considers this to be a continuous event where the contact force is a function of the penetration between the surfaces. This leads to the continuous force contact model, proposed by Lankarani and Nikravesh [8], and brieﬂy described here. Let the contact force between two bodies be written as (23) fs,i = Kδ n + Dδ˙ u

73

Multibody Dynamics: Bridging for Multidisciplinary Applications

where δ is the pseudo-penetration, δ˙ is the pseudo-velocity of penetration, K is the equivalent stiﬀness, D is a damping coeﬃcient and u is a unit vector normal to the impacting surfaces. Using the hysteresis dissipation model and the equivalent stiﬀness, calculated for instance for Hertzian elastic contact [18], the nonlinear contact force is 3 1 − e2 δ˙ n u (24) fs,i = Kδ 1 + 4 δ˙ (−) where δ˙ (−) is the initial contact velocity and e is the restitution coeﬃcient. Note that K is a function of the geometry and material properties of the impacting surfaces.

Application to Railway Dynamics – The Wheel-Rail Contact Problem One of the interesting applications of multibody dynamics with contact mechanics is the description of the wheel-rail contact in railway dynamics, represented in Fig. 12. The stability of the running vehicle depends ultimately on the rail-wheel contact and on the vehicle primary suspension. Therefore, methodologies that provide accurate models to represent the phenomena are of particular importance. In a general case of a railway vehicle one or two points of each wheel are in contact with the rail, as shown in Fig. 12. The diametric section that contains the wheel ﬂange contact point makes an angle sfRw with the diametric section that contains the wheel tread contact point. The possibility of detecting contact in diﬀerent diametric sections allows predicting derailment and it is, therefore, of utmost importance. Let the generalized geometry of the rail and wheel be described by generalized surfaces resulting from sweeping the rail proﬁle along the rail Fla (Le

ct ct) e Tre Tr con

Figure 12.

Two points of contact in the rail and wheel surfaces: lead contact.

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centerlines and the wheel proﬁles around the base circle of the wheel. In order to ensure that the search for the contact points is between convex surfaces, the wheel proﬁle is divided in treat and ﬂange proﬁles. The contact between the rail and one of the wheel surfaces is described generically in Fig. 13, where the mating surfaces are represented as free surfaces.

x Figure 13.

Candidates to contact points between two parametric surfaces.

The geometric conditions for contact between the convex surfaces are deﬁned by vector products deﬁned between the surfaces. The ﬁrst condition is that the surfaces normals ni and nj at the candidates to contact points have to be parallel. This condition means that nj has null projections over the tangent vectors tui and tw i : ⎧ ⎨nTj tui = 0, (25) nj × ni = 0 ⇔ ⎩nT tw = 0. j i The second condition is that the vector d, which represents the distance between the candidates to contact points, has to be parallel to the normal vector ni . This condition is mathematically written as: ⎧ ⎨dT tui = 0, (26) d × ni = 0 ⇔ ⎩dT tw = 0. i The geometric conditions (25) and (26) provide four nonlinear equations with four unknowns, the four parameters u, w, s and t that deﬁne the two surfaces. This system of equations provides solutions for the location of the candidates to contact points that have to be sorted out.

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The coordinates of the candidates to contact points are determined by solving an optimal problem and the distance between such points is calculated in the process. The points are in contact if dT nj 0.

(27)

When contact is detected, the normal force is calculated using Eq. (24) and the tangential forces are evaluated using the Kalker theory, the Polach formulation or the Heuristic nonlinear creep model. It has been found that the Polach formulation provides the best approach for the tangent forces, and it is used hereafter [13]. The wheel-rail contact model outlined here is used to model the ML95 trainset, shown in Fig. 14, which is used by the Lisbon subway company (ML) for passengers’ traﬃc. The multibody model of the trailer vehicle of the train, developed in the work by Pombo [13], is composed of the car shell suspended by a set of springs, dampers and other rigid connecting elements on the bogies. This assembly of connective elements constitutes the secondary suspension, sketched in Fig. 15, which is the main one responsible for the passenger’s comfort. The connections between the bogies chassis and the wheelsets, also achieved by another set of springs, dampers and rigid connecting ele-

Figure 14.

Figure 15.

Schematic representation of the ML95 trainset.

Secondary suspension model of the ML95 trailer vehicle.

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Axlebox

Three-dimensional spring-damper elements a)

Wheelset b)

Figure 16. Primary suspension model of the ML95 trailer bogie: a) Threedimensional spring-damper elements; b) Suspension model with springs and dampers.

Figure 17. Lift of the right wheel of the leading wheelset for vehicle forward velocities of 10 and 20 m/s, using the Kalker linear theory.

ments, constitute the primary suspension represented in Fig. 16. The primary suspension is the main suspension responsible for the vehicle running stability. The simulation results of the vehicle, running in a circular track with a radius of 200 m with velocities of 10 and 20 m/s, show that the prediction of ﬂange contact is of fundamental importance. Fig. 17 shows that contact forces obtained with the Kalker linear theory originate the lift of the outer wheel of the front wheelset at the entrance of the curve. Despite this wheel lift, derailment does not occur and the analysis proceeds up to end. Nevertheless, such results are not realistic since the existence of ﬂange contact involves high creepages, which makes the Kalker linear theory inappropriate to compute the creep forces. Therefore only the Heuristic and the Polach creep force models must be considered. Another aspect to note is that ﬂange contact is detected with all creep force models. Even when running at the speed of 10 m/s, where the centrifugal forces eﬀect is balanced by the track cant, ﬂange contact occurs. Lateral ﬂange forces develop on the wheels of both wheelsets of the front bogie as presented in Fig. 18 for a vehicle forward velocity of 10 m/s and

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Multibody Dynamics: Bridging for Multidisciplinary Applications

using the Polach creep force model. During curve negotiation, the outer wheel of the leading wheelset and the inner wheel of the rear wheelset have permanent ﬂange contact. Referring to Fig. 19, for the velocity of 10 m/s, the ﬂange contact occurs on the outer and in the inner wheels of the vehicle. For the velocity of 20 m/s, only the outer wheels have ﬂange contact. This is explained by the fact that, when running at 20 m/s, the vehicle negotiates the curve with a velocity higher than the balanced speed.

4.

Flexible Multibody Dynamics with Plastic Hinges

Many applications of multibody dynamics require the description of the ﬂexibility of its components. For structural crashworthiness it is 25 000

Le ft W s 3 ( P ol ach) R ig ht W s 3 ( P olach )

20 000

La te ra l Flang e F orc e [N ]

Le ft W s 4 ( P ol ach) R ig ht W s 4 ( P olach )

15 000

10 000

5 000

0

-5 0 0 0 0

3

6

9

12

15

18

21

24

27

30

T im e [s ]

Figure 18. Lateral ﬂange forces on the wheels of both wheelsets of the front bogie for a vehicle forward velocity of 10 m/s, using the Polach creep force model. Flange contact Front wheelset (Ws 4)

Rear wheelset (Ws 3)

Flange contact

Figure 19.

Contact conﬁguration during curve negotiation.

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often unfeasible to use large nonlinear ﬁnite element models. The use of multibody dynamics with plastic hinges is an alternative formulation that allows building insightful models for crashworthiness.

Formulation of Plastic Hinges In many impact situations, the individual structural members are overloaded giving rise to plastic deformations in highly localized regions, called plastic hinges. These deformations, presented in Fig. 20 for structural bending, develop at points where maximum bending moments occur, load application points, joints or locally weak areas [19]. Multibody models obtained with this method are relatively simple, which makes the procedure adequate for the early phases of vehicle design. The methodology described herein is known in industry as conceptual modeling [20].

Figure 20.

Localized deformations on a beam and a plastic hinge.

The plastic hinge concept has been developed by using generalized spring elements to represent constitutive characteristics of localized plastic deformation of beams and kinematic joints to control the deformation kinematics [21], as illustrated in Fig. 21. The characteristics of the spring-damper that describes the properties of the plastic hinge are obtained by experimental component testing, ﬁnite element nonlinear analysis or simpliﬁed analytical methods. The plastic hinge constitutive equation can be modiﬁed to account for the strain rate sensitivity of some materials. A dynamic correction factor is used to account for the strain rate sensitivity given by [21]. Ps = 1 + 0.07V V00.82 , Pd /P

(28)

where Pd and Ps are the dynamic and static forces, respectively, and V0 is the relative velocity between the adjacent bodies. The force or moment

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Multibody Dynamics: Bridging for Multidisciplinary Applications

Analytical Test

Moment ( kNm) 15 30 45 0 Figure 21.

0

79

.05

.10

.15 .20 .25 Rotation ( Rad )

Plastic hinge bending moment and its constitutive relationship.

to apply due to the plastic hinge is multiplied by the ratio calculated in Eq. (28) before it is used in the force vector of the multibody equations of motion.

Application of the Plastic Hinge Approach to Crashworthiness of Surface Vehicles The multibody of an oﬀ-road vehicle with three occupants, shown in Fig. 22, is used to demonstrate the plastic hinge approach to complex crash events. The model includes all moving components of the vehicle, suspension systems and wheels, and a tire model [16]. The biomechanical models for the occupants are similar to those described in Fig. 7. The three occupants, with a 50%tile, integrated in the vehicle are seated. Two occupants in the front of the vehicle have shoulder and lap

Figure 22.

Initial position of the vehicle and occupants for the rollover.

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seat belts while the occupant seated in the back of the vehicle has no seatbelt. The rollover situation for the simulation is such that the initial conditions correspond to experimental conditions where the vehicle moves on a cart with a lateral velocity of 13.41 m/s until the impact with a waterﬁlled decelerator system occurs. The vehicle is then ejected with a roll angle of 23 degrees. The results of the simulation are pictured in Fig. 23 by several frames of the animation. The vehicle ﬁrst impacts the ground with its left tires. At this point the rear occupant is ejected. The rollover motion of the vehicle proceeds with an increasing angular velocity, mainly due to the ground – tire contact friction forces. The occupants in the front of the vehicle are held in place by the seat belts. Upon continuing its roll motion, the vehicle impacts the ground with its rollbar cage, while the ejection of the rear occupant is complete. Bouncing from the inverted position, the vehicle completes another half turn and impacts the ground with the tires again. The HICs for all occupants largely exceed 1000, which indicates a very high probability of fatal injuries for the occupants under the conditions simulated. An experimental test of the vehicle was carried out at the Transportation Research Center of Ohio [22], being an overview of the footage obtained shown in Fig. 24. The outcome of the experimental test, which is rather similar to the outcome of the simulation, is further used to validate the vehicle model [21].

Figure 23. occupants.

Views of the outcome of the rollover simulation of a vehicle with three

Figure 24.

View of the experimental test for the truck rollover.

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5.

Flexible Multibody Dynamics with Finite Elements

General Equations of Motion of a Single Body Let the principle of the virtual works be used to express the equilibrium of the ﬂexible body in the current conﬁguration t+∆t and let an updated Lagrangean formulation be used to obtain the equations of motion of the ﬂexible body [23]. Let the ﬁnite element method be used to represent the equations of motion of the ﬂexible body. Referring to Fig. 25, the assembly of all ﬁnite elements used in the discretization of a single ﬂexible body results in its equations of motion written as [6] ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ gr ¨ r sr 0 Mrr Mrf Mrf ⎣Mφr Mφφ Mφf ⎦ ⎣ω ˙ ⎦ = ⎣g φ ⎦ − ⎣s φ ⎦ − ⎣0⎦ Mf r Mf φ Mf f u ¨ g f s f f ⎤⎡ ⎤ ⎡ 0 0 0 0 ⎦ ⎣ 0 ⎦ (29) 0 − ⎣0 0 u 0 0 KL + KN L

⎡

where ¨ r and ω ˙ are respectively the translational and angular accelerations of the body-ﬁxed reference frame and u ¨ denotes the nodal accelerations measured in body ﬁxed coordinates. The local coordinate frame ξηζ attached to the ﬂexible body, is used to represent the gross motion of the body and its deformation.

t

ζ

t

η t

b ∆t

ζ

ted updated ation

t ∆

ζ

0 0

η

t ′

t

Figure 25.

t

h

∆ b

General motion of a ﬂexible body.

b

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Linear Deformations of Flexible Bodies In many situations it is enough to consider that the components of the multibody system experience only linear elastic deformations. Furthermore, assume that the mode superposition technique can be used. Then, the ﬂexible part of the body is described by a sum of selected modes of vibration as (30) u = Xw where the vector w represents the contributions of the vibration modes towards the nodal displacements and X is the modal matrix. Due to the reference conditions, the modes of vibration used here are constrained modes. Due to the assumption of linear elastic deformations the modal matrix is invariant. The reduced equations of motion for a linear ﬂexible body are [5] gr sr Mrf X Mr 0 q ¨r = (31) − − Λw w ¨ I X T gf XT sf XT Mf r where Λ is a diagonal matrix with the squares of the natural frequencies associated with the modes of vibration selected. For a more detailed discussion on the selection of the modes used the interested reader is referred to [5]. The methodology is demonstrated through the application to the simulation of the unfolding of a satellite antenna, the Synthetic Aperture Radar (SAR) antenna that is a part of the European research satellite

Figure 26. The European satellite with the folded and unfolded conﬁgurations of the antenna.

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ERS-1, represented in Fig. 26. During the transportation the antenna is folded, in order to occupy as little space as possible. After unfolding, the mechanical components take the conﬁguration shown in Fig. 26(a). The SAR antenna consists in two identical subsystems, each with three coupled four-bar links that unfold two panels on each side. The central panel is attached to the main body of the satellite. Each unfolding system has two degrees of freedom, driven individually by actuators located in joints A and B. In the ﬁrst phase of the unfolding process the panel 3 is rolled out, around an axis normal to the main body, by a rotational spring-damper-actuator in joint A, while the panel 2 is held down by blocking the joints D and E. The second phase begins with the joint A blocked, next the panels 2 and 3 are swung out to the ﬁnal position by a rotational spring-damped-actuator. The model used for one half of the folding antenna, schematically depicted Fig. 27, is composed of 12 bodies, 16 spherical joints and 3 revolute joints. The central panel is attached to the satellite, deﬁned as body 1, which has much higher mass and inertia. The data for this antenna is reported in the work of Anantharamann [24]. Panel 3 (B3) Panel 2 (B2 (B2))

1.3 Actuator (1) (1)

a)

Figure 27. model.

Panel 1 (B1)

b)

c)

The SAR antenna: a) half unfolded state b) folded antenna; c) multibody

In the ﬁrst phase of the unfolding antenna, the rotational springdamped-actuator is applied in the revolute joint R3 . For the second phase, the revolute joint R3 is blocked and the system is moved to the next equilibrium position by a spring-actuator-damped positioned in joint R1 . The unfolding processes for rigid and ﬂexible models are shown in Fig. 28, only for its ﬁrst phase. The diﬀerent behavior between the rigid and the ﬂexible models is noticeable in Fig. 28. Though not shown here, the rotational actuator moment responsible for the start of the unfolding is not correctly predicted by the rigid multibody model. Being a very light and ﬂexible structure, the discrepancies, if not identiﬁed during the design stage, would lead to the failure of the unfolding process.

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Figure 28. models).

First phase of the unfolding of the SAR antenna (rigid and ﬂexible

Nonlinear Deformations in Multibody Systems For ﬂexible multibody systems experiencing nonlinear geometric and material deformations, the equations of motion for a ﬂexible body are given by Eq. (29). However, due to the time variance of all its coeﬃcients, Eq. (29) is not eﬃcient for computational implementation. Instead, by considering a lumped mass formulation for the mass matrix and referring the nodal accelerations to the inertial frame, the equations of motion for a single ﬂexible body take the form of [6] ⎡ ¯T ¯ ∗A mI + AM ⎣ − AM ¯ ∗S T 0

⎤⎡ ⎤ ¯ ∗S −AM 0 ¨ r ˙ ⎦ J + ST M∗ S 0 ⎦ ⎣ω q ¨f 0 Mff ⎤ ⎡ ¯ δ fr + AC = ⎣n − ω˜J ω − ST C δ − ¯ IT C θ ⎦ (32) g f − f − (KL + KN L ) u

where the absolute nodal displacements are written as

T − x ¨ ˙ ¨ r d ˜ + δ A k k =u ¨ k + q ¨kf ≡ α ¨ k ω˙ 0 I ˜ (xk + δ k ) + 2˜ ω δ˙ k ω ˜ ω (33) + ω ˜ θ˙ k with xk being the position of node k in the reference conﬁguration. In Eq. (32) M∗ is a diagonal mass matrix containing the mass of the n boundary nodes, T T T ¯ T = [A . . . A]T , S = x A ... x ˜n + δ˜n ˜1 + δ˜1

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Multibody Dynamics: Bridging for Multidisciplinary Applications

and ¯ I = [I . . . I]T where A is the transformation matrix from the body ﬁxed to global coordinate coordinates and xk denotes the position of node k. Vectors C δ and C θ represent respectively the reaction force and moment of the ﬂexible part of the body over the rigid part, given by C δ = g δ − Fδ − (KL + KN L )δδ δ − (KL + KN L )δθ θ ,

(34)

C δ = g θ − Fθ − (KL + KN L )θδ δ − (KL + KN L )θθ θ .

The coupling between the rigid body motion and the system deformations is fully preserved. For a more detailed description of the formulation, and the notation, the interested reader is referred to reference [6]. As an application example of the nonlinear formulation for ﬂexible multibody systems, a sports vehicle with a front crash-box is analyzed for various impact scenarios, represented in Fig. 29, where the angle of Angle 10º no friction

Angle 20º no friction

Angle 20º friction = 0.5

Angle 10º friction = 0.5

10 cm ramp

(a)

(d)

Figure 29.

(e)

Diﬀerent impact scenarios for the sports vehicle.

Figure 30. Motion of the vehicle for a 20◦ oblique impact without contact friction and for impact with an oblique surface for a vehicle traveling over a ramp.

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impact and the topology of the road are diﬀerent. The simulations are carried until the vehicle reaches a full stop. The vehicle motion, for the oblique impact scenario presented in Fig. 29, is characterized by a slight rotation of the vehicle during impact. This rotation is more visible in the case of frictionless impact. At the simulated impact speed the inﬂuence of the car suspension elements on the deformation mechanism is minimal.

6.

Conclusions

The multibody dynamics formalisms provide an extremely eﬃcient framework to incorporate diﬀerent disciplines. The behavior of a good number of phenomena in diﬀerent problems can be represented by kinematic constraints (e.g., contact, muscle action, guidance) or by contact forces (e.g, impact phenomena, control, general interactions). However, diﬀerent disciplines use diﬀerent preferred numerical methods to solve their equilibrium equations which lead to diﬃculties in the co-simulation of diﬀerent systems. The use of multibody formalisms in biomechanics presents a strong increase due to the suitability to model contacts, muscles, anatomical joints, data processing, etc. The treatment of structural components with large rotations or of rotating bodies with structural deformations ﬁnds in the ﬂexible multibody dynamics eﬃcient methods to deal with the problem. A continued eﬀort to close the gap between the ﬂexible multibody dynamics and the nonlinear ﬁnite element method is required. The need for more robust and eﬃcient numerical methods to handle the speciﬁc forms of the MBS equations and the discontinuities associated to intermittent and ‘fast’ behaviors are still required.

Acknowledgements The contents of this work result from a team eﬀort and collaborations with many co-workers among which the contribution by Miguel Silva, Joao ˜ Gon¸calves, ¸ Jo˜˜ao Pombo, Manuel Seabra Pereira, Jo˜ ao ˜ Abrantes, Augusta Neto and Rog´ ´erio Leal are gratefully acknowledged.

References [1] P. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliﬀs, New Jersey 1988. [2] J. Garcia de Jalon, E. Bayo, Kinematic and Dynamic Simulation of Mechanical Systems – The Real-Time Challenge, Springer-Verlag, Berlin 1994. [3] P. Nikravesh and G. Gim, Systematic construction of the equations of motion for multibody systems containing closed kinematic loops, Journal of Mechanical Design, Vol. 115, No.1, pp.143–149, 1993.

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[4] C.W. Gear, Numerical solution of diﬀerential-algebraic equations, IEEE Transactions on Circuit Theory, Vol. CT-18, pp.89–95, 1981. [5] J. Gon¸calves and J. Ambr´ ´ osio, Complex ﬂexible multibody systems with application to vehicle dynamics, Multibody System Dynamics, Vol. 6, No.2, pp.163–182, 2001. [6] J. Ambr´ ´ osio and P. Nikravesh, Elastic-plastic deformation in multibody dynamics, Nonlinear Dynamics, Vol. 3, pp.85–104, 1992. [7] F. Pfeiﬀer and C. Glocker, Multibody Dynamics with Unilateral Contacts, John Wiley and Sons, New York 1996. [8] H. Lankarani and P. Nikravesh, Continuous contact force models for impact analysis in multibody systems, Nonlinear Dynamics, Vol. 5, pp.193–207, 1994. ˇ [9] M. Valasek, Z. Sika, Evaluation of dynamic capabilities of machines and robots, Multibody System Dynamics, Vol. 5, pp.183–202, 2001. [10] H. Møller and E. Lund, Shape Sensitivity Analysis of Strongly Coupled FluidStructure Interaction Problems, [in:] Proc. 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA. AIAA Paper No.2000–4823, 2000. [11] H. Møller, E. Lund, J. Ambr´ ´ osio and J. Gon¸¸calves, Simulation of ﬂuid loaded ﬂexible multiple bodies, Multibody Systems Dynamics, Vol. 13, No.1, 2005. [12] J. Pombo and J.Ambr´ ´ osio, General spatial curve joint for rail guided vehicles: kinematics and dynamics, Multibody Systems Dynamics, Vol. 9, pp.237–264, 2003. [13] J. Pombo and J. Ambr´ ´ osio, A multibody methodology for railway dynamics applications, Technical Report IDMEC/CPM-04/002, IDMEC, Instituto Superior T´ ´ecnico, Lisboa, Portugal, 2004. [14] M. Silva and J. Ambr´ ´ osio, Kinematic data consistency in the inverse dynamic analysis of biomechanical systems, Multibody System Dynamics, Vol. 8, pp.219– 239, 2002. [15] D. Tsirakos, V. Baltzopoulos and R. Bartlett, Inverse Optimization: Functional and Physiological Considerations Related to the Force-Sharing Problem, Critical Reviews in Biomedical Engineering, Vol. 25, Nos.4&5, pp.371–407, 1997. [16] M. Silva and J. Ambr´ ´ osio, Human Motion Analysis Using Multibody Dynamics and Optimization Tools, Technical Report IDMEC/CPM-04/001, IDMEC, Instituto Superior T´ ´ecnico, Lisboa, Portugal, 2004. [17] G.T.Yamaguchi, Dynamic Modeling of Musculoskeletal Motion, Kluwer Academic Publishers, Boston 2001. [18] H. Hertz Gesammelte Werke , Leipzig, Germany 1895. [19] N.W. Murray, The static approach to plastic collapse and energy dissipation in some thin-walled steel structures, [in:] Structural Crashworthiness, N. Jones and T. Wierzbicki [eds.], pp.44–65, Butterworths, London 1983. [20] C.M. Kindervater, Aircraft and helicopter crashworthiness: design and simulation, [in:] Crashworthiness Of Transportation Systems: Structural Impact And Occupant Protection, J.A.C. Ambrosio, ´ M.S. Pereira and F.P Silva [eds.], NATO ASI Series E. Vol. 332, pp.525–577, Kluwer Academic Publishers, Dordrecht 1997.

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[21] P.E. Nikravesh, I.S. Chung, and R.L. Benedict, Plastic hinge approach to vehicle simulation using a plastic hinge technique, J. Comp. Struct. Vol. 16, pp.385–400, 1983. [22] 30 mph Rollover Test of an AM General Model M151-A2 1/4 Ton Jeep, The Transportation Research Center of Ohio, Test Report, 1985. [23] K.-J. Bathe and S. Bolourchi, Large displacement analysis of three-dimensional beam structures, Int. J. Num. Methods in Engng., Vol. 14, pp.961–986, 1979. [24] M. Anantharaman and M. Hiller, Numerical simulation of mechanical systems using methods for diﬀerential-algebraic equations, Int. J. Num. Meth. Eng., Vol. 32, pp.1531–1542, 1991.

RAPID FORMATION OF STRONG GRADIENTS AND DIFFUSION IN THE TRANSPORT OF SCALAR AND VECTOR FIELDS Konrad Bajer Institute of Geophysics Warsaw University [email protected]

Abstract

An important issue in the theory of transport by moving ﬂuids is the role of dissipation when the medium is nearly ideal. The central problem of this nature is understanding of the viscous dissipation at very large Reynolds numbers. We will discuss a few problems in the same category but linear and therefore more promising although, as it turns out, surprisingly rich and far from being solved. Their common denominator is the interplay between diﬀusion and advection. In a typical ﬂow the latter tends to decrease the characteristic length scales of the spatial variations of the transported quantity, thus increasing the rate of diﬀusion. Depending on a particular conﬁguration, either this rapid diﬀusion prevails and eﬃciently annihilates all gradients, or a kind of balance is reached and a quasi-steady dissipative structure emerges.

Keywords: Accelerated diﬀusion, shear dispersion, current sheets, passive scalar, vortex dynamics, Poiseuille ﬂow.

Introduction Dissipative processes are, in general, strongly aﬀected by the motion of the medium. In a stationary ﬂuid, a passive quantity like temperature or concentration of a contaminant would change its distribution only through molecular processes that, in a wide range of physical problems, may be well modelled as ‘simple’ Fickian diﬀusion. When the ﬂuid is in motion, the diﬀusive process in the Lagrangian frame of every material element may result in great complexity of the Eulerian distribution of the transported quantity. In particular, the interplay of advection and diﬀusion typically increases the rate at which the latter smoothes out the spatial variations of that quantity’s distribution. The archetypal 89 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 89–101. © 2005 Springer. Printed in the Netherlands.

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formula describing this interplay is the advection-diﬀusion equation for an intensive quantity, say temperature, ∂T + u · ∇T = κ∇2 T. ∂t

(1)

The dynamics of various physical processes, other than advectiondiﬀusion of a passive scalar, often reduces to the Eq. (1) or to another linear equation of similar type that exhibits the same salient feature of enhanced rate of diﬀusion associated with non-uniform ﬂuid motion. In simple geometries the evolution of magnetic ﬁelds in electrically conducting liquids and plasmas can sometimes be reduced exactly to an advection-diﬀusion problem with important implications in all branches of magnetohydrodynamics whether cosmic, solar, geophysical or that applied in the theory of nuclear fusion devices and in metallurgy. The evolution of weak background vorticity in presence of a strong coherent structure – an important mechanism in two-dimensional turbulence – is described by a linearised vorticity equation that, in spite of some extra complications, is a linear equation (scalar when the ﬂow is twodimensional) containing both advection and diﬀusion ingredients. The same equation also describes the evolution of small perturbations induced in a strong vortex by a weak external irrotational ﬂow. We shall illustrate the phenomenon, called accelerated diﬀusion, with simple solutions of the Eq. (1) for two basic steady ﬂows u(x).

1.

Accelerated Diﬀusion

Let us consider an initial value problem for the advection-diﬀusion equation (1) in an unbounded two-dimensional domain. The problem is linear, so every intial distribution of the scalar, T (x, t = 0), can be decomposed into independently evolving Fourier modes. For simplicity we take the initial distribution to be harmonic in the x direction (see Fig. 1b), (2) T (x, t = 0) = T0 eik0 x , i.e. the initial wave vector is k0 = (k0 , 0). We shall consider two steady ﬂows u(x) (see Fig. 1a) - a stagnation point ﬂow (irrotational linear strain), u(x) = (−αx, αy, 0) ,

(3)

u(x) = (αy, 0, 0) .

(4)

and a linear shear ﬂow

91

Strong Gradients & Diﬀusion in the Transport . . .

(a)

(b)

Figure 1. (a) Two types of the ﬂow considered — a stagnation point ﬂow (irrotational linear strain, top) and linear shear (bottom); (b) Initial distribution of the scalar ﬁeld subject to advection and diﬀusion.

Both ﬂows impose a characteristic advection time scale determined by the value of strain/shear coeﬃcient α, tA = α−1 .

(5)

The initial scalar distribution has a characteristic length scale, k0−1 , and the associate diﬀusion time scale −1 . (6) tD = κk02 The ratio of the two time scales is called Schmidt number when the transported scalar ﬁeld T is the mass concentration of some material admixture (an extensive quantity) and P´ ´eclet number in the case of temperature (an intensive quantity), Pe =

α tD = 2 . tA k0 κ

(7)

In both cases the solution has the general form of a single Fourier mode but with time-dependent wave vector and amplitude, T (x, t) = T0 F (t)eik(t)·x .

(8)

For the stagnation point ﬂow (3) we obtain ∂T ∂T − αx = κ∇2 T ∂t ∂x

(9)

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and the solution is 1

k(t) = k0 eαt , 0 , −1 e2αt

F (t) = e− 2 Pe

(10a)

= exp (− exp(2αt − ln Pe − ln 2)) .

(10b)

The wavelength in the x-direction decreases exponentially. That in the y-direction would, in general, increase exponentially but with the initial condition (2) it started being already inﬁnite (cf. Fig. 2a). The decay of the amplitude F (t) is super-exponential, i.e. extremely fast. The initial variations of the scalar ﬁeld distribution are therefore eliminated by a very rapid process of strain diﬀusion on the time scale tstrain = (ln Pe) tA

(11)

which is much shorter than that of ordinary diﬀusion, in fact only slightly longer than the advection time scale, even when the P´eclet number is very large. For the linear shear (4) we have ∂T ∂T + αy = κ∇2 T ∂t ∂x

(12)

and the corresponding solution is k(t) = k0 (1, −αt) , F (t) = exp −

(a)

1 t − tD 3

t

(13a) 3

.

−2/3 tD Pe

(13b)

(b)

Figure 2. The distribution of the scalar subject to advection and diﬀusion in a stagnation point ﬂow (a) and in a linear shear ﬂow (b) (cf. Fig. 1a). The initial distribution is shown in Fig. 1b.

Strong Gradients & Diﬀusion in the Transport . . .

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Here the wavenumber in the x-direction remains unchanged while that in the y-direction increases linearly in time. This linear growth, considerably slower than the exponential one in the straining ﬂow, accounts for slower decay of the amplitude F (t). For times shorter than tA we have F (t) ∼ e−t/tD ,

t tA = Pe−1 tD .

(14)

Therefore, initially, for a short time, the dissipative process is simple diﬀusion. Later we have 1

3

F (t) ∼ e− 3 (t/tS ) ,

t tA ,

(15)

which means that the process considerably accelerates. Firstly, the exponent changes from t1 to t3 . Secondly, the time scale of the decay becomes tS = Pe−2/3 tD = Pe1/3 tA . (16) This intermediate shear diﬀusion time scale, considerably shorter than that of the ordinary diﬀusive process, but longer than tA , was identiﬁed early in the context of scalar diﬀusion (Rhines & Young 1983) and in the context of magnetic ﬁeld transport (Moﬀatt & Kamkar 1983). The two special linear ﬂows chosen here, i.e. a stagnation point ﬂow and linear shear, may seem to be too simple and lacking all the complications encountered in real transport problems. However, there are at least two reasons to single them out. Firstly, they are often good local approximations of the large-scale, time-dependent ﬂows when the initial spatial scale of the transported scalar is small compared with that of the ﬂow itself and the local time-scale, say tS , is short compared with that of the ﬂow variations. Secondly, these ﬂow conﬁgurations, possibly with a modiﬁed geometry, are often found in diﬀerent physical problems of special interest. We will discuss the transport in a stagnation point ﬂow in the context of ﬂux- and current-sheet formation, common in magnetohydrodynamics, where a ﬂow of this kind is essential. Further, we will consider shear that is always associated with a strong, concentrated vortex, both in its core and outside and aﬀects any passive scalar or weak vorticity that may be present.

2.

Current Sheets and Flux Sheets

The evolution of the magnetic ﬁeld B(x, t) penetrating an electrically conducting ﬂuid moving with velocity u(x, t) is governed by the induction equation, ∂B = ∇ × (u × B) + η∇2 B, (17) ∂t

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where the magnetic diﬀusivity of the ﬂuid, η, is inversly proportional to its electrical conductivity. This approximation of the Maxwell’s equation is used within the framework of magnetohydrodynamics (MHD) to describe processes much slower than electromagnetic waves. This is a kind of advection-diﬀusion equation for a vector quantity B(x, t). In general it describes a much richer variety of phenomena than its scalar counterpart, the kinematic dynamo problem possibly being the most spectacular one (Childress & Gilbert 1995), but in some special symmetric conﬁgurations the two are actually equivalent1 . We will now consider two simple steady solutions of the induction equation (17), both with u(x, t) corresponding to the stagnation point ﬂow (3), that illustrate an important MHD phenomenon, namely the formation of large, localised gradients of the magnetic ﬁeld (Bajer 2004). Strong electric currents, j(x, t) = ∇ × B, are associated with such gradients implying intense Ohmic heating. The structures of this type, called current sheets, play an important role in various MHD processes like, for example, the heating of the solar corona. The ﬁrst solution describes the magnetic ﬁeld lying in the same plane as the ﬂow, B(x) = (Bx (x, y), By (x, y), 0), in which case (17) reduces to the advection-diﬀusion equation (1) for the only component of the vector potential, ∂A ∂A − αx = η∇2 A, ∂t ∂x

B = ∇ × A,

A = (0, 0, A(x, y, t)) .

(18)

As we could see in the previous section, individual Fourier modes ‘travel’ across the Fourier space towards smaller scales (larger k). However, an imposed boundary condition may act as a ‘source’ of one or more Fourier modes and a steady state may be established. In particular, forcing the k = 0 mode, A(x, t) = A(x)ˆ ez ,

A(x) −−−−→ ±A0 ,

we obtain A(x) = A0 erf

α 2η x

,

(19)

x→±∞

B(x) = −A (x) = −A0

2α πη

α 2 − 2η x

e

.

(20)

This solution represents a ﬂux sheet, i.e., a layer of strong magnetic ﬁeld with Gaussian proﬁle, similar in nature to a Burgers vortex layer in which stretching is in balance with diﬀusion. This is also a double current 1 With

linear stretching in the direction of the ﬁeld B the Eq. (17) reduces to a scalar advection-diﬀusion equation with an extra ‘source term’ (Bajer & Moﬀatt 1997).

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sheet with two adjacent layers of strong current in opposite directions. The Ohmic heating Dm is strong, in fact for a given ﬂux Dm −−−→ ∞, η→0

but the magnetic energy Em stored in such sheets also grows without bounds as η → 0, so they cannot be easily created in ﬂows with ﬁnite energy supply (Bajer 2004). They are, therefore, more relevant to the solar dynamo than to the coronal heating problem. When the ﬁeld B = B(x)ˆ ey is perpendicular to the stagnation point ﬂow u = (−αx, 0, αy), we also obtain an advection-diﬀusion equation but this time for a single componet of the ﬁeld, B(x), rather than its vector potential. Imposing B(x) → ±B0 for x → ±∞ we now obtain a steady solution describing a current sheet, B(x) = B0 erf

α 2η x

,

j(x) = B (x) = B0

2α πη

α

2

e− 2η x .

(21)

Ohmic dissipation in such current sheets is small, Dm −−−→ 0, so the η→0

direct heating eﬀect of an individual sheet is weak. However, in a highly conducting medium the magnetic energy stored in such current sheets is small, Em → 0 as η → 0, so they can be easily created even in weak ﬂows. The cumulative direct heating eﬀect may be augmented when many such sheets are created (Parker 1963a). This is likely to be the case when a weakly perturbed integrable ﬁeld relaxes towards a new equilibrium and current sheets appear on the whole range of spatial scales with their distribution, at least in some regions, becoming dense in the limit of inﬁnite electric conductivity of the medium (Bajer 2004). Besides their energetics in a steady or quasi-steady state, the details of the dynamic formation process of strong ﬁeld gradients are also important. The collapse of an X-type neutral point is probably the generic mechanism (Parker 1957, 1963a; Moﬀatt 1990; Linardatos 1993). In spite of the relatively low heat release, the associated magnetic reconnection process locally lifts the topological constraints of ideal MHD thus enabling further relaxation towards yet lower energy states. The role of topological constraints and their violation is a challenging open problem from the physical and from the mathematical point of view, both in linked and in open-ended, braided ﬂux systems (Moﬀatt 1985; Moﬀatt & Ricca 1992). In this section we have discussed some fundamental, ubiquitous MHD processes involving stagnation point ﬂows. Now we will consider a category of archetypal transport processes ruled by the shear ﬂows.

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Figure 3. The formation of a current sheet as a result of the collapse of an X-type neutral point. The cylindrical cavity, surrounded by perfectly conducting solid is ﬁlled with viscous, perfectly conducting ﬂuid penetrated by non-equilibrium magnetic ﬁeld initially containing an X-type neutral point (left panel). Under the evolution governed by the full, nonlinear MHD equations, the neutral point collapses and the system relaxes towards an ideal magnetostatic equilibrium containing a tangential discontinuity of the ﬁeld across which the ﬁeld direction reverses. In a perfectly conducting ﬂuid this corresponds to a singular, delta-like current while in a ﬂuid of ﬁnite conductivity the singular layer would be replaced by a quasi-steady current sheet of ﬁnite length and with ﬁnite current.

3.

Passive Scalar Outside and Inside a Vortex

The dynamics of many ﬂows is best understood in terms of the evolution of coherent vortices and their interactions. For some time this is known to be the case with two-dimensional turbulence (McWilliams 1984) where vortices are sizeable compared with the length scale of the forcing (driven ﬂow) or with the size of the domain (decaying ﬂow). In three-dimensional turbulence the vortices are prominent at small scales (Bajer & Moﬀatt 2002), their diameters seem to be of the order of the Kolmogorov scale (Moﬀatt, Kida & Ohkitani 1994). It is therefore of interest to understand the advection-diﬀusion processes in the presence of concentrated vortices. Locally, the ﬂow near a vortex is simple shear. An initial blob of passive scalar which is small compared with the streamline curvature experiences shearing by a ﬂow similar to (4) (Flohr & Vassilicos 1997) and the initial-value problem for its evolution can be decomposed into Fourier modes evolving according to (8). However, if the initial spatial variations of the scalar are much larger than the size of the vortex, then, on the scale of the vortex, the initial scalar isolines are, to a good approxmation, straight. They are then wrapped around the vortex in the manner shown in Fig. 4. Should the

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Figure 4. The evolution of two stretches of diﬀerent material lines, AB and CD, in the ﬂow due to a point vortex located at the centre. Dashed lines in the left panel mark the streamlines of the ﬂow. Each of the two intervals, originally rectilinear, is wound into a double spiral (right panel). As time progresses, the spirals are increasingly tight and the separation of the two material lines decreases like t−1 .

scalar be non-diﬀusive, this spiral wind-up would continue forever. In Fig. 4 we see that the distance between given isolines decreases with time. It is proportional to t−1 , just like for a single Fourier mode in linear shear (cf. eqn. 13). Such decrease in the radial length scale boosts diﬀusion which eventually prevails. The details of this process inside and outside the vortex are somewhat diﬀerent, but the essential ingredient, i.e. shear, is present in both regions. Looking at the outer region we can approximate the ﬂow by that of a point vortex and obtain both a steady solution (when one mode is forced by a distant boundary condition) or an unsteady, similarity solution in an unbounded domain (Bajer 1998). The steady solution describes the balance inside a cylindrical cavity of radius R ﬁlled with inviscid liquid of ﬁnite thermal diﬀusivity. The motion is irrotational but for a line vortex of circulation Γ on the axis. Uniform temperature gradient, T = T0 y = T0 r sin θ, is imposed on the solid surface of the cylinder. This surprisingly simple solution of Eq. (1) with u = Γ/(2πr)ˆ eθ , T = T0 Re f (r, t)eiθ , (22a) r √1+iP e

Γ , Pe = , (22b) R κ describes, in particular, a thermal boundary layer on the cylinder surface with isotherms wound into a double logarithmic spiral and, also, the f (r, t) = f (r) = −i

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Concentration, Pe = 100

0.09 0.8 1

1

0.08 0.6 0.07 0.4 0.5

0.5

0.06 0.2 0.05 0

0

0 0.04

−0.2

−0.4

−0.5

0.03 −0.5 0.02

−0.6

−0.8

−1

−1

−0.5

0

0.5

1

0.01 −1

−1

−0.5

0

0.5

1

0

Figure 5. Point vortex in the initialy uniform temperature gradient. Boundary condition ﬁxes one mode which makes a steady state possible. The left panel shows isotherms being wound into double logarithmic spirals in a thermal boundary layer on the surface of the cylinder (lighter colours correspond to warmer ﬂuid). The right panel shows the heat ﬂux expelled from the central region (solid lines - integral curves of the heat ﬂux; grey scale - magnitude of the heat ﬂux).

expulsion of the heat ﬂux from the central region (Fig. 5). A straightforward MHD analogue of this solution describes the magnetic ﬂux expulsion, an important phenomenon discovered some fourty years ago (Parker 1966; Weiss 1966). Corresponding similarity solutions, f (r, t) = f (r2 /t), for a point-vortex ﬂow (Bajer 1998) and for the Lamb vortex (Pearson & Abernathy 1984; Moore 1985) describe spiral structures propagating outwards in an unbounded domain. Spiral structures appear in diﬀerent contexts in the theory of turbulence and vortex dynamics (Gilbert 1988; Moﬀatt 1993; Kimura & Herring 2001; Pullin & Lundgren 2001). A gigantic magnetic structure of this kind encompassing the entire solar system is called Parker’s heliospheric spiral (Parker 1963b). The wind-up and accelerated diﬀusion are direct consequences of shear in a vortical ﬂow. This is present both outside a vortex and in its core. However, the radial proﬁles of the shear in the two regions are diﬀerent. The shear, deﬁned as the angular velocity gradient, is proportional to r−3 in the outer region and to r near the centre of the vortex. The details of the gradient annihilation will thus be diﬀerent. In particular, the shear vanishes at r = 0. Interestingly, there is another exact solution eθ (where of (1) with parabolic core vorticity proﬁle, i.e. u = (α0 r+α1 r3 )ˆ α0 , α1 = 0 are arbitrary constants) of the same general form (22a) with (Lighthill 1966; Brunet & Haynes 1995; Bajer, Bassom & Gilbert 2001) 2

f (r, t) = g(t)e−iα0 t−ih(t)r ,

(23a)

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µ = (1 + i) 2α1 /P e. (23b) This solution describes the three phases of annihilation of the gradient of a passive scalar inside a vortex: spiral wind-up, accelerated (shear) diﬀusion moving in from the region of stronger shear towards the centre and the survival of longer-living remnant around the origin where the shear vanishes. The latter is an ‘eigenmode’ decaying without change of shape on a time scale tI = tD P e−1/2 , intermediate between tS and tD (cf. Eq. (16)) (Bajer, Bassom & Gilbert 2001). g(t) = (cosh µt)−2 ,

4.

h(t) = α1 µ−1 tanh µt,

Vortex-Background Interaction

The interplay between advection and diﬀusion resulting in the formation of strong gradients and in accelerated diﬀusion is also a feature of transport of physical quantities that cannot be regarded as passive. Vorticity in a two-dimensional ﬂow is, for example, an ‘active’ scalar ﬁeld governed by the vorticity equation. How important is its coupling to the transporting velocity ﬁeld depends on particular circumstances. When vorticity ﬁeld can be naturally separated into a strong coherent part, Ω and weak background ω (Kevlahan & Farge 1997; Farge, Holschneider & Colona 1990; Beta et al. 2003), the vorticity equation for the background can be linearised and becomes, in some sense, an extension of the advection diﬀusion equation for ω in the ﬂow U associated with Ω, ∂ω + U · ∇ω = ∂(ψ, Ω) + F + ν∇2 ω, ∇2 ψ = −ω. (24) ∂t Compared with the Eq. (1) there is the source term, F , that corresponds to a possible external forcing of the ﬂow. There is also an extra term, ∂(ψ, Ω), corresponding to the diplacement of the coherent vorticity Ω by ˆz associated with the weak background vorticity ω. the ﬂow u = ∇ψ × e This is an Eulerian eﬀect which needs to be separated from the ‘genuine’ evolution of the background. If we choose for Ω the Lamb vortex (diﬀusing Gaussian vorticity dis˙ tribution) which is moving with prescribed velocity (X(t), Y˙ (t)) and for the initial ω – a distribution with locally uniform gradient, we ﬁnd that there is only one choice of (X(t), Y (t) consistent with the far-ﬁeld boundary conditions. We then obtain both the evolution of the background and the motion of the strong vortex. An important conclusion is that a single, strong coherent vortex immersed in a non-uniform weak background vorticity is set in motion by the ﬂow associated with a perturbation of the background that the strong vortex creates (Bajer, Bassom & Gilbert 2004). The vortex also homogenises the background around itself, thus making a ‘hole’ in the

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background vorticity gradient. Therefore, the viscous solution of the linearised vorticity equation conﬁrms Batchelor’s general prediction about uniformisation of vorticity in ﬂows with closed streamlines (Batchelor 1956).

Acknowledgement Much of the material presented in this review relates to the collaborative work with Andrew Bassom and Andrew Gilbert that I enjoyed over several years. I am indebted to Keith Moﬀatt for drawing my attention to a range of focal physical problems and for sharing his insights. The support from the Centre of Excellence in Small Scale Atmospheric Research, CESSAR (EU grant no EVK2-CT-2002-80010), is gratefully acknowledged.

References K. Bajer, Flux expulsion by a point vortex, Eur. J. Mech. B/Fluids 17(4):653–664, 1998. K. Bajer, Abundant singularities, Fluid Dyn. Res. Submitted, 2004. K. Bajer, A.P. Bassom, and A.D. Gilbert, Accelerated diﬀusion in the centre of a vortex, J. Fluid Mech. 437:395–411, 2001. K. Bajer, A.P. Bassom, and A.D. Gilbert, Vortex motion in a weak background shear ﬂow. J. Fluid Mech. 509:281–304, 2004. K. Bajer, H.K. Moﬀatt, On the eﬀect of a central vortex on a stretched magnetic ﬂux tube, J. Fluid Mech. 339:121–142, 1997. K. Bajer, and H.K. Moﬀatt, Tubes, Sheets and Singularities in Fluid Dynamics, Kluwer, 2002. G.K. Batchelor, On steady laminar ﬂow with closed streamlines at large Reynolds number, J. Fluid Mech. 1:177–190, 1956. C. Beta, K. Schneider, M.Farge, and H. Bockhorn, Numerical study of mixing of passive and reactive scalars in two-dimensional turbulent ﬂows using orthogonal wavelet ﬁltering, Chem. Eng. Sci. 58 (8):1463–1477, 2003. G. Brunet, and P.H. Haynes, The non-linear evolution of disturbances to a parabolic jet, J. Atmos. Sci. 52:464–477, 1995. S. Childress and A.D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo, Springer, 1995. M. Farge, M. Holschneider, J.F. Colona, Wavelet analysis of coherent structures in two-dimensional turbulent ﬂows, [In:] Topological Fluid Mechanics (ed. H.K. Moffatt & A. Tsinober), Cambridge University Press, 1990. P. Flohr, J.C. Vassilicos, Accelerated scalar dissipation in a vortex, J. Fluid Mech. 348:295–317, 1997. A.D. Gilbert, Spiral structures and spectra in two-dimensional turbulence, J. Fluid Mech. 193:475–497, 1988. N.K.R. Kevlahan, M. Farge, Vorticity ﬁlaments in two-dimensional turbulence: creation, stability and eﬀect, J. Fluid Mech. 346:49–76, 1997.

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Y. Kimura and J.R. Herring, Gradient enhancement and ﬁlament ejection for a nonuniform elliptic vortex in two-dimensional turbulence, J. Fluid Mech. 439:43–56, 2001. M.J. Lighthill, Initial development of diﬀusion in a Poiseuille ﬂow, J. Inst. Maths. Applics. 2:87–108, 1966. D. Linardatos, Determination of 2-dimensional magnetostatic equilibria and analogous Euler ﬂows, J. Fluid Mech., 246:569–591, 1993. J.C. McWilliams, The emergence of isolated coherent vortices in turbulent ﬂow, J. Fluid Mech. 146:21–434, 1984. H.K. Moﬀatt, Magnetostatic equilibria and analogous Euler ﬂows with arbitrary complex topology, J. Fluid Mech. 159:359–378, 1985. H.K. Moﬀatt, Structure and stability of solutions of the Euler equations: a Lagrangian approach, Phil. Trans. Roy. Soc. A 333:321–342, 1990. H.K. Moﬀatt, Spiral structures in turbulent ﬂows, [in:] New Approaches and Concepts in Turbulence (Monte Verit` ` a), Birkhauser 1993. H.K. Moﬀatt and H. Kamkar, On the time-scale associated with ﬂux expulsion, [in:] Stellar and Planetary Magnetism, pp.91–97. Gordon & Breach, 1983. H.K. Moﬀatt, S. Kida, and K. Ohkitani, Stretched vortices – the sinews of turbulence: high Reynolds number asymptotics, J. Fluid Mech. 259:241–264, 1994. H.K. Moﬀatt and R. Ricca, Helicity and the Cˇ ˇ alugˇ ˇ areanu invariant, J.Proc. Roy. Soc. A, 439:411–429, 1992. D.W. Moore, The interaction of a diﬀusing vortex and an aligned shear ﬂow, Proc. R. Soc. Lond. A 399:367–375, 1985. E.N. Parker, Sweet’s mechanism for merging magnetic ﬁelds in conducting ﬂuids, J. Geophys. Res. 62:509–520, 1957. E.N. Parker, Dynamics of the interplanetary gas and magnetic ﬁelds, Astrophys. J. 128:664–676, 1958. E.N. Parker, The solar-ﬂare phenomenon and the theory of reconnection and annihilation of magnetic ﬁelds. Astrophys. J. Suppl. 8:177–211, 1963a. E.N. Parker, Interplanetary Dynamical Processes, Wiley-Interscience, New York 1963b. R.L. Parker, Reconnection of lines of force in rotating spheres and cylinders, Proc. Roy. Soc. A, 291:60–72, 1966. C.F. Pearson and F.H. Abernathy, Evolution of the ﬂow ﬁeld associated with a streamwise diﬀusing vortex, J. Fluid Mech. 146:271–283, 1984. D.I. Pullin and T.S. Lundgren, Axial motion and scalar transport in stretched spiral vortices, Phys. Fluids, 13(9):2553–2563, 2001. P.B. Rhines, W.R. Young, How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133:133–145, 1983. N.O. Weiss, The expulsion of magnetic ﬂux by eddies, Proc. Roy. Soc. Lond. A 293:310–328, 1966.

WAVE-VORTEX INTERACTIONS IN THE ATMOSPHERE, AND CLIMATE PREDICTION Onno Bokhove Numerical Analysis and Computational Mechanics, Department of Applied Mathematics IMPACT, University of Twente, P.O. Box 217, Enschede, The Netherlands [email protected]

Abstract

Can we construct an accurate atmospheric climate model with a balanced model representing its ﬂuid mechanics, and with dissipative as well as non-dissipative parameterization schemes for the gravity-wave activity? To address this question, we focus our attention on a 1 21 -layer atmospheric model with an isentropic troposphere and isothermal stratosphere. We investigate parcel dynamics in a hybrid Eulerian-Lagrangian formulation, potential vorticity conservation, static stability, linear modes and the concept of balanced ﬂow; and brieﬂy discuss wave-vortex interactions and recent advances in numerical solution techniques.

Keywords: hybrid Eulerian-Lagrangian ﬂuid parcel dynamics, linear modes, balanced models, gravity waves, weather and climate prediction

Introduction Numerical weather and climate prediction is complicated because only the ﬂow scales larger than at least ∼ 10 × 10 × 1 km3 can be resolved to date. When we use the (inviscid) primitive Navier-Stokes equations on these scales, the commonly used (semi-Lagrangian) numerical schemes implicitly ﬁlter all acoustic waves and some of the gravity-wave (GW) motion. The rapid small-scale three-dimensional turbulence is then certainly not resolved. Consequently, also the feedback of the unresolved wave and (quasi-two-dimensional) turbulent motions on the large-scale dynamics requires parameterization. A lot of attention has been paid to simpliﬁed or balanced versions of the primitive equations, in which preservation of the conservation laws (of the inviscid dynamics) such as mass, energy and potential vorticity (PV) has been advocated to enhance the stability of these so-called ba103 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 103–116. © 2005 Springer. Printed in the Netherlands.

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lanced models. The large-scale ﬂow is generally close to hydrostatic and geostrophic balance; the former due to the anisotropy of horizontal and vertical scales and the latter due to the rapid rotation of the Earth, both in these balanced models (by default) and the numerical integration schemes used. Consequently, one has a choice to use either the primitive equations or balanced models. Birner et al. (2002) observed that vertical temperature proﬁles are nearly constant in the stratosphere with a distinct kink at the tropopause between the troposphere and stratosphere. To analyze several properties of atmospheric ﬂows, we therefore derive a conceptual model of the atmosphere with an isentropic troposphere and isothermal stratosphere, where the entropy or potential temperature θ and temperature T are constant, respectively. Subsequently, we illustrate the concept of balance by deriving a balanced model describing only the vortical motion from this so-called “θ-T -model”. A novel derivation of this θ-T -model from the three-dimensional Euler equations, using a combination of asymptotic methods and physical simpliﬁcations, is given in the framework of a hybrid Eulerian-Lagrangian description of a ﬂuid parcel (Section 1). This hybrid formulation of the Euler equations (Dixon and Reich, 2004) describes the Hamiltonian dynamics of each parcel as a dynamical system with six degrees of freedom with the internal and potential energy as function of space and time. The formulation is passive when this function is given. In contrast, an integral equation for the density using the Jacobian between Eulerian and Lagrangian space links the dynamics of all ﬂuid parcels into a dynamically consistent continuum. In the linearized θ-T -model, three time derivatives (or four in the parcel framework) in the model give rise to a pair of fast GW modes and one slow geostrophic mode, whose eigen-periods are separated in time on the f -plane. However, the dynamics are nonlinear and there may be a conversion of energy and momentum between these slow and fast modes. In Section 2 (i.e., Fig. 4), this is illustrated in simulations of the nonlinear dynamics initialized by a linear mode at ﬁnite amplitude, in which a simple hydraulic, dissipative wave-breaking parameterization is used. In the nonlinear dynamics, the slow modes survive approximately on a slow manifold of lower dimension. Balanced models of vortical dynamics describe the slow motion on a slow manifold (Section 3), on which the dimension of phase space is reduced by two thirds (or half in the parcel framework) due to the removal of the pair of (fast) GW modes. Within the Eulerian-Lagrangian framework of parcel dynamics, we illustrate the derivation of (Hamiltonian) balanced models using two velo-

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city constraints which arise in asymptotic expansions in a relevant small parameter (such as the Rossby or Froude number). These constraints deﬁne the reduction of the phase space to the slow or slaving manifold. We compare balanced and unbalanced trajectories in simpliﬁed simulations in Fig. 1(b). The generation of gravity waves by instabilities of a balanced ﬂow, or the absorption of gravity waves by the nearly balanced mean ﬂow through wave-vortex interactions indicate, however, that the slow manifold is not an exact manifold (B¨ u ¨hler and McIntyre, 2003; Vanneste and Yavneh, 2004). In balanced models, these unbalanced GW eﬀects can only be included by explicitly parameterizing the gravity waves. Likewise, in numerical weather and climate prediction small-scale, unresolved gravity waves require parameterization. We ﬁnish by brieﬂy discussing idealized wavevortex interactions and some recently developed numerical schemes for geophysical ﬂows (Sections 4 and 5), and their relevance to General Circulation Models (GCMs).

1.

Eulerian-Lagrangian Dynamics of Fluid Parcels

Three-dimensional compressible Euler equations Consider Newton’s equations of motion for a ﬂuid parcel with position x = (x, y, z)T and velocity u = (u, v, w)T [(·)T denotes the transpose] in a rotating reference frame with rotation vector Ω ∂H H3D dx =u= , dt ∂u

dθ = 0 and dt

∂H H 3D ∂H H3D du = −θ∇Π − ∇φ − 2Ω × u = − − 2Ω × dt ∂x ∂u

(1) (2)

with the parcel energy (extending Frank and Reich, 2003) H3D (x, u, θ, t) = |u|2 /2 + θ Π(x, t) + φ(x),

(3)

three-dimensional gradient ∇; external potential φ, e.g., φ = gz; potential temperature θ = T (p/pr )−κ ; temperature T (x, t); pressure p(x, t) and reference pressure pr ; and Exner function Π = cp (p/pr )κ for an ideal gas p = ρRT with density ρ(x, t), gas constant R, speciﬁc heat at constant pressure cp , and κ = R/ccp . Note that θ∇Π = (1/ρ)∇p, so that (2) attains its usual form (Dixon and Reich, 2004). We can write (1)–(3) as a non-canonical Hamiltonian system dq/dt = J∂H H3D /∂q with state vector q = (x, y, z, u, v, w, θ)T and a skew-symmetric tensor J. The state vector q is a function of time and ﬂuid labels a = (a, b, c)T =

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(a1 , a2 , a3 )T , so q = q(a, t). If the parcel energy is a function of q and t, then (1)–(3) is non-autonomous. The continuum character of the ﬂuid, albeit hidden in the representation (1)–(3), appears through the density ρ(x, t) =

ρ(x , t)δ(x − x )dx dy dz D = ρ0 (a)δ(x − x (a, t))dadbdc (4) D0

with x = x (a, t) in a domain D or D0 , since the Jacobian between label and position space is proportional to the inverse density ρ0 (a)/ρ(x, t) = det |∂x/∂a|.

(5)

Hence, an element of mass dm relates to the density as follows: dm = ρ(x, t)dxdydz = ρ0 (a)dadbdc.

(6)

A common choice is ρ(x, 0) = ρ0 (a) and x(a, 0) = a. The system (1)–(4) is closed and represents the ﬂuid as a continuum. From (1), (4) and (5), we can derive the continuity equation dρ(x, t)/dt = ∂t ρ(x, t) + u · ∇ρ(x, t) = −ρ(x, t)∇ · u(x, t).

(7)

Similarity to 2D vorticity dynamics. We note that this hybrid description is akin to the (more familiar) situation in inviscid, incompressible, two-dimensional vorticity dynamics, where the passive or kinematic advection of each ﬂuid parcel is described by a given stream function ψ(x, y, t) as Hamiltonian with horizontal coordinates xh = (x, y)T and time t. Thus, dx/dt = u = −∂ψ/∂y and dy/dt = v = ∂ψ/∂x. In contrast, a dynamically consistent formulation appears when the vorticity ω = ∇2h ψ is conserved on each ﬂuid parcel and linked to the continuum of parcels using ω(x, y, t) = ω0 (a, b)δ(xh − xh (a, b, t))dadb D0

with domain D0 , delta function δ(·), parcel position xh (a, b, t), and ω0 (a, b) denoting the initial distribution of vorticity on parcels identiﬁed by labels a and b. Given ω on each parcel, we calculate ψ. Hence, the dynamical description is closed, since incompressibility yields dxdy = dadb. With v = (u, v)T and gradient ∇h in the horizontal direction, we ﬁnd dω/dt = ∂ω/∂t + v · ∇h ω = 0.

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Static stability. To illustrate the formulation of the hybrid parcel dynamics, parcel oscillations in a static atmosphere with given parcel energy H3D = |u|2 /2 + θΠ(z) + g z are shown in Fig. 1(a). We choose Π = Π(z) with potential temperature θ = θg (z) in its thermodynamics to satisfy hydrostatic balance θg (z)∂Π/∂z = −g. Hence, we ﬁnd d2 z /dt2 = −N 2 z for small amplitude oscillations with z = z − zr and a reference a¨is¨¨ala¨ frequency N , level zr . Oscillations are then stable with Brunt–V¨ 2 when N = [g(dθg /dz)/θg ]z=zr > 0, neutral when N 2 = 0, and unstable when N 2 < 0. Note that θ is conserved on each parcel of air and generally diﬀerent from θg (z).

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Figure 1. (a) Three trajectories are shown of parcel oscillations in the atmosphere for a given parcel energy. The stratiﬁcation of the atmosphere is statically stable (dθ/dz = cst. > 0) for x < −10 km, unstable (dθ/dz = cst. < 0) for −10 km< x < 10 km, and neutral (constant θ) for x > 10 km and z < 10 km, and isothermal and stable for z > 10 km. The stable oscillations have a period of 10.84 min. When the atmosphere is hydrostatic, these oscillations disappear as the thin lines at zr = 5 km illustrate. (b) 41.7 days of (dimensionless) geostrophically balanced and unbalanced Hamiltonian motion of a particle in a simple, given Montgomery potential M2 (x, y) starting at (x, y) = (1, 1). The predictability horizon lies around 14 days whereafter the balanced (dashed lines) and unbalanced (solid lines) trajectories depart from one another signiﬁcantly.

Hydrostatic primitive equations The atmosphere is shallow for larger scales, and the aspect ratio δ between vertical and horizontal length and velocity scales (D, W and L, U ) arises as a small parameter δ = W/U 1. At leading order in δ, we ﬁnd from the scaled version of system (1)–(4) the dynamics and

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hydrostatic balance ∂H Hp dxh =v= , dt ∂v

(8)

∂H Hp ∂H Hp dv ˆ×v =− ˆ× = −θ∇h Π − f z , − fz dt ∂xh ∂v dθ = 0 and dt

(9)

0 = −θ∂Π/∂z − g

(10)

ˆ the unit vector in the vertical direction, and the hydrowith f = 2 Ω3 , z static parcel energy Hp (x, y, z, u, v, θ, t) = (u2 + v 2 )/2 + M (x, y, z, t)

(11)

with Montgomery potential M = θΠ(x, y, z, t)+g z. The vertical velocity dz/dt follows by insisting hydrostatic balance persists in time. We use these interim results next in the derivation of the conceptual 1 12 - and 2-layer models.

A 1 12 - and 2-layer atmosphere Birner et al. (2002) measured the vertical temperature proﬁles which suggest a conceptual model with an isentropic troposphere and an isentropic or isothermal stratosphere. We therefore simplify the stratiﬁcation of the atmosphere into an isentropic tropospheric layer and an isentropic or isothermal stratospheric layer, see Fig. 2(a). In this ﬁgure, we deﬁne the variables and constants used subsequently and denote their dependence, if any, on x, y and t. (a)

(b)

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fast f

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Figure 2. (a) Sketch of a simpliﬁed atmosphere with an isentropic troposphere and isentropic or isothermal stratosphere. p0 is a passive and constant pressure, and p1 and p2 are active pressures. (b) The slow manifold sketched has a third (or half) of the dimension of the entire Eulerian (or Lagrangian) phase space, with fast and slow variables f and s. Constraining forces, “the hand”, place the full dynamics on the manifold. When s and f are small, the dynamics is linear and separated in time such that f → 0, as it is sketched.

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Integrating hydrostatic balance θ∂Π/∂z + g = 0 in the tropospheric layer from z = z2 = hb to z with z2 < z < z1 , we obtain the Montgomery potential M2 (p2 ) = θ Π + g z = cp θ(p/pr )κ + g z = cp θ2 (p2 /pr )κ + g hb .

(12)

In an isentropic stratospheric layer one ﬁnds likewise, by integrating from z1 to z with z1 < z < z0 and using (12) at z = z1 with p(x, y, z1 , t) = p1 , that M1 (p1 , p2 ) = θΠ + gz = g(z0 − Z0 ) = cp θ(p/pr )κ + gz = cp (θ1 − θ2 )(p1 /pr )κ + cp θ2 (p2 /pr )κ + g(hb − Z0 ), (13) while in an isothermal stratospheric layer, one obtains similarly M1 (p1 , p2 ) = θΠ + gz = g(z0 − Z0 ) = cp θ(p/pr )κ + gz = RT T1 ln(p1 /p0 ) + cp θ2 ((p2 /pr )κ − (p1 /pr )κ ) + g(hb − Z0 ). (14) Note that, without any loss of generality, we have added a constant reference level Z0 to which we can ﬁx the top of the stratospheric layer z0 at a later stage. Any initial z-independence in each layer remains intact, so only two parcels in a vertical column of ﬂuid suﬃce for closure. Hence, the two-layer tropospheric-stratospheric model is the hydrostatic model (8)–(9) applied in each layer ∂H Hα dxα = vα = , dt ∂xα

(15)

dvα ∂H Hα ∂H Hα ˆ × vα − ∇h Mα = −f z ˆ× = −f z − dt ∂vα ∂xα

(16)

with α = 1, 2; x = (x, y)Tα and parcel energy Hα (xα , yα , uα , vα , t) = (u2α + vα2 )/2 + Mα (xα , yα , t).

(17)

Closure of these two-layer equations is reached via the layer pseudodensity σ0 (a, b)δ(x − xα (a, b, t))dadb (18) σα (x, y, t) = DH0

relating the horizontal label and position spaces with (using ∂p/∂z = −ρg) dm = ρdxdydz = −dxdydp/g d = dadbdc ∆m2 = σ2 dxdy = [(p2 − p1 )/g]dxdy = σ20 (a, b)dadb ∆m1 = σ1 dxdy = [(p1 − p0 )/g]dxdy = σ10 (a, b)dadb ∂y b − ∂y a∂ ∂x b. σα0 /σα = ∂x a∂

(19) (20) (21) (22)

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We emphasize that in each layer the Eulerian velocity is independent of depth, so vα = vα (x, y, t). Hence ∂z vα remains zero once it was initially so. Again, we can derive continuity equations ∂t σα + ∇h · (σα vα ) = 0 and a materially conserved PV: dQα /dt = 0 with ˆ · ∇ × vα )/σα . Qα = (f + z 1 12 -layer models. When the stratospheric layer is much deeper than the tropospheric layer, e.g., 45 km versus 15 km, we approximate the top z0 to Z0 and neglect the motion in the stratospheric layer. Thus, from (13) or (14) one ﬁnds M1 (p1 , p2 ) = 0. The stratospheric pressure p1 remains active, but the dynamics [(15) and (16) for α = 2] is evolved in the tropospheric layer. The bottom pressure p2 used to deﬁne M2 [(12)] in the tropospheric momentum equations (16) is then determined from σ2 = (p2 − p1 )/g and M1 (p1 , p2 ) = 0.

Static stability. Static stability means that a ﬂuid parcel perturbed in the vertical oscillates around a certain height with the Brunt-V¨ a¨is¨¨al¨ a frequency N rather than taking oﬀ. The eigen-values a of the 1 12 -layer equations [(15) and (16) for α = 2 with σ2 = (p2 −p1 )/g and M1 (p1 , p2 ) = 0] are (θ2 –θ1 -model) cp κ(θ1 − θ2 )(p1 /pr )κ−1 (p2 − p1 )/pr 2 a ∝ (23) κ−1 T1 /p1 − cp θ2 κ(p1 /pr ) ) (θ2 –T T1 -model). (p2 − p1 )(pr RT These eigen-values are real when the atmosphere is statically stable or dθ/dz > 0: this occurs when θ1 > θ2 in the θ2 -θ1 -model, and when T1 model. While the θ2 -θ1 model remains T1 > θ2 (p1 /pr )κ in the θ2 -T T1 statically neutral or stable if it is initially so, the stability of the θ2 -T model thus depends on p1 (x, y, t).

2.

Linear Modes

Linearized around a “rest depth” H with σ = σ2 = H(x, y) + η, the 1 12 -layer models [(15) and (16) for α = 2] become ∂t v = −f zˆ × v − g ∇h η

and ∂t η + ∇h · (Hv) = 0

(24)

with velocity v = v2 and eﬀective gravity g . This linearized system is akin to the classical, linearized shallow water equations with (for hb = 0) ⎧ (θ1 −θ2 )P P1κ−1 ⎪ ⎨ (θ2 –θ1 -model) κ−1 (θ −θ )P P +θ2 P2κ−1 1 2 1 (25) g ∝ (P P1 /pr )κ−1 ⎪ ⎩ pr RTT1 /PP1 −cp κθ2κ−1 (θ –T T -model) 2 1 κ−1 RT T /P P +c κθ (P P −P P ) 1

1

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18 16 14

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Figure 3. model.

Dispersion relation of linear modes in a periodic channel for the θ2 -θ1 -

and the rest state H = (P P2 − P1 )/g, which is consistent with (23). In a periodic channel with constant H, the linear modes using η ∝ ei(kx+ωt) (i2 = −1) consist of: a vortical or geostrophic mode with frequency ω = 0, Poincar´ ´e GW modes with ω 2 = f 2 + g H (k 2 + l2 ), and counterclockwise propagating boundary-trapped Kelvin modes with ω 2 = g H k 2 (for constant f > 0). We observe in the dispersion diagram in Fig. 3 that there is a time-scale separation between the vortical and GW modes, except perhaps for the lowest-order Kelvin modes and the geostrophic solution. As usual, a linear mode analysis is limited in scope. First, the dynamics is nonlinear, so there is no clear notion of a time scale separation anymore. Nonlinear “slow” dynamics can have high-frequency overtones triggering resonances or interactions with “fast” dynamics. Second, approaching the equator, the eﬀective Coriolis parameter f → 0 , giving rise to equatorial Kelvin waves and mixed Rossby-gravity waves or mixed slow-fast linear modes. Consequently, linear Kelvin or gravity mode solutions of larger amplitude used as initial condition, can develop vortex motion and, vice versa, linear geostrophic or Rossby modes can develop GW motion from instabilities. Mixed fast-slow motion emerges in simulations, see Fig. 4, of the nonlinear evolution of a linear Kelvin mode solution in a zonally periodic channel used as initial condition. In particular PV is constant (in time and/or space) for a Kelvin or Poincar´e mode, ˆ · ∇ × v)/σ = f /H Q2 = Q = (f + z

and ∂t Q + (v · ∇h )Q = 0, (26)

before the occurrence and parameterization of wave breaking. These constant PV regions are then distinguished, ideally, from regions where

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Figure 4. Top: contour plots of σ in simulations of the 1.5-layer θ2 -θ1 -model with shock-capturing numerics which, however, do not conserve absolute vorticity σQ. Bottom: simulations displaying σ (left) and 100× PV anomaly (Q − f /H) (right) at the ﬁnal time. Non-dimensional quantities are displayed, for example in a domain of 4000 × 2000 km with z1 ≈ 15 km and Z0 ≈ 60 km.

a wave breaking parameterization generates non-constant PV anomalies Q − f /H (cf., Peregrine and Bokhove, 1998).

3.

Balanced Dynamics

The concept of balanced large-scale ﬂow arises from the observation that at mid-latitudes the atmosphere and oceans are in approximate geostrophic balance, and near the equator the Earth’s rotation remains inﬂuential. Locally — due to topography, strong (tropical) convection, dissipative and non-dissipative (GW) instabilities — balance often fails. The notion of balance may be formalized in various ways: small Rossby and Froude numbers are identiﬁed from measurements, observations or simulations, and then used in scaling arguments. Subsequently, a perturbative or iterative approach is applied to approximate the full or parent

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model. The resulting dynamics evolves on a slow or slaving manifold of reduced dimensionality, see the sketch in Fig. 2(b). The preservation of certain conservation laws, or the variational or Hamiltonian structure, may be imposed heuristically in these balanced approximations. Whether the conservative or non-conservative approach to balanced dynamics is better, remains undecided and depends on the, perhaps subjective, value placed on (point-wise) accuracy, and long-term stability. Geostrophic balance denotes the alignment of the wind vectors along the pressure or Montgomery potential isobars. To derive this leading order balance, we rewrite the 1 12 -layer equations [(16) for α = 2] and drop the layer subscripts f 1 f ∂H 1 ∂H dui = ij uj − ∂xi M = ij − dt R R R ∂ui R ∂xi

(27)

with the permutation symbol ij , v = (u1 , u2 )T and i, j = 1, 2. The Rossby number R = U/(f L) 1 is placed in (27) at the relevant locations, as the ratio of the GW time scale 1/f and the vortical time scale L/U with typical length and velocity scales L and U . At leading order in R, we ﬁnd geostrophic balance from (27) as a constraint on the velocity with M/f being a stream function in the balance relations u = −∂ ∂y M/f and v = ∂x M/f . In general, (higher-order) velocity constraints obtain the form e.g.

φi = ui − uC i [σ(x)] = ui +

1 ij ∂xj M, f

(28)

in which uC [σ] operates (non-locally) on σ and, hence, through σ on the parcel coordinates x and y. Next, we use these constraints to derive balanced models.

Conservative balanced models: slaved Hamiltonian approach We illustrate the derivation of Hamiltonian balanced models in the hybrid parcel framework. The variables (xi , ui ) are transformed to (xi , φi ) using (28), and a constrained variational derivative is introduced $ ∂H ∂H ∂uC ∂H $$C j = + , $ ∂xi ∂xi ∂uj ∂xi

(29)

where (·)|C denotes that φi = 0 in derivatives of x and y. The evolution on the slow manifold of reduced dimensionality becomes, using (15)

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and (16), dxi ∂H = = ui dt ∂ui

and

0=

∂H C dφi =− + ij σQC uj dt ∂xi

(30)

with σ QC = f + ∂v C /∂x − ∂uC /∂y. The slaved Hamiltonian dynamics on the slow manifold is concisely written as ∂H C dxi = (L−1 )ij dt ∂xj

or

dF C ∂F C −1 ∂H C = (L )ij dt ∂xi ∂xj

(31)

[cf. Dirac (1958)] with skew-symmetric matrix Lij = ij σQC and arbitrary function F C = F C (x, y) and H C = H(x, y, uC , v C ). Simpliﬁed numerical integrations are explained in Fig. 1(b). It is unclear whether the parcel balanced dynamics (31) presented is a didactic simpliﬁcation, or equivalent to the results for the Eulerian balanced equations in Vanneste and Bokhove (2002).

4.

Wave-Vortex Interactions and Numerical Schemes

The parameterization of unresolved gravity waves is a critical component in numerical GCMs. Gravity waves can inﬂuence the large-scale dynamics in various ways: (i) breaking gravity waves dissipate energy to small scales and deposit momentum to drive the mean, large-scale ﬂow (McFarlane, 1987); (ii) instabilities of balanced vortical ﬂows locally excite gravity waves, which can transport energy and momentum away (Vanneste and Yavneh, 2004); and (iii) non-dissipative wave-vortex interactions, such as remote recoil, can lead to a cumulative forcing of the mean vortical ﬂow (B¨ u ¨hler and McIntyre, 2003). The crucial question is how to parameterize these unresolved GW-eﬀects, studied hitherto in isolation, in numerical models for large-scale ﬂows on advective time scales, given the resolved large-scale ﬂow. New numerical schemes have emerged with a focus on improved meshes without pole problem, conservation properties and advection-dominated time integration. Based on gas dynamics and novel ﬁnite-element discretizations (Bokhove, 2005; Fig. 4), an impulse formulation of the θT -model with 3 prognostic equations can be used, which are shockcapturing but with explicit time stepping limited by the largest GW speed. In atmospheric dynamics, the velocity formulation with 3 prognostic equations is often preferred (Ringler and Randall, 2002). The GW speed is then still the limiting factor. Hamiltonian Particle Mesh methods (HPM) involve (15)–(18) with 4 prognostic equations and 1 integral equation (Frank and Reich, 2003). By smoothing the pseudo-density,

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time step restrictions can be lifted. To emphasize the vortical dynamics, a mass-divergence-vorticity formulation is used by Thuburn (1997) and Ringler and Randall (2002) resulting in 3 prognostic and 2 elliptic equations. The advective time step is then used after some numerical stabilization. The approaches by Ringler and Randall conserve mass, energy, potential enstrophy and vorticity. Mass or PV conserving balanced models consist of a prognostic equation and 2–4 elliptic equations of the ﬁrst and second order. These elliptic inversions are time consuming and require special (multi-grid) techniques.

5.

Conclusions

The HPM and related semi-Lagrangian numerical schemes, as well as the ones using vorticity-divergence variables (Frank and Reich, 2003; Thuburn, 1997; and Ringler and Randall, 2002) seem to be most advantageous as they use the larger advective time step, at the expense of introducing an artiﬁcial numerical GW-vortex parameterization. It may be a good strategy to test GW-vortex parameterizations in both the balanced models and high-resolution (in space and time) primitive equations. Otherwise, it is unclear to what extent the (artiﬁcial) numerical GW parameterizations in the numerical schemes jeopardize the physical ones. Clearly, the potential interplay between physical and (hidden) numerical parameterizations of gravity waves is a research question with important implications for GCMs. Finally, a thorough answer to the initial question whether a balanced model can provide accurate climate predictions needs to be postponed, although Olaguer’s (2002) results seem to be encouraging.

Acknowledgments The criticism of J. Frank and B.J. Geurts has been much appreciated. The θ-T -model originates from an unpublished work with W.T.M. Verkley.

References T. Birner, A. D¨ ¨ ornbrack and U. Schumann. How sharp is the tropopause at midlatitudes? Geophys. Res. Lett. 29: 10.1029, 2002. O. Bokhove. Flooding and drying in ﬁnite-element discretizations of shallow-water equations. Part 1: One dimension. J. Sci. Comput. 22, To be published, 2005. O. Bokhove and W.T.M. Verkley. Constrained isentropic models of tropospheric dynamics. Submitted to Quart. J. Roy. Meteor. Soc., 2004. ¨ and M.E. McIntyre. Remote recoil: a new wave-mean interaction eﬀect. J. O. Buhler Fluid Mech. 492, 207–230, 2003.

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P.A.M. Dirac. Generalized Hamiltonian dynamics. Proc. Roy. Soc. Lond. A 246, 326– 332, 1958. M. Dixon and S. Reich. Symplectic time-stepping for particle methods. GAMM, M to appear, 2004. J. Frank and S. Reich. Conservation properties of smoothed particle hydrodynamics applied to shallow water equations. BIT 43, 40-54, 2003. N.A. McFarlane. The eﬀect of orographically excited gravity wave drag on the general circulation of the lower stratosphere and troposphere. J. Atmos. Sci. 44, 1775– 1800, 1987. E.P. Olaguer. An eﬃcient 3-D model for global circulation, transport and chemistry. IMA Vol. Math. Appl. 130, 205–276, 2002. D.H. Peregrine and O. Bokhove. Vorticity and surf zone currents. Proceedings of the 26th International Conference on Coastal Engineering 1998, ASCE, Copenhagen. 745–758, 1998. T.D. Ringler and D.A. Randall. A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations on a geodesic grid. Mon. Wea. Rev. 130, 1397–1410, 2002. J. Thuburn. A PV-based shallow-water model on a hexagonal-icosahedral grid. Mon. Wea. Rev. 125, 2328–2347, 1997. J. Vanneste and O. Bokhove. Dirac-bracket approach to nearly-geostrophic Hamiltonian balanced models. Physica D 164, 152–167, 2002. J. Vanneste and I. Yavneh. Exponentially small inertia-gravity waves and the breakdown of quasi-geostrophic balance. J. Atmos. Sci. 61, 211–223, 2004.

NEAR-CRITICAL POINT HYDRODYNAMICS AND MICROGRAVITY Daniel A. Beysens CEA, Service des Basses Temp´ ´eratures, Grenoble & ESPCI, PMMH, 10, rue Vauquelin, 75015, Paris Cedex 05, France [email protected]

Abstract

Near their critical point, ﬂuids exhibit anomalous behavior of thermodynamic parameters (divergence of speciﬁc heat, compressibility and expansion coeﬃcients) and transport coeﬃcients (heat conductivity, thermal diﬀusivity). Weightlessness (”microgravity”) environment permits to go very close to the critical point, thus allowing key tests of the Renormalization Group theory to be made. It also results in a very particular hydrodynamics of dense and hyper-compressible gases, where weightlessness experiments play a key role. For instance, a very fast thermalization eﬀect (”Piston eﬀect”) is evidenced, where a thermal boundary layer expands and adiabatically heats the whole ﬂuid, leading in some cases to an (apparent) violation of the laws of thermodynamics. Another one is concerned with the use of critical slowing down and microgravity to investigate the dynamics of phase separation with no gravity-induced sedimentation. The key role of the coalescence of domains makes valid only two simple growth laws; they can be successfully applied to a quite diﬀerent situation, the evolution laws in the well-known biological problem of sorting of the embryonic cells. Other situations are concerned with the eﬀect of vibrations. The investigation of the above thermal and phase transition problems suggest that a periodic excitation can act as a kind of artiﬁcial gravity, which induces thermal convection, speeds up phase transition and localizes the liquid and vapor phases perpendicular to it. Some of these phenomena still persist at higher temperature and pressure. Fluids in such supercritical conditions are very appealing to the industry as non-polluting solvents or hosts of chemical reactions with high yield.

Keywords: Critical point, supercritical ﬂuids, phase transition, thermalization, cell sorting, Piston eﬀect, microgravity, vibrations, cell sorting

117 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 117–130. © 2005 Springer. Printed in the Netherlands.

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Introduction

It is widely believed that a ﬂuid can exist only as a gas or a liquid. However, there is another state, the ”supercritical state”, that ﬂuids can exhibit (Fig. 1a). Since Baron Charles Cagniard de La Tour discovered in 1821 that the liquid and gas phases of a carbon dioxide sample became undistinguishable after crossing a ”critical” temperature of 31◦ C and a ”critical” pressure of 72 bar, the intriguing properties of this very particular state has motivated a great number of studies. The critical point co-ordinates vary according to the particular ﬂuid under study. For instance, the critical point of carbon dioxide (CO2 ) is observed at 31◦ C and 72 bar, that of water (H2 O) at 375◦ C and 225 bar and that of hydrogen (H2 ) at 33 K and 13 bar. Above the critical temperature and pressure, supercritical ﬂuids exhibit a number of speciﬁc properties (large density, low viscosity, large diﬀusivity) which make them intermediate between liquids and gases [1]. In addition, their isothermal compressibility and thermal expansion can become very large, especially when they approach the critical point. The highly variable properties of near-critical ﬂuids make them very attractive for studying many phenomena that hold for all ﬂuids because of the critical universality. Supercritical ﬂuids are increasingly used by the food and waste management industry [3] for their solubilization properties (e.g. supercritical CO2 ), as host of ”cold” combustion (e.g. supercritical water), in energetics (supercritical thermal or nuclear plants), and in astronautics (e.g. storage of cryogenic ﬂuids).

Figure 1. (a): Phase diagram of a pure substance. The supercritical ”state” corresponds to a compressed gas that exhibits the density of a liquid. (b): Critical anomaly of the speciﬁc heat at constant volume (C Cv ) measured under zero-gravity in SF6 (Spacelab D2, 1993). (From [2]).

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Near-critical Fluids and Microgravity Fluids in their near-critical or supercritical state are strongly aﬀected by gravity. As they are highly compressible, gravity compresses them under their own weight. The density varies in the sample, thus preventing a close approach to the critical point to be made, cf. [4]. They exhibit anomalies in the transport of heat so that convection and buoyancy phenomena, often turbulent, appear for even minute temperature gradients. We show in the following that space experiment have enabled new phenomena to be discovered thanks to a close approach to the critical point and the removal of convection and buoyancy. The eﬀects of gravity can be removed in space thanks to spacecrafts and satellites. On earth, microgravity conditions can be obtained during a short time period in a free fall tower (a few s) and in parabolic ﬂights of planes (20 s) and sounding rockets (2–12 min.). Some other means can be used with liquid mixtures. Liquid mixtures near their consolute critical point exhibit a number of common features with pure ﬂuids, when concentration is replaced by density as an order parameter of the transition. Some aspects can be then studied with binary liquid mixtures that have been made density-matched by partial deuteration [5]. The compensation of gravity forces in simple ﬂuids is more diﬃcult. The large diamagnetism susceptibility of H2 was used to compensate gravity by a magnetic force, proportional to its density. By using a superconducting coil it became possible to study, without gravity eﬀects, the process of solidiﬁcation and gas-liquid phase transition over a wide range, from the triple point (13.8 K) to above the critical point (33 K) [6]. A vibration device has also been implemented using a special sapphire sample.

2.

Universality and Scaling Laws

An important aspect of the critical region is that most of the anomalies of the thermodynamic and transport properties can be set in the form of scaled, universal (power law) functions with respect to the critical point (CP) coordinates. Then, any results obtained with one single ﬂuid can be immediately generalized to a whole class of systems, the ”class of ﬂuids”, to which belong also liquid mixtures, including polymer melts and solutions, microemulsions, molten salts, monotectic liquid metals, etc. [1]. This scaling is of fundamental nature and stems from the universal behavior that the free energy must asymptotically obey to fulﬁll the conditions of a 2nd order phase transition – the CP. In a sense, scaling is generic to CP phenomena. By permitting the measurements to be extremely close to the critical point, zero-g experiments

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have made possible the precise measurements of important, weak powerlaw divergence, such as that of the speciﬁc heat at constant volume Cv (Fig. 1-b). From space experiments, the temperature divergence of the speciﬁc heat has been determined with a very high precision. With the Tc (T is temperature, Tc is the abreduced temperature τ = (T − Tc )/T solute critical temperature), the speciﬁc heat diverges as Cv ∼ τ −α near the critical point. The ”critical” exponent α is universal. Its precise determination was a key test of the ’Renormalization Group’ theory, which has been developed in order to try to improve the classical macroscopic description of ﬂuid behavior close to the critical point [7]. The value deduced from the space experiments, α = 0.1105 ± 0.027 [2], indeed appears to be very close to the result of the Renormalization Group theory, α = 0.110 ± 0.005.

3.

Phase Transition

Kinetics and Morphology Let us describe a typical phase separation experiment (Fig. 2a) and what has been learned from the space experiments [9]–[10]. The supercritical ﬂuid is thermally quenched from a region of the phase diagram where it is homogeneous (at temperature Ti ) to a region where it is thermodynamically stable as two phases (at temperature Tf ). Droplets nucleate and their development is limited by coalescence events. When the volume fraction of the new phase that has nucleated is low (Fig. 2b), the droplets collide by Brownian motion and coalesce. The average radius of the drops R or the average distance Lm between them evolves as Rφ−1/3 ∼ Lm = 2π(kB T /6πη)t1/3 ,

(1)

where t is time, η is the shear viscosity, T is absolute temperature and kB is the Boltzmann constant. It is worth noting that this law is practically independent of the distance (T − Tc ) from the critical point. To a given φ corresponds a typical interaction length (≈ R/3) between domains. When φ > 0.3, it is found [10] that the ﬂow generated by a coalescence event is able to move a neighboring drop and thus induces another coalescence. Such a process therefore creates a chain reaction of coalescence. In the viscous limit, the pattern looks to be interconnected. Growth is limited by the balance between the capillary pressure gradient σ/R (σ is the gas-liquid surface tension) and the friction due to the shear viscosity, so that (late times) Lm = b(σ/η)t.

(2)

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Figure 2. (a): Schematic phase diagram for simple ﬂuids and liquid mixtures in the plane T − M . CP: critical point. T : temperature. M : order parameter; M = (ρ/ρc ) − 1 for simple ﬂuids and M = c − cc for liquid mixtures. ρ(ρc ) is density (critical density). c (cc ) is concentration (critical concentration). The coexistence curve is described by M = Bτ β with B being a system-dependent amplitude and β = 0.325 an universal exponent. (b): Growth laws when gravity eﬀects are absent. Fluids (SF F6 , CO2 ): all data points except open squares. Liquid mixtures (partially deuterated Cyclohexane and Methanol): letters and open squares. The evolution of the average distance between the domains (Lm ) is expressed in the scaled units ∗ = 2πξ/Lm and t∗ = t/tξ (see text). Lines are theoretical predictions. The Km lower curve (φ > 0.3) corresponds to a ” fast ” growth law and an interconnected morphology (pattern in insert b1) and the reduced upper curve ( φ < 0.3) refers to a ” slow ” growth and a disconnected morphology (pattern in insert b2). (From Ref. [9]).

Here b ≈ 0.03 is a universal constant. All experiments can be rescaled by the natural lengthscale and timescale: the correlation length ξ of density ﬂuctuations (ξ diverges as τ −ν ), and the associated diﬀusion time tξ (tξ diverges as τ −3ν ) .

Application to Biological Tissues The development of domains by coalescence events, as reported above, is very general. The universality of behavior, which is observed in ﬂuids and liquid mixtures, can be extended to other areas of science. In particular, it can be applied to developmental biology where tissues can be considered as very viscous liquids (viscosity η ≈ 106 Po), with a surface tension arising from the balance of adhesion sites between the tissue cells (eﬀective interfacial tension σ ≈ 10 dyn.cm−1 ). When analyzing both the kinetics and morphology of cell sorting in embryonic chicken tissues, [11](Fig. 3), the development of the pattern can be interpreted as

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Figure 3. Upper sequence: gas and liquid phase ordering in SF6 under reduced gravity, after a thermal quench to Tc − 0.7 mK. Cell diameter: 12 mm. Gas and liquid eventually order with the liquid phase wetting the container wall and surrounding the gas phase, corresponding to σwl < σwg . Here σwl and σwg are the wall-liquid and wallgas interfacial tensions, respectively. The three pictures from the left correspond to states at 120 s, 275 s and 3960 s after quench, respectively. Lower sequence: sorting out of chicken embryonic pigmented epithelial cells (dark) from chicken embryonic neural retinal cells (light). Aggregate size: 200 µm. At the end of sorting, neural retinal cells preferentially wet the external tissue culture medium surrounding the aggregates. Here σtn (=1.6 dyn/cm)< σtp (=12.6 dyn/cm), where σtn and σtp are the tissue culture medium-neural retina and the tissue culture medium-pigmented epithelium interfacial tensions, respectively. The three pictures from the left correspond to 17 h, 42 h and 73 h after initiation of sorting, respectively. (From [11]).

a result of the coalescence of domains that rearrange like droplets. The domains continuously coalesce and form a network, showing the same linear evolution as the fast growth in liquids. The pseudo-period between domains can indeed be ﬁtted to a linear growth law Lm = bt. The value of parameter b turns out to be comparable to that extracted from Eq. (2), although surface tension and viscosity values diﬀer by factors as large as 108 .

4.

Thermalization

The ”Piston” Eﬀect The thermal diﬀusivity of ﬂuids vanishes near the critical point and a simple calculation [12] shows that it would need more than one month to reach thermal equilibration in a sample of 1 cm3 at T − Tc = 1 mK.

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At the beginning of the microgravity experiments, it thus seemed hopeless to the scientists to try to homogenize in temperature and density a sample of ﬂuid close to the critical point in a reasonable experiment time. However, in 1986, a preliminary experiment by Straub in a sounding rocket [8] indicated that thermalization might be much faster than expected. In a workshop in 1989 on thermal equilibration near the critical point, Onuki pointed out the importance of ”adiabatic heating”. We proposed a hydrodynamic mechanism of thermalization that was based on hydrodynamics where, at the sample wall the hot diﬀuse boundary layer expands and compresses adiabatically the whole ﬂuid. Thermalization proceeds at the velocity of sound. As a result, a spatially uniform heating of the bulk ﬂuid should be observed. There should be a real ﬂow at the boarder between the bulk ﬂuid and the expanding diﬀuse layer, later observed under microgravity as real ”jet” ﬂows. Onuki and Ferrell and the Gammon team proposed a similar process, an ”adiabatic heating”. In the Physical Review A issue of December, 1990 [12]-[14] Onuki and Ferrell, the Gammon team and our group exposed their views on the subject and all concluded a fast thermalization by this eﬀect. Other reports (Meyer in [15]) came out nearly at the same time. We coined in [12] this adiabatic heating phenomenon the ”Piston Eﬀect” (PE), a name which has been accepted by the scientiﬁc community.

Can Heat Flow Backwards? This eﬀect is at the origin of a very particular behavior [17] when the vapor is in equilibrium with liquid below the critical point (Fig. 4b). While heating the cell, the temperature of the vapor becomes greater than that of the wall. That heat ﬂow could seemingly ﬂow from cold to hot contradicts the laws of thermodynamics. However, as we are here in presence of a thermo-mechanical conversion where the hot boundary layer compresses more the gas than the liquid, the violation is only apparent. These results concerning the PE have been adapted and modiﬁed to the earth’s environment, where gravity couples to the PE-induced ﬂows and the geometry of the phases. Paradoxical phenomena, such as the cooling of the bulk ﬂuid after a heat pulse, have also been obtained [18]. Also, the accelerations of the shuttle have been used (rotation and maneuvering the shuttle) to investigate the eﬀect of density destratiﬁcation, a study performed with Air Liquide Company to validate the codes that are now used in the pressurization of the reservoirs of the Ariane 5 rocket.

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(a) L

δ

ρ

T

T

Fast & homogeneous F The «Piston»:

ρ

Figure 4. (a): The Piston Eﬀect mechanism [12]-[16]: a thin hot boundary layer expands and compresses the bulk ﬂuid. The corresponding temperature proﬁle shows a thin zone of strong gradients near the heated boundary (thermal boundary layer δ) and a homogeneous rise in the rest of the ﬂuid, that settles at the speed of sound. (b): overheating of nearly 20% obtained in the gas phase of a SF F6 sample at 10 K below the critical point (ALICE in MIR, 1999). A temperature rise of Tw = 0.1K is TG ), that of the imposed at the cell wall. The temperature evolution of the gas ( δT liquid ( δT TL , at two locations) are shown. In the insert is reported the sample with the thermistors. The fact that heat can ﬂow from “cold” to “hot” apparently violates thermodynamics. It is a spectacular demonstration of the reality of the “piston” in the “Piston Eﬀect”. (From Ref.[17]). 0.06

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Figure 5. Yield Y = transmitted power/ incident power vs time (s) at various Tc = 33 K). At time t0 , power (7.5 mW) is sent for 300 s at one T − Tc . Fluid is H2 (T end of the cell. The transmitted power is measured at the other end. The conduction Ywall by the wall of the sample cell (curve ”wall”) has been subtracted.

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Since temperature outside the thermal boundary layers is homogeneous, the bulk ﬂuid acts as a thermal short circuit. A question arises whether it is possible to use this Piston eﬀect as a kind of heat pipe to carry heat on long distances. Experiments and simulation have been performed with H2 [19] in a magnetic gravity compensation set-up [6]. The results (Fig.5) show that the heat transfer is very fast, in contrast to the usual conduction process. In particular, the yield Y (see Fig. 5 caption) shows a ﬁnite slope at initial time, in contrast with conduction in the cell walls where the slope is zero. The maximum yield increases when T approaches Tc . Its value corresponds to the heat transfer in the steady conduction state (at inﬁnite time).

5.

High Frequency Vibrations

At high frequency excitation - i.e. frequencies larger than the inverse typical hydrodynamics times - the time average of the Bernoulli pressure, which is proportional to the ﬂuid square velocity, is non-zero. The pressure gradient that appears in a non-homogeneous ﬂuid can thus induce ﬂows perpendicular to the vibration direction. However, at low frequency, vibration acts by its instantaneous acceleration and can induce ﬂows parallel to the vibration (as usually gravity does). In the following, a will denote the vibration amplitude, f the frequency and ω = 2πf the angular frequency.

Vibrational Thermal Eﬀects When a ﬂuid is submitted to a vibrational acceleration in a thermal gradient in the Rayleigh-B´ ´enard conﬁguration, convection is able to start at conditions corresponding to a vibrational Rayleigh number [20] Rav =

[aω(∂ρ/∂T )p ∆T e]2 2ηD

(3)

larger than a few thousands. Here ∆T is the temperature diﬀerence between two ﬂuid layers separated by the distance e and D is the thermal diﬀusivity coeﬃcient. As the ﬂuid temperature becomes closer to the critical temperature, Rav diverges as (T − Tc )−1.9 . The ﬂuid then becomes extremely sensitive to vibration as the critical point is approached. Measurements of ﬂow velocities performed in the MIR station in CO2 and in SF6 conﬁrm this expectation [22]-[23] (Fig. 6). A heat ﬂux was sent into the ﬂuid from a point-like source (thermistor). Depending on the oscillation velocity, two regimes of heat propagation are observed: (i) at low frequency, heat is convected during one oscillation period to form plumes parallel to vibration (Fig. 6b); (ii) at high frequency, heat

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c

ω

ω

Figure 6. Interferometer image of the hot boundary layer around a heating thermistor T h1, supported by a thread. Fluid: SF F6 at Tc +0.5 K. (a) no vibration (hot region underlined in white); (b) under low frequency; (c) under high frequency vibration. Here convection rolls form. (ALICE in MIR, 1999, from [27]).

is convected by convection rolls perpendicularly to the direction of oscillation (Fig. 6c). A numerical simulation and analysis of the convection has been performed by Jounet in [21], emphasizing the role of vortices.

Vibrational Phase Ordering A plane liquid-vapor layer vertically vibrating parallel to gravity displays two diﬀerent regimes [24]. Far from the critical point, a square wave-pattern deformation arises (the usual Faraday instability). At Tc − T0 ≈ 20 mK a temperature T0 close to the critical temperature (T for CO2 ), a transition to a new pattern comprised of lines occurs. This transition is due to the increase of dissipation near the critical point. This is a rather unique example of a strong coupling between two diﬀerent critical point phenomena: the critical point of interface instability and the thermal critical point of the liquid-vapor phase transition. When acceleration is perpendicular to gravity, a Kelvin-Helmholtzlike instability is observed [25], with the interface modulated as a ”frozen” roll wave pattern (Fig. 7a). The mechanism of the instability results from the relative motion of the two ﬂuids induced by vibration. A perturbation becomes unstable if the cell velocity (aω) is larger than the threshold velocity [26] (ρL + ρG )3 σg . (4) (aω)0 = ρL ρG ∆ρ ∆ρ Here ρL (ρV ) is the liquid (vapor) density and ∆ρ = ρL −ρG is the liquidvapor density diﬀerence. This destabilization is due to the increasing eﬀect of the Bernoulli - type pressure arising from the velocity diﬀerence between gas and liquid.

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Figure 7. (a-b): CO2 gas-liquid phases show up under 1-g as gas phase (G) above the liquid phase (L) separated by a ﬂat meniscus. When submitted to vibration, the phases can order in a diﬀerent way: (a) under 1 − g, the interface exhibits ”frozen waves”; (b) under zero-g, gas-liquid phases order in layers. Cell diameter: 10 mm (sounding rocket Maxus 5, April, 2003). (c): Typical phase separation in H2 at T c − 1.06 mK with a = 0.3 mm and f = 20s−1 . From [28].

Three samples of diﬀerent gas volume fractions and various T −T Tc were vibrated at several amplitudes and frequencies ranging from 0.1 mm to 5 mm and 0.1 Hz to 60 Hz, respectively [27]. Although the initial state of the sample was either an emulsion of vapor drops, or a single drop, the ﬁnal state remains the same: vapor and gas phases are forming alternate layers perpendicular to the direction of acceleration. An instability, similar to Kelvin-Helmotz’ with inviscid, zero surface tension, develops as liquid ﬁngers from the cell walls. The ﬁngers coalesce with the droplets in the bulk and/or with the ﬁngers that have grown from the opposite side. Here the viscous boundary layer λ = (2η/ρω)1/2 is the natural lengthscale of the phenomenon.

Phase Transition under Vibration A study of phase separation was performed in H2 under magnetic compensation of gravity (Fig. 7c), for volume fraction φ > 0.3 [28]. The domains are interconnected. When Lm is lower than the viscous boundary layer λ, liquid and gas domains have the same velocity and their growth is unaﬀected. From Eq. (2), the growth velocity U = dLm /dt = bσ/η. When Lm > λ, the domains exhibit diﬀerent velocities whose diﬀerence is proportional to the gas-liquid diﬀerence. The corresponding shear ﬂow

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between the domains speeds up the growth (see Fig. 7c where the crossover is denoted L0 ) as U /U = 1 + ∆U/U , with ∆U/U ∼ (∆ρ)(a/λ). These observations suggest that a periodic excitation can act as a kind of artiﬁcial gravity, which speeds up phase transition and localizes the liquid and vapor phases perpendicular to it.

6.

Conclusion

Supercritical ﬂuids are of both the fundamental interest (universality of phase transition, supercritical hydrodynamics) and industrial interest (supercritical solubilization, nucleation of nanomaterials, oxidation, thermalization, storage). The ﬁeld of Critical Point Phenomena has achieved some major breakthroughs during the last 15 years thanks to microgravity research; in particular, a new thermalization process has been discovered, the ”Piston Eﬀect”, that reveals novel hydrodynamics in such near-critical ﬂuids. The study has strongly modiﬁed our vision of critical point phenomena and even of hydrodynamics: the very unusual hydrodynamics of these supercritical ﬂuids, compressible, dense, and weakly viscous, makes their behavior quite particular when compared to gas or liquids. Future experiments will certainly lead to the discovery of new and unexpected phenomena that will be of interest for both the fundamental and applied science.

Acknowledgments This review has been made possible thanks to the friendly help and contribution of Y. Garrabos, B. Zappoli, J. Hegseth, P. Evesque and V. Nikolayev. The ﬁnancial support of CNES is gratefully acknowledged.

References [1] See e.g. H.E. Stanley, Introduction to phase transitions and critical phenomena, Clarendon Press, Oxford, New York, 1971; D. Beysens, J. Straub, D. Turner, [in:] ”Fluid Sciences and Materials Science in Space”, H.U. Walter [Ed.], pp.221–256, Springer, Berlin, 1987. [2] A. Haupt, J. Straub, Phys. Rev. E 59, pp.1795–1802, 1999. [3] F. Cansell, P. Beslin, B. Berdeu, Environmental Progress 17, pp.258–263, 1998; S. Yesodharan, Current Science 82, pp.1112–1122, 2002. [4] M.R. Moldover, J.V. Sengers, R.W. Gammon, R.J. Hocken, Rev. Mod. Phys. 51, pp.79–99, 1979. [5] D. Beysens, Acta Astron. 12 525-530, 1985; C. Houessou, P. Guenoun, R. Gastaud, F. Perrot, D. Beysens, Phys. Rev. A 32, pp.1818–1833, 1985. [6] R. Wunenburger, D. Chatain, Y. Garrabos, D. Beysens, Phys. Rev.E 62, pp.469– 476, 2000; see http://www.spaceflight.esa.int.

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[7] K.G. Wilson, J. Kogut, Phys. Reports C 12, 75, 1974. [8] K. Nitsche, J. Straub, Naturwissenschaften 73, 370, 1986; J. Straub, L. Eicher, A. Haupt, Phys. Rev. E 51, pp.5556–5563, 1995. [9] D. Beysens, Y. Garrabos, Physica A 281, 361-380, 2000 and refs. therein. [10] V. Nikolayev, D. Beysens, P. Guenoun, Phys. Rev. Lett. 76, pp.3144–3147, 1996; V. Nikolayev, D. Beysens, Physics of Fluids 9, pp.3227-3234, 1997. [11] D.A. Beysens, G. Forgacs, J.A. Glazier, P.N.A.S. 97, pp.9467-71, 2000; ibid., Networks of droplets induced by coalescence: application to cell sorting, in: Dynamical Networks in Physics and Biology, D. Beysens and G. Forgacs [Eds.], pp.161–169, Springer & EDP Sciences, Berlin & Les Ulis, 1998. [12] B. Zappoli, D. Bailly, Y. Garrabos, B. Le Neindre, P. Guenoun, D. Beysens, Phys. Rev. A 41, pp.2264–2267, 1990. [13] A. Onuki, H. Hao, R.A. Ferrell, Phys. Rev.A 41, pp.2255-2259, 1990; A. Onuki, R.A. Ferrell, Physica A 164, pp.245–264, 1990. [14] H. Boukari, J.N. Shaumeyer, M.E. Briggs, R.W. Gammon, Phys. Rev. A 41, pp.2260–2263, 1990. [15] R.P. Behringer, A. Onuki, H. Meyer, J. Low Temp. Phys. 81, pp.71–102, 1990. [16] See e.g. Y. Garrabos, M. Bonetti, D. Beysens, F. Perrot, T. Fr¨¨ohlich, P. Carl´es, B. Zappoli, Phys. Rev. E. 57, pp.5665-5681, 1998. [17] R. Wunenburger, Y. Garrabos, C. Chabot, D. Beysens and J. Hegseth, Phys. Rev. Lett. 84, pp.4100–4103, 2000; M. Sincell, Science 288, pp.789–791, 2000. [18] T. Fr¨ ¨ olich, D. Beysens, Y. Garrabos, V. Nikolayev, A heat ﬂux can cool a nearcritical ﬂuid, preprint, 2004. [19] D. Beysens, D. Chatain, V. Nikolayev, Y. Garrabos, 4th International Conference on Launcher Technology Space Launcher Liquid Propulsion, 3-6 December 2002 – Liege, Belgium. [20] G.Z. Gershuni, D.V. Lyubimov, Thermal Vibrational Convection, John Wiley & Sons, New-York, 1998. [21] A. Jounet, Phys. Rev. E, 65, pp.37301–37304, 2002 [22] S.V. Avdeev, A.I. Ivanov, A.V. Kalmykov, A.A. Gorbunov, S.A. Nikitin, V.I. Polezhae, G.F. Putin, A.V. Zuzgin A.V., V.V. Sazonov, D. Beysens, Y. Garrabos, T. Frohlich, ¨ B. Zappoli, Experiments in the far and near critical ﬂuid aboard the MIR station with the use of the ’Alice-1’ instrument, in Proceedings of the Joint Xth European and VIth Russian Symposium on Physical Science in Microgravity, St. Petersburg, Russia, 15-21 June 1997, Vol.1, pp.333–340, V.S. Avduyevsky and V.I. Polezhaev [Eds.], Institute for Problems in Mechanics, RAS, Moscow. [23] Y. Garrabos, D. Beysens, C. Chabot, R. Wunenburger, V. Polezhaev, V. Emelianov, A. Ivanov, A. Kalmykov, Thermo-Convectional Phenomena Induced by Vibrations in Supercritical SF6 Under Low Gravity, preprint, 2004. [24] S. Fauve, K. Kumar, C. Laroche, D. Beysens, Y. Garrabos, Phys. Rev. Lett. 68, pp.3160–3163, 1992. [25] R. Wunenburger, P. Evesque, C. Chabot, Y. Garrabos, S. Fauve, D. Beysens, Phys. Rev. E 59, pp.5440–5445, 1999.

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[26] D.V. Lyubimov, A. Cherepanov, Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza 6, 8-13, translated in Fluid Dynamics 86, pp.849–854, 1987. [27] D. Beysens, C. Chabot, Y. Garrabos, Microgravity Sci. Technol. 11, pp.113–118, 1998. [28] D. Beysens, D. Chatain, P. Evesque, Y. Garrabos, Phase separation under vibrations in near-critical hydrogen free of gravity eﬀects, submitted, 2004.

FLAW TOLERANT NANOSTRUCTURES OF BIOLOGICAL MATERIALS Huajian Gao∗ , Baohua Ji, Markus J. Buehler, and Haimin Yao Max Planck Institute for Metals Research, Heisenbergstrasse 3, 70569 Stuttgart, Germany

Abstract

1.

Bone-like biological materials have achieved superior mechanical properties through hierarchical composite structures of mineral and protein. Gecko and many insects have evolved hierarchical surface structures to achieve superior adhesion capabilities. We show that the nanometer scale plays a key role in allowing these biological systems to achieve such properties, and suggest that the principle of ﬂaw tolerance may have had an overarching inﬂuence on the evolution of the bulk nanostructure of bone-like materials and the surface nanostructure of gecko-like animal species. We demonstrate that the nanoscale sizes allow the mineral nanoparticles in bone to achieve optimum fracture strength and the spatula nanoprotrusions in Gecko to achieve optimum adhesion strength. Strength optimization is achieved by restricting the relevant dimension to nanometer scale so that crack-like ﬂaws do not propagate to break the desired structural link. Continuum and atomistic modeling have been conducted to verify this concept.

Introduction

New challenges in materials science in the 21st century will include the development of multi-functional and hierarchical materials systems. Nanotechnology promises to enable mankind to design materials using a bottom-up approach, that is, to construct multi-functional and hierarchical material systems by tailor-designing structures from atomic scale and up. However, to this date, there is almost no theoretical basis on how to design a hierarchical material system to achieve a particular set of functions. One strategy is to look among solutions in nature for hints on advanced materials design.

∗ [email protected]

131 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 131–138. © 2005 Springer. Printed in the Netherlands.

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(a)

(b)

Figure 1. Nanostructure of bones (a) and the toe of geckos (b) that consists of a terminal nanostructure called spatula of about 200–500 nm in diameter.

Biological materials, such as bone [1] exhibit many levels of hierarchical structures from macroscopic to microscopic length scales. The smallest building blocks in such materials are generally on the nanometer length scale. For instance, the nanostructure of bone (Fig. 1a) consists of mineral crystal platelets with thickness around a few nanometers embedded in a collagen matrix [1,2]. Interesting nanostructures of biological systems for superior mechanical properties are not just limited to the nanocomposite structure of bone. Gecko and many insects have evolved elaborate hierarchical surface structures in their foot hair to achieve extraordinary adhesion capabilities. These animals possess ability to adhere to vertical surfaces and ceilings. A gecko is found to have hundreds of thousands of keratinous hairs or setae on its foot; each seta is 30 ∼ 130µm long and contains hundreds of protruding nanoscale structures called spatula (Fig. 1b). We attempt to address the following questions. Why is nanoscale is so important to biological systems? What are the basic mechanisms and principles behind biological nanostructures?

2.

The Protein-mineral Bulk Nanostructure of Bone-like Biocomposites

Experimental observations (e.g. [1,3] and further references in [4]) have shown that, at the most elementary structure level, biological materials exhibit a generic structure consisting of staggered mineral platelets embedded in a soft matrix (Fig. 2a). Under an applied tensile stress, the path of load transfer in the mineral-protein biocomposites can be represented by a tension-shear chain model [4] where the mineral platelets carry tensile load and the protein transfers load between mineral crystals via shear (Fig. 2b). In this tension-shear chain model, the mineralprotein composite is simpliﬁed to a one-dimensional chain consisting of tensile springs (mineral) interlinked by shear springs (protein). The integrity of the composite chain structure is hinged upon the strength of mineral platelets since breaking of the platelets would destroy the cri-

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(a)

(b)

Figure 2. A simple tension-shear chain model of biocomposites. (a) Schematic of staggered mineral crystals embedded in a soft (protein) matrix. (b) Tension-shear chain model showing the path of load transfer in the mineral-protein composites.

tical structural links in the composite, leading to disintegration of the protein-mineral network. The strength of mineral platelets plays a crucial role in the fracture energy of the composite. In order to achieve high fracture energy, the mineral platelets must be able to sustain large tensile stress without fracture. How to optimize the strength of the mineral platelets? The Griﬃth theory of fracture [5] and common engineering experiences have shown that the strength of brittle solids is determined by pre-existing ﬂaws. It was pointed out that the nanometer scale is the key to optimizing mineral strength [4]. At the simplest level, this can be understood from the following consideration. A perfect, defect-free mineral particle should be able to sustain mechanical stress near the theoretical strength σth of the material. However, we assume that the particle contains crack-like ﬂaws. For example, protein molecules trapped within the mineral crystals during the biomineralization process are mechanically equivalent to embedded microcracks. Considering all potentially existing cracks in a thin strip, the largest crack, and hence the most dangerous one, will be a crack about half the strip width. The key idea of ﬂaw tolerance [4,6] is that cracks conﬁned in a small structure do not propagate until the material around the crack uniformly reaches the theoretical strength. This can also be demonstrated with the crack conﬁguration shown in Fig. 3. In this conﬁguration, the strength of the material can be calculated from the Griﬃth theory as σf = 4γE ∗ /h for a mineral platelet ∗ 2 width h and fracture surface energy γ, where E = E/ 1 − ν , E being the Young’s modulus and ν the Poisson ratio. According to this expression, the strength of the material approaches inﬁnity when h goes to zero. This is physically impossible since the largest stress a material can sustain is limited by an upper bound (theoretical strength) σth . This suggests that there exists a transition between crack propagation governed by the Griﬃth criterion and uniform rupture of atomic bonds

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at theoretical strength at a critical length scale [4] hcr =

4γE ∗ 2 . σth

(1)

Taking a rough estimate γ = 1 J/m2 , Em = 100 GPa, and σth = Em /30, we found hcr to be around 30 nm for a half-cracked platelet [4]. The nanometer scale not only allows the strength of mineral particles to be optimized near theoretical strength but also renders these particles insensitive to crack-like defects (ﬂaw tolerance). This concept has so far also been studied by atomistic simulations (details see [6]). Figure 3(a) plots the critical failure stress normalized by the theoretical strength, indicating a smooth transition between crack propagation governed by the Griﬃth condition for thick layers (hcr /h < 1) to uniform rupture at theoretical strength for thin layers ( hcr /h > 1). This result is fully consistent with the continuum mechanics analysis [4]. Figure 3(b) plots the distribution of normal stress ahead of the crack. As the strip width is decreased, stress concentration at crack tip disappears and the stress distribution becomes uniform near the crack tip, and thus the solid has become insensitive to ﬂaws. Further analysis of the protein-mineral bulk nanostructure of bone on stiﬀness (discussion of the interplay of the soft protein matrix and the stiﬀ mineral platelet material, and the impact of the aspect ratio of mineral platelet) and fracture energy (including a discussion on sacriﬁcial Ca++ bonds) can be found in [6]. The interested reader is referred to references [4,6-9] for further details of our group on this topic.

(a)

(b)

Figure 3. (a) Fracture strength as a function of layer width h, and (b) stress distribution ahead of the crack for diﬀerent layer widths h.

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3.

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Flaw Tolerant Surface Nanostructure of Gecko for Adhesion

The concept of nanoscale ﬂaw tolerance can be discussed in a more general context to include the surface nanostructure of gecko. Among the hairy biological attachment systems, the density of surface hairs (setae) increases with the body weight of animal, and gecko has the highest density among all animal species that have been studied [10]. The most terminal (smallest) structure of gecko’s attachment mechanism is called spatula (Fig. 1b) which is about 200–500 nanometers in diameter. Why is the spatula size in the nanometer range? To understand this, we have modeled the spatula as an elastic ﬂat-ended cylindrical hair in adhesive contact with a rigid substrate [11]. The radius of the cylinder is R. To test the ability of the ﬂat cylinder to adhere in the presence of adhesive ﬂaws, imperfect contact between the spatula and substrate is assumed such that the radius of the actual contact area is a = αR, and 0< α 0), and an additional input of energy is required to stabilize the protein cluster. For this thermodynamically unfavorable growth process, increasing the force per unit length, f , leads to smaller adhesions. FA can only grow when the overall free energy of Eq. (10) is negative, corresponding to e > 0 – i.e. favorable aggregation energy and an exothermic, local chemical interaction. However immobilization of the extracellular matrix (k → 0) or very large forces can lead to positive value for ∆E, and, if large enough, may arrest the growth process, even if e > 0. This analysis can be generalized to predict the kinetics of growth of FA [30].

Acknowledgments The authors acknowledge very fruitful experimental collaborations with L. Addadi, N. Balaban, A. Bershadsky, B. Geiger, D. Riveline, and theoretical discussions with I. Bischofs and M. Kozlov. This work has been supported by the Israel Science Foundation, the U.S. – Israel Binational Science Foundation, the Minerva Foundation, the German – Israel Foundation and an EU Network Grant. USS is grateful for the support of the Emmy Noether Program of the German Science Foundation.

References [1] B. Alberts, D. Bray, J. Lewis, M. Raﬀ, K. Roberts, J. Watson, Molecular Biology of the Cell, Garland Publishing, New York, 1994. [2] D. Boal, Mechanics of the Cell, Cambridge University Press, Cambridge, 2002. [3] B. Geiger, A.D. Bershadsky, Cell, Vol. 110, pp.139–142, 2002. [4] N.Q. Balaban, U.S. Schwarz, D. Riveline, P. Goichberg, G. Tzur, I. Sabanay, D. Mahalu, S.A. Safran, A.D. Bershadsky, L. Addadi, B. Geiger, Nature Cell Biol., Vol. 3, pp.466–472, 2001.

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[5] U.S. Schwarz, S.A. Safran, Phys. Rev. Lett., Vol. 88, 048102, 2002. [6] I.B. Bischofs, U.S. Schwarz, Proc. Natl. Acad. Sci. USA, Vol. 100, pp.9274–9279, 2003; I.B. Bischofs, S.A. Safran, U.S. Schwarz, Phys. Rev. E, Vol. 69, 021911, 2004. [7] P.C. Dartsch, H. H¨ ¨ ammerle, E. Betz, Acta Anat., Vol. 125, pp.108, 1986. J.H.-C. Wang and E.S. Grood, Connect.Tissue Res., Vol. 41, 29, 2000. [8] K. Jakab, A. Neagu, V. Mironov, R.R. Markwald, G. Forgacs, PNAS, Vol. 101, pp.2804, 2004. [9] B. Geiger, Science Vol. 294, pp.1661–1663, 2001. [10] F.G. Giancotti, E. Ruoslahti, Science, Vol. 285, pp.1028–1032, 1999. [11] R. Zaidel-Bar, C. Ballestrem, Z. Kam, B. Geiger, J. Cell Sci., Vol. 116, pp.4605– 4613, 2003. [12] R.J. Jr Pelham, Y.-L. Wang, Proc. Natl. Acad. Sci. USA, Vol. 94, pp.13661– 13665, 1997. [13] C.-M. Lo, H.-B. Wang, M. Dembo, Y.-L. Wang, Biophys. J., Vol. 79, pp.144–152, 2000. [14] D. Riveline, E. Zamir, N. Q. Balaban, U. S. Schwarz, T. Ishizaki, S. Narumiya, Z. Kam, B. Geiger, A.D. Bershadsky, J. Cell Biol., Vol. 153, pp.1175–1185, 2001. [15] K.A. Beningo, Y.-L. Wang, Trends Cell Biol., Vol. 12, 79, 2002. [16] A.K. Harris, P. Wild, D. Stopak, Science, Vol. 208, pp.177–179, 1980; A.K. Harris, D.Stopak, P.Wild, Nature, Vol. 290, pp.249–251, 1981. [17] M. Dembo, T. Oliver, A. Ishihara, K. Jacobson, Biophys.J., Vol. 70, pp.20082022, 1996; M. Dembo, Y.-L. Wang, Biophys.J., Vol. 76, pp.2307-2316, 1999. [18] R. Merkel, P. Nassoy, A. Leung, K. Ritchie, E. Evans, Nature, Vol. 397, 50, 1999. [19] L. D. Landau, E. M. Lifshitz, Theory of Elasticity, Pergamon, Oxford, 1970. [20] U.S. Schwarz, N.Q. Balaban, D. Riveline, B. Geiger, S.A. Safran, Biophysical Journal, Vol. 83, pp.1380, 2002. [21] R. Siems, Phys. Stat. Sol., Vol. 30, pp.645, 1968. [22] H. Wagner, H. Horner, Adv. Phys., Vol. 23, pp.587, 1974. [23] J. P. Butler, I. M. Tolic-Norrelykke, B. Fabry, J. J. Fredberg, Am. J. Physiol. Cell Physiol., Vol. 282, pp.C595, 2002. [24] J. Y. Wong, A. Velasco, P. Rajagopalan, Q. Pham, Langmuir, Vol. 19, pp.1908, 2003. [25] F. Grinnell, Trends in Cell Biol., Vol. 10, pp.362, 2000. [26] P.F. Davies, , A. Robotewskyj, M.L. Griem, J. Clin. Invest., Vol. 93, pp.20312038, 1994. [27] A. Nicolas, S.A. Safran, Phys. Rev. E, Vol. 69 pp.051902-1–051902-7, 2004. [28] B.Z. Katz, E. Zamir, A.D. Bershadsky, Z. Kam, K.M. Yamada, B. Geiger, Mol. Cell Biol., Vol. 11, pp.1047–1060, 2000. [29] D. Choquet, D.P. Felsenfeld, M.P. Sheetz, Cell, Vol. 88, pp.39–48, 1997. [30] A. Nicolas, B. Geiger, S. A. Safran, PNAS, Vol. 101, pp.12520–12525, 2004.

ELECTROKINETIC FLOW INSTABILITIES IN MICROFLUIDIC SYSTEMS Hao Lin, Michael H. Oddy and Juan G. Santiago Mechanical Engineering Department, Stanford University Stanford, CA 94305, USA [email protected]

Abstract

The stability of electrokinetic ﬂow in a rectangular cross-section microﬂuidic channel with transverse conductivity gradients and driven by streamwise electric ﬁelds was explored. Such a system exhibits a critical electric ﬁeld above which the ﬂow is highly unstable, resulting in ﬂuctuating velocities and rapid stirring. The problem was studied using theoretical and numerical analyses, as well as experimental observations. It was found that the internally generated electric body force was responsible for the instability, whereas the diﬀusion of ion species provided a stabilizing mechanism. Various models including two-dimensional and depth-averaged formulations were studied; modeling results compare well with experimental observations. These results have application to the design and control of on-chip assays that require high conductivity gradients, and provide a rapid mixing mechanism for low Reynolds number ﬂow in microchannels.

Keywords: Electrokinetics, electrokinetic instability, critical electric ﬁeld, electric conductivity gradient, microﬂuidics, microchannel, micromixing

1.

Introduction

Over the past decade there has been an extensive research into the design of devices that perform chemical analysis in micro-fabricated ﬂuidic channel structures. Often referred to as Micro Total Analysis Systems (µTAS), these systems exhibit a mass transport regime that is often diﬀerent from that of macro-scale ﬂow devices. Many of these devices apply electrokinetic liquid-phase, bioanalytical techniques including capillary electrophoresis and isoelectric focusing, and often manipulate the samples having poorly characterized or unknown electrical conductivi343 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 343–354. © 2005 Springer. Printed in the Netherlands.

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ties. As a result, conductivity mismatches often occur between the sample/reagent species and the background electrolyte. In the presence of applied electric ﬁelds, conductivity gradients can induce electrohydrodynamic coupling, which can in turn generate complex, unstable ﬂowﬁelds. Flows exhibiting these physics have been reported in the classical electrohydrodynamics literature (see for example, the seminal paper by Hoburg and Melcher, 1976, and a later work by Baygents and Baldessari, 1998). In this paper, we review ﬂow instabilities due to electric ﬁeld and conductivity gradients coupling in electrokinetic systems. Electrokinetic ﬂows are a subset of electrohydrodynamics characterized by the presence of an electrical double layer and regimes, where transport due to molecular diﬀusion is important. Although desirable for rapid-mixing applications, the electrokinetic instabilities are unwanted in microﬂuidics applications such as sample injection, separation, and controlled diﬀusion-limited reaction processes where minimal sample dispersion is required. This motivates research toward a better understanding of the conditions necessary for the onset of electrokinetic ﬂow instability. In 2001 Oddy et al. ﬁrst reported observation of electrokinetic instability (EKI) in a microchannel system. These experiments were performed in 4-mm-long glass capillaries with rectangular cross-sections, and the instabilities were in general of temporal nature (Oddy et al., 2001). In a slightly diﬀerent geometry (microﬂuidic T-junction), Chen and Santiago also reported spatial ampliﬁcation of disturbances which was later identiﬁed as convective instability (Chen and Santiago, 2002, Chen et al., 2004). In all of these experiments, conductivity gradients were in the spanwise direction (perpendicular to the electric ﬁeld), and there existed critical values of the applied streamwise electric ﬁeld above which instabilities and signiﬁcant stirring occurred. Following these initial experimental observations, there has been a development of models for electrokinetic ﬂow instabilities. Models are useful in predicting threshold conditions for instability onset as well as other ﬂow features including coherent wave structures and mixing rate. Lin et al. (2004) and Chen et al. (2004) showed that a generalized EHD modeling framework (derived from the so-called “leaky dielectric” model ﬁrst developed by Melcher and Taylor, 1969) can be used to describe both the low-conductivity, non-diﬀuse charge dynamics of classical EHD, and the more recently reported ﬂow instabilities of high-conductivity electrolyte in electrokinetic microsystems. Lin et al. (2004) analyzed the temporal stability properties based on a two-ion model, comprising the conductivity transport equation along with the conservation of momentum and electromigration current. They showed that the model provided good qualitative and fair quantitative agreement with regard to

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the threshold electric ﬁelds and critical wavenumbers. Lin et al. (2004) also presented non-linear simulations of their set of governing equations that capture the high Peclet (or the so-called electric Rayleigh) number stirring events observed in experiments. Using a convective framework, Chen et al. (2004) showed that EKI could manifest itself convectively in the presence of a strong electroosmotic ﬂow. In the latter analysis, EKI is modeled using a linearized, thin-layer limit of the Navier-Stokes equations coupled with conservation equations for electrical conductivity and current. The model reveals both convectively and absolutely unstable eigenmodes. More recently, Storey et al. (2004) presented a depth-averaged version of the governing equations used by the Lin et al. (2004) model. Their depth-averaged model compared favorably with a complete three-dimensional model for thin channel geometries. In this paper we present our experimental, analytical, and computational results and some progress in the pursuit of modeling and understanding of EKI in electrokinetic microchannels. We shall brieﬂy introduce the experimental results, followed by a general theoretical formulation developed in Lin et al. (2004). Using these equations we show the results from various linear analyses as well as nonlinear simulations, and assess their qualitative and quantitative agreements with experimental data. We conclude the paper by introducing our latest development in a depth-averaged framework suitable for the study of generalized electrokinetic ﬂows in microchannels with thin channel geometry.

2.

Experimental Observations

Here we show a few exemplary results from our experiments. The microchannel consisted of a borosilicate glass capillary (Wilmad-Labglass, NJ) with a rectangular cross-section. The channel was 4 mm long (in the x or streamwise direction), 1 mm wide (in the y or spanwise direction), and 0.1 mm deep (in the z or depth direction). Two buﬀer streams initially occupied the upper and lower halves of the microchannel, resulting in a diﬀuse conductivity gradient along the spanwise, y-direction; an electric ﬁeld was subsequently applied along the streamwise, x-direction. The conductivity of the buﬀer streams were 50 and 5 µS/cm, respectively, resulting in a conductivity ratio of γ = 10. For ﬂow visualization, an electrically neutral, heavy-molecular-weight dye (70 kDalton) composed of a dextran-rhodamine B conjugate (Molecular Probes, OR) was added to the high-conductivity buﬀer stream. The imposed electric potential initiated an electroosmotic ﬂow in the channel and, for electric ﬁelds above a threshold value, electrokinetic instabilities.

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A representative set of images from experiments conducted at 1, 2, and 3 kV applied potentials are shown in Fig. 1. The potentials were applied over a distance of 40 mm, such that they were equivalent to applied ﬁelds of 25,000, 50,000 and 75,000 V/m, respectively. In each case, the top ﬁgure of each series shows the initial, undisturbed interface between the dyed and undyed buﬀer streams in the channel (t = 0). The successive images in each column show the temporal evolution of the imaged dye under a constant, DC potential. In this color scheme, blue corresponds to the undyed, low-conductivity stream, and red to the dyed high-conductivity stream. For an applied ﬁeld of 25,000 V/m, the interface was only slightly perturbed and only slight ﬂuctuations are apparent in the images captured at 4.0 s and 5.0 s. At the two higher applied voltages, the interface exhibited a rapidly-growing wave pattern within the ﬁrst 1 s. The unstable ﬂuid motion in the channel buckled the interface and proceeded to stretch and fold the material lines. The transverse

Figure 1. Sample images from the experiment, shown for applied ﬁelds of 25,000, 50,000, and 75,000 V/m, corresponding to the ﬁrst, second, and third column. Images obtained at various times are shown for each column. The electric ﬁeld and bulk ﬂow directions were from left to right. High voltage was applied as a Heaviside function at t = 0 s. Each image corresponds to a physical area 1 mm wide (y) and 3.6 mm long (x). The depth of the channel is 100 µm along the z-direction (into the page). Small amplitude waves at t = 1 s grew and led to rapid stirring of the initially distinct buﬀer streams. The instability stretched and folded material lines and, after about 4 s for the 75,000 V/m applied ﬁeld, resulted in a well-stirred, relatively homogeneous dye concentration ﬁeld. The time of the images in each row are shown in the ﬁgure.

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and ﬂuctuating velocities associated with this unstable motion resulted in rapid mixing of the two streams. At the 75,000 V/m applied ﬁeld, the channel reached a well-stirred state and with nearly-homogeneous concentration ﬁelds observable within 5 s.

3.

Formulation

The description of experiments given above serves as an introduction to the problem and describes the observed features of electrokinetic ﬂow instability. We now turn to a theoretical formulation of the ﬂow. In this section, we summarize the governing equations and discuss the parameters of interest in our experiment. The equations of our model are derived from the conservation laws for a dilute, two-species electrolyte solution (Probstein, 1994), and we have adopted (with justiﬁcation) a relaxation assumption to simplify the equations. The scaling analysis and derivations are discussed in detail in Lin et al. (2004) and should not be repeated here. The (dimensionless) equations read 1 ∂σ + v · ∇σ = ∇2 σ, ∂t Rae ∇ · (σ∇Φ) = 0, ∇2 Φ = −ρE , Re

∇v = 0, ∂v + v · ∇v = −∇p + ∇2 v − ρE ∇Φ, ∂t

(1) (2) (3) (4) (5)

where σ is the conductivity, v is the bulk ﬂuid velocity, Φ is the electric ﬁeld (which includes both the applied and generated components), and p is pressure. The electric Rayleigh number (similar to the Peclet number) and the Reynolds number that arise from the nondimensionalization are deﬁned as Uev H Uev H , Re ≡ . Rae ≡ D ν Here H is half-channel width, D is an eﬀective diﬀusivity of the conductivity, and ν is the kinematic viscosity of the electrolyte solution (usually aqueous). The velocity Uev , the so-called electroviscous velocity, is velocity scale derived by setting equal the electric body force and the viscous force in the momentum equation. (See Hoburg and Melcher, 1976, Lin et al., 2004 and Chen et al., 2004; more discussions on these parameters can also be found in these references.)

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Two-dimensional Model

We ﬁrst present a model where we assume that the ﬂow exists only in the x − y plane, and that there is no dynamics in the z-direction. This analysis will capture the basic physics of the instability mechanisms due to the conductivity gradient. We ﬁrst use a linear stability analysis to predict the regimes where we would expect rapid mixing to occur. The base states are a diﬀused, spanwise conductivity proﬁle σ0 = σ0 (y) with a (high-to-low) conductivity ratio of 10, and a shear electroosmotic ﬂow u0 = u0 (y). Note that the spanwise dependence of the latter was induced by that of the former via the dependence of zeta potential on conductivity. We assume that disturbance is periodic in the x direction, and the growth of amplitude is exponential in time. We have obtained, for each streamwise wave number k and applied ﬁeld E0 , a set of eigenvalues (the exponential growth rates), together with their respective eigenfunctions. In Fig. 2 we show a contour plot of the growth rates of the most unstable eigenfunction in the wave number-Rayleigh number (electric ﬁeld) parameter space. Symbol s denotes the real and dimensional growth rate. The neutral stability curve is obtained by setting s = 0. A threshold electric ﬁeld is successfully captured from the minimal value of E0 on the neutral stability curve. As originally reported by Baygents and Baldessari (1998), we found that the inclusion of the diﬀusive term ∇2 σ/Rae in Eq. (1) is crucial for the existence of the neutral stability curve. 5

10

s = 40 sec−1

Rae

s = 4 sec−1 4

10

Eo ((V/m)

s = 20 sec−1

5

10

4

s = 1 sec−1

10

3

10

Neutral 0

10

1

k

10

Figure 2. Contour plot of growth rates (s) versus wave number and Rayleigh number. Dimensional applied electric ﬁeld is provided on the right-hand axis. For the case plotted here, the ratio of the conductivity between the two streams is 10.

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Figure 3. Snapshots of the simulated dye ﬁeld at various instances in time for diﬀerent driving electric ﬁelds. In this color scheme red corresponds to the high conductivity buﬀer, and blue to the low one. Each column indicates a diﬀerent applied ﬁeld and the rows within each column present the selected snapshots in time. The image corresponds to a physical domain of 3.6 mm×1 mm. The time for noticeable waves to develop is decreased as the ﬁeld is increased.

In addition to the two-dimensional linear stability analysis presented above, we have also solved the full nonlinear governing Eqs. (1-5) numerically to capture the nonlinear evolution of the instability observed in the experiments. The initial conditions are the base states plus a small white noise perturbation. The solution details are documented in Lin et al. (2004). Figure 3 shows the nonlinear evolution of the simulated dye at various instances in time and for three diﬀerent electric ﬁelds. The model reproduces many of the essential features observed in the experiments, including the shape and initial break-up dynamics of the interface, the transverse growth of a wave pattern in the interface, and the roll-up of scalar structures observed at later times. Note the similarity in the most unstable (and most apparent) wave number at later times between the simulation and experiments. Despite the similarities between the wave number and dynamics of the interface breakup, the threshold imposed ﬁelds from both the linear and nonlinear predictions are lower than those shown for the experiment in Fig. 1. For example, compare the evolution of the dye at 25,000 V/m from the experiments (Fig. 1, column 1) and the simulation (Fig. 3, column 3). We see that the simulation at 25,000 V/m predicts

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a well-stirred ﬂow ﬁeld in less than three seconds, while experiments show that the ﬂow is stable on the time-scale of the experiments. The simulation of 25,000 V/m is qualitatively similar to the experimental ﬂow at 75,000 V/m (Fig. 1, column 3). Despite the discrepancy in the magnitude of the applied ﬁeld, our simulation captures a threshold ﬁeld and scalar features qualitatively similar to the experiment. In the following section we address possible causes for the under-prediction of the threshold electric ﬁeld by including three-dimensional eﬀects. In comparison with the temporal instability analysis presented here (which is consistent with the experimentally observed instability in the previous section), Chen et al. (2004) analyzed the instability in a convective framework which is consistent with the spatial growth that was observed in T-junction microﬂuidic channels. Among other contributions, the authors found that an important dimensionless group Rν , deﬁned as the ratio of the electroviscous to electroosmotic velocity, is critical in demarcating the absolute/convective instability boundary. Interested readers are referred to Chen et al. (2004).

5.

Depth-averaged Model

In the previous sections we have provided a two-dimensional framework which appears to capture the primary physics of our ﬂow. However, the primary ﬂaw in that model is the two-dimensional assumption, for a channel with an aspect ratio of δ ≡ d/H = 0.1, where d denotes the channel half-width. Such a thin channel geometry was also used in the experimental work of Chen et al. (Chen et al., 2002; Chen et al., 2004). In three dimensions, an EDL forms not only on the top and bottom walls of y = ±1, but also along the side walls (z = ±1), and strongly drives the ﬂow due to the small depth of the channel. The three-dimensional nature of a thin channel has the added dynamics that the ﬂuid motion in the interior of the channel is directly coupled to the top and bottom wall boundary condition (determined in part by the local value of ion density). In Lin et al. (2004) we presented a 3D linear analysis and found that the predicted threshold ﬁeld was one order-of-magnitude higher than that from the 2D linear analysis, in much closer agreement with the experimental observations. However, a depth-averaging approach is preferred since in general, the 3D analysis (simulations) are computationally more expensive, and considering that the microchannels of our interest are “shallow” in the depth (z) direction. In Chen et al. (2004) a depth-averaged analysis was performed on a set of linearized governing equations and the resulted linear equation system was used for convective

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instability analysis. In Storey et al. (2004) a more complete Hele-Shaw type of integrated momentum equation was used. Both models yielded favorable quantitative results when compared with experimental data. Here we extend and complete the ideas that were ﬁrst developed by Chen et al. (2004) and Storey et al. (2004). We develop a generalized, nonlinear depth-averaged model suitable for the study of electrokinetic microchannel ﬂows with thin channel geometries. We accomplish this through a complete asymptotic analysis of the full threedimensional equations based on a smallness parameter which is the channel cross-sectional aspect ratio δ. Our general methodology follows a combined lubrication (for the momentum equations) and Taylor-Aris (for the convective-diﬀusion of the conductivity ﬁeld) approach. Without presenting the details of the derivation, we list the ﬁnal equations as 1 2 ∂σ ¯ 2 2 2 ¯ ¯ +u ¯ · ∇H σ ∇H σ Ra δ ∇H · [U (U · ∇H σ ¯= ¯+ ¯ )] , (6) ∂t Rae 105 e

Reδ 2

∂u ¯ +u ¯ · ∇H u ¯ ∂t

¯ = 0, σ ∇H Φ) ∇H · (¯

(7)

∇H · u ¯ = 0,

(8)

¯ HΦ ¯ − 3U ¯ + δ 2 ∇2H u = −∇H p¯ + ∇2H Φ∇ ¯.

(9)

Here the overbar denotes depth-averaged quantity, the operator ∇H denotes the in-plane gradient (to distinguish from the full three-dimen¯ ≡ u sional gradient), and U ¯ − u∞ is the diﬀerence between the total depth-averaged velocity and the electroosmotic velocity. The main contributions of this new equation set are the Taylor-dispersion-type term in the conductivity equation, and the Darcy-BrinkmanForchheimer (DBF) type of momentum equation which is of second-order consistency in δ (Chen et al., 2004; Liu and Masliyah, 1996). We present preliminary results in the assessment of the validity and accuracy of the model. Figure 4 compares the linear stability results for growth rate of disturbances versus wave number at a single Rayleigh number of Rae = 5.000. We perform linear analyses using the following three models: 1. The linear integrated momentum equations used by Storey et al. (2004). 2. The linearized three-dimensional equations (Lin et al., 2004). 3. The depth-averaged DBF formulation presented here.

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Integrated momentum

0.35

3D linear

0.3

s

0.25 0.2

DBF Formulation

0.15 0.1 0.05 0

2

4

6

8

k

10

12

14

16

18

Figure 4. Comparison of growth rates of disturbances as predicted by three models. Shown here are the real part of the growth rates versus the wave number for Rae = 5.000 as computed with the DBF momentum equation presented here, the integrated momentum equation (δ 0 approximation, Storey et al., 2004), and the three-dimensional equation set. The DBF formulation represents in-plane viscous stresses that quench the unphysically high wave number growth and is in agreement with the three-dimensional result.

Note that at the linear regime, the Taylor dispersion in Eq. (6) drops as a higher-order term, and the only diﬀerence between the models are within the momentum equations. We ﬁnd that all three models are in good agreement for wave numbers below about 4. However, when compared with the more accurate three-dimensional analysis, the DBF momentum equation provides signiﬁcantly better results at higher wave numbers than the lower-order, integrated momentum approximation. The characteristics of the DBF momentum equations also make it more advantageous to use in nonlinear simulations when compared with the integrated momentum equation. In particular, the inclusion of in¯ preserves a mathematical structure similar to the plane diﬀusion δ 2 ∇2H u original Navier-Stokes equations, and enables reproduction of the boundary eﬀects (e.g., at y = ±1 walls) that are not captured by lower-order approximations. This will be discussed further in a future work, and here we simply show some sample results of the full nonlinear, depth-averaged simulations with Eqs. (6–9). That is, a model with the combined eﬀects of Taylor dispersion and the DBF momentum equation. We try to reproduce the experimental image presented in Fig. 1 at the two lower voltages (25,000 and 50,000 V/m); the result is shown in Fig. 5. Again the model reproduces essential features observed in the experiments such

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Figure 5. Nonlinear simulation of the depth-averaged equation system. This model includes the combined eﬀects of the Taylor dispersion and the DBF momentum equation. In the strongly nonlinear regime the Taylor dispersion acts as an extra smoothing mechanism. The results compare favorably with the experimental data presented in Fig. 1.

as the fastest growing wave numbers and the growth rates of the interface disturbance amplitude. However, note that the computations are now at exactly the same ﬁeld strength as those applied in the experiments (as opposed to the unnaturally low ﬁelds used for comparison with the simple 2D model results of Fig. 3). Future work will also include the application of the model to diﬀerent ﬂow conﬁgurations such as those used in ﬁeld ampliﬁed sample stacking (FASS).

6.

Summary

In this work we have presented experimental, numerical, and analytical results that explain the basic mechanisms behind an electrokinetic mixing phenomenon observed in microﬂuidic channels. We have presented analysis and computations based on various sets of assumptions for electrokinetic ﬂows in a long, thin channel with a transverse conduc-

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tivity gradient. Our models are able to predict general trends in the data, as well as many of the qualitative and quantitative aspects of the observed ﬂow ﬁeld. Ongoing work includes the development of a generalized, depth-averaged model for a wide class of electrokinetic ﬂows (such as FASS) in thin microﬂuidic channels. The models presented in this work are useful in optimization studies, as parameter space can be spanned in simulations more quickly than in the laboratory. The results described by Oddy et al. (2001) has demonstrated that oscillatory electric ﬁeld can potentially drive even more vigorous mixing. The models presented here can be used to optimize the form of the forcing function, to design the shape of a micro-mixer, and to develop optimal control strategies for both the micro-mixing and the suppression of instabilities.

Acknowledgments This work was sponsored by DARPA (Contract Number F30602-002-0609) with Dr. Anantha Krishnan as contract monitor and by an NSF CAREER Award (J.G.S.) with Dr. Michael W. Plesniak as contract monitor.

References [1] J. Baygents, F. Baldessari, Electrohydrodynamic instability in a thin ﬂuid layer with an electrical conductivity gradient, Phys. Fluids, Vol.10, 1, 301–311, 1998. [2] C.-H. Chen, J.G. Santiago, Electrokinetic instability in high concentration gradient microﬂows, Proceedings of IMECE-2002, CD Vol.1, #33563, 2002. [3] C.-H. Chen, H. Lin, S.K. Lele, J.G. Santiago, Convective and absolute electrokinetic instability with conductivity gradients, J. Fluid Mech., in press, 2004. [4] J.F. Hoburg, J.R. Melcher, Internal electrohydrodynamic instability and mixing of ﬂuids with orthogonal ﬁeld and conductivity gradients, J. Fluid Mech., Vol.73, 333, 1976. [5] H. Lin, B.D. Storey, M.H. Oddy, C.-H. Chen, J.G. Santiago, Instability of electrokinetic microchannel ﬂows with conductivity gradients, Phys. Fluids, Vol.16(6), 1922–1935, 2004. [6] S. Liu, S. Masliyah, Single ﬂuid ﬂow in porous media, Chem. Eng. Comm. Vol.148–150, 653-732, 1996. [7] J.R. Melcher, G.I. Taylor, Electrohydrodynamics: a review of the role of interfacial stresses, Annu. Rev. Fluid. Mech., Vol.1, 111–146, 1969. [8] M.H. Oddy, J.G. Santiago, J.C. Mikkelson, Electrokinetic instability micromixing, Anal. Chem., Vol.73, 5822–5832, 2001. [9] R.F. Probstein, Physicochemical Hydrodynamics, John Willey & Sons, New York, 1994. [10] B.D. Storey, B.S. Tilley, H. Lin, J.G. Santiago, Electrokinetic instabilities in thin microchannels, Phys. Fluids, in review, 2004.

MOLECULAR MECHANICS OF CYTOSKELETAL COMPONENTS M. Atakhorrami Vrije Universiteit Amsterdam, Dept. Phys. Amsterdam, The Netherlands

K.M. Addas Rowland Institute at Harvard Cambridge, MA, USA

M. Buchanan, G.H. Koenderink, F.C. MacKintosh Vrije Universiteit Amsterdam, Dept. Phys. Amsterdam, The Netherlands

J.X. Tang Brown University, Dept. Phys. Providence, RI, USA

Christoph F. Schmidt Vrije Universiteit Amsterdam, Dept. Phys., Amsterdam, The Netherlands [email protected]

Abstract

Semiﬂexible polymers are of great biological importance in determining the mechanical properties of cells. We have used optical tweezers to trap pairs of micron-sized silica spheres in solutions of semiﬂexible polymers, and laser interferometry to detect their thermal motions with high bandwidth. Frequency-dependent complex shear moduli were extracted from the auto- and cross-correlated bead motions, with the response functions being derived from position-ﬂuctuation data using dispersion relations from linear response theory.

Keywords: Cytoskeleton; semiﬂexible polymers; microrheology; optical tweezers; laser interferometry

355 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 355–364. © 2005 Springer. Printed in the Netherlands.

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Introduction

One of the characteristic lengths that describes a polymer in a solution is the persistence length Lp , which is the length traversed along the ﬁlament contour before the tangent vector thermally randomizes its orientation. Lp is proportional to the bending stiﬀness of the polymer. Flexible polymers are characterized by Lp > L. Semiﬂexible polymers are in the intermediate regime, characterized by Lp ≈ L >> d, with ﬁlament diameter d. The dynamics of polymer solutions and networks greatly depends on the degree of ﬂexibility of the individual polymers. Recent theoretical treatments have addressed semiﬂexible polymer networks [1-4]. Such polymers are found particularly in biology where they form networks that determine the mechanical properties of cells. We report here on rheology experiments on such polymer systems, and in particular on a new microrheology method. There are clear advantages to the miniaturization of a rheology experiment: only small amounts of material are necessary, spatial inhomogeneities can be studied, and the bandwidth of the measurement can be high. We have developed a microrheology technique using laser interferometric tracking of the Brownian motion of micron-sized beads embedded in viscoelastic materials [5]. The simplest implementation of the technique consists of tracking one probe particle at a time (1-bead microrheology) [5]. A further development, avoiding local artefacts due to the insertion of the probe particles into the system, consists of measuring the correlated ﬂuctuations of pairs of probe particles (2-bead microrheology). We have studied the dynamics of several systems and have compared 1-bead and 2-bead results for wormlike micelle, semiﬂexible fd virus and actin solutions. Particularly in the actin systems, which is a ﬁrst approximation to the cytoskeleton of cells, we explore the rich multitude of length and time scales in the dynamic behavior of these networks, which are not accessible to conventional macrorheology.

2.

Principle of the Technique

The microrheology (MR) technique used here is a passive one in which the thermal ﬂuctuations of pairs of micron-sized beads are observed. Complex auto- and cross-correlation response functions of the beads are calculated using the ﬂuctuation-dissipation theorem [5,6]. The complex (n,m) (n) response function αij (f ) relates the Fourier transform ri (f ) of the (n)

(m)

displacement of the nth bead ri (t) to the Fourier transform Fj the force

(m) Fj (t)

(f ) of

acting on the mth bead (i and j are either 1 or 2 for

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the parallel and perpendicular directions with respect to the line joining the bead centers, respectively), (n)

(n,m)

ri (f ) = αij

(m)

(r, f )F Fj

(f ).

(1)

The measured response function must ﬁrst be corrected for the trapping eﬀect of the optical tweezers [7]. The ﬂuctuation-dissipation theorem [5] provides the link between the single-sided power spectral density (PSD) [8] (n,m) (n) (m)∗ (f ) = lim (2/t)rt i (f )rt j (f ) (2) Sij t →∞

(n)

and the imaginary part of the response function, where rt i (f ) is the (n) Fourier transform of the position ri (t), deﬁned over a ﬁnite time t . A Kramers-Kronig integral can then be used to calculate the real part of the response function, provided that the imaginary part is known over a large enough frequency range. The connection between the auto(n,m) correlation response αii (f ) of a bead and the corresponding complex shear modulus (n,m)

Gii

(n,m)

(f ) = Gii

(n,m)

(f ) + iGii

(f )

(3)

of the viscoelastic medium surrounding the bead is assumed to be provided by the generalized Stokes-Einstein relation (GSER) [5], (n,m)

Gii

(f ) =

1 (n,m) [6πaαii (f )]

,

(4)

where G and G are the elastic and loss modulus respectively, and a is the radius of the bead. The cross-correlation shear modulus of the (1,2) solution is derived from the parallel a ≡ α11 response by [9] Gcross (f ) =

1 , [4πrα (f )]

(5)

where r is the distance between the centers of the beads.

3.

Experimental Setup

Two focused laser beams were used to form optical tweezers and trap pairs of probe particles imbedded in solutions of semiﬂexible polymers. Position ﬂuctuation data was recorded with a 200 kHz sampling rate, using laser interferometry and quadrant photodiode detection as described in detail elsewhere [5,6,10]. A near-infrared (1064 nm) laser was used for one trap and a 830 nm laser for the second trap. The beams

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Figure 1. Schematic illustration of the 2-bead microrheology experiment. Two spheres of radius a, are held in two optical tweezers, created by focusing lasers of diﬀerent wave lengths λ = 830 nm and λ = 1064 nm at a separation distance r. The displacements of both particles are detected separately in x and y direction with interferometry.

are brought to a focus in the sample chamber by a high numerical aperture objective. The laser light emerging from the condenser lens, after passing through the sample, is projected onto two quadrant photodiodes in such a way that the back-focal plane of the condenser is imaged [10]. The signals from the four quadrants of each photodiode are combined to obtain the X- and Y -voltages corresponding to the displacements of each bead in these directions in the plane normal to the propagation direction of the laser. The output voltages are, after analog ampliﬁcation and pre-processing, recorded using an A/D interface and digital data acquisition.

4.

Results

Worm-like Micelle Solutions Figure 2 presents the loss and the viscous shear modulus of a worm-like micelle solutions for 1wt % concentration. The worm-like micelle solutions were made of binary mixtures of cetylpyridium chloride (CPyCL) and sodium salicylate (NaSal) dissolved in brine (0.5 M NaCl). A characteristic feature of worm-like micelle solutions is that they relax lowfrequency applied stress with a single dominant relaxation time, reﬂecting breakage and reptation of the ﬁlaments [11]. The system we used

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G', G'' [Pa]

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1

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1000

10000

100000

f [HZ] Figure 2. Elastic modulus and loss modulus for a 1wt% worm-like micelle solution. Figure shows the comparison of 1-bead and 2-bead microrheology. The diameter of the beads was D = 2.1 µm.

produces micelles with typical diameters of 2-3 nm, a persistence length Lp 10 nm, and an average length between 100 nm and 1 µm. We have performed 1-bead MR and 2-bead MR for diﬀerent concentrations of micelles. Bead diameters were 1.16 µm and 2.1 µm. For this system 1-bead and 2-bead MR results show excellent agreement (Fig. 2). More experimental noise is usually found in 2-bead results since the cross-correlation signal between the displacement of two beads is always weaker than the auto-correlation signal of one bead. The abrupt decrease in both G and G at high frequncies is artefactual and due to anti-alias ﬁltering at the Nyquist frequency and to the ﬁnite frequency cut-oﬀ eﬀects on the result of the Kramers-Kronig integral. The eﬀect on G is stronger than on G , reﬂected in about a decade less usable data for G [5]. It is expected that 1-bead measurements could be sensitive to the local medium properties in the direct vicinity of the probe bead, which could be diﬀerent from bulk properties due to surface chemistry or steric (entropic) eﬀects. The unmodiﬁed bulk viscoelastic properties of the solutions should, on the other hand, be reported by the 2-bead results [9]. Even in the absence of speciﬁc chemical or electrostatic interactions, steric depletion of polymer in the vicinity of the probe particles is expected to cause diﬀerences in the 1- and 2-bead results, provided that probe size and network length scales are comparable. In the case of

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the worm-like micelle solutions we studied here, network length scales are at least a decade smaller than bead radius and we therefore did not expect diﬀerences due to surface artefacts. The data nicely conﬁrms this expectation and conﬁrms the validity of both methods. It should be noted that independently measured parameters enter the data evaluation for the two cases, bead diameter for 1-bead results and bead distance (only) for 2-bead results. These results further establish microrheology, both 1-bead and 2-bead methods, as a highly sensitive and high-bandwidth technique to measure shear elastic moduli of viscoelastic materials. Good agreement between more conventional macrorheology and microrheology on similar samples is reported elsewhere [12].

Fd Virus Solutions Fd virus is a ﬁlamentous DNA bacteriophage [6], which is here used as a monodisperse model system for semiﬂexible polymers. Filaments have a diameter of 7 nm, persistence length Lp = 2.2 µm and contour length L = 0.9 µm. Fd solutions were probed with a bead diameter of D = 1.16 µm. Fig. 3(a) shows the concentration dependence of the 1bead (circles) and 2-bead (triangles) elastic moduli. The loss modulus is shown in Fig. 3(b). Data are displayed for 2, 5, 10, and 14 mg/ml fd concentrations. The high-frequency slope of the loss modulus is consistent with single ﬁlament dynamics predictions [1,4]. At such high frequencies, the entanglement of the polymer in its network does not contribute to the modulus. The modulus is proportional to concentration and expected to scale with a power law exponent of 3/4. The elastic modulus is expected to exhibit the same behavior but due to the cut-oﬀ in the Kramers-Kronig integral, the reliably measured elastic modulus extends to about 10 kHz, a frequency apparently still in the transition regime between collective dynamics and high-frequency single ﬁlament dynamics because a true scaling regime would require both moduli to scale with the same power law. The high-frequency behavior is attributed to tension in the ﬁlaments due to the shear-induced extension. At lower frequencies, the tension contributes less, and the dominating contribution to the stress is that of bending and orientational dynamics [1,2]. The data demonstrate that for this system 1-bead MR and 2-bead MR are in good agreement. This is remarkable since the length scales of the system, contour length and persistence length, are comparable with the probe bead diameter. If there was a steric depletion eﬀect, it has evidently led to a negligible eﬀect on the 1-bead results.

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3

Shear Loss Modulus,G" [Pa]

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Frequency [HZ]

Figure 3. (a) The concentration dependence of 1-bead and 2-bead elastic modulus data of 1.16 µm diameter beads in 2, 5, 10 and 14 mg/ml fd solutions. (b) The same for the loss modulus. Pairs of 1-bead (circles) and 2-bead (triangles) moduli are higher for higher concentrations.

Entangled Actin Solutions Actin is a protein of 42 kD molecular weight and a major component of the cytoskeleton of most eukaryotic cells. It forms double-helical ﬁlaments of about 7 nm diameter, with a persistence length Lp = 17 µm, and with a polydisperse length distribution with an average length of approximately L = 17 µm in our in vitro reconstituted model systems.

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Actin stress relaxation is essential for cytoskeleton dynamics and is in vivo strictly regulated by a large number of associated regulatory proteins. We study here simpliﬁed model systems of just pure actin without additional proteins to explore the fundamental properties and dynamic characteristics of the networks it forms. In contrast to the two systems presented above, characteristic length scales in actin networks are large compared to bead size and we expect interesting diﬀerences between 1bead and 2-bead results, which would report on local network structure near the probes. Figures 4(a-b) show the results of 1-bead and 2-bead MR for entangled actin networks at 1 mg/ml actin concentration as a function of bead size. Figure 4(a) shows the elastic shear modulus for 1-bead MR, recorded with various probe bead sizes in comparison with the 2bead results. Both the 1- and 2-bead curves show the 3/4 power-law scaling at high frequencies, which is expected for semiﬂexible polymer networks [1,2,4,5]. This regime connects to more complex dynamics with no clear power law characteristics at lower frequencies. There are also clear diﬀerences among the results on identical samples, with 1-bead MR systematically underreporting the bulk moduli of the sample in the high frequency regime. These diﬀerences between 1-bead and 2-bead MR as well as the systematic trend with probe bead size in the 1-bead results are consistent with the formation of an entropic depletion layer around the beads. The extent of the depletion layer is, roughly speaking, dominated by the shortest of the three characteristic lengths: bead radius, entanglement length and persistence length [1,2,5]. As long as bead size is the shortest length, this depletion layer will dominate the bead response and thereby the 1-bead results. This appears to be the case in the actin networks, and the trend towards less of an inﬂuence of local environment with increasing bead size reﬂects the fact that the other length scales, particularly entanglement length which is estimated to be on the order of 5-10 µm, begin to compete with bead radius. The depletion eﬀect nevertheless does not entirely disappear even at the largest probe size, D = 5 µm. This behavior is visible in both the elastic and the loss modulus (Figs. 4(a-b)). A diﬀerent type of discrepancy is observed at low frequencies within the 1-bead results for the elastic modulus with varying bead size, as well as between 1-bead results with smaller beads and 2-bead results. 2-bead results were independent of bead size and bead distance (after conversion to shear moduli) and the curve shown is the result of averaging data taken at the diﬀerent bead distances (data for D = 1.16 µm bead diameter is shown). We hypothesize that this phenomenon is due to local non-aﬃne deformations of the actin ﬁlaments forming the network and will publish further research on these dyna-

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Molecular Mechanics of Cytoskeletal Components 2

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G' [Pa]

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-1

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f [HZ]

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-1

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-1

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f [HZ]

Figure 4. (a) Elastic modulus and (b) loss modulus for 1 mg/ml entangled actin solutions. Comparison of the moduli from 1-bead MR for bead diameters D = 0.5µ m, D = 1.16µ m and D = 5µ m. 2-bead MR data taken at diﬀerent distances all collapse after scaling by bead distance. Data for bead diameter D = 1.16µ m is shown.

mics elsewhere. It is important to note that video-based microrheology methods that have also been applied to actin networks will mainly test this regime and may therefore be diﬃcult to interpret.

5.

Conclusions

We have applied a passive microrheology technique, evaluating both single-probe particle ﬂuctuations and correlated two-particle ﬂuctuations to measure the complex shear moduli of solutions of various semiﬂexible

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polymers. Results with fd solutions and worm-like micelle solutions show good agreement between 1-bead and 2-bead MR. This strongly conﬁrms that one can measure bulk viscoelastic properties of these polymer systems with 1-bead MR, which is the simpler and less noisy technique compared to 2-bead MR. Such a conﬁrmation has been lacking up to now. More complex behavior was observed for actin solutions, providing strong evidence for local depletion eﬀects near the bead surface as well as indications of non-aﬃne ﬁlament dynamics visible at low frequencies. The results together advocate 1-bead MR as a reliable high resolution and high bandwidth technique to study simple systems far surpassing conventional rheology both in sensitivity and bandwidth. A combination of 1-bead and 2-bead MR, on the other hand, provides a way to explore more complex scale-dependent and local dynamic properties in various polymers, biological and colloidal systems.

Acknowledgments We thank David Morse, Alex Levine and Matteo Pasquali for extensive discussions. Erwin Peterman helped with technical advice. Joost van Mameren, Fredrick Gittes, Mark Buchanan and Joanna Kwiecinska helped at various stages with software for data processing. Karen Vermeulen puriﬁed actin. This project was supported by NSF DMR 9988389 (J.X.T.) and the Dutch Foundation for Fundamental Research on Matter (FOM).

References [1] D.C. Morse, Macromolecules, Vol.31, pp.7030, 1998. [2] D.C. Morse, Macromolecules, Vol.31, pp.7044, 1998. [3] A.C. Maggs, Phys. Rev. E, Vol.55, pp.7396, 1996. [4] F. Gittes, F.C. MacKintosh, Phys. Rev. E, Vol.58, pp.1241, 1998. [5] B. Schnurr, F. Gittes, F.C. MacKintosh, et al., Macromolecules, Vol.30, pp.7781, 1997. [6] K. Addas, J.X. Tang, C.F. Schmidt, Phys. Rev. E, Vol.70, 021503, 2004. [7] A. J. Levine and F. Mackintosh, pp.manuscript in preparation. [8] F. Gittes, C.F. Schmidt, Method. Cell. Biol., Vol.55, pp.129, 1998. [9] A. J. Levine, T.C. Lubensky, Phys. Rev. E, Vol.63(4), 041510, 2001. [10] M. W. Allersma, F. Gittes, M. J. deCastro, et al., Biophys. J., Vol.74, pp.1074, 1998. [11] J. F. Berret, J. Appell, G. Porte, Langmuir, Vol.9, pp.2851, 1993. [12] M. Buchanan, M. Atakhorrami, J.F. Palierne, F.C. Mackintosh, C. F. Schmidt, (manuscript in preparation).

TOPICS IN ASTROPHYSICAL FLUID DYNAMICS Edward A. Spiegel Astronomy Department Columbia University, New York, NY, USA [email protected]

Abstract

This brief description of some ﬂuid dynamical problems of astrophysical interest focuses on two eﬀects that are characteristic of the subject: selfgravity and radiative forces. Self-gravity is important in determining the basic structures of cosmic bodies as well as producing some intriguing instabilities. Radiation, by which we observe these bodies, produces forces that may be disruptive to their basic structures and be a source of vigorous ﬂuid dynamical activity in the form of photon bubbles and radiatively driven vortices.

Keywords: Gravitational instability, solitary waves, astrophysical vortices, photon bubbles

This sampler of problems in AFD begins with the classical problem of gravitational instability and continues with some current problems that I ﬁnd very intriguing. Given the imposed space limitations, I can only touch on these and I need to be very stingy with references (not of my own, of course) and will assume that I need not provide many ﬂuid references for the expected readership of this volume. (But feel free to use the email address above.)

1.

Gravitational Instability

Massaged Models The observable part of our universe looks homogeneous in the large, but it is clearly rather lumpy on smaller scales. There are stars, clusters of stars, galaxies and clusters of galaxies. Possibly there are even larger structures but here we come to a less clearly deﬁned topic where people may yet argue about how (or whether) to study such issues as the dimension of the point set that approximates the distribution of galaxies. But it is undoubtedly interesting to inquire into the origin of the inho365 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 365–377. © 2005 Springer. Printed in the Netherlands.

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mogeneities of the universe. Our current understanding of cosmology is that the early universe was dense and hot and very dissipative. Hence it is thought to have been rather homogeneous in the earliest times that we can reasonably think about and the lumpiness is considered to have been caused by gravity. The breakup of a homogenous ﬂuid into reasonably discrete structures is believed to be caused by an instability whose qualitative nature was already imagined by Newton. Jeans [13] was apparently the ﬁrst to formulate this problem. He assumed a perfect barotropic ﬂuid with selfgravity and used these simple equations of motion: ∂t (ρu) + ∇ · (ρuu) = −∇p − ρ∇V, ∂t ρ + ∇ · (ρu) = 0, ∇2 V = 4πGρ, p = KρΓ .

(1) (2) (3) (4)

Jeans posited a static homogenous solution to these equations (though one presumes that he knew better) and studied perturbations on this “solution”. But there is no inﬁnite, homogenous, static, self-gravitating medium in cosmology, not even in Einstein’s original theory of gravity [11]. Einstein added a term to his equation for the curvature of spacetime that allowed him to ﬁnd a static solution, unstable though it is. Similarly, we may add a term to the right side of the Euler equation to provide our Newtonian universe with a static solution. This term, ρλr, represents a repulsive force, as Eddington [10] pointed out. Now we may introduce V˜ = V − 12 λr2 and replace (1) and (3) by ∂t (ρu) + ∇ · (ρuu) = −∇p − ρ∇V˜ ,

(5)

∇2 V˜ = 4πG(ρ − ρλ )

(6)

where ρλ = λ/G. This recalls the device used in plasma physics to maintain charge neutrality and we may now ﬁnd a homogenous static solution and study its stability to clumping. Einstein’s extra term, “the cosmological term”, intrigued people from its beginning, but its status was uncertain until recently. The recent discovery that the expansion of the universe is accelerating has brought this extra term into vogue, though its physical meaning remains unclear. Does the cosmological term mean what the simple statement given here suggests — the existence of matter with negative gravitational mass [21]? If so, we should perhaps add a ﬂuid equation for this odd material

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as in the two-ﬂuid plasma model. Many cosmologists look on the added term as a negative pressure that can drive the expansion velocity ever upward. Here it merely serves Einstein’s original purpose for it: to produce a static solution.

The Dispersion Relation To study the stability of the homogenous medium, we let ρ = ρ0 + δρ, V˜ = V˜0 + δ V˜ and so on. With the extra term in place, we may assume that the basic ﬁelds are constant. If we then linearize the equations about the static state, we are led by standard manipulations to a KleinGordon equation for δρ: 2 (7) ∂t − ∇2 δρ = (4πGρ0 )δρ. For plane waves with δρ ∝ exp(iωt − ik · x), we obtain the dispersion relation (8) ω 2 = c2 (k 2 − kJ2 ) with kJ2 = (4πGρ0 )/c2 , and c2 = Γp0 /ρ0 . The quantity kJ is called the Jeans wavenumber and, when the wavenumber of the perturbation is much larger than this, we recover ordinary sound waves. But for k < kJ , ω 2 becomes negative and we have instability. Because of the diﬀerence in sign from the electrostatic case, we get instability rather than oscillation. The Jeans length 1/kJ tells us the scale on which the gravity just balances the pressure gradient much as the balance of surface tension and pressure dictates the size of a liquid drop. When the length scale of a perturbation is larger than this critical size, collapse occurs. For a perfect gas with temperature T , we ﬁnd from −1 the number these formulae that kJ ≈ 100 (T /n) light years where n is √ density and the collapse time for very small k is ≈ 5 × 107 / n yr.

Cosmology’s Fictitious Forces To study the formation of inhomogeneities in the universe properly, we need to take account of its expansion. This was ﬁrst done for the linear stability problem by Lifshitz in 1946 in the context of relativistic cosmology, but the Newtonian case conveys the idea [5]. Even that story is on the long side so, to indicate how the expansion reduces the degree of instability, a brief look at the kinematics in an expanding medium may suﬃce. For a universe that is iniﬁnite and homogeneous in the large, it does not matter where we put the origin of coordinates, so let us presume that one has been chosen. The position with respect to the arbitrary origin

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of a ﬂuid element is r. We then transform to isotropically expanding coordinates such that r = R(t)x (9) where R(t) is a nondimensional function that tells us how the global scale of the universe changes in time. For some suitable origin of time we let t0 be the present and take R(t0 ) = 1. We ﬁnd ˙ + Rx˙ = Hr + Rx˙ r˙ = Rx

(10)

where v = Hr is the Hubble ﬂow or global expansion and H0 = H(t0 ) is called the Hubble constant. The acceleration of the ﬂuid element is then ˙ + H 2x . ¨r = R x ¨ + 2Hx + Hx (11) In the expanding coordinates, we acquire three additional terms or ﬁctitious forces reminding us of those gained in going into a rotating frame. The scalar H plays the role of the rotation rate in that comparison, hence the analogue of the Coriolis force is a drag term. Material particles moving through an expanding medium are slowed down with respect to the background. This is an eﬀect analogous to the cosmological redshift of photons. (You may think of this as a stretching of the de Broglie wave˙ corresponding lengths.) Since H is not a constant, we get a force Hx to the Euler force. Finally there is the analogue of the centrifugal force, H 2 x . These extra terms appear in the Euler equation when we go to expanding coordinates. The main point is that the cosmological drag term inhibits the development of gravitational instability but it does not kill it completely in standard cosmological models. Rather, it converts the exponential growth to an algebraic growth. It appears that this feeble instability may suﬃce to produce the structures we see in the universe around us according to many simulations. Still, if you want to get your hands analytically on the way gravitational instability develops, as one may do for weak instabilities, it is best to consider another static conﬁguration of the mass.

Polytropic Slabs If, in equation (4), you treat Γ as a parameter and not necessarily the ratio of speciﬁc heats, you have what is called the polytropic gas law. The spherically symmetric, static, self-gravitating solutions of Eqs. (1)– (4), served as models of stars in the nineteenth century and they remain qualitatively useful even now. The disks in spiral galaxies may also be modeled as polytropes to good eﬀect and these are useful for studying their gravitational instability.

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As a prelude to studying ﬂuid dynamics in disks, one may study simpler ﬂattened objects such as polytropic layers or slabs. The simplest case is a static conﬁguration with the layer extending inﬁnitely in two directions that we may call horizontal. The third, or vertical, direction is given the designation z. Just as for a stratiﬁed atmosphere, we may write the hydrostatic condition, the only diﬀerence being that the gravity is not speciﬁed but is governed by the Poisson equation. Static solutions have ρ = ρ0 (z) with ρ0 (z) = ρ0 (0)CΓ (z)

(12)

where CΓ (z) is given by a simple integral that, for general Γ, is a beta function. Several special cases may identiﬁed, notably C1 = sech2 (kJ z),

Here kJ2

4πGρ0 = c2

C2 = cos(kJ z).

= z=0

4πG [ρ0 (0)]2−Γ . ΓK

(13)

(14)

The characteristic thickness of the slab is ∼ 1/kJ as before but now we have a static solution with no artiﬁce. The model is admittedly simpliﬁed but it has scope for interesting dynamics. For instance, the distribution of density on the midplane controls the layer thickness and this may vary in the rotating case so that we may ﬁnd Rossby waves propagating through disks. Ledoux [14] derived the marginal stability condition for linearized perturbations on the isothermal slab (Γ = 1). He found two horizontal wavelengths that are marginal, with khor = 2kJ and 0. Here is a situation in which we can make use of asymptotic approaches for the modes of long wavelength where the instability ﬁrst arises weakly. This is unlike the homogenous case, for which the maximum growth rate occurs at k = 0. The real case is even more favorable to this approach since disks of galaxies are embedded in very massive halos that make the disk thicknesses even less than the Jeans lengths of the disks. The halos themselves, though not visible, are generally believed to exist on the basis of their gravitational eﬀects. Their inﬂuence strengthens the validity of the the thin-layer approximation. Linear theory reveals that there are acoustic modes and gravity modes, just as in models of standard atmospheres. The surprise is that the instability occurs in the gravity modes and not in the acoustic modes as in the original Jeans problem. The gravity waves combine both thickness variations and true density variations, so have all that is needed. Since the largest scales are nearly marginal, one can assume slow times

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and long lengths to develop shallow layer theories [18]. With the additional restriction to small amplitudes, as in weakly nonlinear theories, one may derive nonlinear wave equations in the manners of Boussinesq and Korteweg and de Vries [19]. The familiar form of the theory is modiﬁed by a term representing the eﬀect of self-gravity that comes in through the Poisson equation. For instance, in the case of nonlinear waves of small amplitude in a thin layer with Γ = 2 (the easiest case) one ﬁnds for the surface deformation that: 1 (15) ηT − 3ηηX + ηXXX = µH[η] 2 where the Hilbert transform is 1 H[η] = P π

∞ −∞

η(Y ) dY Y −X

(16)

and X and T are suitably stretched variables. This equation has pole solutions, but we cannot say whether it is completely integrable. However, there are certainly solutions resembling solitons. The interest in such a result is that it has sometimes been thought that the highly dispersive nature of the waves in this kind of problem would not allow the formation of coherent structures. In fact, the nonlinearity in this problem leads to the formation of long-lived nonlinear waves. What happens in the more realistic case of a rotating disk? This is an issue that is unresolved as yet in the nonlinear case. The ﬁniteness of the disk brings discrete modes into play and they behave chaotically [4]. Still, some global order may emerge, perhaps with the help of other eﬀects.

2.

Astrophysical Vortices

The masses of stars range from 60-80 times the mass of the sun (4 × 1033 gm) down to a few tenths of a percent of the solar mass, that is, down to the masses of the giant planets. The massive stars have very hot atmospheres, (tens of thousands of Kelvins), while the low mass ones have much cooler atmospheres as a rule (6000 K for the sun). At both ends of this spectrum, atmospheric turbulence is observed. At the cool end, this is caused by thermal convection resulting from the lowering of the thermal conductivity through the raising of opacity by partially ionized hydrogen. The ionization of hydrogen also favors convection by raising the speciﬁc heat. In the atmospheres of the hottest stars, turbulence is detected through broadening of the spectral lines and it is often supersonic. The origin of this turbulence is not agreed upon, though

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there is no shortage of possible sources. Hot atmospheres are fully ionized and are not subject to thermal convection in the usual way, but they are rapid rotators and they pulsate. When the pulsation is vigorous, thermal convection may be driven parametrically. Moreover, hot stars have high radiation pressure that provokes instability and complicates the dynamics. Already at the qualitative level there are some interesting ﬂuid dynamical issues. We see spots on the sun and these are caused by magnetic ﬂux tubes that protrude from the solar surface. The ﬁelds are locally strong enough to inhibit the convective transport of heat outward and so relatively cool (but still quite warm) spots are produced. By contrast, in Jupiter’s atmosphere, a much cooler place, we ﬁnd evidence for vortex tubes at the surface. Since there is a full range of masses between the two limits (sun and Jupiter) we may someday be able to observe the transition between the two kinds of coherent structures, but there is no reason why this transition could not be studied theoretically at present, perhaps numerically. This is a potentially revealing instance of the transition between the purely ﬂuid and the magnetoﬂuid regimes. Related questions arise in the study of the accretion disks that form around condensed objects on many scales. These are rather diﬀerent from the disks in spiral galaxies. Accretion disks represent inﬂows of mass from various sources such as companion stars in the case of binary stars, to the ambient stars around massive black holes in the centers of galaxies. The primitive nebula that might have preceded the formation of our solar system, as Kant and Laplace ﬁrst suggested, are cooler examples of this kind of structure. As the matter ﬂows toward the central object, its net angular momentum makes itself felt and a disk is formed. Before it can settle into the central object, the inﬂowing matter must get rid of its angular momentum. Various mechanisms have been proposed for expelling the angular momentum, mostly calling on some form of turbulence though waves and magnetic ﬁelds have been considered. Vortices could also play a part in the process. There are some similarities here to the ﬂow around the polar vortex on earth that is central to the ozone problem, though accretion disks frequently are magnetized. The large scale ﬂow in accretion disks is governed mainly by Kepler’s laws so that the circular velocity √ around the central object in an axially symmetric disk varies like 1/ r. This represents a linearly stable shear and it is not yet decided whether nonlinear instability can occur in such a ﬂow. (I ﬁrst heard people arguing about that some thirty-ﬁve years ago.) However, as Chandrasekhar and others showed, magnetic ﬁelds can catalyze the conversion of the energy in the rotational ﬂow of disks

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into turbulence, but it was some time till the importance of this result for disks was appreciated [3]. Can the resulting disorder lead to the formation of vortices? Twenty-ﬁve years ago I put a drawing of a disk with whorls in it in an article on turbulence for an encyclopaedia aimed at twelve-year olds. The hope was that in ten years one of them would appear in my ofﬁce with a fully completed simulation that revealed what the disk really looked like. This did not happen, so the next message in a bottle was sent as a remark at the end of a paper on vortices in stars and planets [9]. This got a response from P.A. Yecko whose thesis revealed that largescale spiral vortices formed. (At the suggestion of A. Ingersoll, Yecko adapted a code written by E. Chassignet for simulating the oceanic thermocline.) Several astronomers objected that the Keplerian shear in disks would shred a vortex. The shredding is avoided by anticyclonic vortices which shield themselves with protective cocoons of reduced shear. Several subsequent simulations with higher resolution have shown their robustness [6, 12, 15]. A recent, 2-D, compressible simulation at high resolution by G. Murante and colleagues in Torino strikingly shows how a single anticyclonic vortex survives in the Keplerian shear, at least for the ten disk rotations they followed. That vortex also generated largescale spiral extensions in line with Yecko’s results. The issue of what magnetic ﬁelds do to these processes needs clariﬁcation as do theoretical questions about how vortices form. But their existence would play a role in forming observable inhomogeneities on disks [1]. An example of a disk simulation in a two-dimensional Keplerian ﬂow with slowly decaying turbulence is shown after [7] in Fig. 1.

Figure 1.

Vortices on a Keplerian Disk

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Photoﬂuiddynamics

The radiation that permits us to observe cosmic bodies also plays a role in their formation, structure and evolution. The thermal aspects of the radiation are familiar to ﬂuid dynamicists, at least qualitatively. What may be of more interest in an introduction to AFD is the way that the force of radiation on matter may inﬂuence the dynamics of the ambient material medium. The phenomena arising in this subject seem suﬃciently diﬀerent from ordinary ﬂuid dynamics that I have followed the advice of M. E. McIntyre and sought a suitably distinctive terminology. The name used here for the subject is inspired by Lighthill’s “Bioﬂuiddynamics.” What follows is a very brief introduction to some aspects of this subject. For some background on the equations see [16].

The Radiative Fluid The most direct derivation of the basic equations of ﬂuid dynamics subject to stresses from a coexisting radiation ﬁeld is by way of the transport equations for radiation and matter. These are kinetic equations that contain terms representing the interaction of the matter with the radiation. This somewhat technical problem is best formulated by treating the radiation ﬂuid as a gas of photons. Nevertheless, there are some real complications depend on the frequencies of the photons and the state of the matter. Such details are out of place here, so we simply assume that the matter is grey, that is, indiﬀerent to the frequencies of the photons passing through it. We shall also omit the details of the state of the matter such as the degree of ionization, which we shall take to be complete for very hot objects. So we may go straight to the ﬁrst two moments of the transfer equation which, for a gas of photons, are equations for the moments of the distribution function or speciﬁc intensity. Let E, F and IP be the energy density, energy ﬂux and pressure tensor of the radiation ﬂuid. These satisfy the two moment equations ∂t E + ∇ · F = interactions with matter

(17)

and (18) ∂t F + c2 ∇ · IP = −ρκc F −1 where (ρκ) is the mean free path of a photon. We see that the latter equation may be written as ρκ (19) ∇ · IP = − F + O(c−2 ). c Naturally, we face the usual problem of closing oﬀ the moment hierarchy and here we take the simplest closure, IP = 13 EIII . Thus we ignore

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the (sometimes important) radiative contribution to the viscous stress. In some limits, we may argue from (17) that the divergence of the ﬂux is quite small. (In eﬀect, we are leaving out retardation terms with this approximation.) This is about as simple as we can make this problem, yet much of interest remains as we see when we write the equation for the material ﬂuid as 1 ρ(∂u + u · ∇u) = −∇p − ∇E − ρ∇V. 3

(20)

To this we add the mass conservation equation and the radiative equations which have been reduced to the diagnostic conditions 1 ∇ · F = 0 and ∇E = ρκcF . 3

(21)

Radiatively Induced Instabilities To see in what kind of conditions radiative eﬀects may become important dynamically, consider the simple case of a plane-parallel medium stratiﬁed under gravity. The foregoing equations show that the hydrostatic condition is 1 dp 1 1 = − ∇E − ρ∇V = κcF − ρ∇V ρ dz 3 3

(22)

where z is the upward vertical coordinate. In the typical case where the ﬂow of radiation is outward, hence upward, the radiative force is one of levitation and it balances the gravitational attraction downward when the right side of this equation vanishes. That condition is called the Eddington limit. Near this condition, we ﬁnd instabilities of both sound waves and gravity waves [22]. These are induced by both the thermal and dynamical eﬀects of radiation and there is still room for a better physical understanding of these processes. Those familiar with ﬂuidized beds will recognize a commonality between the Eddington limit and the onset of ﬂuidization, although the medium being traversed by the radiation is a ﬂuid even below the critical condition. As in ﬂuidization, the material layers are rendered unstable by the traversing ﬂuid, though the details of the instability mechanisms do diﬀer. As in those cases where voids form in ﬂuidized beds, the radiative ﬂuid is much less dense than the particles of the medium. This and other arguments suggest that photon bubbles will form in hot stars near to the Eddington limit [17]. Approximate solutions for photon bubbles can be constructed in the way that this is done in the theory of ﬂuidization [20]. Related discussions have been given in the context

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of convection in the chimerical supermassive hot objects [23] and polar caps of pulsars where the magnetic ﬁelds are strong (1012 gauss) [2]. An interesting aspect of this process bears on the question of the lifetimes of objects that exceed the Eddington limit. In the case of ﬂuidized beds, ﬂow through the bed in excess of the value needed for ﬂuidization is observed to escape in voids, or bubbles. We may similarly expect cosmic bodies to survive above the Eddington limit. A modiﬁed limit needs to be calculated but another feature of this problem needs to be addressed ﬁrst.

Photovorticity The hottest objects are relatively rare and there are very few nearby. We see them because they are intrinsically luminous but it is hard to know whether they are like other rotating, turbulent cosmic bodies in forming vortices or concentrated magnetic ﬂux tubes. However, it is clear that the hot objects do rotate rapidly and are turbulent in their outer layers. Moreover, there are observational grounds for supposing that there are spots on hot stars [8] and disks [1]. What can cause spots in such conditions? Hot stratiﬁed media are unstable and, as for ﬂuidized beds, we may expect the formation of (photon) bubbles in objects near the Eddington limit. If we make a vortex in such conditions, we anticipate that, as in many laboratory experiments on rotating turbulence, bubbles are attracted into vortices. Indeed, it is the practice to use small bubbles as markers of vortices in such experiments. On the other hand, bubbles ﬂowing into a vortex will bring with them angular momentum and, when this is of the right sign, the vortex will be intensiﬁed. We have in such a situation the makings of an instability for vortex production analogous to what has been seen in laboratory experiments [24]. Vortex formation in hot media is of interest since it would lead to strong inhomogeneities in the emerging radiation ﬁeld that would have diagnostic implications. It would also be important in the ﬂuid dynamics of hot objects since a vortex of the right kind is a conduit through which radiation may escape from a hot object or disk without disrupting it. A simple calculation reveals the nature of this process. Let us omit the complications of global rotation and consider an isolated vortex in a stratiﬁed polytropic ﬂuid with radiation coming from below [9]. A standard vortex with gravity balancing the pressure gradient in the vertical direction and the centrifugal force in the (horizontal) radial direction is readily constructed. Then, in radiative equilibrium, we have E = aT 4 , where T is the temperature. For radiative prob-

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Figure 2.

Flow through a phortex

lems the parameter value Γ = 4/3 is frequently adopted. This value has the advantage of making ρ3 /T a constant and that greatly simpliﬁes the second of (21). That pair of equations is then readily solved and the streamlines of the radiative energy ﬂux F are as shown in Fig. 2; the contours of T are also indicated. This vortex provides not only a safety valve by which the radiation may escape but such a beam should make itself apparent in observations of hot stars and disks.

References [1] M.A. Abramowicz, A. Lanza, E.A. Spiegel, and E. Szuszkiewicz, Vortices on accretion disks, Nature, Vol. 356, 41, 1991. [2] J. Arons, Photon bubbles – Overstability in a magnetized atmosphere, Astrophys. J., Vol. 399, pp.561–578, 1987. [3] S.A. Balbus and J.S. Hawley, Instability, turbulence and enhanced transport in accretion disks, Rev. Mod. Phys., Vol. 70, pp.1–53, 1998. [4] N.J. Balmforth and E.A. Spiegel, Nonlinear ringing of polytropic disks, Physica D, Vol. 97, pp.1–28, 1996. [5] W.B. Bonnor, Jeans’ formula for gravitational instability, Mon. Not. Roy. Ast. Soc., Vol. 117, 104, 1957. [6] A. Bracco, A. Provenzale, E.A. Spiegel, and P.A. Yecko, Spotted Disks, [in:] Theory of Black Hole Accretion Disks, M. Abramowicz, G. Bjornsen & J. Pringle, [eds.], Cambridge Univ. Press, 1998. [7] A. Bracco, P.-H. Chavanis, A. Provenzale, and E.A. Spiegel, Particle Aggregation in Keplerian Flows, Phys. Fluids Vol. 11, 2280, 1999. [8] J.P. Cassinelli, [in:] The origin of Nonradiative Heating/Momentum in Hot Stars, A.B. Underhill and A.G. Michalitsianos, [eds.], (NASA 2358), pp.2–23, 1985.

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[9] T.E. Dowling and E.A. Spiegel, Stellar and jovian vortices, Ann. N.Y. Acad. Sci., Vol. 617, pp.190–216, 1990. [10] A.S. Eddington, The Mathematical Theory of Relativity, Cambridge Univ. Press., 1924. [11] A. Einstein, Kosmolosiche Betrachtung zur allgemeinen Relativit¨¨atstheorie, Berlin: Sitzungberichte, p.149, 1917. [12] P. Godon and M. Livio, On the nonlinear hydrodynamic stability of thin keplerian disks, Astrophys. J., Vol. 521, 318, 1999. [13] J.H. Jeans, Astronomy and Cosmogony, Cambridge Univ. Press., 1928. [14] P. Ledoux, Sur la Stabilit´ ´e Gravitationelle d’Une Nebuleuse Isotherme, Ann. d’Astrophys., Vol. 14, pp.438–447, 1951. [15] H. Li, S.A. Colgate, B. Wendroﬀ, and R. Liska, Rossby wave instability of thin accretion disks. III. Nonlinear simulations, Astrophys. J., Vol. 551, pp.874–896, 2002. [16] D. Mihalas and B.W. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, 1984. [17] K.H. Prendergast and E.A. Spiegel, Photon Bubbles, Comments on Astrophys. and Space Phys., Vol. 5, 43, 1973. [18] Z.-S. Qian, E.A. Spiegel, and M.R.E. Proctor, The gravitational instability of a gaseous slab, Stability and Applied Analysis of Continuous Media, Vol. 1, 33, 1990. [19] Z.-S. Qian and E.A. Spiegel, Autogravity waves in a polytropic layer, Geophys. & Astrophys. Fluid Dyn., Vol. 74, pp.225–244, 1994. [20] E.A. Spiegel, Photoconvection, [in:] Problems in Stellar Convection, E.A. Spiegel and J.-P. Zahn, [eds.], Springer-Verlag, 1977. [21] E.A. Spiegel, Gravitational screening, On Einstein’s Path: Essays in Honor of Engelbert Schucking, A. Harvey, [ed.] (Springer-Verlag, N.Y.), Chapt. 32, pp.465– 474, 1998. [22] E.A. Spiegel and L. Tao, Photoﬂuid instabilities of hot stellar envelopes, Phys. Rep., Vol. 311, pp.163–176, 1999. [23] V.S. Thorne, Thesis, Univ. of Manchester, 1968. [24] J.S. Turner and D.K. Lilly, The carbonated-water tornado vortex, J. Atmos. Sciences, Vol. 20, pp.468–471, 1963.

MINIATURIZATION OF EXPLOSIVE TECHNOLOGY AND MICRODETONICS D. Scott Stewart University of Illinois Department of Theoretical and Applied Mechanics [email protected]

Abstract

Condensed phase explosives used in conventional explosive systems have a charge size on the order of a meter or a sizable fraction of a meter. We discuss a range of issues, theoretical, computational and experimental, required to scale the size of explosive systems downwards by a factor of one hundred to one thousand, applications and prospects for a ubiquitous new technology.

Keywords: Detonation, shock physics, compressible ﬂow, miniaturization, explosives, new technology

1.

Introduction

A detonation is a chemical reaction driven shock wave in molecularly premixed material called an explosive. The chemical energy released in the reaction zone behind the lead shock is converted into kinetic energy and pressure/volume work done by the reactants. Explosives can be gases, liquids and solids. Detonation pressures in organically based condensed phase explosives (typically made from nitrated hydrocarbons) are in the range of 300–400 Kbar (30–40 GPa), and can potentially induce hundreds of Kbars of pressure in inert materials for fractions of microseconds. Detonation shock speeds are on the order of 3–10 kilometers/sec. The thermodynamic cycle and high pressure, high compression states that can be induced in materials are unlike those that can be obtained with other thermo-mechanical systems, including lasers. Hence detonative processes oﬀer unique methods of altering the state of material surfaces and can serve as a high energy density source for microdevices. Properly engineered, stable explosive detonation fronts work in combination by a principle of synchronicity (i.e. the detonation is a phase-controlled explosion front) and detonations can generate precise 379 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 379–385. © 2005 Springer. Printed in the Netherlands.

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motion-controlled ﬂows that can be used for materials processing and other applications.

2.

Applications

While condensed phase explosives are used in military, mining and demolition applications, other less commonly known applications of explosives include their use for materials processing, precision cutting and pulsed power application. Speciﬁcally, detonation of explosive ﬁlms can be used in surface treatment and hardening of materials. Other material processing applications include cladding and explosive welding, sintering, shock consolidation of powders and shock-induced chemical synthesis. Pulsed power applications include magnetic ﬂux compression, pulsed detonation engines, explosive lasing and the generation of extremely high intensity light pulses. There are biomedical applications for detonation of micro-sized explosive charges that include lithotripsy and localized destruction of pathological tissue [1]. Explosive and pyrotechnic elements pervade satellite and aerospace systems and hence there is interest in the miniaturization of explosive systems for microaerospace and satellite platforms. Suitably controlled detonation fronts represent a basic technology with unique aspects. By establishing the basic parameters of micro-scale explosive systems it should be possible to design micro-scale devices for welding, cladding, pulsed power, surface treatment and so on (as mentioned above) in novel, ubiquitous and unforeseen ways. Micro-explosive systems hold the promise of being a basic enabling technology with wide-spread application. Figure 1 shows a sketch of a experimental conﬁguration being designed at the University of Illinois, in collaboration with Prof. W. Kriven (Material Science and Engineering, UIUC) and Prof. R. Adrian (TAM, UIUC), to study high pressure, temperature synthesis of ceramic materials. The initiator consists of a capacitance discharge unit (CDU) that ﬁres a 10–micron thick wire (typically gold) or metallic ﬁlm embedded in the detonable ﬁlm. The electrical current dump causes the metal to expand from a nominally cylindrical or ﬂat source as plasma and drive a shock wave into the ﬁlm to start the chemical reaction in the ﬁlm. The detonation supported shock wave sweeps across the sample and the detonation shock drives an inert shock into the donor material to do the localized processing of near surface material. Other initiation conﬁgurations include laser driven micro-ﬂyers that induce shocks to start reaction.

Miniaturization of Explosive Technology and Microdetonics

Figure 1.

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Micro-explosive system for materials processing of ceramics

Explosive System Scaling Arguments and Requirements

An explosive system includes the main charge (the secondary explosive), the initiation system (which includes the initiation train and booster made of primary explosives or electrical or optical laser initiators) and the inerts, upon which the explosive products act. Conventional macro-scale explosive system design paradigms exist for explosive systems that have dimensions on the order of a meter or sizable fraction of a meter. The question, what are the scaling principles for smallscale design that are consistent with existing large-scale design, can be addressed by dimensional analysis, based on the Euler equations and consideration of matching shock initiation and propagation experiments of small-scale systems to their to large-scale counterparts. Scaling arguments, [2] show that extreme miniaturization by a scale reduction of current large-scale explosive systems by a factor of 100 to 1000 is possible. To employ existing large-scale design rules, the detonation reaction zone length scale must scale with the device dimension size. Short reaction-zone explosive materials (with small critical diameters) must be used for main charges. This means that one must select the main charge explosives from the list of primary explosives (used in large-scale initiator trains or detonators). Also one might consider using very short

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reaction zone explosives that have never been considered for use in the past, because of safety considerations. The other route to miniaturization uses explosive materials for propagation in sub critical charge dimensions. One expects to experience signiﬁcant transients that do not fall in the existing quasi-steady design paradigms. This route requires a detailed understanding of transient detonation propagation.

4.

A deﬁnition of Microdetonics and the Initiation of Small Systems

This brings us to a deﬁnition of the term “microdetonics” that we attribute to James E. Kennedy of Los Alamos National Laboratory. Kennedy’s deﬁnition of microdetonics is the detonation physics behaviors that are dominated by transient eﬀects such as detonation acceleration, detonation spread and curvature eﬀects that are commonly associated with initiation of explosives by small sources. This deﬁnition includes both small charges and small initiation sources, where transient phenomena is dominant. Reliable and safe initiation systems for miniaturized systems can be built using existing exploding wire and exploding foil initiation systems with existing, well-understood electrical designs. Initiation energy is stored in a standard capacitance discharge unit. It is also possible to build optical initiation systems whereby energy is transmitted through a optical ﬁber to the explosive charge. Initiation system can be placed on chips, [3] and designed with standard photo-lithographic techniques, [4].

5.

New Science Needed to Enable the Technology

In order to deﬁne the properties of the new explosive materials there is a need for a comprehensive linear and nonlinear stability theory for nonideal detonation that can incorporate non-ideal equation of state and realistic reaction rate laws for condensed explosives. Recent eﬀorts are underway to develop novel nano-engineering composite energetic materials and explosives that can be candidates for the miniaturized secondary charge. An entirely new linear stability theory for steady detonation has been developed by us at Illinois, [5] to guide design of miniaturized explosive systems in a rational way that incorporates descriptions of nonideal equation of state and reaction rate kinetics. In order to deﬁne detonation propagation in small dimensions, one must understand the critical conditions required for ignition and propagation of detonation for both ideal and nonideal explosives [6]. This

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includes the development of asymptotic theory for fast and sensitive chemical kinetics. It is important to understand aspects of detonation and shock diﬀraction and how that phenomena aﬀects successful detonation propagation.

High Resolution Multi-Material Simulation Technology is Required Design of integrated systems requires modern high resolution, multidimensional and multi-material, time-dependent simulation. High ﬁdelity simulation is an essential tool that is required to specify the geometry and select materials for miniaturized explosive system. Figure 2 shows a recent simulation of a “corning turning” experiment carried out by E. Ferm of Los Alamos National Laboratory. The corner turning experiment measures the transient eﬀect of the change in conﬁnement of detonation. The initial conﬁguration of Ferm’s experiment has a 6 mm radius, 125 mm long cylindrical stick (donor charge) of explosive PBX-9502 joined to a wider/shorter 25 mm radius, 50 mm long (acceptor charge) cylinder of PBX-9502. The detonation is started and travels as a curved steady detonation in the donor. Once the detonation

–Total length 200 mm - 6mm radius PBX-9502 donor - 150 mm long - 25 mm radius PBX-9502 acceptor - 50 mm long - Density plot near break-out

Detonation shock

"dead zone" with unreacted explosive in diffraction region Figure 2. MULTIMAT Simulation: The density record of a corner turning experiment that shows the eﬀects of detonation diﬀraction and the appearance of dead zones

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in the donor charge enters the acceptor, the detonation must expand into the large acceptor region. Because the lateral boundary of the acceptor charge is perpendicular to the axial propagation direction of the detonation in the donor and that the lateral boundary is unconﬁned, a large depressurization (rarefaction) of the detonation occurs. When the reaction rate of the explosive is pressure sensitive, the depressurization eﬀectively slows or stops the reaction in the region aﬀected by the corner turning diﬀraction event. Hence a large region appears where the explosive does not burn. That region is dubbed a “dead zone”. Instead of reacting, the explosive in the dead zone is simply shocked. For microcharges or for initiation by small sources, the corner turning experiment is a generic conﬁguration that must be studied in detail. Figure 2 is a snapshot of a density record from a simulation carried out by members of our group (D. S. Stewart, B. Wescott and S. Yoo) with our UIUC-code MULTIMAT. Our simulation shows the detonation after the detonation in the donor has entered the acceptor. The explosive charges are initially adjacent to a very low density inert material (shown to the right). As the simulation progresses the material interface between the explosive products and the inert material expands. Our simulation show the appearance of large dead zone preceded by a low pressure shock in the diﬀraction region nearest the corner, connected to a fully emergent detonation in center and conforms closely to Ferm’s experiment. The code MULTIMAT uses high-resolution (4th order in space and 3rd order in time) compressible reactive ﬂow solvers combined with a modern level-set treatment that represent interfaces to enable multi-material simulation required for microdetonic devices.

High Speed Measurement and Other New Areas of Mechanics Research The events of explosive technology take place at the limits of conventional experimental methods that measure mechanical quantities. Microdetonics is an area whose investigation will stimulate the creation of new measurement technologies. Recently our colleague, R. Adrian and his co-workers are working to develop PIV systems, [7] that can make capture motion events generated by shocks in optically accessible solids. These test solids can be used as a measuring instrument and a “full-ﬁeld” witness plate to capture the energy and momentum transfer from adjacent shock materials that are under investigation. Unlike lower speed PIV systems that take full-ﬁeld velocity measurements in water or air, full ﬁeld measurements in the interior of solids have not been available experimentally. This emerging measurement technology should allow

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for unprecedented improvements in understanding load transfer at the interface between materials. The physics of exploding wire and foils involves the coupling of the mechanics of phase transformations and the magneto hydrodynamics. Target ceramic materials require new thermo-mechanical theories for ceramic formation under rapid high pressure/temperature loading. The theory of critical energy and pulse duration for initiation of detonation is at the heart of fundamental questions in reactive ﬂow science. It is very likely microdetonic devices can be made for use for other basic material property investigations of a fundamental nature. In short, the area of microdetonics and the creation of precisely controlled miniaturized explosive systems will surely be coupled to fundamental advances in thermomechanics and have an impact on areas of technology that can use precision, high energy density output sources applied to materials with precision.

Acknowledgments This work is supported by the US Department of Energy, DOE/LANL and the US Air Force Research Laboratory, AFOSR-Mathematics and Munitions Directorate, Eglin AFB, Florida.

References [1] K. Takayama, T. Saito, Shock Wave/Geophysical and Medical Applications, Annual Review of Fluid Mechanics, Vol.36, pp.347-379, 2004. [2] D.S. Stewart, Toward Miniaturization of Explosive Technology, Shock Waves, Shock Waves, Vol.11, pp.467-473, 2002. [3] T.A. Baginski, S.L. Taliaferro, D.W. Fahey, Novel Electroexplosive Device Incorporating a Reactive Laminated Metallic Bridge, Propulsion and Power, Vol.17, No.1, 2001. [4] A.S. Tappan, A.M. Renlund, G.T. Long, S.H. Kravitz, K.L. Erickson, W.M. Trott, M.R. Baer, Microenergetic processing and testing to determine energetic material properties at the mesoscale, Proceedings of the 12th International Symposium on Detonation, San Diego, USA, 2002. Available online at http://www.sainc.com/onr/detsymp/detsymp2002/technicalProgram.htm. [5] A.R. Kasimov, D.S. Stewart, B.L. Wescott, Sunhee Yoo, Linear Instability Analysis of Non-Ideal, Condensed Phase Detonation, University of Illinois, submitted for publication, 2004. [6] A. Kasimov, D.S. Stewart, Asymptotic theory of the evolution and failure of self-sustained detonations, TAM Report No. 1042 UILU-ENG-2004-6003, ISSN 0073-5264, to appear in the Journal of Fluid Mechanics, 2004. [7] M.J. Murphy, R.J. Adrian, D.S. Stewart, G.S. Elliott, K.A. Thomas, J.E. Kennedy, Visualization of blast waves created by exploding wires, submitted to the Journal of Visualization.

FOAMS IN MICROGRAVITY Denis Weaire and Simon Cox Department of Physics, Trinity College, Dublin 2, Ireland [email protected]

1.

Introduction

Ordinary aqueous foam, which is our main subject in this paper, needs no introduction. Who has not taken a few minutes to study its beautiful structure (Fig. 1) and to watch it change? If you do so, you may study it from at least three diﬀerent perspectives:

Figure 1.

An aqueous foam as seen by the photographer-artist Michael Boran.

387 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 387–394. © 2005 Springer. Printed in the Netherlands.

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as a compacted heap of individual bubbles of widely varying shape and size. as a division of space by conjoined soap ﬁlms, all slightly curved. as a network of lines, the so-called Plateau borders where the ﬁlms meet. The ﬁrst description relates to the manner of formation of the foam, which can even be designed to produce highly monodisperse foams. The second is the key to its stability: how long will the ﬁlms survive without rupture? The third description often comes to the fore in theories of physical properties, such as conductivity, drainage or the mechanics of solidiﬁed foam. Plateau(1873) gave us the ﬁrst coherent account of the basic rules to which the structure must conform, particularly for relatively dry foams, of low liquid content. Underlying these rules is the essential principle of the minimization of surface energy (usually under the constraint of ﬁxed bubble volumes). Indeed, most of the static and quasi-static properties of a foam may be explained by arguments which derive from that principle. It entails the Laplace-Young law (which we should celebrate in 2004/5 as this is its bicentenary year), and Plateau’s rules for the junctions of ﬁlms and borders (Weaire and Hutzler(1999)). As the liquid content is increased, some of Plateau’s strictures are relaxed. Whereas only fourfold conﬂuences of borders are possible for the ideal dry foam, higher numbers may come together in stable junctions of the wet foam. We have not yet grasped the complexities of wet foam structures, except perhaps in two dimensions. Even in the carefully chosen special case of the symmetric eight-fold junction, shown in Fig. 2, progress has been slow since Weaire and Phelan(1996) raised questions

Figure 2. This wet eight-fold junction of Plateau borders is stable until the liquid fraction is reduced to an exceedingly small value, as shown by Brakke. Image courtesy of K. Brakke.

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about it, on the basis of experiment (see in het Panhuis et al.(1998)), neatly following the tradition of Plateau. Very recently, Ken Brakke has completed a masterful analysis which leads to the conclusion that a ﬁnite but exceedingly small liquid fraction Φ is required to stabilize this junction.

2.

The Surface Evolver

Brakke is the originator and chief exponent of the Surface Evolver (Brakke(1992)), a suite of software which has been applied to such static problems. Its impact on this ﬁeld is only part of a wider inﬂuence, whenever surface energy is dominant in physics and engineering (e.g. Collicott and Weislogel(2004)). As the crystallographer Alan Mackay has said The Evolver is a spectacular example of the eﬀects of a gift to science which advances a whole ﬁeld.

Figure 3 gives some further examples of the applications of the Evolver undertaken by our group. The most celebrated of these is the 1994 discovery (Weaire and Phelan(1994)) of a structure of monodisperse dry

Figure 3. Examples of the use of the Surface Evolver. (a) The Weaire-Phelan structure which is conjectured to ﬁll space with the lowest surface area. (b) A ﬁnite cluster of bubbles, used to investigate their local structure (Cox and Graner(2004)). (c) The meniscus surrounding a single bubble trapped between a glass plate and a liquid pool (Vaz et al.(2004)). (d) When two drops of oil are squeezed between parallel plates, there is a symmetry-breaking instability (e) at a certain critical separation (Bradley and Weaire(2001)).

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foam that has a lower surface energy than that conjectured by Lord Kelvin (Thomson(1887); Weaire(1994)). This structure has two diﬀerent bubble shapes that ﬁt together to form a structure of overall (body-centred-) cubic symmetry. It is about to be manifested in a spectacular building for the Beijing Olympics – the Water Cube – illustrated in Fig. 4. Its construction is based on a network of steel beams, corresponding to the Plateau borders of the WeairePhelan structure. In analyzing its stability, the engineers must have repeated the kind of exercise undertaken by materials scientists for opencelled solid foams such as polyurethane (Warren and Kraynik(1991)). It should prove to be an inspiring instance of the harmony of scientiﬁc and aesthetic principles.

Figure 4. The designer’s vision for the Beijing National Swimming Centre for the 2008 Olympics. The interior of its transparent walls consist of the Weaire-Phelan structure of Fig. 3(a). Image courtesy of Arup, PTW and CSCEC.

3.

Debates Over Drainage

As anticipated in the closing chapter of the book of Weaire and Hutzler(1999), the focus of foam physics has moved from static to dynamic properties, related to drainage and rheology. Drainage is the passage of liquid through the foam (mainly through the Plateau borders), driven by gravity or by pressure gradients. Its main properties are captured by a simple continuum theory expressed in a nonlinear partial diﬀerential equation. Suppose we pour liquid steadily into the top of a foam: how fast does it travel downwards under gravity? Assuming Poiseuille ﬂow in the

391

Foams in Microgravity

borders, a relation may derived for which the velocity is v ∼ Q1/2 ,

(1)

where Q is the ﬂow-rate of added liquid. Although supported by several experiments (Weaire et al.(1997)), this was challenged (Koehler et al.(1999)), in favour of an index of one-third. It transpired that the discrepancy between the old and the new results lay in the use of diﬀerent surfactants to stabilize the foam. Some of these produce relatively rigid surfaces (hence Poiseuille ﬂow) while others leave the surfaces mobile (Durand et al.(1999)). In future, we will be more cautious in asserting generic properties!

4.

Getting Rid of Gravity

Beyond a certain ﬂow-rate, the steady drainage described above becomes unstable, giving rise to a slow convective motion (Hutzler et al. (1998), Vera et al. (2000), Weaire et al. (2003)). It therefore cannot be used as a proxy for the equilibrium structure of a very wet foam, which was part of the original motivation for its study. How then are we to prepare such uniform wet foams? A static foam under gravity has only a very limited height of wet foam (if any) at the bottom. It may be estimated to extend to the height h=

l02 , d

(2)

where d is the bubble diameter and the capillary length is l02 =

σ . g∆ρ

(3)

Here σ is the surface tension, ∆ρ the density diﬀerence between gas and liquid and g the acceleration due to gravity.

Figure 5.

Strategies for preparing and studying wet foams.

392

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The rest of the foam is essentially dry. Hence the ease of preparing dry foams and our frustration in wishing to study wet samples. Several strategies present themselves, and are summarized in Fig. 5. Of these, density-matched emulsions have been used by Mason et al.(1995) and others to probe static properties which are common to foams and emulsions, in theory. Another strategy is to use small enough bubbles that the wet foam thickness in Eq. (2) above becomes considerable. Thirdly, we may use steady drainage or attempt measurements in a short time-scale, before drainage has developed (Saint-Jalmes and Durian(1999)). All of these suﬀer from limited applicability: hence the appeal of getting rid of gravity altogether, in drop towers, parabolic ﬂights, rockets or space-stations.

5.

Foams under Microgravity

An honoured mention should be made of early space microgravity experiments on foams, particularly by David Noever (1994) and Noever and Cronise (1994), but these isolated eﬀorts did not result in any coherent progress. Following the creation and operation of a Topical Team for this subject by the European Space Agency, there is some hope of a more systematic approach. Currently the subject is being tackled in two ESA-sponsored projects. The ﬁrst aims to utilize microgravity conditions to create homogeneous metallic foams (W¨ u ¨bben et al.(2002)). An example of a metallic foam is shown in Fig. 6. These materials are proving their potential in, for example, the automobile industry. The foamed metal should not suﬀer from drainage while in its liquid state, since this would lead to variations of density within the solid product (Banhart and Weaire(2002); Cox et al.(2001)). Obviously, microgravity sidesteps that limitation. The second project is a study of wet aqueous foams in equilibrium, so that the processes of drainage, rheology and coarsening due to gas diﬀusion can be examined independently (Saint-Jalmes and Langevin(2004)).

6.

Conclusions

For its satisfactory completion, the basic theory of the physics of foams needs to be extended to wet foams, initially in a state of static equilibrium. Describing dynamic wet foams will still be a considerable challenge. But it will reward success: churning, ﬂowing wet foams lie at the heart of the chemical industry. As Weaire and Hutzler(1999) said, throwing down the challenge, a walk by the seaside on a stormy day is

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Figure 6. An example of foamed zinc. The bubbly melt must be solidiﬁed quickly to retain the homogeneous structure, a stricture not required in the microgravity environment. Image courtesy of J. Banhart.

enough to excite curiosity. But it may be that the calmer environment of space is needed for the ﬁrst progress in understanding wet foams.

Acknowledgments Tristram Carfrae (Arup) kindly provided details of the Water Cube (Fig. 4). We wish to thank the European Space Agency for support under ESA Contract 14308/00/NL/SH (AO-99-031) CCN 002 MAP Project AO-99-075.

References [1] J. Banhart and D. Weaire, On the road again: metal foams ﬁnd favor, Physics Today, July:37–42, 2002. [2] G. Bradley and D. Weaire, Instabilities of Two Liquid Drops in Contact, Comp. Sci. Eng., Sept/Oct:16–21, 2001. [3] K. Brakke, The Surface Evolver, Exp. Math., 1:141–165, 1992. [4] S.H. Collicott and M.M. Weislogel, Computing existence and stability of capillary surfaces using surface evolver, AIAA J., 42:289–295, 2004. [5] S.J. Cox, G. Bradley, and D. Weaire, Metallic foam processing from the liquid state: the competition between solidiﬁcation and drainage, Euro. J. Phys: Appl. Physics, 14:87–97, 2001.

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[6] S.J. Cox and F. Graner, Three-dimensional bubble clusters: shape, packing and growth-rate, Phys. Rev. E, 69:031409, 2004. [7] M. Durand, G. Martinoty, and D. Langevin, Liquid ﬂow through aqueous foams: From the Plateau border-dominated regime to the node-dominated regime, Phys. Rev. E, 60:R6307–R6308, 1999. [8] S. Hutzler, D. Weaire, and R. Crawford, Convective instability in foam drainage, Europhysics Lett., 41:461–465, 1998. [10] S.A. Koehler, S. Hilgenfeldt, and H.A. Stone, Liquid ﬂow through aqueous foams: The node-dominated foam drainage equation, Phys. Rev. Lett., 82:4232–4235, 1999. [11] T.G. Mason, J. Bibette, and D.A. Weitz, Elasticity of Compressed Emulsions, Phys. Rev. Lett., 75:2051–2054, 1995. [12] D.A. Noever, Foam Fractionation of Particles in Low Gravity, J. Spacecraft Rockets, 31:319–322, 1994. [13] D.A. Noever and R.J. Cronise, Weightless bubble lattices: A case of froth wicking, Phys. Fluids, 6:2493–2500, 1994. [9] M. in het Panhuis, S. Hutzler, D. Weaire, and R. Phelan, New variations on the soap ﬁlm experiments of Plateau. I. Experiments under forced drainage, Phil. Mag. B, 78(1):1–12, 1998. [14] J.A.F. Plateau, Statique Exp´rimentale ´ et Th´eorique des Liquides Soumis aux Seules Forces Mol´culaires ´ , Gauthier-Villars, Paris, 1873. [15] A. Saint-Jalmes and D.J. Durian, Vanishing elasticity for wet foams: Equivalence with emulsions and role of polydispersity, J. Rheol., 43:1411–1422, 1999. [16] A. Saint-Jalmes and D. Langevin, The foam experimental module for the ISS, Microg. Sci. Tech., in press, 2004. [17] W. Thomson, On the Division of Space with Minimum Partitional Area, Phil. Mag., 24:503–514, 1887. [18] M.F. Vaz, S.J. Cox, and M.D. Alonso, Minimum energy conﬁgurations of small bidisperse bubble clusters, J. Phys.: Condens. Matter, 16:4165–4175, 2004. [19] M.U. Vera, A. Saint-Jalmes, and D.J. Durian, Instabilities in a liquid-ﬂuidized bed of gas bubbles, Phys. Rev. Lett., 84:3001–3004, 2000. [20] W.E. Warren and A.M. Kraynik, The Nonlinear Elastic Behavior of Open-Cell Foams, J. Appl. Mech., 58:376–381, 1991. [21] D. Weaire, editor, The Kelvin Problem, Taylor & Francis, London, 1994. [22] D. Weaire and S. Hutzler, The Physics of Foams, Clarendon Press, Oxford, 1999. [23] D. Weaire, S. Hutzler, S. Cox, N. Kern, M.D. Alonso, and W. Drenckhan, The Fluid Dynamics of Foams, J. Phys.: Condens. Matter, 15:S65–S73, 2003. [24] D. Weaire, S. Hutzler, G. Verbist, and E. Peters, A review of foam drainage, Advances in Chemical Physics, 102:315–374, 1997. [25] D. Weaire and R. Phelan, A counter-example to Kelvin’s conjecture on minimal surfaces, Phil. Mag. Lett., 69:107, 1994. [26] D. Weaire and R. Phelan, Vertex instabilities in foams and emulsions, J. Phys.: Condensed Matter, 8:L 37–L 43, 1996. [27] T. W¨ u ¨ bben, J. Banhart, and S. Odenbach, Production of metallic foam under low gravity conditions during parabolic ﬂights, Microgravity Sci. Tech., 13:36–42, 2002.

Author Index On the following pages authors and co-authors of all presentations are listed. The extended summaries of the presentations can be found on the enclosed CD-ROM. Each name is followed by sequence of entries consisting of session and presentation IDs. For the lectures contained in the present book, the appropriate page number is also given, and the presenting author entry is additionally highlighted by boldface letters. IDs of contributions of presenting authors are printed in upright typeface. Only eligible presentations are listed, i.e. presented during the Congress, excluding multiple presentations and those given by a proxy.

396

Abbott J.R. — FM16 11725

Abdalla H.M. — FSM6 11806 Abdou L. — SM2 12082 Abdul-Latif A. — SM4 10456 Abdunabi T.A. — SM25 11082 Abed F.H. — SM18 10393 Abedi R. — SM1 12441 Abkarian M. — FM1 12423 Aboudi J. — MS1 12011 Acartuerk A. — SM15 11194 Acharya A. — SM18 11820 Acrivos A. — FM16 10361 FM4 10753 Adachi S. — FM25 12041 Adachi T. — FM5 11864 Adams N.A. — FM24 10455 FM24 11256 FM24 11802 FM24 12564 FSM6 10399 Adda-Bedia M. — SM9 10338 Addas K. — MS2 10709 (p. 355) D’Addetta G.A. — SM4 11414 Aderogba K. — SM13 10745 Adrian R.J. — FM25 12161 FM25 12196 MS4 11607 Agapov D. — SM17 11688 Agarwal R.K. — FM11 10872 Aghalovyan L.A. — SM6 11526 Ahcene B. — FSM4 13050 Ahlawat A.S. — SM3 10098 Ahmetolan S. — SM11 10891 Aider A.A. — FM13 10185 Aidun C.K. — FM1 12124 Aifantis K. — SM13 11782 Aitta A. — FM21 11328 Aksel N. — FM14 10642 FM14 10928 Akylas T.R. — FM26 10762 Al-Abduljabbar A. — SM9 12798 Albers B. — SM11 10072 Alderliesten R. — SM8 12217 Alexander J.I.D. — MS5 12447 Alexeev A. — FM14 12293 FSM1 12516 Ali R. — FM2 11887 Alibiglu A. — SM13 13009 Alleborn N. — FM14 11413 Allix O. — SM1 12237 Almeida S.F.M. de — SM13 12148 Altenbach H. — SM27 11368 Altobelli S.A. — FM18 10983 Alvarez J.O. — FSM1 12477 Amalia P. — SM17 11352 Amberg G. — FM21 12312 Ambr´ ´ osio J. — SL1 11040 (p. 61) Ammann M. — SM15 10792 Amyot O. — FM12 11343 Anderson D.M. — FM21 12768

ICTAM04 Andersson L.-E. — SM2 10686 Andreev O. — FM19 11120 Andreeva T.A. — FSM1 12149 Andrejczuk M. — FM9 12592 Andrews M.G. — SM13 11162 Andrianov I.V. — SM13 12588 Antkowiak A. — FM13 12483 Antretter T. — SM14 10088 Anweiler S. — FM20 11745 Arakelian V. — SM16 10553 Arbez P. — FM6 12510 Aref H. — FM25 11676 FSM7 10003 Argyriadi K. — FM14 10557 Arias I. — SM9 12171 Ario I. — SM22 12688 Ariza M.P. — FSM6 11471 Arora M. — FM8 11999 Arroyo M. — SM1 12455 Asai M. — FM13 10489 Aschenbrenner C. — FM13 10188 Ashawesh G.M. — SM25 11082 Ashida F. — MS1 12018 Ashmore J. — FM21 12276 Askes H. — SM12 11277 SM1 12924 Asmolov E.S. — FM16 11480 Astley J. — FSM1 11249 Atakhorrami M. — MS2 10709 (p. 355) Aubouy M. — FSM5 11095 Auregan Y. — FM24 12740 Aurnou J.M. — FM7 11440 Auslender F. — SM13 11507 Aussillous P. — FM21 11320 Avalos-Zuniga R.A. — FM19 12120 Avramov K.V. — SM25 12989 Awrejcewicz J. — SM25 12989 SM2 11481

Baaijens F.P.T. — SM18 11344

Baars A. — FM10 13048 Babinsky H. — FM5 10984 Babushkin I.A. — FM7 12267 Bachorz P. — SM25 12770 Badel P.-B. — SM4 12913 Bai X. — SM9 12174 Bai Y. — FSM7 11765 SM7 11764 Baillis C. — SM22 11174 Bajer K. — FM25 12275 SL2 10042 (p. 89) Balamurugan V. — MS1 12147 Balandin D.V. — SM3 11219 Baldessari F. — FM8 11289 Balint A.M. — FM21 11362 Balint D.S. — MS3 12386

397

Author Index Balint S. — FM21 11362 Baltov A.I. — SM18 11953 Balueva A.V. — SM9 11856 Bambrey R.R. — MS6 11336 Banach Z. — FSM6 11237 Banaszak J. — SM15 13006 Banaszek J. — FM21 11364 Banerjee A. — SM17 11375 Banta R.M. — MS6 10513 Bar-Lev O. — FM17 12308 Bar-Yoseph P.Z. — FM13 11461 Baracho Neto O.G.P. — SM25 11494 Baradat C. — SM16 10553 Barba L.A. — FM6 11937 Barbe F. — SM14 12421 SM14 12747 Barber T.J. — FM14 11477 Bargmann S. — SM1 12924 Barkley D. — FM13 12431 Barmin A.A. — FM9 11703 Barnea D. — FM20 11655 Baroud C.N. — FM1 11013 MS4 12047 SL18 10495 Barral S. — FM19 12363 Barrau J.-J. — SM13 11180 Bartel T. — SM14 12077 Barth´ ´es-Biesel D. — MS2 11961 Batra R.C. — SM10 12594 Baty H. — FM19 11429 Bauer D. — SM15 12614 Bauruelle J.C. — SM22 11174 Baweja M. — SM3 12756 Bayer I.S. — FM4 11603 Bazant M.Z. — MS4 11777 Bazin B. — FM12 10234 Baldyga J. — FM22 12415 Becache E. — SM2 12382 Becker M. — FSM6 12118 Becker W. — SM15 11463 SM9 11484 Beda P.B. — SM12 10314 Bednarz T. — FM7 13022 Behringer R.P. — FM17 12399 Behringer R. — FM12 12617 Belcher S.E. — FM26 10746 Belzons M. — FM17 11751 Ben Dhia H. — SM11 12763 Ben Hamida A. — SM1 12727 Benallal A. — SM12 10265 Benilov E.S. — FM26 11468 Benilov E. — FM25 11467 Bennacer R. — FM15 11386 Bennett D.J. — FM4 10753 Benzerga A.A. — SM1 12737 Berbenni S. — SM18 11498 Beresneva E.N. — FM7 10538

Berezovski A. — SM10 11592 SM14 11393 Berﬁeld T. — MS1 11685 Berg A. van den — FM1 12400 Bergdorﬀ M. — FM6 12583 Bergeon A. — FM22 12731 Bergerot A. — SM13 11180 Bergman L.A. — SM25 10408 Bergmann R. — FM10 11909 FM17 10253 Bergougnoux L. — FM16 11588 FM16 12345 Bernardo M. di — FSM2 11363 Beron-Vera F.J. — MS6 12776 Berton G. — FM17 12883 Bertram A. — SM18 12549 Beskos D.E. — SM1 12845 Bessonnet G. — SM17 12539 Betelu S.I. — FM8 12548 Beysens D.A. — SL3 10880 (p. 117) Bezpalcova K. — FM9 10467 Bhattacharya K. — MS1 11731 SM14 10721 Bi W. — FM17 12883 Bialecki R.A. — SM24 11137 Bidulya A. — SM17 11688 Bielski J. — SM4 11796 Bielski P. — FM8 12311 Bielski W.R. — FM12 11248 Bigoni D. — SM12 10265 Billant P. — FM25 11391 FM25 12108 MS6 12294 Billardon R. — SM4 10974 Bin G. — SM1 10559 Binding D.M. — FM4 12254 Bing-Zheng G. — SM10 11572 Bioul F. — FM15 12591 Biros G. — FSM1 12691 Bisgaard A. — FM23 10247 Blab R. — SM27 11981 Blachut J. — SM22 10378 Blackmore D. — FM25 11683 Blajer W. — SM17 10026 Blanc X. — FSM6 12336 Blasinska A. — FM4 11184 Blekhman I.I. — SM25 11055 SM25 11719 Blekhman I. — SM25 11055 Blachowski B.D. — SM3 10457 Blonski ´ S. — FM17 11169 FM4 11184 FM8 12873 Blume H. — SM2 12086 Blyth M.G. — FM18 10766 Bo W. — SM24 13013 Bobaru F. — SM24 12789 Bobylov A.A. — SM2 11933 Bocciarelli M. — SM18 10797

398 Bochenek B. — SM24 12075 Bodnar A. — SM4 11665 Boeck T. — FM19 12090 FM8 11723 Boehlke T. — SM18 12549 Boehm H.J. — SM13 12321 Boehm R. — SM13 12576 Boer P. de — FM11 12258 Boer R. de — SM15 10198 Boerner E.F.I. — SM1 12584 Bogacz R. — SM22 12931 Bogaert N. van den — FM15 12591 Boguslawski A. — FM3 11914 FM6 12076 Bohatier C. — SM17 11361 Bokhove O. — MS6 10716 (p. 103) Bolotnik N.N. — SM3 11219 Bolotnova R.Kh. — FM8 12190 Bolzon G. — SM18 10797 Bonamy D. — SM9 12378 Bonn D. — FM8 13025 Bonnecaze R.T. — FM16 11991 FM16 12574 Bonnevie Harbitz C. — FM9 12270 Bonsel J.H. — SM25 11783 Bontoux P. — FM6 13051 FM15 11893 FM25 12080 Bontozoglou V. — FM14 10557 Borg U. — SM13 12214 Borges L.S.A. — SM10 13018 Borisov A.V. — SM1 12351 Bornert M. — SM13 11507 SM13 11805 Borodich F.M. — SM2 11135 SM2 11215 Borodulin V.I. — FM24 12675 FM2 10275 Boronska K. — FM7 10914 Boro´ n ´ski P. — FM6 10921 Borrell M. — FM8 11289 Borve S. — FM19 11809 Bos F. van der — FM6 13030 Bostrom A. — FSM7 12603 Bottaro A. — FM11 12602 Bottausci F. — FM22 12761 Bouchet G. — FM24 12740 Bouch V. — FM7 11972 Boudaoud A. — FM8 13025 Boudifa M. — SM4 11885 Bouizi O. — FM15 12636 Boukpeti N. — SM20 11667 Boulanger P.S. — FSM3 11818 SM11 11812 Bouruet-Aubertot P. — MS6 12601 Boussa H. — SM1 12727 Bouvet C. — SM13 11180 SM14 12225

ICTAM04 Bouville M. — FSM6 11476 Boyland P. — FM22 12442 Bozhko A.A. — FM7 10538 Brady J.F. — FM16 12452 SL4 12160 Braescu L. — FM21 11362 Bragov A.M. — SM20 10374 Brancher J.P. — FM13 10185 Brancher P. — FM13 12483 Branicki M. — FM18 12164 FSM6 11862 Braud P. — FM5 13016 Brauer H. — FM19 12605 Braun S. — FM2 11084 Braunbrueck A. — SM11 11620 Bravo-Castillero J. — SM13 11176 Brechet Y. — SM1 12737 Brekelmans W.A.M. — FSM6 11830 SM18 11344 SM18 11963 SM1 12723 Bremond N. — FM8 11317 Brenner H. — FM16 11725 Brenner M.P. — FM8 11330 Breuer K.S. — MS4 11604 Breysse D. — SM18 12634 Brinckmann S. — SM8 11927 Brinson C.L. — MS1 12027 Brizuela E. — FM3 11888 Broeckhoven T. — FM3 11888 Brons M. — FM23 10247 Brovchenko I.A. — FM26 12268 Brown E.N. — MS1 13010 MS1 13011 Browne D.J. — FM21 11364 Bruls O. — SM17 12208 Brun C. — FM24 12740 Brunet E. — MS4 12130 Brunig M. — SM4 11406 Bruyn J.R. de — FM17 12338 Buehler O. — MS6 10977 Buhl T. — SM24 12079 Bulgakova G.T. — FM12 10548 Bulgarelli U.P. — FM8 12547 Burachik R. — SM24 10594 Burczy´ n ´ ski T.S. — SM24 11137 Burde G.I. — FM26 12813 Burde G. — FM2 11984 Burnett D.S. — FSM1 11373 Burton D. — MS1 12027 Buryachenko V. — SM6 12685 Buschmann M.M. — FM2 10948 Busilas A. — SM16 12546 Busse F. — FM19 10931 Byrne H.M. — FM23 11060 Byrtus M. — SM25 11759 Byskov E. — SM22 12297 Byun Y.-H. — FM6 11569

399

Author Index Blawzdziewicz J. — FM8 11411 Buhler ¨ O. — MS6 12157

Caﬀrey J.P. — MS1 13007

Cailletaud G. — SM14 10088 Calhoun R. — FM9 12738 Callegari G. — FM4 10140 Calloch S. — SM14 12225 SM18 12469 Camotim D. — SM22 12558 Campana E.F. — FM8 12547 Campregher R. — FM6 11456 Cannata G. — FM24 11151 Cantat I. — FSM5 12052 Carabineanu A.S. — FM1 11943 Carbonneau X. — FM6 12510 Cardin P. — FM19 12330 Cardon A. — SM13 12818 Cardonne C. — FM22 12761 Carey M. — FM12 12617 Carlotti P. — MS6 10513 Carnasciali M.-I. — MS5 12651 Carneiro C.A.V. — SM10 13018 Carpen I.C. — FM16 12452 Carpenter P.W. — FM2 11887 Carpinteri A. — FSM6 11224 Carriere P. — FM13 11186 Cartraud P. — SM1 12359 Casalis G. — FM13 12508 Casandjian C. — SM4 10011 Case S. — SM23 10242 Casey J. — FSM3 11096 Cassar C. — FM16 11749 Cassardo C. — MS6 13017 Castellanos A. — FM4 12437 Castro F. — FM8 11670 Cataldi-Spinola E. — FSM1 12586 Caughey T.K. — MS1 13007 Caulliez G. — FM13 12233 FM26 11230 Cazacu O. — SM18 11930 SM20 12779 Cazalbou J.B. — FM6 12510 Cebers A. — MS2 11102 Cenedese A. — FM7 11673 Cercignani C. — MS4 10723 Chabert E. — SM13 11805 Chaboche J.L. — SM4 11885 Chadli M. — SM4 10456 Chakraborty P. — FM25 12161 Challamel N. — SM4 10011 Chang C.-C. — FM25 11741 Chang F.-R. — SM16 12712 Chang H.-H. — SM10 10364 Chang J.-T. — SM1 11193 Chang K.T. — FM11 11043 Chang S.-Y. — SM24 11826

Chao G. — SM11 11271 Chaplin J.R. — FSM4 12557 Charrier Mojtabi M.C. — MS5 10980 Chassaing P. — FM6 12510 Chassiakos A.G. — MS1 13007 Chatterjee A. — SM25 12814 Chatzidai N. — FM18 12859 Chatzigeorgiou G. — SM4 12749 Chauve M.-P. — FM24 10506 Chazallon C. — SM18 12634 Chen B. — SM24 11532 Chen C.F. — FM21 11058 Chen F. — FM9 11294 Chen H.-C. — FM25 11741 Chen J. — SM25 10166 Chen L.-Q. — FSM2 12674 Chen P. — FM13 11995 Chen S. — SM14 11395 Chen T. — SM6 11960 Chenchiah I. — SM14 10721 Cheng G.D. — SM24 13013 Cheong K.B. — FM8 12596 Chern M.-J. — FSM4 11668 Cherniy D.I. — FM23 12868 Chernoray V.G. — FM10 11455 FM2 11339 Chernousko F.L. — SM3 11243 Chew H.K. — FM8 12596 Chiba N. — SM18 10652 Chickichev I. — SM20 12364 Chien W.Z. — FM21 11208 Chini G.P. — FM14 12155 Chiskis A. — MS3 12432 Chizhik S.A. — SM2 10999 Chmielniak T.J. — FM6 11578 FM10 11916 Cho J.-H. — SM27 11827 Cho Y.-S. — SM1 11817 Choi C.K. — FM21 13008 Choi H. — FM11 11636 FM19 11637 Choi J. — FM11 11636 Choi S.T. — MS3 12778 Cholet C. — SM17 11361 Chomaz J.-M. — FM13 12425 FM13 12711 FM25 11391 FM25 12108 MS6 12294 Chow C.L. — SM4 10961 Christophorou C. — FM24 12965 Chro´ ´scielewski J. — SM19 11890 Chrzanowski M. — SM4 11665 Chu C.-C. — FM25 11741 Chucholowski C. — SM26 11672 Chung C.A. — FM21 11208 Chung S.-H. — SM1 12441 Churochkin D.V. — FSM1 11508 Chwa M. — MS3 12251

400 Cichocki B. — FM16 10993 FM18 11300 Ciechanowski J. — FM9 12266 Cieslik J. — SM25 12498 Cieszko M. — FM12 12611 Cilliers J.J. — FSM5 12851 Claire D. — SM4 12021 Clamond D. — FM26 11417 Clanet C. — FM8 11317 Clavel M. — SM8 12228 Clayton J.D. — SM18 10955 Clercx H.J.H. — FM25 11964 Clercx H. — FM22 12633 Cleri F. — FSM6 12334 Cliﬀord M.J. — SM13 11387 Cloirec M. — SM1 12359 Cloitre M. — FM16 12574 Cocks A.C.F. — SM15 12903 Cocou M. — SM2 12382 Coelho I. — SM12 12682 Coelho P.J. — FM3 12348 Cohen J. — FM13 11461 Cohen-Addad S. — FSM5 12578 Coiﬀet F. — FM5 13016 Coker D. — SM2 11987 Cola B.A. — FM22 12158 Colette A. — FM25 11391 Collicott S.H. — MS5 12447 Colombo L. — FSM6 12334 Colonius T. — FM11 12387 Coman C.D. — SM22 12967 Combescure A. — SM1 11797 SM22 11174 Comte P. — FM19 11429 FM24 12740 Constantin P. — FM23 10938 Constantinescu D.M. — SM9 10864 Cooker M.J. — FM26 11345 Coons J. — FSM5 12435 Cordeiro Fernandes P. — FM20 12139 Cornwell P. — FSM7 12642 Costa M. — FM3 12348 Costanzo F. — FSM7 12642 Cottet G.-H. — FM6 12583 Cottron M. — SM9 10429 Courrech du Pont S. — FM17 13029 Cox B.N. — SM13 11162 Cox S.M. — FM22 10497 FM23 11060 Cox S. — MS5 10598 (p. 387) Craig W. — FM26 11548 Cramer A. — FM15 12349 Crandall S.H. — SM25 10863 Cristescu N.D. — SM18 11930 SM20 12779 Crochet M.W. — SM15 12771 Croll J.G.A. — FM9 12072

ICTAM04 Cudzilo S. — FM3 12745 Cugy P. — SM18 12469 Cummins S.J. — FM8 12163 Cunningham J.C. — SM15 12917 Currie P.K. — FSM5 12986 Cuypers Y. — FM24 11303 Czarnecki S. — SM24 11760 Czechowski L. — FM21 12743 Czerwi´ n ´ ska J. — FM24 11802 FSM6 10399

D

ahan M. — SM6 11901 Dahl S.R. — FM17 12073 Dail B. — FM15 10363 Dal Pont S. — SM15 12114 Dalziel S.B. — FM25 12051 Dam D.B. van — FM14 11509 Dam R. — SM22 12297 Daneshmand F. — FM9 10733 Daneshmehr A. — SM13 13009 Danishevs’kyy V.V. — SM13 12588 Dankowicz H. — SM17 11020 Danner T. — FM8 11669 Daru V. — FM8 11979 Darve E. — FM16 12005 Daube O. — FM13 10640 Davaille A. — FM7 12126 Davide B. — SM12 11254 Davidson P.A. — FM19 11124 Davis R.H. — FM16 11473 FM8 12329 FM8 12741 Deblaise D. — SM16 10553 Debatin K. — FM5 11864 Debin S. — SM1 10559 Decamp S. — FM9 12410 Degre G. — MS4 12130 Dekajlo K. — FM9 10576 Delannay R. — FM17 12883 FSM5 12052 Delgado A. — MS2 12031 Dell’Aversana P. — MS5 12651 Delsaute B. — FM15 12591 Delville J. — FM5 13016 Dems K. — SM24 11349 Denda M. — MS1 12786 Denier J.P. — FM2 10103 Denis A. — SM18 12634 Denkov N.D. — FM8 11669 Deprince X. — SM18 11110 Derby J.J. — FM15 10363 Derksen J. — FM20 12397 Deshpande V.S. — SM5 11022 Destrade M. — FSM3 11950 Detournay E. — SM15 12713 SM25 12535 Dettmar J. — FSM6 12112 Di G. Sigalotti L. — FM6 10857 Dias F. — FM26 12089

401

Author Index Dick E. — FM6 11602 Didelle H. — MS6 12494 Dijksman J.F. — FM4 12894 Dimakopoulos Y. — FM14 12858 Ding E.-J. — FM1 12124 Dinkler D. — SM25 12785 Dmitrochenko O. — SM17 10893 Dobler W. — FM7 10306 Dobovsek I. — SM12 12419 Dobrolyubov A.I. — FM26 11144 Doche O. — FM11 11394 Dodge A. — MS4 12536 Doerﬀer P.P. — FM5 12997 Doghri I. — SM4 11886 SM18 13038 Dohnal F. — SM25 12026 Dolezel I. — SM2 10211 Dollet B. — FSM5 11095 Domaas U. — FM9 12270 Doma´ n ´ ski W. — FSM3 11840 Domaradzki A.J. — FM24 10149 Domenico C. — SM12 11254 Dominguez J. — SM8 10698 Dominguez J. — SM17 12071 SM9 11321 Dommelen J.A.W. van — SM18 11344 Dongen M.E.H. van — SM11 11271 Dorfman K.D. — FM4 11209 Dorfmann L.A. — FSM3 11227 Dormieux L. — SM15 11941 Doudard C. — SM18 12469 Doval-Montes L. — SM13 11176 Dovgal A.V. — FM2 11339 Dovgiy S.A. — FM23 12868 Dowling A.P. — FSM1 12527 Dragon A. — SM18 11110 Drazer G. — FM16 10361 Dreiden G.V. — SM11 10114 Drescher A. — SM20 11667 Dritschel D.G. — MS6 11336 Drobi´ n ´ ski P. — MS6 10513 Drobniak S. — FM3 11914 Drugan W.J. — SM12 12025 Du J. — SM24 10433 Dual J. — FSM1 12586 MS3 12384 Dubois F. — SM17 11361 Duchemin L. — FM8 12388 Duck P.W. — FM13 10916 FM2 10103 Dudeck M. — FM19 12363 Dufour G. — FM6 12510 Duineveld P.C. — FM4 12894 Duluc M.C. — FM8 11979 Dumais J. — MS2 11581 Dumontet H. — SM1 12727 Dupere I.D.J. — FSM1 12527

Duplat J. — FM22 12326 Dupont R. — FM13 12233 Dupret F. — FM15 12591 Durgin W.W. — FSM1 12149 Duschlbauer D. — SM13 12321 Dusek J. — FM13 12766 Duszy´ n ´ski P. — FM6 12565 Duysinx P. — SM17 12208 Dvorak V. — FM5 12219 Dlugosz A. — SM24 11137

Eberhard P. — SM17 12133

Eberhardsteiner J. — SM27 11981 Eck C. — FSM6 11340 Eckert K. — FM21 12365 Eckert S. — FM21 12365 Eckhardt B. — FM13 10313 Edwards N.R. — FM26 12424 Eggers J. — FM14 11089 Ehlers W. — SM15 10792 SM15 11194 Ehrenstein U. — FM13 12372 Ehrlacher A. — SM15 12114 SM15 12614 Eidelman A. — FM10 12070 Ekiel-Je˙z˙ ewska M.L. — FM16 11409 FM18 11300 El Abd A. — SM18 12634 El G. — FM26 11134 El Ganaoui M. — FM15 11386 FM15 11893 El Maihy A. — FM22 10130 Elaguine D. — SM2 11365 Eldredge J.D. — FSM1 12180 Elias F. — FSM5 11095 Elmabrok A.M. — SM25 11082 Elperin T. — FM10 12070 FM7 11053 Emelin V.K. — SM7 11556 Emmerich H. — FSM6 11340 Engel R.S. — SM27 10906 Engelbrecht J. — SM10 11592 SM11 11813 Engelen R.A.B. — SM4 11415 England A.H. — SM10 12273 Epstein M. — SM6 12122 Eremeyev V.A. — SM19 10287 Erenburg V. — FM3 12199 Erickson J. — SM1 12441 Ermakov M.K. — FSM7 11869 MS5 11843 Ern P. — FM20 12139 Erofeyev V.I. — SM11 12203 Escalona J.L. — SM17 12071 Esveld C. — SM24 11015 Etling D. — MS6 12035 Etnyre J.B. — FM23 11677

402 Evans D.L. — FSM7 12642 Evans G.M. — FM15 11903 Evers L.P. — SM18 11963

F

aciu C. — SM14 11852 Faisst H. — FM13 10313 Faivre M. — FM1 12423 Falcovitz J. — FM5 11049 Falk M. — FSM6 11476 Falkovich G. — FM22 12048 Fan J. — SM27 11506 Fan Y. — SM1 12441 Fang D. — MS1 12016 Fayzrakhmanova I.S. — FM21 12278 Fedenkova A.A. — FM2 10275 Feissel P. — SM1 12237 Felderhof B.U. — FM16 11236 Feng L. — SM24 12287 Feng X.-Q. — MS3 12314 Feng X. — MS1 12016 Fenghuan S. — MS3 12181 Fenili A. — SM3 13004 Feraille T. — FM13 12508 Fernandes A. — SM10 12093 Fernandez-Feria R. — FM9 11346 Fernando H.J.S. — FM9 10576 FM9 12738 Ferrarese S. — MS6 13017 Ferreira M.J.F. — FM20 10690 Ferri F. — FM10 10445 Feuillebois F. — FM16 10248 Fey R.H.B. — SM25 11783 Fidlin A. — SM25 11679 SM25 12694 Finn M.D. — FM22 10497 FM23 11060 Firsova A.D. — SM25 11712 Fischer F.D. — SM14 10088 Flanagan J.D. — FM26 11468 Fleck N.A. — SM13 12680 SM5 11022 Fleck N. — MS2 10988 Floryan J.M. — FM13 10489 Flukiger F. — FM12 11343 Fl´ ´ or J.-B. — MS6 11829 Fochesato C. — FM26 12089 Fomin N.A. — FM10 12213 Fontelos M.A. — FM8 12548 Fornalik E. — FM7 12173 Forterre Y. — FM17 11775 MS2 11581 Foure T.M. — FM13 10188 Fourment C. — FM5 13016 Fowler P.W. — SM1 11482 Foysi H. — FM24 11116 Fraigneau Y. — FM13 10188 Frana K. — FM19 12587 Franciosi P. — SM18 11498

ICTAM04 Francius M. — FM26 11417 Francois M.M. — FM8 12163 Frankel I. — FM7 12850 Franklin J. — FM16 11884 Franz S. — FM24 11256 Fredriksson P. — SM18 11410 Freidin A. — SM14 11837 Freund J.B. — FSM1 10907 Friedrich R. — FM24 11116 Frischmuth K. — SM11 11853 Frohlich ¨ J. — FM6 11348 Frolova N.Yu. — SM7 12272 Frost T. — SM22 12297 Fructus D. — FM26 11417 Fruend E.-T. — FM1 11851 Fu S. — FM24 12652 Fu Y. — SM14 11837 Fuchs P. — SM22 12084 Fuenmayor J. — SM8 10698 Fuetterer C. — FM4 11209 Fujinuma S. — SM18 11295 Fujita K. — SM17 12582 Fukumoto Y. — FM25 11615 SM25 12593 Fukunishi Y. — FM11 12491 Funakoshi M. — FM23 12484 Furutani Y. — SM12 11432

G

abrielli P.G. — FSM6 12334 Gad-el-Hak M. — FM11 10128 FM2 10948 Gadaj S.P. — SM14 10046 SM14 11308 Galaktionov O.S. — FM1 12063 Galambos P.C. — FM4 10753 Galanov B.A. — SM2 11135 Gallaire F. — FM13 12372 FM13 12425 Gallerano F. — FM24 11151 Galperin B. — MS6 11152 MS6 11942 Galtier A. — SM18 12469 Galvanetto U. — FSM2 11241 Galvin J.E. — FM17 12073 Gamallo P. — FSM1 11249 Gambarotta L. — MS2 13049 SM4 12298 Gambaryan-Roisman T. — FM14 12293 Gambin B. — SM13 11895 SM13 12667 Gambin W.L. — SM18 11975 Ganczarski A. — SM4 12335 Gandzha I.S. — FM26 11591 MS2 12059 Ganguly P. — SM1 12458 Ganqing F. — SM24 12287

403

Author Index Gao C.-F. — MS1 11857 Gao H. — SL5 10772 (p. 131) Gao X. — MS1 12027 Garcia-Sanchez F. — SM9 11321 Garcia-Villalba M. — FM6 11348 Gardeniers H. — FM1 12400 Garikipati K. — FSM6 11476 Garstecki A. — SM24 12628 Garstecki P. — MS4 12966 Gaspar Z. — SM22 10915 Gautier P.-E. — SM17 11361 Gavrilov S.N. — SM6 12324 Gavrilova E. — FSM4 10520 Galka A.A. — SM13 12732 Galka A. — SM13 11895 SM13 12667 Geers M.G.D. — FSM6 11830 SM14 11395 SM18 11963 SM1 12131 SM1 12723 SM4 11415 SM4 12341 Geﬀroy P. — FM11 12244 Gelder B. van — FSM5 12373 Gendelman O.V. — SM25 11079 Geoﬀroy S. — FM12 11343 Georgelin M. — FM21 12543 Gerbeth G. — FM11 12068 FM13 12542 FM15 12349 FM19 12613 Gerkema T. — FM26 12087 Gerland P. — SM16 12417 Germay C. — SM25 12535 Gerstmayr J. — SM17 12029 Getling A.V. — FM7 10306 Geubelle P.H. — SM10 11177 Geurts B.J. — FM6 13030 Gharib M. — FM1 12784 Ghicini S. — FM16 11588 FM16 12345 Ghrist R.W. — FM23 11677 Gibson A.N. — FM10 11455 Giessen E. van der — MS2 10627 SM1 12737 SM8 11927 Gilat R. — MS1 12011 Gilewicz J. — FSM3 11642 Gill S.P.A. — MS3 12627 Gilles L. — SM4 12277 Ginalski M.K. — FM1 12409 Girgis I.G. — FM13 10487 Girish J. — SM13 12531 Giusti A. — FM20 13012 Gladden L.F. — FM21 11320 Glema A. — SM5 12949 Glicksman M.E. — FSM5 11240 Gloaguen J.M. — SM18 12339 Glocker C. — FSM2 11183 Gloria A. — FSM6 12078 Godard V. — SM4 12913

Goddard J.D. — FM17 10371 Godoy-Diana R. — FM25 12108 Goektepe S. — SM4 12116 Goetsch M. — FSM1 12586 Goldhirsch I. — FM17 12308 Goldobin D.S. — FM13 12280 Golﬁer F. — FM12 10234 Golinval J.-C. — SM17 12208 Gollub J.P. — FM17 12445 Goncalves P.B. — SM22 12420 Goncharova O.N. — MS5 11757 Gondret P. — FM17 13029 Gong L. — FSM5 11605 Gonik M.A. — FM21 11331 Gontarowskiy P. — SM2 10211 Gonthier K.A. — SM15 12771 Gonthier Y. — SM2 12773 Gonzalez A. — FM4 12437 Goo B. — SM8 11737 Gorbatikh L. — SM2 12412 Goryacheva I. — SM2 11196 Goto T. — FM1 11441 Goujon-Durand S. — FM11 12736 Goulitski K. — FM26 10245 Gourjii A.A. — FM22 12110 FM25 12196 Govindarajan R. — FM2 12631 Govorukhin V.N. — FM12 10199 Graf T. — SM15 10792 Graham A.L. — FM16 11725 Gramstad O. — FSM4 12956 Graner F. — FSM5 11095 Grants I. — FM13 12542 Gravouil A. — SM1 11797 Gray G.L. — FSM7 12642 Grebe R. — SM15 12614 Green M.L. — SM20 12779 Green N.G. — FM4 12437 Grekova E.F. — SM6 12434 Griﬃths R. — MS6 11499 Griggs A.J. — FM8 12741 Grimshaw R.H.J. — SM11 11242 Gristchak V.Z. — SM25 10656 Grondin F. — SM1 12727 Grossman-Clarke S. — FM9 12738 Grue J. — FM26 11417 FSM4 12956 Grunefeld G. — FM10 12070 Gryschka M. — MS6 12035 Gu Y. — SM24 11532 Guazzelli E. — FM16 11588 FM16 11850 FM16 12345 Guba P. — FM21 11270 Guddati M.N. — FSM1 11973 Gudmundson P. — SM18 11410 Guedes de Carvalho J.R.F. — FM20 10690

404 Guegan A. — FM11 12305 Guerses E. — SM1 12115 Guest S.D. — SM1 11482 SM22 12669 Gueydan F. — SM12 12751 Guinovart-Diaz R. — SM13 11176 Guitong Y. — MS3 12181 Gundrum T. — FM19 12613 Guo J.-G. — SM16 10510 Guo L. — SM10 11551 Guo Z. — FSM4 12521 Gupta N.K. — SM5 12982 Guse N. — SM17 11223 Gutﬁnger C. — FSM1 12516 Gutkowski W. — SM3 10457 SM3 10458 Guz I.A. — SM9 10824 Guzina B. — SM20 12364

Haber R.B. — SM1 12441

Haberkorn M. — FM24 12740 Hachemi A. — SM18 12058 Hackl K. — SM14 12077 SM24 12074 Haddad J. — FM4 11648 Hadjiconstantinou N.G. — MS4 11437 Hagedorn P. — SM25 11261 Hahn S. — FM11 11636 Hajzman M. — SM25 11759 Halbedel B. — FM19 11906 Halkjaer S. — SM24 11744 Halley P. — FSM5 12435 Hallworth M.A. — FM17 10524 Hammerton P.W. — FM2 12241 FSM1 12497 Hanada S. — SM11 12418 Hanagud S. — SM10 12968 Hansen J.S. — SM13 12148 Hansen L.V. — FSM1 12533 Hao S. — SM9 10676 Haque A. — MS3 12705 Hara T. — FM26 10746 Harambat F. — FM22 11190 Harrison P. — SM13 11387 Hartmann C. — MS2 12031 FM10 13048 Hasegawa S. — FM5 11560 Hata T. — SM11 10344 Hattori Y. — FM19 11458 FM25 11615 Haughton D.M. — SM6 11220 Hayashi S. — MS1 10821 Hayes M.A. — FSM3 11818 SM11 11812 Haynes P. — MS6 10987 (p. 139) Hazel A.L. — FM1 12295 He J. — FM24 12652

ICTAM04 He Y. — SM27 11506 Healey J.J. — FM13 12150 Heijst G.J.F. van — FM22 12110 FM22 12633 FM25 11964 FM25 12051 Heil M. — FM1 11013 FM1 12295 Heintz A. — FM6 11763 Hellmich C. — SM11 12313 Herakovich C.T. — FSM7 10226 Herczy´ n ´ski A. — FM2 11984 FSM4 11986 Herrada M.A. — FM25 10683 Herrmann K.P. — SM9 10824 SM9 12249 Herskovits J. — SM24 10594 Herve I. — FM11 12244 Hickel S. — FM24 11256 FM24 12564 Hierck B.P. — FM1 11900 Higashi A. — SM25 11066 Hikihara T. — FSM2 12014 Hild F. — SM18 12469 SM4 12021 Hilgenfeldt S. — FM1 12400 FSM5 10597 FSM5 12373 Hinch J. — FM16 11588 FM16 12345 Hino R. — SM24 12870 Ho K.H. — FM21 11208 Ho Y.G. — SM24 12185 Hochlenert D. — SM25 11261 Hodson H.P. — FSM5 12395 Hogan J.S. — FSM2 10499 Hohe J. — SM15 11463 SM9 11484 Hohler R. — FSM5 12578 Hojjati H.M. — SM1 11291 Holeman J.E. — FM9 12738 Holm D.D. — FM6 13030 Holnicki-Szulc J. — MS1 13015 Holobut P. — SM3 10458 Holopainen S. — SM24 12200 Homayoun Heidari A. — FSM1 11973 Homsy G.M. — MS5 11133 Hong Y. — SM8 12391 Hopkins M.M. — FM18 10983 Horanyi S. — FM19 10931 Horimoto H. — FSM1 10294 Hornowski T. — SM15 12567 Hornych P. — SM18 12634 Horst E. — SM25 12026 Hosseini-Godarzi A. — SM9 10646 Hosseini-Tehrani P. — SM9 10646 Howle L. — FM12 12617 Howlin C.P. — FM26 11468 Hoyer K.W. — FM25 12821

405

Author Index Hrenya C.M. — FM17 12073 Hribersek M. — FSM4 10826 Hu X.X. — SM19 12224 Hu X. — FM24 11802 Huang H. — FM26 11126 FM4 12894 Huang I.-D. — SM1 11193 Huang P. — MS4 11604 Huang R.F. — FM11 11043 Huang R. — MS3 10699 Huang X. — FM4 11165 Huang Y. — MS3 12314 Huerre P. — FM11 12305 FM13 12425 Hufenbach W. — SM13 12576 Hughes G. — MS6 11499 Hughes T.J.R. — SL6 13003 (p. 153) Hulbert G.M. — SM11 12772 Hulin J.-P. — FM20 12092 Hulsbaek L. — SM22 12297 Hulsenberg D. — FM19 11906 Hunt G.W. — SM22 12688 Hunter J.K. — FM5 11730 Huppert H.E. — FM17 10524 FM21 11320 FM21 11328 FM9 10366 Hussein M.I. — SM11 12772 Hutchinson J.W. — MS3 12386 Hutter K. — FSM6 12040 Huu Nam T. — SM1 12844 Hwang I.G. — FM21 13008 Hwang K.-C. — MS1 12016 MS3 12314 Hwang P.A. — FM26 11230 Hwang W.-S. — SM24 11858

I

afrati A. — FM8 12547 Ignaczak J. — SM13 10347 Ilison O. — SM11 12568 Im S.H. — MS3 10699 Im S. — SM1 11817 Imai K. — SM15 10790 SM18 11295 Imanishi E. — SM17 12537 Imielowski S. — SM22 12931 Indinger T. — FM24 12564 Inoue T. — SM14 10434 Insperger T. — FSM2 10252 Ionita A. — SM13 11014 Iooss G. — FM26 11290 Ippolito M. — FSM6 12334 Irannejad R.H. — SM2 10718 Irgens F. — FM9 12270 Irschik H. — MS1 10784 SM17 12029 Isermann R. — SM16 12758 Ishihara M. — MS1 11857

Ishii K. — FM25 12041 Isoda H. — FM1 12522 Itihara M. — SM17 12582 Ivanov A.V. — FM2 10275 Ivanov I.B. — FM8 11669 Ivanov M. — FM5 10531 Ivanov T.P. — SM6 10972 Iwai K. — FM15 10860 Iwankiewicz R. — FSM2 12972 Iwnicki S.D. — SM26 10909 Izawa S. — FM11 12491

Jacqmin D. — FM4 10753

Jacques N. — SM18 11309 Jacquin L. — FM11 12244 Jaeger A. — SM27 11981 Jakobsen B. — FM23 10247 Jakubiak B. — FM9 12592 James C.D. — FM4 10753 James R.D. — MS1 10722 Jang G.-W. — SM24 10048 Jankowski R. — SM25 12659 Janour Z. — FM9 10467 Jansen E.L. — SM19 12346 Jarza A. — FM9 12266 Jarz¸ebowski A. — SM20 12848 Jasi´ n ´ ska M. — FM22 12415 Jasi´ n ´ ski M. — SM1 11389 Jasiuk I.M. — SM6 12189 Jaszczur M. — FM22 12610 Javaitis I. — MS2 11102 Jemiolo S. — MS2 12100 Jenkins A.D. — MS6 12463 Jensen H.M. — MS3 11070 (p. 165) Jensen J.S. — SM24 12222 Jensen O.E. — FM14 12155 Jeon W.-P. — FM11 11636 Jiang H. — SM24 12789 Jiang J.-B. — FM24 11488 Jiang L.J. — FM22 12209 Jie M. — SM4 10961 Jinnouchi Y. — SM7 10691 Joensson P.-A. — SM26 11795 John A. — SM1 12562 Johnson M.E. — FSM1 12497 Jolivet L. — SM12 12751 Joly P. — SM2 12382 Jones A.S. — MS1 13011 MS1 13010 Jones N. — SM5 11107 Jones R.B. — FM18 10565 Jones W.P. — FM3 11914 Jop P. — FM17 11775 Jordan P. — FM5 13016 Joseph D.D. — FM20 10843 Joseph P. — MS4 12673 Josserand C. — FM8 11614

406 Joubert P. — FM7 12220 Jullien C. — MS4 12536 Jun S. — SM1 11817 Jung J.-Y. — MS1 10985 Jung S. — SL18 10495 Juric D. — FM8 11979

Kabouya N. — FM13 12766

Kachanov Y.S. — FM24 12675 FM2 10275 Kaczmarek M. — SM15 12566 SM15 12567 SM15 12577 Kaczy´ n ´ ski A. — SM9 11980 Kagemoto H. — FSM4 11758 Kaiser I. — SM25 12322 Kaliszky S. — SM24 12573 Kalliadasis S. — FM14 10220 Kallivokas L.F. — FSM1 12691 FSM1 12717 Kaltayev A. — FM3 11031 Kam Liu W. — SM1 12167 Kambe T. — FM23 11166 Kameyama T. — FM15 10860 Kan K. — FM14 11739 Kanarska Y. — MS6 12408 Kanaun S.K. — SM11 11989 Kaneda M. — FM7 13022 Kaneko Y. — FM17 11876 Kang L.-C. — SM1 10270 Kanno Y. — SM2 12646 Kapania R.K. — FSM7 12752 Kapla´ n ´ ski F.B. — FM25 12032 Kappl K. — SM27 11981 Karagiozova D. — SM5 11107 Karaiwa M. — SM13 11426 SM9 11425 Karajan N. — SM15 11194 Karakasidis T.E. — FSM2 12482 Karapetsas G. — FM18 12859 Karas M.S. — SM1 12340 Karcher C. — FM19 12290 Karihaloo B.L. — FSM6 11806 Karpenko E.E. — SM25 10470 Karpov E.G. — SM1 12167 Kashtalyan M. — SM9 10824 Kasperski G. — FM15 12636 Kataoka T. — FM26 11746 Katsuyama T. — FSM2 11535 Kawamoto A. — SM24 12302 Kawazoe H. — FM23 12484 Kazakevitch M.I. — SM25 12632 Kazemzadeh-Parsi M.J. — FM9 10733 Ka´ ´zmierczak B. — MS2 10838 Ke F. — SM7 11764 Keane R.J. — FM22 12545 Keer L.M. — SM2 11215 Kent E.F. — FM18 11915

ICTAM04 Keppens R. — FM19 11429 Kere P. — SM13 10667 Kerr R.M. — FM24 12430 Kerschen E.J. — FSM1 12477 Keunings R. — SL7 10512 Khabakhpasheva T.I. — FSM4 10911 Kharif C. — FM26 11417 Khasanov M.M. — FM12 10548 Khelidj A. — SM4 12749 Kheradvar A. — FM1 12784 Khoei R.A. — SM2 10718 Khom’yak T.V. — SM17 10779 Khoo B.C. — FM24 11802 Khoshravan M.R. — SM13 10189 Khotyanovsky D.V. — FM6 11485 Khotyanovsky D. — FM5 10531 Khusid B. — FM16 10361 FM4 10753 Khusnutdinova K.R. — SM11 11242 Kiger K.T. — FM1 11900 Kim B.-K. — SM27 11827 Kim J.-H. — MS1 10985 Kim J. — FM11 11636 FM11 12013 FM19 11637 Kim K.-S. — MS3 12778 SM8 12034 Kim Y.-M. — FM1 12655 Kim Y.Y. — FSM7 12752 SM24 10048 Kimura Y. — FM23 11660 Kinell M. — FM5 10789 King G.P. — FM22 12545 King J.R. — FM14 12155 King M.P. — FM7 10438 King R. — FM11 12624 Kireenkov A.A. — SM2 11778 Kishino Y. — SM20 12462 Kit E. — FM26 10245 Kitagawa H. — MS3 11122 Kityk A. — FM8 11462 Kiyono K. — FSM2 11535 Klapp J. — FM6 10857 Klarbring A. — MS2 10670 SM2 10686 Kleeorin N. — FM10 12070 FM7 11053 Klepaczko J.R. — SM11 11360 SM14 10046 SM18 10734 Klingbeil D. — SM4 10865 Kloker M.J. — FM13 12381 Kluwick A. — FM2 11083 FM2 11084 FM5 10789 Knap J. — SM1 11154 SM9 12171 Knight D. — FM5 11560 Kobayashi S. — FM5 11864 Kobiera A. — FM3 12843

407

Author Index Koca´ n ´ da D. — SM8 12914 Kocourek V. — FM19 12290 Koenderinck G.H. — MS2 10709 (p. 355) Koikari S. — FM23 11660 Kolandavel M.K. — FM1 11851 Kolcavova Sirkova B. — FSM3 11247 Kolesnikov Y. — FM19 11120 FM19 12090 Kolk K. — SM9 12044 Kolmychkov V.V. — FM15 11893 Komarova V.Y. — FM2 10275 Kondic L. — FM17 12399 Kondo D. — SM15 11941 Kondrachuk A.V. — MS2 12059 Konieczny P. — FM11 12602 Kononov Y.N. — SM17 10779 Kopiev V.F. — FM11 12156 Koplik J. — FM16 10361 Korczyk P.M. — FM10 12855 Korepanov V.V. — SM6 12289 Kornev K.G. — FM4 10140 Korvink J.G. — SM24 11316 Koseki T. — SM17 12582 Kosi´ n ´ ski W. — SM11 11853 Kositsyn A. — FSM7 11724 Kothe D.B. — FM8 12163 Kotucha G. — SM24 12074 Koudella C. — MS6 12601 Koumoutsakos P. — FM24 11256 FM6 12583 Kouznetsova V. — FSM6 11830 SM14 11395 Kovacs F. — SM1 11482 Kovaleva A. — SM3 11265 Kovaleva L. — FM12 12528 Kowalczyk P. — SM24 12679 Kowalczyk P. — FSM2 11363 Kowalczyk P.G. — FM6 11763 Kowalczyk-Gajewska K. — MS2 11808 SM6 10442 Kowalewski T.A. — FM10 12855 FM14 11477 FM21 11331 FM4 11184 FM7 11160 FM7 12173 FM8 12873 FM9 10576 Kowalewski Z.L. — SM27 11957 Kowalski S.J. — SM15 13006 Kozien M.S. — SM25 12511 Kozlov A.A. — FM7 12267 Kozlov V.V. — FM2 11339 Krasnikovs A. — SM23 10242 Krasnopolskaya T.S. — FM22 12257 Krasovskyy V.L. — SM19 11815 Krauklis A.V. — FM10 12213 Krause E. — FM25 12690 Kravtsova M.A. — FM2 11780 Krawczyk J. — FM9 12592

Kraynik A.M. — FSM5 10597 Krein A. — FM10 12070 Kreuzer E.J. — SL8 10544 (p. 173) FSM4 11912 Krieger U. — FM19 11906 Kristiansen O. — FM26 11417 Krivtsov A.M. — SM11 12481 Kroeger M. — SM2 12086 Kruszka L. — SM20 10374 Kruyt N.P. — SM20 10866 Kruzelecki J. — SM24 11634 Kubacki S. — FM6 12076 Kubair D.V. — SM10 11177 Kubik J. — SM15 12566 SM15 12577 Kucaba-Pi¸etal A. — MS4 10971 Kuczma M.S. — MS1 10136 Kudela H. — FM25 12836 Kudryavtsev A.N. — FM6 11485 Kudryavtsev A. — FM5 10531 Kuhl A. — FM3 12745 Kuhl E. — SM12 11277 SM1 11211 SM1 12924 Kuhn G. — SM9 12044 Kuilekov M. — FM19 12605 Kuistiala A. — SM4 10960 Kujawski D. — SM9 10334 Kulesh M.A. — SM6 12289 Kullander F. — FSM1 12533 Kulvietis G. — SM16 10900 Kumagai T. — FM8 11299 Kumar A. — FM4 10753 Kuna M. — SM9 11045 Kuna-Ciska H. — SM4 11796 Kurashige M. — SM15 10790 SM18 11295 Kurkin A.A. — FM9 11433 Kurnik W. — SM22 12931 SM22 12934 Kuroda M. — FM11 12491 Kurowski M. — FM25 12275 Kurowski P. — SM7 12799 Kurpa L.V. — SM19 12350 Kursa M. — SM24 11760 Kurzydlowski K.J. — SM7 12666 Kurzyna J. — FSM2 12927 Kus W. — SM24 11137 Kwon Y.-I. — FM15 10363 Kyriakides S. — FSM5 11605 SM22 11600

Llorca J. — SM13 11080

Labiausse V. — FSM5 12578 Labuz J.F. — SM20 11845 Lac E. — MS2 11961 Lacaze L. — FM13 11396 Lackner R. — SM27 11981

408 Lacor C. — FM3 11888 Ladd A.J.C. — FM16 11418 Ladd T. — FM12 11994 Ladev´ ´ eze P.J. — SL9 10508 (p. 187) SM1 11769 SM25 11246 Lagha M. — FM13 11647 Lamanna G. — FM10 10445 Lambros J. — SM10 11177 Lampis M. — MS4 10723 Landa M. — MS1 12637 Lane-Serﬀ G.F. — FM9 11182 Lange C. — SM2 12773 Lange U. — FM8 12388 Langevin D. — FSM5 12212 Langkamp A. — SM13 12576 Langre E. de — FSM4 11801 Lanos C. — SM4 10011 Larecki W. — FSM6 11237 Larsson R. — SM15 12518 Larue de Tournemine A. — FM20 12383 Lathrop D.P. — FM19 11681 Lauga E. — FM8 11330 Lavinskaya E.I. — FM10 12213 Lavrenteva O.M. — FM8 11250 Le Bars M. — FM7 12126 Le Bris C. — FSM6 12078 FSM6 12336 Le Dizes S. — FM13 11396 FM13 12509 Le Gal P. — FM13 11396 FM24 10506 Le Grognec P. — SM22 11962 Le Quere P. — FM7 12220 FM8 11979 Le van A. — SM22 11962 Leal G.L. — FM8 11289 Lebensohn R.A. — SM18 11005 Leblanc S. — FM13 10525 Leblond J.-B. — SM4 12913 SM9 12393 Lecomte-Beckers J. — SM1 11298 Lee C. — FM6 11569 Lee D.-H. — SM24 11858 Lee D. — FM11 11636 Lee H.T. — FSM5 12851 Lee J.-W. — FM6 11569 Lee J. — FM16 11418 Lee S.-J. — FM1 12655 Lee S.M. — FM9 12738 Lee W.-S. — SM27 11827 Leﬁk M. — SM13 12970 Legarth B.N. — SM13 11794 Legendre D. — FM8 11911 Legoll F. — FSM6 12336 Leine R.I. — FSM2 11183 Leiner W. — FM7 12173

ICTAM04 Lekszycki T. — MS2 12315 Lele S.K. — FSM1 11561 Lemak M. — SM17 11375 Lenci S. — FSM2 12129 Lengyel A. — SM1 12101 Lenormand R. — FM12 10234 Lentzen S. — MS1 11752 Leonard A. — FM6 11937 Leonardi E. — FM14 11477 FM15 11386 Leong S.S. — FM14 11477 Leorat J. — FM19 11621 Lerbet J. — SM17 11674 Leroy Y.M. — SM12 12751 Lesieur M. — SL10 10731 (p. 203) Letser Y.A. — SM2 11933 Leung R.C.K. — FM6 10163 Leungvichcharoen S. — SM11 12514 Levin V.M. — SM11 11989 Levitas V.I. — SM14 11325 Levitsky S.P. — FM4 11648 Levy Y. — FM3 12199 FM3 12348 Lewandowski J. — FSM1 12555 Lewi´ n ´ski T. — SM24 10052 SM24 11760 Lewis D.M. — FM1 11908 Lewis S.R. — MS6 12494 Lexcellent C. — SM14 12225 Li H. — SM9 10676 Li J. — FM2 10103 FSM7 12752 SM25 10166 SM4 12036 Li K. — SM7 10492 Li Q. — SM9 11934 Li X.M. — FM6 10163 Li Y. — MS6 12194 Liakopoulos A.B. — FSM2 12482 Liang B. — SM26 10909 Liao X. — FM7 12168 MS6 12169 Libersky L.D. — SM1 10829 Liechti K.M. — MS3 12132 (p. 217) Lijuan L. — SM24 12287 Lik Chan C. — FM21 11058 Lim C.W. — SM19 12224 Lim J. — FM11 12013 Lim K.-M. — SM1 12901 Lim K.-W. — FSM1 11973 Lima R. — FM8 11026 Lin C.Y. — SM25 11867 Lin J. — SM11 11099 Lin Y. — SM1 10559 Linden P.F. — MS6 10261 Lindner M. — SM2 12086 Ling Z. — SM13 12242 Lipniacki T. — FM24 11161

409

Author Index Lippermann F. — FSM5 13033 Lipson S. — FM16 11884 Lisitsin Y.V. — SM2 10342 Lisowski W. — SM7 12799 Lister J. — FM8 12388 Litewka A. — SM4 11228 Liu J.T.C. — FM13 10487 Liu T. — SM24 11532 Liu X. — SM6 11998 Liu Y.-C. — FM3 10367 Liu Y.-L. — FM24 11488 Liu Y. — SM18 11005 Lo Jacono D. — FM22 12731 FM3 10918 Loboda V.V. — SM9 12249 Lobov N.I. — FM7 12344 Loefdahl L. — FM2 11339 Loehnert S. — SM1 12534 SM1 12584 Lofdahl L. — FM10 11455 Loginov M.S. — FM24 10455 Loglisci N. — MS6 13017 Logo J. — SM24 11623 SM24 12573 Logvinova K. — FM12 12376 Lohse D. — FM10 11909 FM17 10253 FM1 12400 Loimer T. — FM12 11487 Lomunov A.K. — SM20 10374 Long A.C. — SM13 11387 Long S.G. — SM13 12496 Longere P. — SM18 11110 Longhetto A. — MS6 13017 Longmao Z. — MS3 12181 Lopes S.R.X. — SM22 12420 Lopez-Lopez E. — SM13 11176 Louge M.Y. — MS5 10714 (p. 229) Love B.M. — SM10 12594 Lozia Z. — SM26 11832 Lu J.-Z. — FSM4 11668 Lu K. — FM4 11165 Lu T.J. — FSM1 12527 FSM5 12395 Lu W. — SM12 11432 Lu X. — SM10 12968 Lube G. — FM17 10524 Lubowiecka I. — SM19 11890 Lucey A.D. — FSM4 11736 Luczak M. — SM25 11273 Luding S. — SM17 12133 Lukes V. — SM10 11671 Lukomsky V.P. — FM26 11591 MS2 12059 Lund E. — SM24 12098 SM24 12916 Lund F. — FSM1 12643 Lundberg B. — SM7 12551

Lusseyran F. — FM13 10188 Lvov G.I. — SM19 10071 Lyakh V.V. — SM6 12575 Lyly M. — SM13 10667 Lysenko V.V. — SM25 10656 Lyubimov D.V. — FM13 12280 FM7 12344 MS5 12402 Lyubimova T.P. — FM21 12278 FM7 12344 MS5 12402

L

obocki L. — FM9 12592 Lodygowski T. — SM5 12949 Lukasiewicz S.A. — SM1 11291

M

a J. — SM8 11686 Ma L. — SM10 11551 Mac Sithigh G. — SM6 12354 MacKintosh F.C. — MS2 10709 (p. 355) MacMartin D.G. — FM11 12387 Maciejewski J. — SM20 12848 Mader M. — FM1 10741 Maderich V.S. — FM26 12268 Maderich V. — MS6 12408 Maeda T. — SM11 10422 SM25 12593 Magara Y. — SM9 11425 Magariyama Y. — FM1 11441 Magatti D. — FM10 10445 Magnaudet J. — FM20 12139 FM8 11911 Mahadevan L. — MS2 11581 Maier G. — SM18 10797 Mailybaev A.A. — SM25 11536 Maj M. — SM18 11185 Majewski J. — FM6 12618 Majewski T. — SM25 12984 Majkut M.M. — FM10 11916 Majorkowska-Knap K. — SM8 11371 Makhovskaya Y. — SM2 11196 Makin V.K. — FM26 11230 Makinde O.D. — FM3 10747 Makipelto J. — SM1 12310 Makowski K. — FM19 12363 Malinowski S.P. — FM10 12855 Malinowski S. — FM9 12592 Maliwan K. — MS5 10980 Malki-Epshtein L. — FM9 10366 Maluleke G.H.-S. — SM6 12039 Mamaev I.S. — SM1 12351 Mandal P. — SM19 10233 Mandelis A. — FM10 12719 Manela A. — FM7 12850 Mang H.A. — SM11 12313 SM19 11021 Manneville P. — FM13 11647 Mansukh M. — MS1 12786 Mansur S.S. — FM6 11456

410 Manucharyan G.V. — SM25 12989 Marah D. — MS6 10261 Marchioli C. — FM20 13012 Margulies S.S. — MS2 10933 Marian J. — SM1 11154 Marie Habraken A. — SM1 11298 Marinho W. — FM6 11456 Marinova D. — SM3 11036 Markert B. — SM15 11194 Markine V.L. — SM24 11015 Marmottant P. — FM1 12400 Marquillie M. — FM13 12372 Marsavina L. — SM9 10864 Marsik F. — FM11 12258 FM2 12062 MS1 12637 Martinand D. — FM13 11186 Martins J.A.C. — SM12 12682 SM2 12646 Maruschak P.O. — SM8 11920 Marvalova B. — SM1 12844 Marze S. — FSM5 12212 Maslov B.P. — SM27 10018 Mason D.P. — SM6 12039 Masri R. — SM5 12043 Masri S.F. — MS1 13007 Massabo R. — SM13 11162 Massart T.J. — SM4 12341 Masuda M. — FSM1 10294 Matalon M. — FM3 10918 Matas J.P. — FM16 11850 Matioukevitch S.I. — SM9 10897 Matsuda H. — SM19 12224 Matsui R. — MS1 10821 Matsumoto Y. — MS2 12192 Mattioni A. — FSM6 12334 Matveyenko V.P. — SM6 12289 Matvienko A. — FM10 12719 Matysiak S.J. — SM9 11980 Matyukhin Y. — SM2 10211 Maugin G.A. — FSM3 11347 SM10 11592 SM14 11393 SM6 12434 Maurel A. — FM24 11303 FSM1 12643 Maurine P. — SM16 10553 Mavletov M. — FM12 12528 Maxworthy T. — FM13 11995 Ma´ ´ zdziarz M. — MS2 11808 SM20 11584 Mazeika D. — SM16 10900 Mazhorova O.S. — FM15 11893 Mazza E. — MS2 12064 Mazzilli C.E.N. — SM25 11494 McElwaine J.N. — FM9 12396 McFarland M.D. — SM25 10408 McGlashan S. — FSM5 12435

ICTAM04 McIntyre M.E. — MS6 10977 MS6 12157 McPhee J. — SM2 12773 McPhie M.G. — FM16 11431 Mebarki A. — SM2 12082 Medeiros M.A.F. — FM13 12381 Medyanik S.N. — SM1 12167 Meer D. van der — FM10 11909 FM17 10253 Meftah F. — SM2 12082 Meftah S. — SM14 12747 Megaridis C.M. — FM4 11603 Mehandia V. — FM16 10603 Mehshikov Y.L. — SM25 11716 Meiburg E. — FM13 11995 FM13 12425 Meijaard J.P. — SM25 11338 Meinhart C. — FM22 12761 Meironke H. — FM7 11905 Mejak G. — SM1 11334 Meleshko V.V. — FM22 12110 FM22 12257 FM22 12633 FM23 12868 FM25 12196 Melnik O.E. — FM9 11703 Melo J. — FM3 12348 Melville W.K. — MS6 12722 Men S. — FM19 12605 Mendez C. — FM8 11670 Mendonca M.T. — FM13 12381 Mendoza G. — FM6 10857 Mendrok K. — SM7 12799 Menetrier L. — MS4 12047 Menon N. — FM5 11072 Menshykov O.V. — SM9 11195 Menzel A. — FSM3 12392 Mercier J.F. — FSM1 12643 Mercier S. — SM18 11309 Mergheim J. — SM1 11211 Merodio J. — SM12 11705 Mestel J.A. — FM19 12979 Meunier P. — FM22 10532 Meysman F.J.R. — FM1 12063 Mezic I. — FM22 12439 FM22 12761 FSM2 12468 Mezyk A. — SM25 12770 Micciche P. — FM10 10445 Michaltsos G.T. — SM22 11172 Michalek T. — FM7 11160 Michalowski R.L. — FM17 12790 Michel U. — FM21 12365 Micunovic M. — SM18 10383 Middelburg J.J. — FM1 12063 Miehe C. — FSM6 12112 FSM6 12113 FSM6 12118 SM1 12115 SM4 12116 Miettinen A. — SM2 10459 Mikhalchenko G. — SM26 11589

411

Author Index Mikhayalov D.N. — FM12 11115 Mikkelsen R. — FM17 10253 Millet C. — FM13 12509 Miloh T. — FM20 12342 Min B.-Y. — FM6 11569 Minc N. — FM4 11209 Minier J.-P. — FM24 10564 Miozzi M. — FM7 11673 Miskiewicz M.A. — SM7 12666 Mitsumoji T. — FM13 12291 Miura K. — SM17 12464 Miyamoto H. — FM9 11164 Miyazaki T. — FM24 10937 MS6 12194 Mizerski K.A. — FM19 12107 Mizuguchi F. — SM25 11066 Mizuno M. — SM18 11350 Moctezuma M. — FM8 11026 Modarres-Sadeghi Y. — FSM4 11078 Moe A. — FM9 12270 Moes N. — SM1 12359 Moﬀatt H.K. — FM18 12164 FM19 11458 FSM7 10016 Moﬀatt K. — FSM6 11862 Mohan J.A. — FM5 11072 Moisy F. — FM24 11814 Mojtabi A. — MS5 10980 Mokos V.G. — SM1 10665 Molenaar D. — FM25 11964 Molinari A. — SM14 11852 SM18 11309 Moller H.T. — SM24 12916 Mondy L.A. — FM18 10983 Monetto I. — SM4 12298 Moniuk W. — FM8 12311 Monkewitz P.A. — FM13 11186 FM24 11454 FM3 10918 Monkewitz P. — FM24 12965 Moo Koh H. — SM8 12034 Morch K.A. — FM8 11999 Moreau J.-J. — SM17 11361 Moreau R. — FM15 10860 Morel A. — MS2 11961 Moreno M.J. — FM20 11189 Morgan H. — FM4 12437 Morita C. — SM19 12224 Morize C. — FM24 11814 Morland L.W. — FM18 10297 Moroni M. — FM7 11673 Morozov N.F. — SM9 10897 Morris J.F. — FM16 11850 Morro A. — SM11 11596 Morzy´ n ´ski M. — FM11 12624 FM11 12736 FM24 11454 SM24 12413 Mouaze D. — FSM4 12557 Moulia B. — FSM4 11801

Moulin F. — MS6 11829 Mounajed G. — SM1 12727 Mousavi S. — SM7 12551 Movchan A.B. — FSM1 11896 Movchan N.V. — FSM1 11896 Mr´ o ´z Z. — SM18 10064 SM24 11349 SM2 11074 Muc A. — MS3 12251 SM13 11457 SM13 12560 Mueller A. — SM9 11484 Mueller U. — FM19 10931 Mullarney J. — MS6 11499 Muller-Slany H.H. — SM1 10423 Murai M. — FSM4 11758 Musalimov V.M. — SM2 10342 Muth B. — SM17 12133 Mutschke G. — FM11 12068 Myagkov N.N. — SM11 10426

Na S.-W. — FSM1 12691

FSM1 12717 ¨ G. — FM16 11431 Nagele FM18 11300 Nagata M. — FM13 12291 Nagib H. — FM24 12965 Nagy P. — MS5 12651 Nait Abdelaziz M. — SM18 12339 Nakamura M. — FM1 12522 Nakano H. — MS6 11152 Nakas A.A. — SM25 11082 Nakata K. — FM1 11441 Nakatani A. — MS3 11122 Nalepka K. — MS3 11444 Nalepka P. — MS3 11444 Narayanan S. — MS1 12147 Narita F. — SM13 11426 SM13 11610 SM9 11425 Naumavicius R. — SM16 12546 Naumenko K. — SM27 11368 Nava A. — MS2 12064 Navarro C. — SM8 10698 Nayak H.V. — FM16 11991 Nazarenko L. — SM13 10555 Needleman A. — SM1 12737 SM2 11987 SM9 10910 Neel M.-C. — FM12 12376 Neethling S.J. — FSM5 12851 Neimark A.V. — FM4 10140 Neishtadt A. — FM22 12439 SL11 10551 (p. 241) Neitzel G.P. — MS5 12651 Nepomnyashchy A.A. — FM14 10543 Nerinckx K. — FM6 11602 Nesteruk I. — FM2 11017 Netto T.A. — SM22 11600 Newsom R.K. — MS6 10513 Ngan A.H.W. — FSM5 11663

412 Nguyen A.V. — FM15 11903 Nguyen Q.P. — FSM5 12986 Nicolas A. — MS2 12811 (p. 329) Nicolas M. — FM16 11749 FM17 11751 Nicolleau F. — FM22 10130 Nigmatulin R.I. — FM8 12190 Nihei T. — FM25 12041 Nijmeijer H. — SM25 11783 Nikiﬁrov S. — FM5 10531 Nikiforovich E.I. — FM25 12196 Nikitin N.V. — FM15 11929 Nikitin S.A. — FM15 11929 FSM7 11869 Nikolaevskiy V.N. — FM12 11109 Nikrityuk P.A. — FM21 12365 Niordson C.F. — SM18 11790 Nir A. — FM8 11250 Nishimura J. — SM3 10879 Nishimura M. — FM1 11441 Nishimura T. — SM10 12461 SM17 12464 Niss K. — FM23 10247 Niziol J. — SM25 12511 Noack B.R. — FM11 12624 FM11 12736 FM24 11454 Noda N. — MS1 11857 Nogarede B. — FM11 12602 Noguchi H. — SM1 12177 SM22 10692 Nohguchi Y. — FSM7 12488 Nore C. — FM13 10640 Norman J.T. — FM16 11991 Norris A.N. — SM25 12404 Noskowicz S.H. — FM17 12308 Nott P.R. — FM16 10603 Nouar C. — FM13 12766 Nouy A. — SM1 11769 Novak V. — MS1 12637 Novosiadliy V.A. — FM7 13005 Nowacki W.K. — SM14 10046 SM14 11308 Nowak A.J. — FM1 12409 Nowak M. — SM24 12413 Nowak Z. — SM19 12328 SM4 12333 Nycander J. — MS6 12210

Oberste-Brandenburg C. —

SM14 12121 Obrecht H. — SM22 12084 Oertel H. Jr. — FM5 11864 Ogasawara N. — SM18 10652 Ogden R.W. — FSM3 11227 SM12 11705 SL12 10143 (p. 263) Ohkitani K. — FM23 10938 Ohl C.-D. — FM8 11999

ICTAM04 Ohno N. — SM1 12177 SM22 10692 Okada M. — SM17 11700 Okandan M. — FM4 10753 Okkels F. — MS4 12047 MS4 12130 MS4 12536 Okulov V.L. — FM13 13019 Okumura D. — SM1 12177 SM22 10692 Olascoaga M.J. — MS6 12776 Olhoﬀ N. — SM24 10433 Oliferuk W. — SM18 11185 Olshevsky A.A. — SM2 12559 SM2 12559 Olsson T. — MS2 10670 Omang M. — FM19 11809 Onck P. — MS2 10627 Ong R. — MS1 11685 Ootao Y. — SM10 12465 Orantek P. — SM1 12562 Orlov S.V. — SM2 10342 Oron A. — FM14 10543 Ortiz M. — FSM6 11471 SM1 11154 SM9 12171 Osipov M.N. — SM7 11556 Osipov V. — FSM1 11508 Osiptsov A.A. — FM9 11703 Ospennikov N.A. — FM7 12344 Ostachowicz W.M. — MS1 10680 (p. 275) Ostoja-Starzewski M. — SM11 11682 Ostrowska-Maciejewska J. — SM6 10442 Ostrowski Z. — SM24 11137 Otheguy P. — MS6 12294 Ottino J.M. — FM22 11293 Ovsyannikov S.V. — SM7 12272 Ozawa H. — MS6 11732 Ozoe H. — FM7 12173 FM7 13022

Pagneux V. — FM24 12740

Paidoussis M.P. — FSM4 11078 Pakiela Z. — SM7 12666 Pakleza J. — FM8 12873 Pakula M. — SM15 12566 SM15 12577 Palaniappan D. — FM18 12981 Pamplona D.C. — SM22 12420 Pandolﬁ A. — SM9 12171 Pantelyat M. — SM2 10211 Papageorgiou D. — FM26 12715 Papas P. — FM24 11454 FM3 10918 Papoulia K. — SM1 12458 Parau E. — FM26 11345 Park J. — FM19 11637

413

Author Index Park N. — FM17 12790 Park S.-J. — SM17 10893 Parland H. — SM2 10459 Parnes R. — MS3 12432 Parra M.T. — FM8 11670 Parsons A.T. — FM2 12320 Pasero E. — FM24 11151 Pasol L. — FM16 10248 Pasqualino I.P. — SM1 12607 Passarel W. — SM24 10594 Pastoor M. — FM11 12624 Patterson M.D. — FM25 12051 FM9 12396 Paulino G.H. — SM10 12665 Paulino G. — SM10 10135 Pavlidis M. — FM18 12859 Pavlovskaia E.E. — FSM2 11302 SM11 12481 Pavlovskaia E. — SM25 10470 Pawlowski P. — MS1 13015 Payne D.A. — MS1 11685 P´ a ´czelt I. — SM2 11074 Peacock T. — FM26 11199 Pecquet E. — SM1 11298 Pedersen E.M. — FM1 11851 Pedersen N.L. — SM24 10669 SM24 11936 Pedersen P. — SM24 10669 Pedersen S.L. — SM17 12569 Peerlings R.H.J. — SM4 11415 SM4 12341 Pegushin A.G. — SM11 12203 Pellegrino S. — SM22 11097 Peradzy´ n ´ski Z. — FM19 12363 FSM2 12927 MS4 10971 Perelmuter M.N. — MS3 11599 Perov V. — MS6 11942 Perrin B. — FM17 13029 FM20 12092 Perzyna P. — SM5 12949 Petermann M. — FM22 12610 Petersen R. — FM23 10247 Petit S. — SM13 11180 Petrova V.E. — SM9 10579 Petryk H. — SM12 10996 Pettermann H.E. — SM13 12321 P¸e¸cherski R.B. — MS3 11444 SM18 10064 Pfeiﬀer F.G. — SM16 10809 Phedina M.E. — SM4 11779 Philippidis T. — SM13 12818 Phillips O.M. — FM9 10366 Phillips W.R.C. — FM26 11179 Photiadis D.M. — SM25 12404 Picandet V. — SM4 12749 Picciotto M. — FM20 13012 Pichler B. — SM11 12313

Pichler U. — MS1 10784 Piech´ ´ or K. — MS2 10838 Pieczyska E.A. — SM14 11308 Piedboeuf J.-C. — SM2 12773 Piekarski J. — MS2 11808 Pienkowska I.T. — FM18 11922 Pierard O. — SM4 11886 SM18 13038 Pierre L. — SM4 12277 Pietraszkiewicz W. — SM19 10287 Piiroinen P. — FSM2 11363 Pijaudier-Cabot G. — SM4 12749 Pilgun G.V. — SM19 12350 Pinto da Costa A. — SM12 12682 SM2 12646 Piotrowski Z. — FM9 12592 Pivovarov M. — SM25 11699 Placidi L. — FSM6 12040 Plamenevskii B.A. — SM9 10897 Plotnikov P. — FM26 11290 Plourabou´ ´e F. — FM12 11343 Plourabou F. — FM22 12731 Plunian F. — FM19 12120 Pocheau A. — FM21 12543 FM22 11190 Pocwierz M. — FM6 12565 Podgorski T. — FM1 10741 Pogorelov D. — SM17 10893 SM17 11640 SM17 11688 SM26 11589 Pohorecki R. — FM8 12311 Polach P. — SM17 10647 Polezhaev V.I. — FM15 11929 FSM7 11869 Poloukhina O.E. — FM9 11433 Polyakov N.V. — SM25 11716 Polyzos D. — SM1 12845 Pommier S. — SM8 12228 Poncet S. — FM24 10506 Ponta F. — FM25 11676 Ponte Casta˜ n ˜ eda P. — SM18 11005 Ponter A.R.S. — SM27 11201 Pop O. — SM9 10429 Popov Y.P. — FM15 11893 Popp K. — SM25 12322 SM2 12086 Pouget J. — SM10 12093 Pouliquen O. — FM16 11749 FM17 11751 FM17 11775 Powers T.R. — MS2 11866 Pozorski J. — FM24 10564 Pozorski Z. — SM24 12628 Pozrikidis C. — FM18 10766 Prat M. — FM12 11343 Preumont A. — SL13 10930 (p. 287) Price W.G. — FM2 12320 FSM4 11833 SM25 11834

414 Princevac M. — FM9 12738 Prioris S. — FM5 10984 Proczek M. — FM21 11364 Prokunin A.N. — FM18 11460 Prosperetti A. — FM10 11909 Prunet-Foch B. — FM4 11954 Pukhnachov V.V. — MS5 11757 Pumir A. — FM22 12048 Purini R. — MS6 13017 Putelat T. — SM2 12563 Putin G.F. — FM7 10538 FM7 12267 Puzzi S. — FSM6 11224 Py C. — FSM4 11801 Pyryev Y. — SM2 11481 Pyrz R. — SM13 10244 SM13 12489

Q

iu X. — FM24 11488 Qiu Z. — FM4 10753 Qu J. — SM8 11686 Querin O.M. — SM24 11623 Quey R. — SM14 12421 Quilliet C. — FSM5 11095 Quintard M. — FM12 10234

Rabaud M. — FM17 13029

FM24 11814 Radkowski S. — SM25 12977 Radler K.-H. — FM19 12120 Rafai S. — FM8 13025 Raftoyiannis I.G. — SM22 11172 Raghu Prasad B.K. — SM9 10419 Rajasekhar G.P. — FM17 11876 Rajchenbach J. — FM17 10959 Ramachandra L.S. — SM13 12531 Ramaswamy A. — SM3 10098 Rambod E. — FM1 12784 Ramm E. — SM4 11414 Ramos A. — FM4 12437 Randles P.W. — SM1 10829 Rao R.R. — FM18 10983 Raszillier H. — FM14 11413 Ravasoo A. — SM11 11620 Ravi-Chandar K. — SM9 12378 Razi Y.P. — MS5 10980 Read P.L. — FM22 12545 MS6 12494 Reardon P.T. — FM16 11725 Rebow M. — FM21 11364 Redelsperger J.-L. — MS6 10513 Rega G. — FSM2 12129 Regnier V. — FM15 12591 Regucki P. — FM25 12836 Reinaud J.N. — MS6 11336 Reinelt D.A. — FSM5 10597 Reinl A. — FM4 12254

ICTAM04 Rejniak K.A. — MS2 10689 Rekik A. — SM13 11507 Ren M. — FM7 11251 Ren W. — SM7 10492 Ren Z. — FSM4 10826 Renotte A. — FM11 12602 Resagk C. — FM19 12605 Resseguier T. de — SM18 11110 Rethore J. — SM1 11797 Reusch F. — SM4 10865 Rhines P.B. — SL14 12781 Ribeiro P. — SM25 12759 Ricci S. — SM4 11406 Richard P. — FM17 12883 Richard T. — SM25 12535 Richiardone R. — MS6 13017 Richterova J. — FSM3 11247 Ricken T. — SM15 10198 Ricoeur A. — SM9 11045 Riedel J.J. — SM20 11845 Rill G. — SM26 11672 Rindt C.C.M. — FM7 11251 Ringgaard S. — FM1 11851 Rinoshika A. — FM24 12004 Riou H. — SM25 11246 Ris V.V. — FM7 12730 Risbet M. — SM8 12228 Risso F. — FM20 12139 Rochinha F.A. — SM10 13018 Rodi W. — FM6 11348 Rodriguez M.A. — FM8 11670 Rodriguez-Ramos R. — SM13 11176 Rodzewicz M. — SM8 11371 Rogachevskii I. — FM10 12070 FM7 11053 Rohan E. — SM10 11671 Roig V. — FM20 12383 Rojiani K.B. — FSM7 12752 Rosakis A.J. — SM2 11987 Rosenthal B. — SM22 12084 Rossky P.J. — MS3 12132 (p. 217) Rothenburg L. — SM20 10866 Rother M.A. — FM8 12329 Roumi F. — SM13 10483 Roux S. — SM4 12021 Roy A. — SM18 11820 Roy D. — SM25 10750 Rozhkov A. — FM4 11954 Rozvany G.I.N. — SM24 11623 Ruan H.H. — SM5 11051 Ruban A.I. — FM2 11397 Rubin M.B. — SM1 12584 Rubinstein A.A. — SM9 11626 Rudi Y.A. — FM25 12032 Rudnyi E.B. — SM24 11316 Rueberg T. — SM15 11119 Ruimy C. — SM6 11901

415

Author Index Ruith M.R. — FM13 11995 Ruith M. — FM13 12425 Ruo A.C. — FM9 11294 Ruoﬀ R.S. — MS3 11594 (p. 303) Rusinek A. — SM14 10046 SM18 10734 Rymuza Z. — SM2 10999 Ryzhak E.I. — SM12 10867

Saada R.A. — SM2 12082

Saanouni K. — SM4 11885 Sabina F.J. — SM13 11176 Sackmann E. — SL15 11327 (p. 313) Sadlej K. — FM16 10993 Sadowski T. — SM7 11617 Saez A. — SM9 11321 Safarik P. — FM5 12219 Safran S.A. — MS2 12811 (p. 329) Sahn D. — FM1 12784 Saida S. — FM8 11299 Saif T. — MS3 12705 Saint-Jalmes A. — FSM5 12212 Saintillan D. — FM16 12005 Saito T. — FM5 11049 Sakalo V.I. — SM2 12559 Sakiyama T. — SM19 12224 Saksala T. — SM24 10621 Salalha W. — FM4 12319 Salem A. — FM13 12766 Salin D. — FM20 12092 Salmon J.-B. — MS4 12047 Salupere A. — SM11 11813 SM11 12568 Sam C.-H. — SM1 12458 Sam Han J. — SM24 11316 Samborski S. — SM7 11617 Samsonov A.M. — SM11 10114 Samsonowicz J. — SM25 12977 Sanches C.T. — SM25 11494 Sanchez U. — FM20 11189 Sanchis A. — FSM4 12956 Sano O. — FM17 11876 FM17 11883 Sanomura Y. — SM18 11350 Santaoja K. — SM4 10960 Santiago J.G. — MS4 12961 (p. 343) Sapountzakis E.J. — SM1 10665 Sarkar S. — FM24 11116 Sarler B. — FM7 11160 Sarout J. — SM15 12713 Sartorius D. — FM24 12675 Sato K. — SM15 10790 SM19 10770 Sato M. — SM7 10691 Saussine G. — SM17 11361 Sauzay M. — SM18 12697 Savic L. — FM2 12030

Saville D.A. — SL16 10529 Savova R. — SM6 10972 Sayir M.B. — SM11 11085 Sburlati R. — SM6 12987 Scarella G. — SM2 12382 Schaeﬀer N. — FM19 12330 Schanz M. — SM15 11119 Scheichl B. — FM2 11083 Scheimberg S. — SM24 10594 Schenkel T. — FM5 11864 Schiehlen W. — SM17 11223 SM17 12134 Schjoedt-Thomsen J. — SM13 12489 Schlogl S.M. — SM14 11625 Schmid M. — FM9 12626 Schmid P. — FM11 12305 Schmidt C.F. — MS2 10709 (p. 355) Schmidt M.J. — SM20 12779 Schmidt R. — MS1 11752 Schneider L.C.R. — SM15 12903 Schnerr G.H. — FM5 12066 Schranz C. — SM19 11021 Schreﬂer B.A. — SM15 12114 Schreurs P.J.G. — SM1 12131 Schroeder W. — FM25 12690 Schuette H. — SM4 12088 Schulte H. — SM16 12417 Schulze D. — FM19 12290 Schulze T.P. — FM21 12768 Schumacher J. — FM22 10896 Schuster M. — FM6 12337 Schwarz U. — MS2 12811 Scott R.A. — SM11 12772 Sederman A.J. — FM21 11320 Sedlak P. — MS1 12637 Segev R. — SM6 11523 Seguin P. — SM17 12539 Segurado J. — SM13 11080 Seidel C. — SM25 12785 Seiden G. — FM16 11884 Seifried R. — SM17 12134 Sejnoha M. — SM23 12380 Selezov I. — SM11 11401 Self B.P. — FSM7 12642 Sellier A. — FM19 10025 Semenov Y.A. — FM8 10580 Semenova I.V. — SM11 10114 Semler C. — FSM4 11078 Semma E. — FM15 11386 Seon T. — FM20 12092 Serebryakov V. — FM5 12066 Sergeichev I.V. — SM20 10374 Sergent A. — FM7 12220 Serre E. — FM25 12080 Seto T. — SM10 12461 Seyranian A.P. — SM22 11939 Shakeri M. — SM13 13009

416 Shaqfeh E.S.G. — FM16 12005 Shardakov I.N. — SM6 12289 Sharma P. — MS3 11474 Sharp R.S. — SM26 10701 Shasholko D.I. — SM2 10999 Shchennikov V.V. — SM7 12272 Shemer L. — FM20 11655 FM26 10245 Shen C. — SM1 11580 Sheng P. — FM4 11165 Sherbaum V. — FM3 12199 Shergold O. — MS2 10988 Sheu T.W. — FM1 11187 Shevchenko I.V. — FM12 10199 Shevchenko K.V. — SM2 12559 Shevchuk V. — SM2 11697 Shevtsov I.Y. — SM24 11015 Shi D.-L. — MS3 12314 Shibutani Y. — SM23 12571 Shillor M. — SM2 11452 Shimizu M. — FM8 11299 Shimokawa S. — MS6 11732 Shimomura Y. — FSM6 11862 Shimoyama K. — FM9 11164 Shindo Y. — SM13 11426 SM13 11610 SM9 11425 Shiono Y. — SM17 12582 Shishkina E. — SM25 11055 Shklyaev S.V. — FM7 12344 Shleykel A.L. — FM7 13005 Shodja H.M. — SM13 10483 Shrira V.I. — FM13 12233 FM26 12087 Shrivastava S.C. — SM22 11653 Shtern V. — FM19 10235 FM25 10683 Shulzhenko M. — SM2 10211 Shy S.S. — FM22 12209 Shyshkanova G. — SM2 11611 Sichermann W.M. — FSM4 11912 Sideman S. — MS2 11148 Sidoroﬀ F. — SM14 12747 Sielamowicz I. — FM17 11169 Siemaszko A. — SM1 12579 Sigmund O. — SM24 11744 Silva E.C.N. — SM10 12665 Silveira Neto A. — FM6 11456 Silvestre N. — SM22 12558 Simons G. — MS3 12384 Simpson J.E. — FM25 12051 Simes F.M.F. — SM12 12682 Sinclair G.B. — FSM3 11704 Sinka C.I. — SM15 12903 SM15 12917 Sira E. — FM6 10857 Siso-Nadal F. — FM19 11124 Sittner P. — MS1 12637

ICTAM04 Skali S. — FM13 10185 Skews B.W. — FM5 11072 Skotheim J. — MS2 11581 Skrzypek J.J. — SM4 11796 Sladek J. — SM10 10836 SM19 11238 Sladek V. — SM10 10836 SM19 11238 Slobozhanin L.A. — MS5 12447 Sawianowski J.J. — FSM3 12406 Slowicka A. — FM4 12910 Smas P. — SM24 11634 Smeulders D.M.J. — SM11 11271 Smirnov E.M. — FM7 12730 Smirnov S. — FM3 11888 Smirnovskii A.A. — FM7 12730 Smith F.T. — FM2 11210 Smits A. — SM13 12818 Smyth A.W. — MS1 13007 So R.M.C. — FM6 10163 Soares M.E.S. — SM25 11494 Sobczyk K. — CL 10697 (p. 19) Sobieczky H. — FM5 10945 Sofonea M. — SM2 11452 Sokoowska R. — SM25 12984 Sokoowski J. — SM24 10052 SM24 11486 Soldati A. — FM20 13012 Sommeria J. — FM9 12410 MS6 11547 MS6 12494 MS6 12601 Soomere T. — FM9 12099 Sorensen J.N. — FM13 13019 Sorokin S. — FSM1 11337 Sotera M.R. — FSM2 12972 Sottos N.R. — MS1 11685 MS1 13011 MS1 13010 Sousa J.M.M. — FM3 12348 Souza L.C.G. — SM3 13004 Souza L.F. — FM13 12381 Sparks R.S. — FM17 10524 Spelt P.D.M. — FM8 12106 Spencer A.J.M. — SM10 12273 Spiegel E. — SL17 10158 (p. 365) Spiegl M. — SM27 11981 Squires T.M. — FM16 12452 MS4 11777 Sreenivasan K.R. — FM22 10896 Srigiriraju S.V. — MS2 11866 Staalhand J. — MS2 10670 Stachurski A. — SM4 12333 Staicu A.D. — FSM5 12373 Staquet C. — FM26 12424 Staroszczyk R. — FM18 10297 Stavroulakis G. — SM3 11036 Steblyanko P.A. — SM1 11202 Steen M. — FM12 12617

417

Author Index Steenhoven A.A. van — FM7 11251 Stefanelli R. — FSM1 12586 Stefani F. — FM19 12613 Stegmann J. — SM24 12098 Steigenberger J. — SM25 11699 Steinberg L. — FSM3 11140 Steindl A. — SM22 11816 Steinmann P. — FSM3 12392 SM12 11277 SM1 11211 SM1 12924 Steinrueck H. — FM2 12030 Stepan G. — FSM2 10252 Stepanova L.V. — SM4 11779 Stephan P. — FM14 12293 Stevanovic-Hedrih K. — SM17 10624 Stewart D.S. — MS4 12976 (p. 379) Stichel S. — SM26 11795 Stieglitz R. — FM19 10931 Stijnman M. — FM10 11909 Stiller J. — FM19 12587 Stokes Y.M. — FM8 11855 Stolpe M. — SM24 12302 Stone H.A. — FM16 12056 FM1 12423 MS4 12966 Storakers B. — SM2 11365 Stoychev G.B. — SM18 11953 Stremler M.A. — FM22 12158 Striz B. — FSM3 11247 Strozik M.D. — FM10 11916 Struzewska J. — FM9 12592 Stulov A. — SM1 10728 Stupkiewicz S. — SM2 12553 Styczek A. — FM6 12565 Su A. — FM3 10367 Sudak L. — SM6 12122 Sudhakar V. — SM25 12984 Sugano N. — SM17 12537 Sugii T. — MS2 12192 Sugimoto N. — FSM1 10294 Sugimoto T. — FM1 10132 Sugiura T. — SM11 12418 Sugiura Y. — SM25 12593 Suiker A.S.J. — SM13 12680 SM14 10648 Sukoriansky S. — MS6 11152 MS6 11942 Sullivan J.M. — FSM5 10597 Sun H. — FM10 10445 Sun L. — SM10 10135 Sun M. — FM5 11049 Sun Q. — SM14 12881 Sun Y. — SM8 11686 Sundaresan S. — FM20 12397 Sung Lee H. — SM8 12034 Suponitsky V. — FM13 11461 Suzuki T. — FM11 12387 FSM1 11561

Svendsen B. — SM4 10865 Swaters G.E. — MS6 10634 Swevers J. — SM16 11282 Swift F.J. — SM26 10909 Swinney H.L. — SL18 10495 Symens W. — SM16 11282 Sypeck D.J. — FSM5 12395 Szabo R. — FM21 11362 Szalai R. — FSM2 10252 Sze K.Y. — SM9 11568 Szekrenyes A. — SM13 11747 Szmyd J.S. — FM7 12173 FM7 13022 Szmyd J. — FM22 12610 Szojda L. — SM4 11228 Szumbarski J. — FM6 12565 Szumowski A. — FM11 11918 Szwaba R. — FM5 12997 Szymczak P. — FM12 11994 Szymczyk J.A. — FM21 11331 FM7 11905 Szyszkowski W. — SM3 12756 ´ Swito´ n ´ ski E. — SM25 12770

Tabaei A. — FM26 10762

FM26 11199 Tabeling P. — MS4 12047 MS4 12130 MS4 12536 MS4 12673 MS4 12960 Taberlet N. — FM17 12883 Tadmor G. — FM11 12624 FM24 11454 Tagawa T. — FM7 13022 Taillard K. — SM14 12225 Takada T. — MS1 10821 Takagi M. — SM14 11308 Takagi S. — MS2 12192 Takahashi N. — FM24 10937 Takayama K. — FM5 11049 Takeda T. — SM13 11610 Talagrand O. — MS6 10803 Taleb L. — SM14 12421 SM14 12747 Tamai K. — FM5 11049 Tamuzs V. — FSM7 12259 Tan L.H. — FM14 11477 Tan S.H.N. — SM13 10412 Tan V.B.C. — SM13 10412 Tang J.X. — MS2 10709 Tang M. — FM24 12652 Tang Y. — SM9 11626 Tanigawa Y. — SM10 12465 Tanizawa K. — SM17 11700 SM3 10879 Tanno H. — FM5 11049 Tardu S. — FM11 11394

418 Tarn J.-Q. — SM10 10364 Tarnai T. — SM1 11482 SM25 11384 Tartar M. — FM13 10640 Tatsumi T. — FM24 12151 Tauchert T.R. — MS1 12018 Tay A.A.O. — SM1 12901 Tay T.-E. — SM13 10412 Taya M. — SM10 12978 Tcholakova S.S. — FM8 11669 Tejchman J. — SM15 12358 Telega J.J. — FM12 11248 FSM3 11642 MS2 12100 SM13 11895 SM13 12667 SM13 12732 SM1 12562 SM2 11452 Ten Hagen T.L.M. — FM1 11900 Teodorczyk A. — FM3 12911 Terletska K.V. — SM11 11377 Tesdall A.M. — FM5 11730 Teymur M. — SM11 10891 Thermann K. — SM12 10996 Thess A. — FM19 11120 FM19 11906 FM19 12090 Thiria B. — FM11 12736 Thite S. — SM1 12441 Thivolle-Cazat E. — MS6 11547 Thomas P.J. — MS6 10261 Thomer O. — FM25 12690 Thompson A.F. — FM21 11328 Thomsen J.J. — SM25 12694 SM25 11679 Tian J. — FSM5 12395 Tian Z. — SM9 12174 Tigoiu V.M. — FSM3 11638 Tihon J. — FM14 12656 Tilgner A. — FM19 12125 Timchenko V. — FM15 11386 Ting E.C. — SM1 10270 Ting L. — FM25 11683 Tkachev P.V. — SM12 12223 Tobushi H. — MS1 10821 SM14 11308 Tokarzewski S. — FSM3 11642 SM13 11895 SM13 12667 Toland J.F. — FM26 11290 Toll S. — SM15 12518 Tomita Y. — SM12 11432 Topolnikov A.S. — FM8 12190 Toropov V. — SM24 12870 Touhei T. — SM1 10274 Tran-Cong T. — FSM5 12435 Travnicek Z. — FM11 12258 Tr¸¸ebicki J. — SM11 11682 Trevelyan P. — FM14 10220 Troger H. — SM22 11816 Trostinetsky E. — FM20 11655

ICTAM04 Trotsyuk A. — FM5 10531 Trulsen J. — FM19 11809 Trumel H. — SM18 11110 Trzci´ n ´ ski R. — FM8 12873 Trzebicki M. — SM13 12560 Tsai J.C. — FM17 12445 Tsai P.-S. — SM16 12712 Tsamopoulos J. — FM14 12858 FM18 12859 Tsekhmister Y.V. — FM26 11591 Tsemakh D. — FM8 11250 Tsepoura K.G. — SM1 12845 Tsubota K.-I. — FM1 12522 Tsubota K. — MS2 12453 Tsuru T. — SM23 12571 Tsutahara M. — FM26 11746 Tsypkin G.G. — FM21 11926 Tuck E.O. — FM8 11855 Tucker P.G. — FM24 12430 Tuckerman L.S. — FM13 10640 FM13 12431 Tuckerman L. — FM6 10921 FM7 10914 Tuliszka-Sznitko E. — FM25 12080 Tumin A. — FSM1 12477 Tur M. — SM8 10698 Turner M.R. — FM2 12241 Turska E. — SM24 12679 SM9 11910 Turteltaub S. — SM14 10648 Tutty O.R. — FM2 12320 Tvergaard V. — SM9 10910 Tylikowski A. — SM10 10045 Tyliszczak A. — FM3 11914 Tyrkiel E. — FSM2 11416

U

aliev Z. — FM3 11031 Ualiyev G. — SM17 10533 Ualiyev Z.G. — SM17 10533 Ubachs R.L.J.M. — SM1 12131 Udwdia F.E. — SM3 11163 Ugawa A. — FM17 11883 Uhl T. — SM7 12799 Uj J. — SM13 11747 Ulbrich R. — FM20 11745 FM20 12886 Ulrych B. — SM2 10211 Ungarish M. — FM16 11884 MS6 11585 Urban D. — SM5 12043 Ursem N.T.C. — FM1 11900 Ustohalova V. — SM15 10198

Vadillo J.L. — FM11 10872

Vainchtein D. — FM22 12439 Vakakis A.F. — SM25 10408 Vakhitova N.K. — FM8 12190

419

Author Index Vakhrouchev A.V. — SM1 12054 Valance A. — FM17 12883 Valdek U. — SM7 12551 Valle V. — SM9 10429 Valtorta D. — MS2 12064 Valverde J. — SM17 12071 Van Brussel H. — SM16 11282 Van Hemelrijck D. — SM13 12818 Vanden-Broeck J.-M. — FM26 11345 FM26 12715 Vanneste J. — MS6 12104 Varghese S. — FM9 12980 Varhsney K. — FM15 12349 Varyanychko M.A. — SM19 11815 Vasiljev P. — SM16 10900 Vasudeva Murthy A.S. — FM9 12980 Velasco Fuentes O.U. — FM25 12748 Vennemann P. — FM1 11900 Vereecke B. — SM4 10974 Veres I.A. — SM11 11085 Vereshchaka S.M. — SM19 10071 Veron F. — MS6 12722 Versluis M. — FM17 10253 FM1 12400 Vesenjak M. — FSM4 10826 Viallat A. — FM1 12423 Viatkina E.M. — SM1 12723 Viba J. — FSM7 12259 Vidya Sagar R. — SM9 10419 Vierendeels J. — FM6 11602 Vignes-Adler M. — FM4 11954 Villafruela J.M. — FM8 11670 Villermaux E. — FM22 10532 FM22 12326 FM8 11317 Vimmr J. — FM6 11702 Vinod N. — FM2 12631 Viovy J.-L. — FM4 11209 Vit T. — FM11 12258 FM2 12062 Vitkova V. — FM1 10741 Vlachogiannis M. — FM14 10557 Voigt L.K. — FM23 10247 Volkov K.N. — FM20 10416 Volkova V.E. — SM25 12632 Volles R. — SM1 11298 Vollmann J. — MS3 12384 Vorobev A.M. — MS5 12402 Voyiadjis G.Z. — SM18 10393 Vu-Delcarte C.D. — FM15 12636

Waarsing J.H. — MS2 11808

Waclawczyk M. — FM24 10564 Wada S. — FM1 12522 MS2 12453 Wadley H.N.G. — FSM5 12395 Wagner C. — FM8 11462 Wagner S. — FM24 12675

Wahi P. — SM25 12814 Wajnryb E. — FM16 11409 FM18 11300 FM8 11411 Walenta Z.A. — FM3 12911 FM4 12910 MS4 10971 Walker P.G. — FM1 11851 Wall D.P. — FM13 12291 Walsh A.M. — FM17 12338 Wan F.S. — FM8 12596 Wan Y. — MS1 12016 Wang C.-Y. — SM1 10270 Wang C. — FM8 12596 SM1 11580 Wang D.W. — FM26 11230 Wang H.-T. — SM9 11568 Wang H. — SM1 11720 Wang J. — SM11 11099 Wang L.-S. — SM16 12712 Wang M. — MS3 12132 (p. 217) Wang P. — SM1 11720 SM9 12174 Wang R.-Z. — SM1 10270 Wang Z. — SM9 12174 Weaire D. — MS5 10598 (p. 387) Weber J.E. — FM26 11803 Weber M. — SM16 12758 Wei M. — FSM1 10907 Wei Q. — MS6 12194 Wei Y. — MS3 11279 Weichen S. — SM10 10501 Weichert D. — SM13 12588 SM18 12058 Weidman P. — FM26 11199 FM2 11984 FSM4 11986 Weier T. — FM11 12068 Weigand B. — FM10 10445 Weir G.J. — SM2 11008 Weiss D.A. — FM4 12254 Weitsman Y.J. — SM13 11014 Weiwen L. — SM24 12287 Welch K. — SM7 12551 Wen W. — FM4 11165 Weronko J. — SM22 12934 Wesfreid J.E. — FM11 12736 Westerweel J. — FM10 11088 FM1 11900 Wheeler L. — MS3 11474 White J.M. — MS3 12132 (p. 217) White S.R. — MS1 13010 MS1 13011 Whitesides G.M. — MS4 12966 Widmann R. — SM15 10198 Wiercigroch M. — FSM2 11302 SM11 12481 SM25 10470 Wierschem A. — FM14 10642 FM14 10928 Wieteska R. — FM6 12618

420 Wi¸eckowski Z. — SM20 10231 Wiggins S. — FM22 11293 Wijeyewickrema A.C. — SM11 12514 Wijngaarden L. van — OL 10498 (p. 1) Willers B. — FM21 12365 Williams P.D. — MS6 12494 Willis J.R. — SM13 11782 SM2 12563 Wilma´ n ´ ski K. — SM11 10612 Wilson H.J. — FM16 11473 Wilson M. — FM7 10438 Wingerde A.M. van — SM13 12818 Winter R.M. — MS3 12132 (p. 217) Wi´ ´sniewski K. — SM19 12328 SM24 12679 SM9 11910 Witkowski A.S. — FM10 11916 Witkowski W. — FM3 12911 SM19 11890 Wnuk M.P. — SM9 10250 Wojciechowski J. — FM11 11918 Wojnar R. — SM13 12732 Wola´ n ´ ski P. — FM3 12745 FM3 12843 Wong Y.W. — SM22 11097 Woods A.W. — FM21 11926 Worster M.G. — FM21 11270 FM21 11320 FM21 11328 FM21 12276 Woude D. van der — FM22 12633 Woznica K. — SM18 12339 Wriggers P. — SM1 12534 SM1 12584 Wu D. — SM9 12492 Wu J. — SM4 12036 Wu L. — FM15 12591 SM10 11551 Wuest A. — FM9 12626 Wurz W. — FM24 12675 Wysocki M. — SM15 12518

Xia M. — SM7 11764

Xiao Q.Z. — FSM6 11806 Xing T.J. — FSM4 11833 SM25 11834 Xiong A.-K. — FM11 12491 Xiong P.Y. — SM25 11834 Xu H. — MS5 10714 (p. 229) Xu X. — SM7 11764

Yadav A. — FM3 12348

Yamaguchi E. — MS4 11607 Yamaguchi T. — FM1 12522 MS2 12453 Yamamoto K. — SM17 11700 SM3 10879

ICTAM04 Yamashita K. — FSM1 10294 Yamasue K. — FSM2 12014 Yamazaki Y.H. — MS6 12494 Yang J. — SM1 11580 Yang S. — FM4 11165 SM8 11737 Yang X. — FM24 10149 Yang Y. — FSM4 12521 Yano J.-I. — MS6 12140 Yao Z. — SM1 11720 Yaremchuk V.P. — FSM7 11869 Yarin A.L. — FM4 10932 Yasinskyi A. — SM2 11697 Yasniy P.V. — SM8 11920 Yau J.-D. — SM3 11287 Yazdi S.S.H. — SM25 11904 Yazykov V. — SM26 11589 Yevdokymov D.V. — MS5 10958 Yin H. — SM10 10135 Yokosawa S. — FM1 12522 Yokota K. — FM8 11299 Yoo W.-S. — SM17 10893 Yoon Y. — FM8 11289 Yoshida F. — SM24 12870 SM27 12450 Yoshimura T. — FM24 12151 Yoshinaga T. — FM14 11739 Youn S.-K. — SM24 11826 SM27 11827 Young K.Y. — SM24 12185 Young T.H. — SM25 11867 Yourdkhani A. — SM13 10189 Yu S. — SM9 11934 Yu T.X. — SM5 11051 Yuan T.H. — FM22 12209

Z

abielski L. — FM19 12979 Zachara A. — FM8 12873 Zaera R. — SM18 10734 Zahn M. — FM4 11982 Zairi F. — SM18 12339 Zaj¸ac D. — FM20 12886 Zakharov V.E. — FM26 10861 Zaleski S. — FM8 11614 FM8 11723 Zametaev V.B. — FM2 11780 Zammali C. — SM11 12763 Zampolli M. — FSM1 11373 Zangeneh M.S. — SM25 11904 Zaoui A. — SM13 11507 SM13 11805 Zardecki D. — SM26 11832 Zavala-Garay J. — MS6 12776 Zavaliangos A. — SM15 12917 ´ G. — SM1 11389 Zboinski Zeidis I. — SM25 11699 Zeilstra C. — FM17 10253

Author Index Zelinger Z. — FM9 10467 Zeman J. — SM23 12380 Zeman V. — SM25 11759 Zenit R. — FM8 11026 Zenkovskaya S.M. — FM7 13005 Zhang C. — SM10 10836 Zhang K. — FM7 12168 MS6 12169 Zhang N.-H. — FSM2 12674 Zhang R. — SM8 11686 Zhang R. — SM27 10906 Zhang W.W. — FM8 12429 Zhang X. — FM26 11680 Zhao X. — SM17 11020 Zhao Y.-P. — SM16 10510 Zheltovodov A.A. — FM24 10455 Zheng L. — SM9 12174 Zheng X.J. — MS3 12486 Zheng X. — SM6 11998 Zhou C. — SM8 12391 Zhou M. — MS3 10105 Zhou W. — SM1 12901 Zhou Y.C. — MS3 12486 SM13 12496 SM9 12492 Zhou Y. — FM24 12004 FSM4 12521 Zhu H.X. — FSM5 11292 Zilitinkevich S. — FM7 11053 Zimmermann K. — SM25 11699 Zinchenko A.Z. — FM8 12329 FM8 12741 Ziopaja K. — SM24 12628 Zi´ olkowski M. — FM19 12605 Zitha P.L.J. — FSM5 12986 Zmitrowicz A. — SM2 11613 Zouhar G. — FM21 12365 Zussman E. — FM4 10932 FM4 12319 ˙ Zochowski A. — SM24 11486 ˙ Zukowski J.M. — FM10 11916 ˙ Zyczkowski M. — SM24 12075

421

Mechanics of the 21st Century Proceedings of the 21st International Congress of Theoretical and Applied Mechanics, Warsaw, Poland, 15--21 August 2004

Edited by

WITOLD GUTKOWSKI Polish Academy of Sciences, Warsaw, Poland and

TOMASZ A. KOWALEWSKI Polish Academy of Sciences, Warsaw, Poland

This eBook does not include ancillary media that was packaged with the printed version of the book. A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 ISBN-13 ISBN-10 ISBN-13

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Contents

Committees

viii

Preface

ix

Acknowledgments

xi

Exhibitors

xii

Congress Statistics

xiii

Opening Ceremony

xv

Closing Ceremony

xxiii

Scientiﬁc Program

xxx

Interplay between Air and Water Leen van Wijngaarden

1

Stochastic Dynamics of Engineering Systems Kazimierz Sobczyk

19

Multibody Dynamics: Bridging for Multidisciplinary Applications Jorge A.C. Ambr´sio ´

61

Rapid p Formation of Strongg Gradients and Diﬀusion in the Transport of Scalar and Vector Fields Konrad Bajer

89

Wave-Vortex Interactions in the Atmosphere, and Climate Prediction Onno Bokhove

103

Near-critical Point Hydrodynamics and Microgravity Daniel A. Beysens

117

Flaw Tolerant Nanostructures of Biological Materials Huajian Gao, Baohua Ji, Markus J. Buehler, and Haimin Yao

131

vi

ICTAM04

Transport and Mixing in the Atmosphere Peter H. Haynes

139

Variational and Multiscale Methods in Turbulence Thomas J. R. Hughes, Victor M. Calo, and Guglielmo Scovazzi

153

Mechanics of Thin Film Structures Henrik Myhre Jensen

165

Nonlinear Dynamics in Ocean Engineering Edwin J. Kreuzer, Wolfgang M. Sichermann

173

A Bridge g between the Micro- and Mesomechanics of Laminates: Fantasy or Reality? Pierre Ladev`eze Turbulence and Large-Eddy Simulations Marcel R. Lesieur Nano Mechanical Analysis of IFM Force Proﬁles on Self-Assembled Monolayers Mingji Wang, Kenneth M. Liechti, Vibha Srinivasan, John M. White, Peter J. Rossky Collisional Granular Flows with and without Gas Interactions in Microgravity Michel Y. Louge and Haitao Xu

187 203

217

229

Probability Phenomena in Perturbed Dynamical Systems Anatoly Neishtadt

241

Mechanics of Rubberlike Solids Ray W. Ogden

263

Elastic Wave Propagation p g Development for Structural Health Monitoring Wieslaw Ostachowicz

275

On the damping of a piezoelectric truss Andre Preumont

287

Strength of Nanostructures Rodney S. Ruoﬀ, ﬀ Nicola M. Pugno

303

Micromechanics of Cells Erich Sackmann, Andreas Reuther, and Doris Heinrich

313

Elastic Interactions of Biological Cells Samuel A. Safran, A. Nicolas, U. S. Schwarz

329

Contents

vii

Electrokinetic Flow Instabilities in Microﬂuidic Systems Hao Lin, Michael H. Oddy and Juan G. Santiago

343

Molecular Mechanics of Cytoskeletal Components M. Atakhorrami, K.M. Addas, M. Buchanan, G.H. Koenderink, F.C. MacKintosh, J.X. Tang, Christoph F. Schmidt

355

Topics in Astrophysical Fluid Dynamics Edward A. Spiegel

365

Miniaturization of Explosive Technology and Microdetonics D. Scott Stewart

379

Foams in Microgravity Denis Weaire and Simon Cox

387

Author Index

395

21st International Congress of Theoretical and Applied Mechanics President and Chairman of Local Organizing Committee

Witold Gutkowski

Co-Chairmen

Michal Kleiber Wlodzimierz Kurnik

Secretary-General

Tomasz A. Kowalewski

Members of Local Organizing Committee Konrad Bajer (Warszawa) Romuald B¸edzinski ´ (Wroclaw) Tadeusz Burczynski ´ (Gliwice) Tadeusz Chmielniak (Gliwice) Krzysztof Dolinski ´ (Warszawa) Stanislaw Drobniak (Cz¸¸estochowa) Jozef ´ Giergiel (Krak´ ´ ow) Stanisaw Kocanda ´ (Warszawa) Jozef ´ Kubik (Bydgoszcz) Zenon Mroz ´ (Warszawa) J´ ´ ozef Niziol (Krak´ ´ ow) Wojciech Nowacki (Warszawa) Janusz Orkisz (Krak´ ´ ow) Andrzej Palczewski(Warszawa)

Wojciech Pietraszkiewicz (Gda´ n ´ sk) Stanislaw Radkowski (Warszawa) Czeslaw Rymarz (Warszawa) Kazimierz Sobczyk (Warszawa) Jaroslaw Stefaniak (Pozna´ n) ´ Jacek Stupnicki (Warszawa) Andrzej Styczek (Warszawa) Gwidon Szefer (Krak´ ´ ow) ´ Eugeniusz Switonski ´ (Gliwice) Andrzej Tylikowski (Warszawa) Zenon Waszczyszyn (Krak´ ´ ow) Zbigniew Wesolowski (Kielce) Wadyslaw Wlosinski ´ (Warszawa) Czesaw Wo´zniak (Cz¸¸estochowa)

Piotr Perzyna (Warszawa)

Henryk Zorski (Warszawa) ˙ Micha Zyczkowski (Krak´ ´ ow)

Henryk Petryk (Warszawa)

Members of the IUTAM Congress Committee Hassan Aref (USA) member of XCCC Ted Belytschko (USA) representative of IACM Martin Bendsoe (Denmark) representative of ISSMO Dimitri Beskos (Greece) David Bogy (USA) Dick van Campen (Netherlands) David Durban (Israel) Fernand Ellyin (Canada) representative of ICM Juri Engelbrecht (Estonia) Norman Fleck (UK) Ben Freund (USA) Graham Gladwell (Canada) Peter Gudmundson (Sweden) Michael Hayes (Ireland) representative of ISIMM Tsutomu Kambe (Japan) Bhushan Karihaloo (UK) representative of ICF Alfred Kluwick (Austria) Valery Kozlov (Russia) Edwin Kreuzer (Germany)

Yu Ku (USA) Gary Leal (USA) representative of ICR Peter Lugner (Austria) representative of IAVSD Fernando Lund (Chile) Keith Moﬀatt (UK) Chairman, member of XCCC Peter Monkewitz (Switzerland) Rene Moreau (France) member of XCCC, representative of HYDROMAG Niels Olhoﬀ (Denmark) member of XCCC Timothy Pedley (UK) Secretary, member of XCCC Bernhard Schreﬂer (Italy) member of XCCC Kazimierz Sobczyk (Poland) Pierre Suquet (France) Ernst Tuck (Australia) Manuel Velarde (Italy) representative of CISM Eiichi Watanabe (Japan) Feng-Gan Zhuang (China)

Preface

The 21st International Congress of Theoretical and Applied Mechanics (ICTAM04) took place August 15 – 21, 2004, in Warsaw, Poland. It was organized by Polish National Committee of IUTAM, Institute of Fundamental Technological Research of the Polish Academy of Sciences (IPPT PAN) and Warsaw University of Technology. The Congress venue was the main building of Warsaw University of Technology. The idea of congresses devoted to mechanics, can be traced back to a conference on problems of ﬂuid mechanics in Innsbruck, 1922. It was organized by four individuals, whose names, are and will, remain very well known to next generations of scientists, C. W. Oseen, T. Levi-Civite, T. von K´ a´rm´an, and L. Prandtl. This conference was so fruitful, that the organizers decided to arrange similar meetings in the future, every four years, and to extend the scope of the future meetings to include solid mechanics. Hence a series of meetings started eighty years ago with the 1st ICTAM held in Delft, Netherlands. From the meetings of the Congress Committee sprang the idea of a more permanent organization to look out for the world interests in the mechanical sciences. Thus, IUTAM, the International Union of Theoretical and Applied Mechanics, was formed on September 26, 1946. In 1947 IUTAM became a member of ICSU, the International Council of Scientiﬁc Unions, itself founded in 1931. The highest authority of IUTAM is the General Assembly, with delegates from the Adhering Organizations, each of which is aﬃliated with a national learned society in a given country.

Scientiﬁc Program Contemporary mechanics poses both, the fundamental problems from the area of pure science, and its strong links with modern technology. It spreads over such areas of our knowledge as oceanography, physical chemistry, biology, medicine, geophysics and astrophysics. Hence, any conclusions deduced in the framework of mechanics, are likely to have

x

ICTAM04

a value for other ﬁelds. We may easily ﬁnd prove of it in the scientiﬁc program of the ICTAM04. It consists of plenary opening and closing lectures, sectional lectures, mini-symposia, and contributed papers presented in lecture and seminar presentation sessions. These were intended to cover all aspects of mechanics. All contributed papers were peer reviewed. Recommendations had been received from Pre-selection Committees of the National Committees of the nine countries: Canada, France, Germany, Japan, PR China, Poland, Russia, UK and USA. Moreover, recommendations had been received from the Chairs of the Mini-Symposia and of the Pre-nominated Sessions with classiﬁcations of the papers submitted within the topics of their Symposia/Sessions. Finally, the International Papers Committee paid careful attention to the above recommendations. Accordingly, of the 2086 eligible submissions 1574 contributions were invited by the IPC for their presentation. The total number of submitted and accepted papers represents a quite substantial enhancement relative to the previous congresses, providing evidence of vitality of the contemporary mechanics. This volume of Proceedings consists of a book with full texts of invited talks and attached CD-ROM with Extended Summaries of 1225 papers presented during the Congress by authors1 . We have tried to assemble the paper volume with an extensive index of names and papers collected on the CD-ROM. We hope that this volume – pre-ordered at the Congress in record numbers – will be found useful not only as a document of the event but to assess achievements and new paths of research in mechanics of 21st Century. W. GUTKOWSKI T.A. KOWALEWSKI

1 From 1273 papers presented during the Congress we exclude multiple presentations and those given by proxy.

xi

Acknowledgments We would like to express our thanks to our colleagues from the IUTAM Congress Committee, and from the International Papers Committee. Their advice and eﬀorts have helped us to overcome some problems, connected, as usual, with the organization of world-wide meetings. Special gratitude goes to the Chairs responsible for Mini-Symposiums and Pre-nominated Sessions. Their work in organizing sessions, encouraging papers submission in the area of interest, and later reviewing all papers cannot be overestimated. Smilingly, we are deeply indebted to members of the nine National Committees for their valuable contribution to the reviewing procedure. The organization and execution of the Congress was performed by the Institute of Fundamental Technological Research of Polish Academy of Sciences and Warsaw University of Technology. The detailed work of organization was due to many persons from both institutions, personnel of PCO Mazurkas Travel as well as numerous external co-workers. Without their extremely valuable help organization of such meeting would be impossible. Thank you all so much! W. GUTKOWSKI T.A. KOWALEWSKI

Exhibitors Cambridge University Press Dantec Dynamics EDEN Elsevier Era Business Intelligent Laser Applications International Publishing Service Institute of Fundamental Technological Research of Polish Academy of Sciences Kluwer Polish Tourist Organisation Springer Strategic Test Scandinavia Testlab

Congress Statistics A detailed list of ICTAM04 participants is enclosed on the CD-ROM. Here, for brevity, we give the basic statistics of the Congress. The total number of pre-registered to ICTAM04 participants was 2928. The Congress organizers received 2245 abstracts of contributed papers before the ﬁnal deadline of January 16, 2004. Not all abstracts were followed by an extended summary. Hence, the International Papers Committee invited 1574 contributions reviewing 2186 submitted extended summaries. Finally 1235 contributed papers and 38 invited talks were given in Warsaw. Fluid Mechanics was a subject of 550 presentations, Solid Mechanics was selected as a subject of 611 presentations, and 112 contributions aimed to cover problems involving both areas of mechanics and education in mechanics. The total number of participants, accepted papers and presentations represent a quite substantial enhancement relative to the previous congresses (Table 1), providing evidence of vitality of the contemporary mechanics. Table 2 displays brieﬂy country statistics of the participants and presentations. Table 1.

ICTAM04 compared with four previous congresses

ICTAM

Submitted Papers

Grenoble 1988 Haifa 1992 Kyoto 1996 Chicago 2000 Warsaw 2004

1262 1183 1642 1953 2245 (2186)

Presented papers/ presented from organizers country 573 n. a. 420 n. a. 703 192 1126 445 1273 144

Participants/ participants from organizers country 951 340 525 85 936 332 1430 587 1515 194

xiv

ICTAM04

15 3 5 3 2 1 3 1 1 2

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

9 8 8 8 7 7 7 7 5 5 4 4 4 3 3 3 2 2 2 2 1 1 1 1 1 1 1 1515

8 8 8 6 5 4 5 4 5 5 4 3 2 3 2 3 1 2 1 1 1 1 1 1 1 1 1273

Invited lectures

Country Portugal Estonia Iran Norway Bulgaria Ireland Latvia South Africa Romania Mexico Belarus Singapore Turkey Lithuania Serbia Slowenia Chile Kazakhstan Libya Saudi Arabia Algeria Armenia Croatia New Zealand Nigeria Quatar Slovakia Total:

Presentations

185 144 153 100 98 72 75 41 38 36 27 21 19 15 13 17 15 11 15 14 14 11 7 11 9 7 10 9

Participants

204 194 170 118 107 89 79 54 43 38 37 30 25 19 19 19 18 18 18 17 14 13 12 12 12 10 10 9

Invited lectures

Country USA Poland France Germany UK Japan Russia China Netherlands Ukraine Israel Denmark Italy Belgium Canada S. Korea Austria Sweden Taiwan Czech Republic Spain Brazil Finland India Switzerland Australia Hungary Greece

Presentations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

ICTAM04 country statistics

Participants

Table 2.

1

38

Opening Ceremony2 Professor W. Gutkowski Distinguished Guests, Esteemed Colleagues, Ladies and Gentlemen, It is my great honor and pleasure to welcome you to the 21st International Congress of Theoretical and Applied Mechanics. I welcome you on behalf of the Local Organizing Committee in Poland, here in Warsaw in this historical and beautiful ediﬁce of the Warsaw University of Technology. The ediﬁce, a witness of many scientiﬁc, cultural and political events. Just sixty years ago, in August heavy ﬁghting was going on here during the Warsaw Uprising. Today you are in the country of growing economy, and for several months a member of European Community. Mister President, Distinguished Guests and Colleagues, Ladies and Gentlemen! It is my great honor and privilege to announce the opening of the 21st International Congress of Theoretical and Applied Mechanics! We are here, scientists from 55 countries, joined by a common passion, a passion for Mechanics. Ladies and Gentlemen, let me now introduce our distinguished guests and organizers: Professor Keith Moﬀatt, President of the International Union of Theoretical and Applied Mechanics and President of the Congress Committee of IUTAM, Professor Werner Schiehlen, Vice-President of IUTAM, Professor Ben Freund, Treasurer of IUTAM, Professor Dick van Campen, Secretary General of IUTAM, Professor Tim Pedley, Secretary General of IUTAM, Professor Micha Kleiber, Minister of Science and Information Technology of Polish Government, Chairman of the State Committee for Scientiﬁc Research, Co-chairmen of the Local Organizing Committee, Professor Janusz Lipkowski, Vice-President of Polish Academy of Sciences, Professor Stanisaw Mankowski, ´ President of the Warsaw University of Technology, Professor Wodzimierz Kurnik, Vice-President of the Warsaw University of Technology, Co-Chairman of the Local Organizing Committee, and Professor Tomasz Kowalewski, Secretary-General of the Congress. Ladies and Gentlemen, to start the Congress I will kindly request Professor Keith Moﬀatt of the Cambridge University, President of the International Union of Theoretical and Applied Mechanics and President of the IUTAM Congress Committee, to address the Congress.

2 The unabridged version of the account of the Opening Ceremony can be found on the accompanying CD-ROM.

xvi

ICTAM04

Professor K. Moﬀatt Distinguished Guests, Dear Colleagues, Ladies and Gentlemen, I am honoured to stand before you in this magniﬁcent setting of the Warsaw University of Technology, and to say some words of welcome on behalf of IUTAM, the International Union of Theoretical and Applied Mechanics. This is the 21st Congress in our history, so here we celebrate our ‘coming-of-age’. We also celebrate the 80th anniversary of the ﬁrst Congress of Applied Mechanics (as it was then called), which was held in Delft in 1924. We should pay tribute to the great scientists who had the vision to initiate this series of quadrennial Congresses, particularly J.M. Burgers, Theodore von Karman, Ludwig Prandtl and G.I. Taylor. I note that this year is the centenary of Prandtl’s seminal paper on boundary-layer theory. An IUTAM Symposium on “One Hundred Years of Boundary Layer Research” has been held just last week in G¨ o¨ttingen, in recognition of the crucial role that this branch of mechanics has played in Aerodynamics and many other ﬁelds of application. IUTAM itself grew from the early Congresses, and was formally established in 1948; Poland was one of the earliest members and has been an Adhering Organisation of IUTAM since 1952. The Congress Committee is appointed by the General Assembly of IUTAM, but otherwise retains the autonomy that it enjoyed from the outset. This means that when things go wrong, IUTAM can blame the Congress Committee; of course, when things go right, IUTAM shares in the credit! There were 207 participants at that ﬁrst Congress in Delft; these early pioneers would be gratiﬁed to know that there are over 1500 participants at this 21st Congress, as there were also at the 20th Congress in Chicago four years ago. It is a measure of the continuing vigour of our subject that the Congress attracts such strong and widespread participation. Of course we must be careful not to equate quantity with quality; we do however strive to maintain high quality in the papers selected for presentation whether as lectures or as seminar/poster presentations. Here I pay tribute to the hard work of our International Papers Committee, which, advised by National Committees and by Convenors of Mini-Symposia and Pre-nominated Sessions, had to select the papers from more than 2000 that were submitted; their task was extremely demanding, but the quality and range of the programme before us for this Congress will I hope convince you that they have done an excellent job, in spite of the great diﬃculties. It is highly appropriate that we hold this Congress in Warsaw, a city of grace and beauty, which has emerged from the dreadful traumas of the last century to face the new century with greatly renewed vigour and

Opening Ceremony

xvii

optimism. One cannot visit this land without being deeply conscious of the long years of oppression suﬀered by the Polish people. It is hard now for us to comprehend the tragedy of the Warsaw uprising, which erupted exactly sixty years ago, and which has been commemorated here this month. The long bleak years of the Cold War gave little solace to the Polish people. Yet throughout these years, there remained for our community a ﬂicker of light in the gloom: I refer to the biennial meetings in ﬂuid mechanics that were organised in Poland throughout the 60s and 70s and well into the 80s by our dear colleague Wladek Fiszdon, meetings which enabled scientiﬁc contacts between East and West to be established and nurtured across and despite the Iron Curtain, and which helped to keep alive the universal and apolitical spirit of our subject. Wladek Fiszdon became a member of the General Assembly of IUTAM in 1971; he is now a life-member in recognition of his devoted services. His health unfortunately does not allow him to be with us today, but I am sure you would wish me to convey to him the very good wishes of the whole IUTAM community. IUTAM is an organisation founded on principles of friendly collaboration between scientists irrespective of race, creed or gender. It is a privilege to come to its Congresses, and to renew acquaintance with so many old friends and colleagues. It is also a particular privilege for me to welcome to this Congress younger scientists who may be attending an IUTAM Congress for the ﬁrst time. I remember vividly my own such experience: With the encouragement of my late mentor George Batchelor, I attended the 10th Congress in Stresa, Italy, in 1960. It was an exhilarating experience, which opened my eyes to the great scope and challenges of our vast ﬁeld of endeavour. If this is your ﬁrst IUTAM Congress, I hope that you will ﬁnd it a similarly exhilarating experience, and that you will look forward to succeeding Congresses with ever-keener anticipation. G.I. Taylor once told me that he liked to save his best paper in each four-year period for presentation at ICTAM; we should do well to follow his good example! The time for thanks will come at the end of the Congress on Friday, but I’d like now to say at least a preliminary thank you to the local team who have worked so hard on all the preparations: the Congress President Witold Gutkowski, the Secretary-General of the Congress Tomasz Kowalewski, and the whole Local Organising Committee. They have done a superb job, and I sense that this is going to be a correspondingly superb Congress. Thank you, Witold and Tomasz! Now without further ado, I have much pleasure in conﬁrming the Opening of ICTAM04.

xviii

ICTAM04

Welcome to Warsaw, and we wish you a fruitful Congress; or, as it is said here: Witamy w Warszawie i z˙ yczymy udanego Kongresu! Professor W. Gutkowski Thank you very much, Keith, for your kind words about Polish contribution to international mechanics. Let me also express our great appreciation for you, IUTAM Bureau and Executive Committee of the Congress Committee for the support and valuable advice during four years of preparation for the Congress. Let me ask now Professor Micha Kleiber, Minister of Science and Information Technology of Polish government, Chairman of the State Committee for Scientiﬁc Research and Co-chairman of the Local Organizing Committee to address the Congress. Professor M. Kleiber Mr. President of IUTAM, Very Distinguished Guests of the Congress, Colleagues and Friends, On behalf of the government of the Republic of Poland I have the honour and great pleasure to welcome you to the 21st International Congress of Theoretical and Applied Mechanics here in Warsaw. I hope you will forgive me if I start on a rather personal note but I want to emphasize how special for me is this opportunity of addressing you, dear colleagues, in my current double capacity as a co-chairman of the Congress and as a representative of the Polish public administration. Frankly, I have to admit that because of my other duties I have not done much in terms of the Congress organization, and it is my colleagues at the Institute of Fundamental Technological Research of Polish Academy of Science and the Warsaw University of Technology who at the end of the week will deserve to be congratulated for their exceptional engagement and, I am sure, organizational perfection. The Congress is something very special in every respect – it attempts to summarize what we have accomplished in the last 4 years, it facilitates everyone’s research planning for the future, it stimulates interactions with other ﬁelds of science and technology, it strengthens our professional and personal links, it gives us a marvellous feeling of being a group of people coming from so many, sometimes very distant, places all over the world, but a group of people who share similar interests, similar vision of rationality in all human endeavors, similar perception of ways to better understand and improve the world around us. I am one of those who consider the ﬁelds of mechanics as a true research success story of last decades – contrary perhaps to some people

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from outside of our profession, I have no doubts that the progress we have achieved together deﬁnitely places mechanics on one of the very top locations in the hierarchy of modern research disciplines – with tremendous impact on both our perception of the physical world and the means to implement new technologies so much improving the quality of our life. We are very glad to host you, distinguished delegates, in Poland, in the city of Warsaw. You will ﬁnd here, I am sure, traditional hospitality and openness of the people, great public interest in the Congress debates and interesting encounters with the country and its capital so much rooted in the complex history of Europe – country which has only recently managed to deﬁnitely overcome its politically so much unfortunate and undeserved past, country which, as a new member of the European Union, looks forward with unmatched optimism and aspirations to contributing to the progress in science and its positive implications for all of us. I wish all the participants vivid, instructive presentations and discussions and, at the same time, enjoyable stay in Warsaw. Professor W. Gutkowski Thank you, Professor Kleiber. I would like to express my gratitude for your initiative to organize the Congress in cooperation with the Institute of Fundamental Technological Research and the Warsaw University of Technology, in this beautiful ediﬁce. This has been a great idea. Special thanks also for strong ﬁnancial support of the Congress by the Ministry of Science and Information Technology. May I ask Professor Janusz Lipkowski, Vice President of the Polish Academy of Sciences, representing the President of the Polish Academy of Sciences, to the podium. Professor J. Lipkowski Mr. President, Excellences, Distinguished Guests and Participants of the Congress, The Polish Academy of Sciences consists of three parts. First, it is a corporation of elected members, the most distinguished scientists of our country who represent the Polish science. The Academy also organizes the scientiﬁc life in the country through its scientiﬁc committees. There are more than 100 committees in all scientiﬁc disciplines and scientists from all institutions in Poland, universities, institutes, societies etc. serve as members of the committees. One of these, Polish National Committee of the International Union of Theoretical and Applied Mechanics, is coorganizing the present Congress. Last but not least, the Academy has scientiﬁc institutes performing high level research studies.

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About 9% of all scientiﬁc ‘population’ in Poland are employed in the institutes while their output measured by the number of scientiﬁc publications amounts to more than one quarter of all scientiﬁc output in the country. We are proud of our institutes; please feel invited to visit any of them during your stay in Poland, you will be more than welcome. I am pleased to mention that one of the institutes, the Institute of Fundamental Technological Research, is also a coorganizer of the Congress. By the way, it is the mother institution of our Minister of Scientiﬁc Research and Information Technology who served as its director before joining the government of Poland. Ladies and Gentelmen, I made this brief introduction to our Academy of Sciences in the belief that you would become interested in developing scientiﬁc cooperation with our Academicians and institutions. We do count on international joint research which is an obvious choice in the XXIst century. I wish you a very fruitful conference, and a pleasant and enjoyable stay in Poland! Professor W. Gutkowski Thank you very much, Professor Lipkowski. The Polish Academy of Sciences gave us very strong support when we were proposing the organization of the Congress here in Warsaw. Almost all members of the Local Organizing Committee are members of the Committee of Mechanics of the Academy. Let me now request Professor Stanisaw Ma´ nkowski, ´ President of the Warsaw University of Technology, to address the Congress. Professor St. Ma´ n ´ kowski Professor, Minister, Ladies and Gentleman, Distinguished Guests, The Warsaw University of Technology, with its tradition of teaching in the ﬁeld of technical and exact sciences reaching the year 1826, the symbol of which is the Main Building, erected over a 100 years ago and where the Large Hall is an expression of beauty, but also of application of the theoretical and applied mechanics, would like to extend a warm welcome to the participants of this, already the 21st , congress. The Warsaw University of Technology which teaches approximately 32 000 students is the largest technical university in this part of Europe. Teaching and research are inseparable. Therefore I believe that the 21st International Congress of Theoretical and Applied Mechanics is organised in the right place. I wish the authors of the presented scientiﬁc contributions many practical applications, accurate theoretical descriptions and many quota-

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tions. I would like to express my gratitude to the International Committee for this decision, and the Polish National Committee, the Institute of the Fundamental Technological Research of the Polish Academy of Sciences and the Team of the Warsaw University of Technology led by Vice-President Prof. Wlodzimierz Kurnik – for the passion and eﬀort put into the organisation of the congress. Professor W. Gutkowski Thank you very much. It is my great pleasure to say that Professor Mankowski, ´ together with Professor Wodzimierz Kurnik, Vice-President of the University, have undertaken an enormous organizing task to host us in this historical building. Ladies and Gentlemen, I believe you will like the city of Warsaw, and other places in Poland. There are so many interesting historical and cultural places to visit. Palaces, beautiful gardens, museums, concert halls and theaters. Unfortunately, within a couple of days you will be able to see just a few of them. So, we are bringing to you here a small fragment of our cultural heritage – the music of Chopin. For a quarter of an hour, Mr. Marcin Rudzinski, ´ a student of the Fryderyk Chopin Academy of Music, will present us some of the nicest pieces of this famous composer. ... Ladies and Gentlemen, Dear Colleagues, Mechanics is ﬂourishing! No doubt! Increasing number of submissions from ICTAM to ICTAM is showing theoretical and applied importance of mechanics and its multidisciplinary relations with other sciences. In Kyoto there were 1642 submitted papers, in Chicago 1953, and in Warsaw 2186! Thank you for coming, and for contributing with your papers to the development of Mechanics in the 21st century. We can say that all submitted papers present the latest, up-to-date results of our research. Really latest! Suﬃce it to say that on January 5, four days before the submission deadline, only 600 papers reached our server. I needn’t say that no smile could be seen on our faces. Then during the last four days, the number of incoming papers started to grow exponentially with time. Finally, at midnight on the 9th of January the number of submitted paper reached 21 hundred! The authors were then really anxious to send their latest results. Let me mention an e-mail message we received in the late afternoon on Friday. The author of the message was kindly asking if the deadline at midnight should be considered at Greenwich time or at local time! The authors constitute the core and the essence of the Congress. However, its realization strongly depends on sponsoring. Let me then ac-

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knowledge numerous supporting institutions and sponsors who helped us to start the organization of ICTAM04. First of all, I would like to express our thanks to the Polish Universities and Institutes supporting the Congress, and contributing, at the same time, to the exhibition “Mechanics in Poland”. ... Speaking about Polish Mechanics, I would like to turn your attention to a booklet by Professor Zbigniew Olesiak, which you will ﬁnd in your congress bag. The book shows the 19th century roots of our mechanics embedded in our diﬃcult history of that time3 . Let me next express gratitude to numerous scientiﬁc journals for publishing our announcements. This certainly increased the number of people interested in the congress. Let me express appreciation for our friends, organizers of the Chicago ICTAM, for their support, very much needed at the beginning of the Congress preparations. It would be impossible to mention all our sponsors by name. To all of them presented on these two banners, I am sending, on behalf of the Local Organizing Committee our deepest thanks. Ladies and Gentlemen. Before we start our work I will ask Professor Tomasz Kowalewski, Secretary-General of the Congress to give you some latest information. Professor T. Kowalewski I am very touched to see here, in Warsaw, so many old friends and all those numerous new ones, with whom I had pleasure to exchange thousands of emails during the last four years. Thank you very much for coming . . . – began his talk Secretary-General of the Congress. After passing on the latest technical information concerning organizational matters, Prof. Kowalewski expressed his gratitude to numerous young volunteers who were of great help in the last stages of the congress preparation. Professor W. Gutkowski Thank you, Tomasz. In a few minutes Professor Leen van Wijngaarden will present an Opening Lecture. Presiding the session are Professor Hassan Aref from the USA and Michal Kleiber from Poland. I cordially wish you successful presentations and discussions of your papers. Enjoy your stay in Warsaw! Thank you for your attention. 3 The

text of the book can be found on the accompanying CD-ROM.

Closing Ceremony1

Closing Address by Professor Keith Moﬀatt, President of IUTAM Dear Colleagues, Ladies and Gentlemen, We have now come to the Closing Ceremony of this Congress. We shall announce the Bureau Prizes in the course of this ceremony. But my ﬁrst duty is to call on Professor Tim Pedley, Secretary of the Congress Committee, to present his report on ICTAM04. Congress Report by Professor Tim Pedley, Secretary General of the Congress Committee Well, Ladies and Gentlemen, another ICTAM has come to an end. It has been a wonderful Congress and the biggest yet — even slightly bigger than the Millennium Congress in Chicago four years ago. There were 1445 active participants, including students, up from 1430, from 57 diﬀerent countries (up from 54). That last number is a welcome increase in view of IUTAM’s aim of spreading interest and understanding in mechanics more and more widely in the world. ... This has been a resoundingly successful Congress, both scientiﬁcally and organisationally. On behalf of the IUTAM Congress Committee, and of yourselves, I would like to express our profound thanks to the President of the Congress, Professor Witold Gutkowski, and to the SecretaryGeneral, Professor Tomasz Kowalewski, for their immense contribution to ensuring that success. Possibly most impressive, on the organisational front, was the provision of fully synchronised computer facilities in all the lecture rooms, with the uploading of all computerised presentations in advance. ... On behalf of the Congress Committee, I would also like to thank the members of the International Papers Committee (IPC) for their dedicated hard work in evaluating and selecting the papers to be presented at the Congress. These were Professors Peter Monkewitz, Howard Stone, Bernhard Schreﬂer, Kazimierz Sobczyk and Viggo Tvergaard. A lot of 1 The unabridged version of the account of the Closing Ceremony can be found on the accompanying CD-ROM.

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reading was required in February and March of this year, and culminated in an intense four day meeting in Warsaw at the end of March. The IPC were guided by the recommendations of the chairs of the mini-symposia and the pre-nominated sessions, and by the national committees of nine major countries: Canada, China, France, Germany, Japan, Poland, Russia, UK and USA. All those involved in the paper assessment process deserve our warm thanks. ... Professor Keith Moﬀatt Now, Ladies and Gentlemen, I have great pleasure in announcing that the Congress Committee agreed yesterday on the location of the next IUTAM Congress, which will be held in August 2008. As you will know, we have now held 21 Congresses, all in the Northern hemisphere. The next Congress will, for the ﬁrst time, be in the Southern hemisphere. The Australasian bid to hold the Congress in Adelaide, South Australia, has been successful. The Congress President will be Professor Ernie Tuck, and I have pleasure in now inviting him to say a few words. Professor Tuck expressed his gratitude for electing Adelaide for the next congress, inviting all participants to Australia in 2008. Acknowledgments by Professor Witold Gutkowski, President of ICTAM04 and Chairman of the Local Organizing Committee Mister President, Ladies and Gentlemen, Dear Colleagues, The organization of an ICTAM consists, among other things, in cooperation of many bodies: Congress Committee, International Papers Committee, Chairs of Sessions, and National Committees of IUTAM. I would like to express our gratitude to all those, for their very eﬀective cooperation, which facilitated our complex work of the Local Organizing Committee. Sincere thanks! As you already know, there are three institutions standing behind the organization of ICTAM, here in Warsaw. The idea of organizing an ICTAM in Poland has its long history. It originated in the Committee of Mechanics of the Polish Academy of Sciences and at the some time the Polish National Committee of IUTAM. Without the intellectual encouragement of the whole Polish mechanics community, the organization of the congress in our country wouldn’t be possible. This is then right time and right place to express my sincere appreciation to all members of the Committee of Mechanics,

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for emboldening us in the organization of ICTAM04. Thanks to you we could disseminate information about the Congress to a very broad international scientiﬁc community. Sincere thanks! The Institute of Fundamental Technological Research, known as IPPT, has also been, for years, strongly supporting the organization of an ICTAM in Poland. When the Congress Committee of IUTAM invited us to organize the congress, the institute became the center of preparatory works. All scientists and administration were highly supportive. IPPT hosted meetings of IUTAM Bureau and XCCC, as well as the International Papers Committee. I wish to express my sincere thanks to the directors of the institute, Professors Kleiber and Nowacki, for their help and for their patience, when watching the disorder inevitable in such situations and paying sharply growing telephone bills. Thank you, Professor Kleiber, thank you, Professor Nowacki, thank you, all friends from IPPT. The idea of organizing the Congress in Warsaw found many supporters at the largest university in Poland, the Warsaw University of Technology. The university has oﬀered its beautiful ediﬁce we are now in as a site of the congress. The organization of ICTAM here was strongly encouraged by Professor Stanisaw Mankowski, ´ President of the university and the host of the Welcome Reception. Thanks to the team led by Professor Wlodzimierz Kurnik, Vice President of the University and Co-chairman of the Local Organizing Committee, together with Professor Stanisaw Radkowski, we had this beautiful building prepared for our gathering. (. . . ) Professors Ma´ n ´kowski, Kurnik and all friends from the university — many thanks! Such a great event as our congress couldn’t be organized without the help of a professional bureau organizing large conferences. We asked Mazurkas Travel to: take care of your money, make hotel reservation, serve lunches and coﬀees, prepare poster stands, transportation, organization of excursions and many other things for good comfort of participants. Let me then express my thanks to Mazurkas Travel and its director Mrs. Barbara Zygmunt. The dissemination of information started with the ﬁrst announcement and then continued with the second. The nice booklet of the second announcement is the work by Professor Krzysztof Doli´ n ´ski, member of the Local Organizing Committee. ... It is impossible, to mention by names all young volunteers, mostly PhD students. It has been a real pleasure to see all those strongly motivated young girls and boys working hard and trying to do their

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best. They were led by two extremely active and eﬃcient members ´ eczkowska and Dr. Kamil of the Congress Bureau — Mrs. Izabella Sl¸ Kulesza. ´ e¸czkowska, Dr. Kulesza and all young Professor K. Doli´ n ´ski, Mrs. Sl¸ volunteers, your contribution to the successful Congress cannot be overestimated. Thank you! Last but not least, I am acknowledging the heart, soul, brain and calculator of the Congress organization — Professor Tomasz Kowalewski, Secretary General of ICTAM 2004! You have been meeting him from the beginning, at the congress page, that is since September 2000. Tomasz has loaded for you all update information. For the ﬁrst time in the history of IUTAM Congresses, participants could see the program and read an abstract and extended summary of an arbitrary paper, still being home, before reaching the congress site. This was Tomasz’s idea and his realization. The perfect preparation of the meeting of the International Paper Committee helped much to take right decisions on accepting papers for presentations. Again, this was Tomasz’s idea and realization with the help of several PhD students. Suﬃce it to say that each of 2000 papers could be found within seconds by title, author’s name or country. And with all that, thousands, thousands of e-mails sent and replied to. It is impossible to enumerate all the works he has done, but I should add at least two others. He kept our ﬁnances under strict control and he infected the minds of all young people who worked with him with his enthusiasm for organization of ICTAM 2004. ... There are not enough words to express my deepest thanks to you and I hope all participants will join me in conveying you our sincere thanks for your great contribution to the success of the Congress. Thank you, Tomasz. In order to commemorate your great contribution to organization of the Congress, let me hand you this 19th century statuette of a Bacchanate, a woman, as we know enjoying good wines. The statuette is inscribed in Polish: To Professor Tomasz Kowalewski for his outstanding contribution to organization of ICTAM04. Organizing Committee. IUTAM Bureau Prizes — Professor Keith Moﬀatt (certiﬁcates), Professor Witold Gutkowski (statuettes) The members of the Bureau have this week been attending many lectures and seminar/poster presentations given by young researchers. The word ’young’ is of course a relative term: I regard everyone younger than myself as young! But to be eligible for a Bureau prize, you must be really

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young, and that means under 35 years of age. The Bureau met today over lunch to agree on the winners, and I have to say ﬁrst that the competition was a stiﬀ one — we may well say of Olympic standard; and three ’gold medallists’ have emerged from this competition. The prizes consist of a certiﬁcate, a cheque for $500, and, as a special gesture from the local organisers, a beautiful owl, symbol of wisdom, cast in bronze. The three prizes are awarded as follows: Ingo Kaiser, of the Institute of Robotics and Mechatronics, Wessling, Germany, for his lecture on “The running behaviour of an elastic wheelset”; Ingo gave a beautiful demonstration of the eﬀect of elastic deformation on the stability wheels running on a track; you should be warned of this if you plan to return home by rail after the Congress! Taisuke Sugii, of the University of Tokyo, Japan, for his seminar/poster presentation entitled “Molecular dynamics study of permeation process of small molecules through a lipid bilayer”; biomechanics is a ﬁeld of tremendous potential and rapidly growing importance; this paper provides an excellent example of the application of mechanical principles at the biomolecular level.

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Pantxica Otheguy, of LadHyX, Ecole Polytechnique, Palaiseau, France, for her lecture on “Instability of corotating vertical vortices in a stratiﬁed ﬂuid: why strongly stratiﬁed turbulence is not similar to 2D turbulence”; Pantxica gave a beautiful interpretation of the ‘zig-zag instability’ to which such vortices are subject, illustrated by a video of an experiment showing the nonlinear outcome of this instability. We congratulate these three young scientists on the excellence of their presentations. Ladies and Gentlemen, This week the General Assembly of IUTAM met twice, and as part of its business elected its new Bureau to serve our Union for the next four years from 1st November 2004. The President will be Professor Ben Freund from Brown University, Providence, Rhode Island, USA, who has served IUTAM so successfully as Treasurer for the past eight years. Professor Dick van Campen will continue as Secretary-General for a further four-year term. Dick has been extremely successful in implementing recommendations of a previous Appraisal Committee, both as regards setting up Working Parties in nine diﬀerent priority areas, and in developing an excellent website for IUTAM, at which information about all our activities can be easily accessed. We are most grateful to him for his willingness to continue to carry this heavy responsibility. Professor Juri Engelbrecht, from Estonia, who has served on the Bureau for eight years, has accepted the important position of Treasurer. I myself continue, according to IUTAM rules, as Vice President for the next four years. Four other members of the Bureau have been elected by the General Assembly. These are: Professors Tsutomu Kambe (Japan), Alfred Kluwick (Austria), Niels Olhoﬀ (Denmark) and Zhemin Zheng (China). I wish to thank the retiring members of the Bureau, Carlo Cercignani, Roddam Narasimha, Jean Salencon ¸ and Werner Schiehlen (retiring VicePresident) who have served with great conscientiousness and wisdom. We have also elected four new members-at-large of the General Assembly, in recognition of their distinction in mechanics and their service to our Union: these are Professors Andy Acrivos (USA), Sol Bodner (Israel), Werner Schiehlen (Germany) and Franz Ziegler (Austria). So you see that we have done our best to ensure that IUTAM will continue to be well governed! Now I just have to add some words of thanks myself: ﬁrst to my old friend and colleague, Tim Pedley, who will continue as Secretary of the Congress Committee for a further four years. This is a task of great

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responsibility, which Tim has carried out with ﬂair and eﬃciency. I ask you to join me in thanking him. Second, I wish to thank Witold Gutkowski, who has been such a gracious host to us here in Warsaw. The smooth running of the Congress owes much to his eﬀorts over the last four years. Finally, I wish to echo Professor Gutkowski’s remarks concerning Tomasz Kowalewski, Secretary-General of the Congress. Tomasz has been indefatigable in his control of every detail of this immense organisation, in which he has shown amazing skill, patience, good humour and eﬃciency, and we all owe both Tomasz and Witold a resounding vote of thanks. And even more ﬁnally, I wish to thank all of you, Invited Sectional Lecturers, Convenors of Mini-Symposia, Chairmen of Pre-nominated Sessions, contributors of lectures and seminar/poster presentations, and all of you who by your presence have made this Congress such a brilliant success. I wish you all a safe journey home, and will look forward to seeing you again down under in Adelaide in August 2008, if not before!

Keith Moﬀatt

Scientiﬁc Program On the following pages the contents of the scientiﬁc program of the Congress are listed. The program consists of plenary opening and closing lectures, eighteen sectional lectures, six minisymposia, and sixty prenominated sessions devoted to all aspects of mechanics. OL CL SL

Opening Lecture Closing Lecture Sectional Lectures

Mini Symposia MS1 MS2 MS3 MS4 MS5 MS6

Smart materials and structures Tissue, cellular and molecular biomechanics Mechanics of thin ﬁlms and nanostructures Microﬂuids Microgravity ﬂow phenomena Atmosphere and ocean dynamics

Pre-Nominated Sessions on Fluid Mechanics FM1 FM2 FM3 FM4 FM5 FM6 FM7 FM8 FM9 FM10 FM11 FM12 FM13 FM14 FM15 FM16 FM17 FM18 FM19 FM20 FM21 FM22 FM23 FM24 FM25 FM26

Biological ﬂuid dynamics Boundary layers Combustion and ﬂames Complex and smart ﬂuids Compressible ﬂow Computational ﬂuid dynamics Convective phenomena Drops and bubbles Environmental ﬂuid dynamics Experimental methods in ﬂuid mechanics Flow control Flow in porous media Flow instability and transition Flow in thin ﬁlms Fluid mechanics of materials processing Fluid mechanics of suspensions Granular ﬂows Low-Reynolds-number ﬂow Magnetohydrodynamics Multiphase ﬂows Solidiﬁcation and crystal growth Stirring and mixing Topological ﬂuid mechanics Turbulence Vortex dynamics Waves

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Pre-Nominated Sessions on Solid Mechanics SM1 SM2 SM3 SM4 SM5 SM6 SM7 SM8 SM9 SM10 SM11 SM12 SM13 SM14 SM15 SM16 SM17 SM18 SM19 SM20 SM21 SM22 SM23 SM24 SM25 SM26 SM27

Computational solid mechanics Contact and friction mechanics Control of structures Damage mechanics Dynamic plasticity of structures Elasticity Experimental methods in solid mechanics Fatigue Fracture and crack mechanics Functionally graded materials Impact and wave propagation Material instabilities Mechanics of composites Mechanics of phase tranformations (jointly with IACM) Mechanics of porous materials Mechatronics Multibody dynamics Plasticity and viscoplasticity Plates and shells Rock mechanics and geomechanics Solid mechanics in manufacturing Stability of structures Stochastic micromechanics Structural optimization Structural vibrations Vehicle dynamics Viscoelasticity and creep

Topics Involving Both Fluid Mechanics and Solid Mechanics FSM1 FSM2 FSM3 FSM4 FSM5 FSM6 FSM7

Acoustics Chaos in ﬂuid and solid mechanics Continuum mechanics Fluid-structure interaction Mechanics of foams and cellular materials Multiscale phenomena in mechanics Education in mechanics

The description of the session consists of the session ID, the descriptive name of the session (e.g. ‘Smart materials and structures’) and the list of responsible co-chairs in parentheses. The list of the contributions follows. It is sorted by the unique lecture ID. The Mini Symposia (MS1. . . MS6), apart from the Lecture Presentations and Seminar Presentations contain Introductory Lectures. The latter are distinguished by the boldface letters of the author names.

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OL – Opening Lecture 10498 Leen van Wijngaarden (Netherlands): Interplay between Air and Water

CL – Closing Lecture 10697 Kazimierz Sobczyk (Poland): Stochastic Dynamics of Engineering Systems

SL – Sectional Lectures 10042 Konrad Bajer (Poland): Rapid Formation of Strong Gradients and Diﬀusion in the Transport of Scalar and Vector Fields 10143 Raymond W. Ogden (UK): Mechanics of Rubberlike Solids 10158 Edward Spiegel (USA): Problems in Astrophysical Fluid Dynamics 10495 Harry L. Swinney (USA): Scaling in Quasi-2D Turbulence Experiments in a Rotating Flow 10508 Pierre J. Ladev´eze (France): A Bridge Between the Micro- and Mesomechanics of Laminates: Fantasy or Reality? 10512 Roland Keunings (Belgium): Non-Newtonian Fluid Mechanics Using Molecular Theory 10529 D.A. Saville (USA): Electrokinetics & Electrohydrodynamics in Microﬂuids 10544 Edwin Kreuzer (Germany): Nonlinear Dynamics in Ocean Engineering 10551 Anatoly Neishtadt (Russia): Probability Phenomena in Perturbed Dynamical Systems 10731 Marcel Lesieur (France): Turbulence and Large-Eddy Simulations 10772 Huajian Gao (Germany): Nanoscale Mechanics of Biological Materials 10880 Daniel A. Beysens (France): Near-Critical Point Hydrodynamics and Microgravity 10930 Andre Preumont (Belgium): Some Issues in Active Vibration Control of Smart Structures 11040 Jorge Ambr´ ´ osio (Portugal): Multibody Dynamics: Bridging for Multidisciplinary Applications 11327 Erich Sackmann (Germany): Microviscoelasticity of Cells: Cells as Viscoplastic Bodies 12160 John F. Brady (USA): Suspensions: From Micromechanics to Macroscopic Behavior 12781 Peter B. Rhines (USA): Ocean Circulation and its Inﬂuence on Climate 13003 Thomas J.R. Hughes (USA): Variational and Multiscale Methods in Turbulence

MS1 – Smart materials and structures (J. Holnicki-Szulc, Poland & N. Sottos, USA) 10680 Wieslaw M. Ostachowicz (Poland): Elastic Wave Propagation Development for Structural Health Monitoring 10722 Richard D. James (USA): A Way to Search for Smart Materials with Unprecedented Physical Properties 13010 Scott R. White (USA): Autonomic Healing of Polymers and Composites 10136 Mieczyslaw S. Kuczma (Poland): Composite Plates with Active Fibres 10784 Hans Irschik (Austria): Transient Eigenstrains Without Incremental Displacements in a Hyperelastic Body 10821 Hisaaki Tobushi (Japan): Shape Fixity and Shape Recovery of Shape Memory Polymer and their Applications 10985 Ji-Hwan Kim (S. Korea): Vibration Control of Stiﬀened Plates with Integrated Piezoelectrics 11685 Nancy R. Sottos (USA): Stress Eﬀects on Ferroelectric Thin Film Patterning, Properties and Performance 11731 Kaushik Bhattacharya (USA): A Novel Approach to the Application of Ferroelectric Thin Films to Micro-actuation

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11752 Sven Lentzen (Germany): A Geometrically Non-Linear Finite Shell Element with Piezoelectric Layers 11857 Cun-Fa Gao (Japan): Thermal-Induced Fracture of Electroded Piezoelectric Composites 12011 Rivka Gilat (Israel): Thermal Buckling of Active Composite Plates with Shape Memory Alloy Fibers 12016 Daining Fang (China): Study of Non-Linear Magnetomechanical Constitutive Relations of Ferromagnetic Materials 12018 Fumihiro Ashida (Japan): Optimum Control of Thermoelastic Deformation in a Smart Composite Disk 12027 Catherine L. Brinson (USA): SMA Hybrid Composites: Self-healing, Self-Stiﬀening and Shape Control Simulations 12147 Sadagopan Narayanan (India): Active Control of FGM Plates Using Distributed Piezoelectric Sensors and Actuators 12637 Michal Landa (Czech Republic): Ultrasonic Characterization of Phase Transformation in NiTi Wire During Thermomechanical Loading 12786 Mitsunori Denda (USA): Upper and Lower Bounds of Electric Induction Intensity Factors for Multiple Piezoelectric Cracks by the BEM 13007 Andrew Smyth (USA): Direct Identiﬁcation of the State Equation in Complex Nonlinear Systems 13011 Alan Jones (USA): Self-Healing Polymer Composites for Extended Fatigue Life 13015 Piotr Pawlowski (Poland): The Concept of Multifoldig and Its Experimental Validation

MS2 – Tissue, cellular and molecular biomechanics (D. Barth´ ´es-Biesel, France & A. Hoger, USA) 10709 Christoph F. Schmidt (Netherlands): Molecular Mechanics of Cytoskeletal Components 10933 Susan S. Margulies (USA): Tissue Mechanics 12811 Samuel A. Safran (Israel): Elastic Interactions of Biological Cells 10627 Erik van der Giessen (Netherlands): Micromechanics of Cytoskeletal Actin Networks 10670 Tobias Olsson (Sweden): Residual Stress Fields in Soft Tissues 10689 Katarzyna A. Rejniak (USA): From Individual Cells To Complex Tissues – an Immersed Boundary Approach 10838 Kazimierz Piech´ or (Poland): Travelling Waves in a Model of Skin Pattern Formation 10988 Norman A. Fleck (UK): Mechanics of Deep Penetration of Soft Solids 11102 Andrejs Cebers (Latvia): Mechanics of Elastic and Viscous Magnetic Filaments 11148 Samuel Sideman (Israel): Intracellular Control Mechanisms of Cardiac Contraction & Energetics 11581 Yoel Forterre (France): Mechanics of Venus’ ﬂytrap Closure 11808 Jaroslaw Piekarski (Poland): Approximations of Stiﬀness Tensor of Bone – Determining and Accuracy 11866 Thomas R. Powers (USA): Theory of Polymorphism in Bacterial Flagella 11961 Dominique Barth´ ´es-Biesel (France): Hydrodynamic Interaction Between Two Bioartiﬁcal Capsules in Shear Flow 12031 Christoph Hartmann (Germany): Stress and Strain in a Yeast Cell under High Hydrostatic Pressure 12059 Alexander V. Kondrachuk (Ukraine): Models of Hair Cell Bundle Functioning 12064 Edoardo Mazza (Switzerland): Measuring the Mechanical Properties of Soft Biological Tissues 12100 Stanisaw Jemiolo (Poland): Anisotropic Hyperelastic and Pseudo-Hyperelastic Materials and Applications to Soft Tissue Modelling 12192 Taisuke Sugii (Japan): Molecular Dynamics Study of Permeation Process of Small Molecules Through A Lipid Bilayer

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12315 Tomasz Lekszycki (Poland): Modeling of Periodic Load Eﬀects in Bone Tissue Adaptation 12453 Ken-ichi Tsubota (Japan): A Particle Method Computer Simulation of Blood Flow 13049 Luigi Gambarotta (Italy): Wrinkling and Buckling of Isotropic Biological Tissues

MS3 – Mechanics of thin ﬁlms and nanostructures (H.M. Jensen, Denmark & Z. Suo, USA) 11070 Henrik M. Jensen (Denmark): Mechanics of Thin Film Structures 11594 Rodney S. Ruoﬀ (USA): Mechanics of Nanostructures 12132 Kenneth M. Liechti (USA): A Hybrid Molecular/Continuum Analysis of IFM Experiments on a Self-assembled Monolayer 10105 Min Zhou (USA): Thermomechanical Continuum Representation of Atomistic Deformation at Arbitrary Time and Size Scales 10699 Rui Huang (USA): Ratcheting-induced Wrinkling of an Elastic Film on a Metal Layer Under Cyclic Temperatures 11122 Akihiro Nakatani (Japan): Atomistic Study of Size Eﬀect in Torsion Tests of Nanowire 11279 Wei Yang (China): Microstructural and Atomistic Simulation for Deformation of Nano-grained Metals 11444 Kinga Nalepka (Poland): Energy-Based Approach to Limit States in Nanostructures. Calculation of the Critical Values of Energy from Firt Principles 11474 Pradeep Sharma (USA): Size-dependent Elastic State of Embedded Nano-inclusions & Quantum Dots 11599 Mikhail N. Perelmuter (Russia): Fracture Criterious for Bridged Crack: from Macro to Nanoscale 12181 Fenghuan Sha (China): Investigation of Wave Propagation in Multiwall 12251 Malgorzata Chwa (Poland): Homogenisation Models of Carbon Nanocomposites Mechanical Properties 12314 Xi-Qiao Feng (China): Micro- and Nano- mechanics of Carbon Nanotubes Composites 12384 Jurg Dual (Switzerland): Characterization of MEMS Materials 12386 Daniel S. Balint (UK): An Analytical Model of Oxide Rumpling as the Mechanism Leading to Failure in Thermal Barrier Coatings 12432 Raymond Parnes (Israel): Instabilities of Composite Materials Reinforced by NanoFibres: a Re-examination of Elastic Buckling 12486 Xuejun J. Zheng (China): Interfacial Adhesion of PZT Ferroelectric Thin Films Determined by Nano-Indentation Method 12627 Simon P.A. Bill (UK): A Cellular Automaton for Modelling Evolution of Heteroepitaxial Systems 12705 Taher Saif (USA): Mechanical Behaviour of Nano Grained Metals 12778 Kyung-Suk Kim (USA): Nano-scale Planar Field Projection of Atomistic Decohesion of Crystalline Solids

MS4 – Microﬂuids (R.J. Adrian, USA & J. Santiago, USA) 12960 Patrick Tabeling (France): Slip, Patterns, and other small Things in Microﬂuidic Systems 12961 J. Santiago (USA): Electrokinetic Flow Instabilities in Microﬂuidic Systems 12976 D. Scott Stewart (USA): Miniaturization of Explosive Technology and Microdetonics 10723 Carlo Cercignani (Italy): Plane Poiseuille Flow in a Rareﬁed Gas with General Boundary Conditions 10971 Anna Kucaba-Pi¸etal (Poland): Water Flows in Copper and Quartz Nanochannels 11437 Nicolas G. Hadjiconstantinou (USA): A Second-Order Slip Model for Early-TransitionRegime Flows 11604 Kenneth S. Breuer (USA): Direct Measurement and Simulation of Apparent Slip Velocities in Sub-Micron-Scale Flows

Scientiﬁc Program

xxxv

11607 Eiichiro Yamaguchi (USA): Theoretical and Experimental Study of Microchannel Blockage Phenomena 11777 Todd M. Squires (USA): Induced-Charge Electro-Osmosis: Theory and Microﬂuidic Applications 12047 Laure Menetrier (France): Using Microﬂuidics to Investigate Reaction-diﬀusion Phenomena in Simple Flows 12130 Guillaume Degr´e (France): Magnetic Particles Aggregation in the Presence of a Hydrodynamic Shear 12536 Marie Caroline Jullien (France): Chaotic Mixing and Resonances in a Microﬂuidic System 12673 Pierre Joseph (France): An Accurate Velocity Proﬁle Measurement System for Microﬂuidics: A Direct Measurement of the Slip Length 12966 Piotr Garstecki (USA): Tunable Microﬂuidic Bubble Generator

MS5 – Microgravity ﬂow phenomena (J.I.D. Alexander, USA & P. Neitzel, USA) 10598 Denis Weaire (Ireland): Foams, Films and Surfaces in Microgravity 10714 Michel Y. Louge (USA): Collisional Granular Flows with and without Gas Interactions in Microgravity 11133 George M. Homsy (USA): Microgravity and Microscale Fluid Mechanics 10958 Dmytro V. Yevdokymov (Ukraine): Hydrodynamic Eﬀect of Slow Phase Transitions in Microgravity 10980 M.C. Charrier Mojtabi (France): Heat Transfer Due to High Frequency Vibration: a New Approach for Achieving Thermally Optimum Geometry Under Microgravity Conditions 11757 Vladislav V. Pukhnachov (Russia): Mathematical Models of Microconvection for Isothermally Incompressible and Weakly Compresible Liquids 11843 Michael K. Ermakov (Russia): Onset of Oscillations in High-Prandtl Thermocapillary Liquid Bridges: Linear-stability Analysis vs. Experiment 12402 Tatyana P. Lyubimova (Russia): Spherical Two-phase Interface in a Near-critical Fluid. Gradient Approach 12447 Iwan Alexander (USA): Capillary Pressure of a Liquid Between Uniform Spheres Arranged in a Square-packed Layer 12651 G. Paul Neitzel (USA): Recent Advances in Permanent Noncoalescence and Nonwetting

MS6 – Atmosphere and ocean dynamics (M.E. McIntyre, UK & J. Sommeria, France) 10716 Onno Bokhove (Netherlands): Wave Vortex Interactions in the Atmosphere and Oceans; with Applications to Climate 10803 Olivier Talagrand (France): Assimilation of Observations into Numerical Models 10987 Peter Haynes (UK): Transport and Mixing in the Atmosphere 10261 Peter J. Thomas (UK): Modelling Oceanographic Coastal Currents in Small-scale and Large-scale Laboratory Experiments 10513 Pierre Carlotti (France): Near-surface Turbulence in a Neutral Planetary Boundary Layer: Comparison of LES with the CASES’99 Experiment Observations 10634 Gordon Swaters (Canada): Meridional Flow of Source Driven Grounded Abyssal Flow in a Wind Driven Basin with Topography 10977 Michael E. McIntyre (UK): Remote Recoil and Wave Capture: Wave–vortex Interactions in Atmosphere-Ocean Models 11152 Boris Galperin (USA): Anisotropic Large-Scale Turbulence on Giant Planets and in the Ocean 11336 Jean Noel Reinaud (UK): Strong Vortex Insteractions in Quasi-Geostrophic Flows 11499 Ross Griﬃths (Australia): Turbulent Horizontal Convection and the Global Thermohaline Circulation of the Oceans

xxxvi

ICTAM04

11547 Joel Sommeria (France): Instability of Gravity Driven Coastal Current in a Turntable Experiment 11585 Marius Ungarish (Israel): Intrusive Gravity Currents in a Stratiﬁed Ambient – Novel Theoretical Results and Insights 11732 Shinya Shimokawa (Japan): Irreversible Transition to a State with Higher Entropy Production in Oceanic General Circulation 11829 Jan-Bert Fl´ or (France): Interactions of Planar Waves with a Baroclinic Vortex 11942 Semion Sukoriansky (Israel): A New Spectral Closure Model of Turbulent Flows with Stable Stratiﬁcation and Its Application to Atmospheric SBLS 12035 Dieter Etling (Germany): Roll Vortices in the Atmospheric Boundary Layer 12104 Jacques Vanneste (UK): Spontaneous Generation of Inertia-Gravity Waves by Balanced Motion 12140 Jun-Ichi Yano (France): The Energy Cycle of the Tropical Madden-Julian Oscillations seen through Wavelets 12157 Oliver Buhler (USA): Wave Capture and Wave-vortex Duality 12169 Keke Zhang (UK): A New Theory For Convection In Rapidly Rotating Spherical Systems 12194 Takeshi Miyazaki (Japan): Vortex-based Models of Quasigeostrophic Turbulence 12210 Jonas Nycander (Sweden): Generation of Internal Waves in the Deep Ocean by Barotropic Tides 12294 Pantxika Otheguy (France): Instability of Corotating Vertical Vortices in a Stratiﬁeld Fluid: Why Strongly Stratiﬁeld Turbulence is not Similar to 2D Turbulence 12408 Yuliya Kanarska (Ukraine): Laboratory and Numerical Modelling of Exchange Flows 12463 Alastair D. Jenkins (Norway): Wave-Mean Flow Interaction in Coupled AtmosphereIce-Ocean Systems 12494 Peter L. Read (UK): Multiple Jet Formation in a Convectively Driven Flow on a Betaplane 12601 Pascale Bouruet-Aubertot (France): Intermittency in Stratiﬁed Turbulence Produced by Breaking Internal Gravity Waves 12722 Fabrice Veron (USA): Measurements of the Inﬂuence of Ocean Surface Kinematics on Air-sea Heat Fluxes 12776 Francisco J. Beron-Vera (USA): Linear Waves and Baroclinic Instability in an Inhomogeneous-Density Layered Primitive-Equation Ocean Model 13017 Silvia Ferrarese (Italy): Simulation of Sea Surface Temperature Trends Under Severe Wind Forcing With a Full Atmosphere-Ocean Coupled Model

FM1 – Biological ﬂuid dynamics (M. Gharib, USA & F. van de Vosse, Netherlands) 10132 Takeshi Sugimoto (Japan): Mechanics of the Bounding Flight Revisited 10741 Thomas Podgorski (France): Deformation of Vesicles Flowing Through a Capillary 11013 Charles N. Baroud (France): How to Breathe in a Liquid-Filled Lung: Symmetry of Airway Reopening 11187 Tony W.H. Sheu (Taiwan): Computational Exploration of Liver Acinus Microstructure 11441 Tomonobu Goto (Japan): Bacterium Swimming Motion Close to a Wall 11851 Maheshwaran K. Kolandavel (UK): A CFD Study of the Eﬀects of Physiological Vessel Wall Motion on Oxygen Transport in Coronary Arteries 11900 Peter Vennemann (Netherlands): In Vivo PIV Measurement in the Embryonic Chicken Heart 11908 David M. Lewis (UK): A Model of Plankton Dynamics Coupled with a LES of the Surface Mixed Layer 11943 Adrian S. Carabineanu (Romania): Self-Propulsion of an Oscillatory Wing 12063 Oleksiy S. Galaktionov (Netherlands): Bioirrigation in Marine Sediments: Ecological Conclusions from Numerical Modelling

Scientiﬁc Program

xxxvii

12124 Cyrus K. Aidun (USA): Direct Numerical Simulation of Red Blood Cell Flow and Aggregation 12295 Andrew L. Hazel (UK): Three-Dimensional Airway Reopening – Finite-ReynoldsNumber Eﬀects 12400 Philippe Marmottant (Netherlands): Cell Permeabilisation and Transport Focused Around Oscillating Microbubbles 12409 Maciej K. Ginalski (Poland): Computational Model of Selected Transport Processes in an Infant Incubator 12423 Manouk Abkarian (USA): Red Blood Cell Dynamics, Deformation and Rheology via Microﬂuidic Experiments 12522 Masanori Nakamura (Japan): Flow in an Integrated Model of Heart and Aorta 12655 Sang-Joon Lee (S. Korea): In Vivo Visualization of the Water Reﬁlling Process in Xylem Vessels Using Synchrotron X-Ray Micro-Imaging 12784 Mory Gharib (USA): On the Issue of Optimal Trans-Mitral Flow

FM2 – Boundary layers (P.W. Duck, UK & A. Kluwick, Austria) 10103 James P. Denier (Australia): The Development (and Suppression) of Very Short-Scale Instabilities in Buoyant Boundary Layers 10275 Yury S. Kachanov (Russia): 3D Distributed Boundary-Layer Receptivity to NonStationary Free-Stream Vortices in Presence of Surface Roughness 10948 Matthias H. Buschmann (Germany): Extending the Generalized Logarithmic Law to the Wall 11017 Ihor Nesteruk (Ukraine): Sub- and Supersonic Shapes without Separation and Cavitation 11083 Bernhard Scheichl (Austria): Non-Unique Quasi-Equilibrium Turbulent Boundary Layers 11084 Stefan Braun (Austria): Near Critical Unsteady Three-Dimensional Triple Deck Flows 11210 Frank T. Smith (UK): Long Layers Exhibiting Local Jumps, in Industrial and Biomedical Applications 11339 Victor V. Kozlov (Russia): Secondary Instability of Stationary Vortex Packets in a Swept Wing Boundary Layer 11397 Anatoly I. Ruban (UK): Discontinuous Solutions of the Boundary-Layer Equations 11780 Vladimir B. Zametaev (Russia): New Numerical Method for Complex Interacting Flows 11887 Peter W. Carpenter (UK): Why do Dolphins Have Cutaneous Ridges? 11984 Patrick D. Weidman (USA): Two-Fluid Jets and Wakes 12030 Herbert Steinrueck (Austria): The Trailing Edge Problem for Mixed Convection Flow Past a Horizontal Plate 12062 Tomas Vit (Czech Republic): Experimental and Theoretical Study of Heated Coanda Jet 12241 Matthew R. Turner (UK): A Combined Numerical and Asymptotic Approach to Boundary Layer Receptivity Problems 12320 Owen R. Tutty (UK): Flow Along a Long Thin Cylinder 12631 Narayanan Vinod (India): Aspects of the Laminar-Turbulent Transition in Axisymmetric Boundary Layers

FM3 – Combustion and ﬂames (N. Peters, Germany & P. Wola´ n ´ ski, Poland) 10367 Ay Su (Taiwan): Enhancement of the Impinging Diﬀusion Flame by Splash Plate 10747 Oluwole Daniel Makinde (South Africa): Exothermic Explosions in a Slab: a Case Study of Series Summation Technique 10918 David Lo Jacono (Switzerland): A Nearly 1-D Non-Premixed Flame Near Extinction. Cell Formation and the Eﬀect of the Direction of Bulk Flow 11031 Aidarkhan Kaltayev (Kazakhstan): Simulation of Flame Propagation in a Tube with Obstacles

xxxviii

ICTAM04

11888 Tim Broeckhoven (Belgium): Large Eddy Simulation of Piloted and Bluﬀ – Body Diﬀusion Flame 11914 Artur Tyliszczak (Poland): Inﬂuence of the Subgrid Models on Combustion Modelling 12199 Yeshayahou Levy (Israel): Chemical Aspects of the Flameless Oxidation Applied for GasTurbine Combustor 12348 Pedro J. Coelho (Portugal): Experimental and Numerical Investigation of a Flameless Oxidation Combustor 12745 Piotr Wola´ n ´ ski (Poland): Detonations of Hexane Vapor/Droplets-Air Mixtures 12843 Arkadiusz Kobiera (Poland): Simulation of Ram Accelerator with PETN Layer 12911 Zbigniew A. Walenta (Poland): Simple Model of a Detonating Gas for use with the Direct Monte-Carlo Simulation Technique

FM4 – Complex and smart ﬂuids (B. Khusid, USA & A. Yarin, Israel) 10140 Konstantin G. Kornev (USA): Capillary Microﬂuidics for Viscoelastic Fluids 10753 Boris Khusid (USA): Field-Induced Dielectrophoresis and Phase Separation in Suspention 10932 Alexander L. Yarin (Israel): Electrospinning of Nanoﬁbers from Polymer Solutions 11165 Ping Sheng (China): The Giant Electrorheological Eﬀect in Suspensions of Nanoparticles 11184 Sawomir Blo´ nski ´ (Poland): Electrospinning of Liquid Jets 11209 Kevin D. Dorfman (France): Modeling DNA Separations in Self-Assembled Magnetic Arrays: Comparison of Theory and Experiment 11603 Ilker Bayer (USA): Contact Angle Dynamics of Droplets Impacting on Flat Substrates 11648 Semyon P. Levitsky (Israel): Dissipation Features at Nonlinear Pulsations of Bubbles in Viscoelastic Fluids 11954 Aleksey N. Rozhkov (Russia): Break up of Polymer Solution Drop Impacting a Small Target 11982 Markus Zahn (USA): Ferrohydrodynamic Hele-Shaw Cell Flows and Instabilities with Simultaneous DC Axial and In-Plane Rotating Magnetic Fields 12254 Daniel A. Weiss (Germany): Spray Impact on Solid Walls of Non-Newtonian Fluids, Including Yield Stress and Thixotropic Behavior 12319 Eyal Zussman (Israel): Nanowires Assembly Using Microﬂuidic: an Experimental Investigation 12437 Antonio Castellanos (Spain): Particle Manipulation in Microﬂuidics: the Role of Dielectrophoresis, Electrohydrodynamics and AC Electrokinetics 12894 Paul C. Duineveld (Netherlands): Non-Newtonian Eﬀects of Ink-Jet Printed Droplets 12910 Agnieszka Sowicka (Poland): Conditions for Creating Thin Liquid Layers at the Contact Surface of Two Other Liquids

FM5 – Compressible ﬂow (H.G. Hornung, USA) 10531 Mikhail S. Ivanov (Russia): Hysteresis-Related Phenomena in Shock Wave Reﬂection 10789 Alfred Kluwick (Austria): Nonclassical Dynamics of Laminar Dense Gas Boundary Layers 10945 Helmut E. Sobieczky (Germany): Analytical Models for Shocks in Compressible Flow 10984 Holger Babinsky (UK): LDA Investigation of a Transonic Bump Flow 11049 Kazuyoshi Takayama (Japan): Unsteady Drag Force Measurements of Shock Loaded Bodies Suspended in a Vertical Shock Tube 11072 Beric W. Skews (South Africa): Shock Wave Reﬂection in a Non-Circular Inlet 11560 Susumu Hasegawa (Japan): Numerical Optimization of 2D Scramjet Inlets 11730 John K. Hunter (USA): The Mach Reﬂection of Weak Shocks 11864 Susumu Kobayashi (Japan): Eﬀect of Surface Roughness on Mach Reﬂection 12066 Vladimir V. Serebryakov (Ukraine): On the Theory for Subsonic, Transonic and Supersonic Flows in Water with Supercavitation 12219 Vaclav Dvorak (Czech Republic): Interaction of Supersonic Flows in an Ejector

Scientiﬁc Program

xxxix

12997 Ryszard Szwaba (Poland): Shock Wave-Boundary Layer Interaction Control by Streamwise Vortices 13016 Joel Delville (France): Correlation of Nearﬁeld Pressure with Mixing Layer Velocity in a Supersonic Jet

FM6 – Computational ﬂuid dynamics (L. Kleiser, Switzerland & W. Schroeder, Germany) 10163 Randolph C.K. Leung (China): Non-Reﬂecting Boundary Condition for Direct Aeroacoustic Computation 10857 Jaime Klapp (Mexico): Treesph Simulations of Choked Flow Systems Using Smoothed Particle Hydrodynamics 10921 Piotr Boro´ n ´ ski (France): Poloidal-Toroidal Decomposition in Cylindrical von Karman Flow 11348 Manuel Garcia-Villalba (Germany): On Inﬂow Boundary Conditions for Large Eddy Simulation of Turbulent Swirling Jets 11456 Rubens Campregher (Brazil): Numerical Simulation of the Flow over a BackwardFacing Step in a Beowulf-Class Cluster 11485 Alexey N. Kudryavtsev (Russia): Development and Paractical Application of WENO Schemes for Compressible Fluid Flow Computations 11569 Jae-Woo Lee (S. Korea): Numerical Analysis and Design Optimization of Lateral Jet Controlled Missile 11578 Tadeusz Chmielniak (Poland): Numerical Prediction of Energy Dissipation in Condensing 11602 Erik R. Dick (Belgium): A Pressure-Correction Method for All Mach Numbers 11702 Jan Vimmr (Czech Republic): Numerical Computation of Compressible Viscous Flow Through a Male Rotor-Housing Gap of Screw Compressors 11763 Piotr G. Kowalczyk (Poland): Fast Numerical Method for the Boltzmann Equation on Nonumiform Grids 11937 Lorena A. Barba (UK): Computation of Viscous Vortices with Fully Meshless Method 12076 Sawomir Kubacki (Poland): Dirichlet/Dirichlet and Neumann/Neumann Parallel Non-Overlapping Domain Decomposition Method 12337 Milan Schuster (Czech Republic): CFD Methods in Industrial Applications Vehicle External Aerodynamics and Aerodynamic Interaction of Moving Vehicles 12510 Guillaume Dufour (France): Numerical Error Evaluation for Tip Clearance Flow Calculations in Centrifugal Compressor 12565 Andrzej Styczek (Poland): Simulation of a Viscous Flow Past a Three Dimensional Obstacle Using Vortex Particles 12583 Petros Koumoutsakos (Switzerland): Multiscale Simulations Using Particles 12618 Jerzy Majewski (Poland): Investigation of WENO Scheme for 3D Unstructured Grids 13030 Fedderik van der Bos (Netherlands): Commutator – Errors in Large-Eddy Simulation of Turbulent Flow 13051 Patrick Bontoux (France): Three-Dimensional Rayleigh-Benard Instability in a Supercritical Fluid by Direct Numerical Simulation

FM7 – Convective phenomena (G. de Vahl Davis, Australia & K. Zhang, UK) 10306 Alexander V. Getling (Russia): Cellular Compressible Magnetoconvection: a Mechanism for Magnetic-Field Ampliﬁcation and Structuring 10438 Martin P. King (Italy): Scaling Laws for Thermal Convections 10538 Aleksandra Aleksandrovna Bozhko (Russia): On Features of Magnetic Convection in Ferroﬂuid 10914 Katarzyna Boro´ n ´ ska (France): Multiplicity of Patterns in Cylindrical Convection 11053 Igor Rogachevskii (Israel): Large-Scale Semi-Organized Structures in Geophysical Turbulent Convection

xl

ICTAM04

11160 Tomasz Michalek (Poland): Natural Convection for Anomalous Density Variation of Water — Numerical Benchmark 11251 C.M. Rindt (Netherlands): 3D Flow Transition Behind a Heated Cylinder 11440 Jonathan M. Aurnou (USA): Experimental Studies of Planetary Core Convection and Dynamo Processes 11673 Antonio Cenedese (Italy): Penetrative Convection in Stratiﬁed Fluids: Velocity Measurements by Image Analysis 11905 Heiko Meironke (Germany): Experimental and Numerical Studies of Convection Flow in a Cylindrical-Conical Fermenting Tank 11972 Vanda Bouch´e (Italy): Sea Convective Motions Driven by Random Buoyancy Inputs 12126 Michael Le Bars (UK): Thermochemical Convection in Two Superimposed Miscible Viscous Fluids 12168 Xinhao Liao (China): Nonlinear Convective Patterns in Spherical Rayleigh-Benard Systems 12173 Elzbieta ˙ Fornalik (Poland): Visualization of the Flow Structure and Temperature Field in the Region of Mixed Convection 12220 Anne Sergent (France): Large Eddy Simulation of Rayleigh-Benard Convection in an Inﬁnite Fluid Layer 12267 Aleksander Alekseevich Kozlov (Russia): The Inﬂuence of Translational Vibration of Circular Polarization on Fluid Convection Stability and Flow Patterns 12344 Dmitry V. Lyubimov (Russia): Thermal Buoyancy Convection in Systems with Deformable Interfaces 12730 Alexander A. Smirnovskii (Russia): Convective Phenomena in Rotating Annuli Heated on Periphery 12850 Avshalom Manela (Israel): On the Rayleigh-Benard Problem in the Continuum Limit 13005 Vasiliy A. Novosiadliy (Russia): The Inﬂuence of Vibration on the Onset of Marangoni Convection in Horizontal Fluid Layer 13022 Hiroyuki Ozoe (Japan): Air Convection in a Cubic Enclosure with Laterally Shifted Electric Coil without a Gravity Field.

FM8 – Drops and bubbles (J. Eggers, UK & A. Prosperetti, USA) 10580 Yuriy A. Semenov (Ukraine): Method for Solving Nonlinear Problems on Unsteady Free-Boundary Flows 11026 Miguel F. Moctezuma Sanchez (Mexico): Bubble Wall Interaction and Bubble Pairs Motion Using Potential Flow Theory 11250 Olga M. Lavrenteva (Israel): Locomotion of a Viscous Drop, Induced by the Internal Secretion: Boundary Eﬀects 11289 Gary L. Leal (USA): Theoretical Studies of Flow-Induced Coalescence 11299 Teruo Kumagai (Japan): Occurrence of Micro-Bubbles During the Coalescence of Two Bubbles 11317 Nicolas Bremond (Netherlands): Atomization of an Undulating Liquid Sheet 11330 Eric Lauga (USA): Evaporation-Driven Assembly of Colloidal Particles 11411 Eligiusz Wajnryb (Poland): High-frequency Linear Viscosity of Emulsions Composed of Two Viscoelastic Fluids 11462 Christian Wagner (Germany): Molecule Conﬁgurations in a Droplet Detachment Process of a Semdilute Xanthan Solutions 11614 Christophe Josserand (France): Spreading and Retraction of Impacting Drops 11669 Slavka S. Tcholakova (Bulgaria): Main Factors Controlling the Emulsiﬁcation Process under Turbulent Conditions. Experiment and Data Interpretation 11670 Teresa Parra (Spain): Water Mist Behavior as Flame Supressant 11723 Stefan Zaleski (France): Numerical Simulation of Liquid-Gas Interfaces with Applications to Atomization 11855 Ernest O. Tuck (Australia): Viscous Extensional Flow and Drop Break-Oﬀ Under Gravity

Scientiﬁc Program

xli

11911 Jacques J. Magnaudet (France): Evolution of a Pair of Spherical Bubbles Rising Side by Side at Moderate Reynolds Number 11979 Patrick Le Qu´ ´er´e (France): On the Numerical Simulation of Two Phase Liquid-Vapor Phenomena 11999 Manish Arora (Netherlands): Cavitation Inception on Micro-particles: a Self Propelled Particle Accelerator 12106 Peter D.M. Spelt (UK): Level-Set Simulations of Shear Flow with Inertia Pas a Droplet Adhering to a Wall with Moving Contact Lines 12163 Marianne Francois (USA): Modelling Surface Tension Using a Ghost Fluid Technique within a Volume of Fluid Formulation 12190 Andrei S. Topolnikov (Russia): Dynamics of Bubble Supercompression in Organic Liquids 12311 Ryszard Pohorecki (Poland): Hydrodynamics of Gas Bubbling through Organic Liquids 12329 Michael A. Rother (USA): Surfactant Eﬀects on Buoyancy-Driven Coalescence of Spherical Drops 12388 Laurent Duchemin (UK): Static Shapes of Levitating Viscous Drops 12429 Wendy W. Zhang (USA): A Long-Wavelength Model of Viscous Entrainment 12547 Ulderico P. Bulgarelli (Italy): Entrainment of Air Bubbles During Strong VorticityFree-Surface Interaction 12548 Marco A. Fontelos (Spain): Spreading of Charged Microdroplets 12596 C. Wang (Singapore): Multiple Bubbles Dynamics Using Level Set Indirect Boundary Element Method 12741 Andrew J. Griggs (USA): Low-Reynolds-Number Motion of a Drop Beween Two Parallel Plane Walls 12873 Andrzej Zachara (Poland): Thermodynamic Parameters of Vapour Bubble Growth by Image Analysis 13025 Salima Rafai (France): Singular Droplets

FM9 – Environmental ﬂuid dynamics (H.E. Huppert, UK & R. Narasimha, India) 10366 Owen M. Phillips (USA): The Growth and Structure of Double-Diﬀusive Cells Adjacent to a Side-Wall in a Salt-Stratiﬁed Enviroment 10467 Zbynek Janour (Czech Republic): Flow and Dispersion in the Atmospheric Boundary Layer Investigation by Physical Modelling 10576 Krzysztof Dekajo (Poland): Experiments on Up-slope to Down-slope Transition in an Inclined Box Filled with Water 10733 Mohammad J. Kazemzadeh-Parsi (Iran): Analysis of Double-Free Surface Flow through Gates Using Element-Free Galerkin Method 11164 Hitoshi Miyamoto (Japan): Free Surface Behavior in Turbulent Open-Channel Flows 11182 Gregory F. Lane-Serﬀ (UK): Integral and Laboratory Modelling of Sedimentation from Turbulent Buoyant Jets 11294 Falin Chen (Taiwan): Modiﬁed Shallow Water Equations for Inviscid Gravity Currents 11346 Ramon Fernandez-Feria (Spain): Dam-Break Flow for Arbitrary Slope of the Bottom 11433 Oksana E. Poloukhina (Russia): Extended Nonlinear Theory for Topographic Rossby Waves 11703 Andrew A. Osiptsov (Russia): The Propagation of Viscous Gravity Currents over a Rigid Conic Surface 12072 James G.A. Croll (UK): An Alternative Model for “Pingo” Formation in Permafrost Regions 12099 Tarmo Soomere (Estonia): Fast Ferry Traﬃc as a New Forcing Factor of Enviromental Processes in Non-Tidal Sea Areas 12266 Jaroslaw Ciechanowski (Poland): Dynamics of Separation Zone behind the 2D Hill in Oscillating Incident Wind. 12270 Arne Moe (Norway): Eﬀects of Curvature in Avalanche Deﬂecting Dams

xlii

ICTAM04

12396 Jim N. McElwaine (UK): Lobe and Cleft Formation at the Head of a Gravity Current 12410 Sabine Decamp (France): Experimental and Numerical Simulation of Dense Water Overﬂows on a Continental Slope 12592 Szymon P. Malinowski (Poland): High Resolution Modelling of Atmospheric Flow over Southern Poland 12626 Alfred J. W¨ u ¨ est (Switzerland): Formation and Rapid Expansion of Double Diﬀusive Layering in Lake Nyos 12738 John E. Holeman (USA): Joint Urban 2003 Surface Energy Budget Measurements and Analysis 12980 A.S. Vasudeva Murthy (India): Nocturnal Temperature Inversions Under Calm Clear Conditions: an Analytical Study

FM10 – Experimental methods in ﬂuid mechanics (A. Leder, Germany & J. Westerweel, Netherlands) 10445 Grazia Lamanna (Germany): On the Evaporation of a Monodisperse Droplet Stream at High-Pressure 11088 Jerry Westerweel (Netherlands): HPIV using Polarization Multiplexing Holography in Bacteriorhodopsin (bR) 11455 Valery Chernoray (Sweden): Time-resolved Wall Shear Stress Measurements using MEMS 11909 Raymond P.H.M. Bergmann (Netherlands): Void Collapse and Jet Formation: The Impact of a Disk on a Water Surface. 11916 Andrzej S. Witkowski (Poland): Comprehensive Experimental and Computational Investigations of the Unsteady Flow in an Axial Flow Low Speed Compressor Stage 12070 Tov Elperin (Israel): Experimental Detection of the New Phenomenon of Turbulent Thermal Diﬀusion 12213 Nikita A. Fomin (Belarus): 3D Vortices Structure Monitoring in Turbulent Flows by Digital Speckle Photography 12719 Anna Matvienko (Canada): Thermal-Wave Resonator Cavity: Modelling and Applications for Water Mixtures 12855 Piotr M. Korczyk (Poland): Particle Image Velocimetry (PIV) for Cloud Droplets – Laboratory Investigations 13048 Albert Baars (Germany): Optical Diagnosis Systems for Measuring Thermoﬂuiddynamicals Phenomena in Liquid Biosystems Under Ultra High Pressure

FM11 – Flow control (J.B. Freund, USA & M. Gad-el-Hak, USA) 10128 Mohamed Gad-el-Hak (USA): Liquids: The Holy Grail of Microﬂuidics Modeling 10872 Ramesh K. Agarwal (USA): Active Control of Shock/Boundary Layer Interaction in Transonic Flow Over Airfoils 11043 Rong F. Huang (Taiwan): Manipulating a Vee-Shaped Bluﬀ Body Wake Using a Fluidic Oscillator 11394 Sedat F. Tardu (France): Response on the Near Wall Turbulence to Localized Unsteady Blowing Periodical and Dissymetric in Time 11636 Haecheon Choi (S. Korea): Passive Control of Turbulent Flow behind a Model Vehicle for Drag Reduction Using Wake Disrupter 11918 Andrzej Szumowski (Poland): Control of Internal Supersonic Flow Separation 12013 Junwoo Lim (USA): A Singular Value Analysis of Boundary Layer Control 12068 Tom Weier (Germany): Separation Control by Stationary and Time Periodic Lorentz Forces 12156 Victor F. Kopiev (Russia): On the Possibility and Prospects of Turbulent Flow Noise Control 12244 Herv´ ´ e Illy (France): Control of Flow Oscillations over a Cavity by Means of a Spanwise Cylinder

Scientiﬁc Program

xliii

12258 Zdenek Travnicek (Czech Republic): Synthetic Jet Actuation at the Resonance Frequency 12305 Alan Guegan (France): Optimal Energy Growth and Optimal Control of the Swept Attachment – Line Boundary Layer 12387 Tim Colonius (USA): Feedback Control of Vortex Shedding in a Separated Diﬀuser 12491 Seichiro Izawa (Japan): Reduction of Aerodynamic Noise Induced by Flow over a Cavity 12602 Philippe Konieczny (France): Control of Turbulent Streaks by Active Wall Movement 12624 Mark Pastoor (Germany): Model-Based Control of Shear Flows Using Low-Dimensional Galerkin- and Vortex Models 12736 Marek Morzy´ n ´ ski (Poland): Numerical Analysis of the Wake Control behind a Circular Cylinder with Oscillatory Rotation

FM12 – Flow in porous media (Abder Kader Mojtabi, France & V. Nikolaevskiy, Russia) 10199 Vasiliy Govorukhin (Russia): Numerical Investigation of Convective Regimes in a Planar Filtrational Convection Problem 10234 Michel Quintard (France): Dissolution in Porous Media: Upscaling, Instabilities and Heterogeneity Eﬀects 10548 Guzel T. Bulgakova (Russia): Instability and Dynamic Chaos of Non-equilibrium Filtration of Gaseous Liquid 11109 Victor N. Nikolaevskiy (Russia): Plastic Mass Flow of Sand Under Action of Pore Pressure Gradient 11115 Dmitry Nikolaevich Mikhaylov (Russia): P-Waves Behavior at Transition from Liquid to Gas-Saturated Porous Media 11248 Wlodzimierz Bielski (Poland): Nonstationary Flow of Stokesian Fluid through Elastic Skeleton with Hierarchical Structure 11343 Franck Plouraboue (France): Conﬁned Air-liquid Drainage: Local Analysis and Invasion Percolation Model 11487 Thomas Loimer (Austria): A Joule-Thomson Process of a Wetting Fluid Near Saturation 11994 Piotr Szymczak (Poland): Microscopic Simulations of the Dissolution of Rock Fractures 12376 Marie-Christine N´ ´eel (France): Fractional Model for Solute Spreading in Randomly Heterogeneous Porous Media 12528 Liana Kovaleva (Russia): Oscillatory Modes of Adsorption in the Flow of Multicomponent Systems 12611 Mieczyslaw Cieszko (Poland): Extended Description of Pore Space Structure and Fluid Flow through Anisotropic Porous Materials 12617 Robert P. Behringer (USA): Onset of Convection for a Miscible Fluid in a Porous Medium

FM13 – Flow instability and transition (P. Huerre, France & P.A. Monkewitz, Switzerland) 10185 Aomar Ait Aider (France): Instabilities in a Taylor-Dean Open Flow 10188 Fran¸¸cois Lusseyran (France): Shear Layer Instability and Frequency Modes Inside an Open Cavity 10313 Bruno Eckhardt (Germany): Travelling Waves and Transition to Turbulence in Pipe Flow 10487 Joseph T.C. Liu (USA): Nonlinear Mechanics of Wavy Instability of Steady Longitudinal Vortices and Drag Rise in Boundary Layer Flow 10489 J.M. Floryan (Canada): Stability of Flow in a Rough Channel 10525 St´ ´ephane Leblanc (France): Stability of Lagrangian Ideal Flows 10640 Laurette Tuckerman (France): Instability Thresholds of Flow Between Exactly CounterRotating Disks

xliv

ICTAM04

10916 Peter W. Duck (UK): Transient Growth in Developing Plane and Hagen Poiseuille Flow 11186 Denis Martinand (UK): Three-dimensional Global Modes in Spatially Varying RayleighBenard-Poiseuille Convection 11396 Laurent Lacaze (France): Elliptical Instability in a Rotating Spheroid 11461 Jacob Cohen (Israel): The Instability of a Localized Vortex Disturbance in Uniform Shear Flow 11647 Maher Lagha (France): Subcritical Transition to Turbulence in Plane Couette Flow 11995 Eckart Meiburg (USA): Three-Dimensional Vortex Breakdown in Swirling Jets and Wakes 12150 Jonathan J. Healey (UK): A New Convective Instability with Growth Normal to a Boundary Layer 12233 Guillemette G. Caulliez (France): By-pass Laminar-Turbulent Transition of the WindDriven Free Surface Flow 12280 Denis S. Goldobin (Russia): Inﬂuence of Swirl Vibrations on Flow in Long Cylinder 12291 Masato Nagata (Japan): Nonlinear Stability of Rotating Channel Flow 12372 Uwe Ehrenstein (France): On Instability Mechanisms in a Separating Boundary-layer Flow 12381 Leandro Franco de Souza (Brazil): Gortler Vortex Secondary Stability: Varicose Mode 12425 Fran¸¸cois Gallaire (France): Spiral Vortex Breakdown as a Global Mode 12431 Dwight Barkley (UK): Computational Study of Turbulent-Laminar Bands in Couette Flow 12483 Arnaud Antkowiak (France): A Generic Mechanism for By-Pass Transition in Vortices 12508 Thierry Feraille (France): Global Stability of the Flow Induced by Wall Injection 12509 Christophe Millet (France): Acoustic Field Generated by Instability Waves in the Transonic Regime 12542 Ilmars Grants (Latvia): Nonlinear Transition of a Flow Driven by a Rotating Magnetic Field 12711 Jean-Marc Chomaz (France): Fully Nonlinear Global Modes and Transition to Turbulence in Open Flows 12766 Cherif Nouar (France): Stability of Plane Poiseuille Flow and Energy Growth in the Case of a Bingham Fluid 13019 Jens Norkaer Sørensen (Denmark): Instability of the Far Wake Behind a Wind Turbine

FM14 – Flow in thin ﬁlms (N. Aksel, Germany & V. Shkadov, Russia) 10220 Seraﬁm Kalliadasis (UK): Dynamics of a Reactive Falling Film at Large Peclet Numbers 10543 Alexander Oron (Israel): Long-Wave Marangoni Instability in Binary-Liquid Films with Soret Eﬀect 10557 Vasilis Bontozoglou (Greece): Solitary Waves on Liquid Film Flowing Along a Periodic Wall 10642 Nuri Aksel (Germany): Eﬀect of Bottom Undulations on the Stability of Film Flow Down Inclined Planes 10928 Andreas Wierschem (Germany): Hydraulic Jumps and Resonance in Gravity-Driven Flows of Liquid in Inclined Wavy Channels: Transition and Hysteresis 11089 Jens G. Eggers (UK): Hydrodynamic Theory of De-Wetting 11413 Norbert Alleborn (Germany): Linear Response of a Viscous Liquid Sheet 11477 Le Han Tan (Australia): Experimental and Numerical Study of Marangoni-Natural Convection 11509 Dirkjan B. van Dam (Netherlands): Layer Thickness Distribution of Thin-Film InkJet Printed Structures. 11739 Takao Yoshinaga (Japan): Ampliﬁcation of Nonlinear Disturbances on a Falling Liquid Sheet

Scientiﬁc Program

xlv

12155 Gregory P. Chini (USA): Thin Film Flows Near Isolated Humps and Interior Corners 12293 Tatiana Gambaryan-Roisman (Germany): Gravity- and Shear- Driven Thin Films Flow on Heated Hicrostructured Walls 12656 Jaroslav Tihon (Czech Republic): Hydrodynamics of the Solitary Waves Travelling Down a Liquid Film 12858 John Tsamopoulos (Greece): Transient Displacement of Viscoelastic Liquids by Air

FM15 – Fluid mechanics of materials processing (F. Dupret, Belgium & R. Moreau, France) 10363 Jeﬀrey J. Derby (USA): Analysis of Flow-Induced, Step-Bunching Instabilities During the Growth of Crystal from Liquid Solutions 10860 Ren´ ´ e J. Moreau (France): Relevance of Alfven Waves in Process Metallurgy under a High Magnetic Field 11386 Mohammed El Ganaoui (France): Eﬀect of Thermal Boundary Modulation in a Restricted Fluid Domain of a 3D Vertical Bridgman Apparatus 11893 Viatcheslav V. Kolmychkov (Russia): 3D Computer Simulation of Time-Depended Solutal Convection 11903 Geoﬀrey M. Evans (Australia): Liquid and Gas Jets Impinging on a Moving Wetted Surface 11929 Vadim I. Polezhaev (Russia): Convective Instabilities in Czochralski Model 12349 Andreas Cramer (Germany): New Possibilities for Velocity Measurements and Model Experiments in Liquid Metal Processing 12591 Fran¸¸cois Dupret (Belgium): Dynamic Simulation of the Entire Crystal Growth Process: Multi-Scale Analysis of Melt Flow Transients 12636 Othman Bouizi (France): Sensitive Regions and Optimal Perturbations in the Floating Zone Using the Adjoint System

FM16 – Fluid mechanics of suspension (R. Bonnecaze, USA & E. Guazzelli, France) 10248 Fran¸cois Feuillebois (France): Eﬀective Viscosity of an Inhomogeneous Dilute Suspension Flowing Along a Wall 10361 Andreas Acrivos (USA): Velocity Fluctuations in Non-Brownian Suspensions Undergoing Simple Shear Flows 10603 Vishwajeet Mehandia (India): The Collective Dynamics of Self-Propelled Particles 10993 Krzysztof Sadlej (Poland): Microstructure of a Dilute Sedimenting Suspension 11236 B.U. Felderhof (Germany): Sedimentation of Dilute Suspensions 11409 Maria L. Ekiel-Je˙z˙ ewska (Poland): Relaxation Time for Sedimenting Spheres of a Suspension with Periodic Boundary Conditions 11418 Anthony Ladd (USA): Numerical Simulations of Particle Suspensions in a Rotating Flow 11431 Gerhard Naegele (Germany): A Mode-Mode Coupling Scheme of Colloidal Electrolyte Friction 11473 Helen J. Wilson (UK): The Eﬀect of Diﬀerent Particle Contacts on Suspension Rheology 11480 Evgeny S. Asmolov (Russia): Evolution of Suspension Sedimenting in a Container Bounded by Horizontal Walls 11588 Elisabeth Guazzelli (France): Spreading Fronts and Fluctuations in Sedimentation: Part I Experiments 11725 Alan L. Graham (USA): Constant Force and Constant Velosity Momentum Tracers in Concentrated Suspensions 11749 Cyril Cassar (France): Flow of a Concentrated Suspension Down a Rough Plane 11850 Jeﬀrey F. Morris (USA): Inertial Migration of Rigid Spherical Particles in Poiseuille Flow 11884 Gabriel Seiden (Israel): Segregation of Suspended Particles in a Rotating Fluid-Filled Horizontal Cylinder – Experiment and Theory

xlvi

ICTAM04

11991 Roger T. Bonnecaze (USA): Migration of Buoyant Mono- and Bi- Disperse Suspensions in Low Reynolds Number Pressure-Driven Pipe Flow 12005 David Saintillan (USA): Dynamic Simulations of the Instability of Sedimenting Fibers 12056 Howard Stone (USA): Mobility of Membrane-Trapped Particles: Protrusion into the Surrounding Fluid 12345 E. John Hinch (UK): Spreding Fronts and Fluctuations in Sedimentation: Part II Computer Simulations 12452 Ileana C. Carpen (USA): Single Particle Motion in Colloidal Dispersions 12574 Michel Cloitre (France): Slip and Flow in Pastes

FM17 – Granular ﬂows (R. Behringer, USA & I. Goldhirsch, Israel) 10253 Detlef Lohse (Netherlands): Impact 10371 Joe D. Goddard (USA): Maximum-Entropy Estimates and Virtual Thermomechanics for Granular Assemblies 10524 Herbert E. Huppert (UK): Granular Column Collapse 10959 J. Rajchenbach (France): Gravity Flow of a Densely-Packed Granular Material 11169 Irena Sielamowicz (Poland): Particle Image Velocimetry Analysis of Granular Material Flows 11751 Maxime Nicolas (France): Pore Pressure Relaxation During Granular Compaction 11775 Pierre Jop (France): Granular Flows on a Heap 11876 Osamu Sano (Japan): Collapse, Growth and Merging of Cavity Regions in a Granular Material Due to Viscous Flow 11883 Akiko Ugawa (Japan): Undulations and Ripples of a Thin Granular Layer Due to Vertical Vibration 12073 Christine Hrenya (USA): Species Segregation Driven by a Granular Temperature Gradient 12308 Isaac Goldhirsch (Israel): Kinetics of Weakly Frictional Granular Gases 12338 John R. de Bruyn (Canada): Morphology and Scaling of Impact Craters in Granular Media 12399 Lou Kondic (USA): Extended Granular Temperature 12445 J.C. Tsai (USA): Evolution of Internal Structure of Sheared Dense Granular Flows: Crystallization and History-Dependent Final States 12790 Radoslaw L. Michalowski (USA): Arching in Granular Media 12883 Renaud L. Delannay (France): Transverse Motion, Segregation and Rotations in 2D Granular Flows 13029 Sylvain Courrech du Pont (UK): Velocity Proﬁles During Granular Avalanches

FM18 – Low-Reynolds-number ﬂow (R.H. Davis, USA & C. Pozrikidis, USA) 10297 Ryszard Staroszczyk (UK): Radially Symmetric Polar Ice Sheet Flow with Evolving Anisotropic Fabric 10565 Robert B. Jones (UK): Hydrodynamic Interaction of a Spherical Particle in Poiseuille Flow Between Planar Walls 10766 Mark G. Blyth (UK): Two-Layer Stagnation Point Flows 10983 Lisa A. Mondy (USA): Free Surface Deformation in Suspensions Near a Rotating Rod 11300 Bogdan Cichocki (Poland): Particles Located on a Planar Free-Surface-Hydrodynamic Interactions in Quasi-Two-Dimensional System 11460 Alexander Prokunin (Russia): Microcavitation and Detachment of a Stokes Particle in Near-Wall Slow Motion 11915 Emin Fuad Kent (Turkey): Flow Visualization Experiments of Cellular Stokes Flows Induced by Rotation of a Cylinder Variously Positioned Inside Channels 11922 Izabella Pie´ n ´ kowska (Poland): Many-Sphere Hydrodynamic Interactions: Weak Convective Inertia Eﬀects 12164 Michal Branicki (UK): Viscous Eddy Structures in an Oscillating Cylinder with Sharp Corners

Scientiﬁc Program

xlvii

12859 George K. Karapetsas (Greece): Transient Squeeze Flow of Viscoplastic Liquids 12981 Devanayagam Palaniappan (Qatar): Slow Rotation of a Double Sphere in a Viscous Fluid

FM19 – Magnetohydrodynamics (P.A. Davidson, UK & A. Thess, Germany) 10025 Antoine Sellier (France): Migration and Interaction of two Conducting Particles Freely Immersed in a Liquid Metal 10235 Vladimir Shtern (USA): Bifurcation of Conical Magnetic Field 10931 Ulrich M¨ u ¨ller (Germany): Complementary Experiments at the Karlsruhe Dynamo Test Facility 11120 Jurij B. Kolesnikov (Germany): Liqiud Metal Flow Under Inhomogeneous Magnetic Field 11124 Peter A. Davidson (UK): Small-Scale Motion in the Core of the Earth 11429 Hubert Baty (France): Magnetohydrodynamic Instabilities of Astrophysical Jets 11458 Yuji Hattori (Japan): Magnetohydrodynamic Motion of Toroidal Magnetic Eddies 11621 Jacques L´ ´eorat (France): Fluid Dynamos and Precession Driving 11637 Jungwoo Kim (S. Korea): Large Eddy Simulation of Magnetic Damping of Jet 11681 Daniel P. Lathrop (USA): Observations of the Magnetorotational Instability in Spherical Couette Flow 11809 Steinar Borve (Norway): Simulating the Orszag-Tang vortex using RSPH 11906 Uwe Krieger (Germany): Homogenisation of Electrically Heated Glass Melts by Lorentz Forces 12090 Andre D. Thess (Germany): A Model for Liquid Metal Current Limiters 12107 Krzysztof A. Mizerski (Poland): The Magnetohydrodynamic Couette Flow in a Plane and Spherical Geometries with Singular Hartmann Boundary Layers 12120 Avalos-Zu˜ n ˜iga (France): Mean Electromotive Force for a Ring of Helical Vortices 12125 Andreas Tilgner (Germany): Numerical Simulations of Dynamo Experiments 12290 Vaclav Kocourek (Germany): Stability of Liquid Metal Drops Aﬀected by HighFrequency Magnetic Fields 12330 Nathanael ¨ Schaeﬀer (France): Quasi-Geostrophic Dynamos 12363 Serge Barral (Poland): Model of Gas Flow Inside a Plasma Thruster 12587 Jorg ¨ Stiller (Germany): Numerical Study of the Flow in a Finite Cylinder Driven by a Rotating Magnetic Field 12605 Hartmut Brauer (Germany): Interface Reconstruction in Cylindrical Two-Compartment-Systems Using Magnetic Field Tomography 12613 Frank Stefani (Germany): Contactless Inductive Flow Tomography: Theory and Experiment 12979 Jonathan A. Mestel (UK): Dynamo Action in Steady Helical Pipe Flow

FM20 – Multiphase ﬂows (S. Balachandar, USA & J. Magnaudet, France) 10416 Konstantin Volkov (UK): Large-Eddy Simulation of Particle Dispersion in the Duct with Fluid Injection 10690 Maria Joseﬁna Ferreira (Portugal): Sings of Flooding Instability in Inclined Liquid Films at High Pressure and Mass Transfer in High Density Gas Slugs 10843 Daniel Joseph (USA): Viscous and Viscelastic Potential Flow 11189 Urbano J. S´ ´ anchez Dominguez (Spain): Separation and Sorting of Heavy Particles Suspended in a Fluid by Settling in a Periodic Vorticity Field 11655 Elena Trostinetsky (Israel): Gas-Liquid Interfractial Distribution in Inclined Downward Pipe Flow 11745 Stanislaw Anweiler (Poland): Videogrametry in Fluidized Bed Reactors 12092 Thomas Seon (France): Gravity Induced Mixing of Miscible Fluids in Vertical and Inclined Tubes 12139 Frederic Risso (France): Oscillatory Motion of Freely-Moving Light Bodies: from Cylinders to Disks

xlviii

ICTAM04

Touvia Miloh I. (Israel): Non-Uniform Flow Hydrodynamics of Deformable Shapes Veronique Roig (France): Mean Motion Induced in a Liquid by Rising Bubbles Jacobus J. Derksen (USA): Plane Couette Flow of Dense Liquid-Particle Suspensions Daniel Zaj¸ac (Poland): Image Processing Method in Estimation of Bubble Column’s Work 13012 Cristian Marchioli (Italy): Statistics and Preferential Distribution of Micro-Particles in Turbulent Boundary Layer: Implications for Resuspension Mechanisms

12342 12383 12397 12886

FM21 – Solidiﬁcation and crystal growth (M. Glicksman, USA & M.G. Worster, UK) 11058 Chuan F. Chen (USA): Experimental Observations of Hydrate Formation in a Convection Tank 11208 Chih-Ang Chung (Taiwan): Morphological Stability of Directional Solidiﬁcation under Temperature Modulations 11270 Peter Guba (UK): Nonlinear Oscillatory Convection in Mushy Layers 11320 Pascale Aussillous (UK): Magnetic Resonance Imaging of Structure and Convection in Solidifying Mushy Layers 11328 Andrew Thompson (USA): Solidiﬁcation and Compositional Convection of a Ternary Alloy 11331 Michael A. Gonik (Russia): AHP Setup for Low Laminar Melt Flow Study in Crystal Growth 11362 Liliana Braescu (Romania): Optimization of the Growth Conditions of a Nd:YVO4 Cylindrical Bar 11364 Jerzy Banaszek (Poland): Front Tracking Technique on a Fixed Grid in Modelling of Binary Mixture Soldiﬁcation with Natural Convection 11926 George G. Tsypkin (Russia): Salt Precipitation in Geothermal Reservoirs 12276 Jacqueline Ashmore (UK): Boundary-Layer Analysis of Chimney Structures in Mushy Layers 12278 Irina Fayzrakhmanova (Russia): Travelling Magnetic Field Inﬂuence on Crystal Growth by Bridgman Method 12312 Gustav Amberg (Sweden): A Semi-Sharp Phase Field Method for Quantitative Phase Change Simulations 12365 Sven Eckert (Germany): Directional Solidiﬁcation of Pb-Sn Alloys Aﬀected by a Rotating Magnetic Field 12543 Marc Georgelin (France): Cell Shapes in Directional Solidiﬁcation: a Global Study 12743 Leszek Czechowski (Poland): Convection Driven by Tidal Heating: Numerical Model and Parameterized Theory 12768 Daniel M. Anderson (USA): Ternary Alloy Convection in Mushy Layers 13008 Chang Kyun Choi (S. Korea): Convective Instabilities During Solidiﬁcation of a Mushy Layer

FM22 – Stirring and mixing (H. Aref, USA & E. Villermaux, France) 10130 Frank C.G.A. Nicolleau (UK): Development of the Fractal Dimension of Material Elements in Homogeneous Isotropic Turbulence Using Kinematic Simulation 10497 Stephen M. Cox (Australia): Chaotic Advection in a Mixer with Changing Geometry 10532 Patrice Meunier (France): Enhanced Mixing by Vortices 10896 Jorg ¨ Schumacher (Germany): Geometric Features of High-Schmidt Number Scalar Mixing 11190 Alain Pocheau (France): Front Propagation in Laminar Cellular Flows: an Experimental Study 11293 Stephen Wiggins (UK): On the Design of 3D Micromixers Having the Bernoulli Property 12048 Alain Pumir (France): Intermittent Distribution of Heavy Inertial Particles in Turbulent Flows

Scientiﬁc Program

xlix

12110 Alexandre Gourjii (Ukraine): Chaotic Stirring of Passive Fluid by a Vortex Pair in Circular Domain 12158 Mark A. Stremler (USA): Chaotic Advection and Mixing in Pulsed Source-Sink Systems 12209 Shenqyang Shy (Taiwan): On Dissipative Structures of Stirring-Grids Turbulence 12257 Tatyana S. Krasnopolskaya (Ukraine): Evaluation of Transport Properties by Exchange Matrix Method 12326 Emmanuel Villermaux (France): Mixing Is an Aggregation Process 12415 Jerzy Baldyga (Poland): Stirring and Mixing Eﬀects in Agglomerative Precipitation 12439 Dmitri L. Vainchtein (USA): Resonances and Mixing in Stokes Flows 12442 Philip Boyland (USA): Mixing in Multiconnected Planar Domains 12545 Richard J. Keane (UK): Eulerian Measures for Lagrangian Stirring in a Thermally Driven Flow 12610 Marek Jaszczur (Poland): An Analysis of Mixing Process in a Static Mixer 12633 Dennis van der Woude (Netherlands): Stirring by Blinking Rotlets in a Bounded Stokes Flow 12731 Alain Bergeon (France): Weak Inertia and Mixing Between Rough Surfaces 12761 Frederic Bottausci (USA): Active Shear Superpositon Micromixer

FM23 – Topological ﬂuid mechanics (P.L. Boyland, USA & K. Ohkitani, Japan) 10247 Morten Brons (Denmark): Streamline Topology of the Nearwake of a Circular Cylinder at Low Reynolds Numbers 10938 Koji Ohkitani (Japan): Eulerian-Lagrangian Analysis of Navier-Stokes Turbulence 11060 Matthew D. Finn (UK): Topological Chaos in Simple Mixers 11166 Tsutomu Kambe (Japan): Gauge Principle for Ideal Fluids and Variational Principle 11660 Yoshi Kimura (Japan): Particle Transport by a Vortex Soliton 11677 Robert W. Ghrist (USA): Generic Hydrodynamic Instability 12484 Mitsuaki Funakoshi (Japan): Relation Between Mixing Eﬃciency and Geometrical Property of Stable Manifolds 12868 Dmytro I. Cherniy (Ukraine): Topological Aspects of the Tornado Problem

FM24 – Turbulence (Olivier Metais, France & R.D. Moser, USA) 10149 Julian Andrzej Domaradzki (USA): Large Eddy Simulations of Decaying Rotating Turbulence 10455 Maxim S. Loginov (Germany): Large-Eddy Simulation of Shock-Wave / TurbulentBoundary-Layer Interaction 10506 Sebastien Poncet (France): Experimental Study of Rotor-Stator Flows with Centripetal Fluxes 10564 Marta Waclawczyk (Poland): PDF Computation of Turbulent Flows with a New Near-Wall Model 10937 Naoya Takahashi (Japan): Interaction Between a Columnar Vortex and External Turbulence 11116 Rainer Friedrich (Germany): Turbulence Scalings in Supersonic Channel Flow 11151 Enrico Pasero (Italy): On the Scale Similarity in Large Eddy Simulation 11161 Tomasz Lipniacki (Poland): Two Scale Approach to Anisotropic Turbulence in Hel II 11256 Stefan Hickel (Germany): Optimization of an Implicit Subgrid-Scale Model for LES 11303 Agnes Maurel (France): Study of the Turbulent Energy Spectrum Build Up in an Experimental Vortex Burst 11454 Bernd R. Noack (Germany): Empirical Galerkin Models for Incompressible Flow — Pressure-Term and ’Subgrid’ Turbulence Representations 11488 Yu-L. Liu (China): Orthonormal Wavelet Analysis of CGT in Fully Developed Asymmetric Turbulent Channel Flow 11802 Xiangyu Hu (Germany): The Cellular Structure and Its Tracks of a H2 /O2 /Ar Detonation Waves

l

ICTAM04

11814 Frederic Moisy (France): Energy Spectrum in Rotating Turbulence 12004 Akira Rinoshika (Japan): Three-Dimensional Turbulent Structures of Diﬀerent Scales 12151 Tomomasa Tatsumi (Japan): Inertial Similarity of Velocity Distributions in Homogeneous Isotropic Turbulence 12430 Robert M. Kerr (UK): A New Mixed Nonlinear LES Models for Boundary Layers 12564 Thomas Indinger (Germany): 3D-Measurements in an Adverse-Pressure-Gradient Turbulent Boundary Layer over Smooth and Ribbed Surfaces 12652 Song Fu (China): POD Analysis of Coherent Structures in Turbulent Flows 12675 Vladimir I. Borodulin (Russia): Resonant Interactions of 3D Instability Waves in an Airfoil Boundary Layer for Harmonic and Broadband Perturbations 12740 Pierre Comte (France): Compressibility Eﬀects and Sound Propagation in Turbulent Channel Flow 12965 Hassan M. Nagib (USA): Impact of Pressure-Gradient Conditions on High Reynolds Number Turbulent Boundary Layers

FM25 – Vortex dynamics (G.J.F.van Heijst, Netherlands & E. Krause, Germany) 10683 11391 11467 11615 11676 11683 11741 11964 12032 12041 12051 12080 12108 12161 12196 12275 12690 12748 12821 12836

Miguel A. Herrada (Spain): New Means of Vortex Breakdown Control Paul Billant (France): Instabilities of a Vortex Pair in a Stratiﬁed and Rotating Fluid Eugene Benilov (Ireland): Stability of Oceanic Vortices: a Solution to the Problem Yasuhide Fukumoto (Japan): Curvature Instability of a Vortex Ring Fernando L. Ponta (USA): Numerical Experiments on Vortex Shedding From an Oscillating Cylinder Denis Blackmore (USA): Bifurcation of Motions of Three Vortices and Applications Hung-Cheng Chen (Taiwan): Strong Cyclonic Vortices over Topography on a BetaPlane Gert Jan F. van Heijst (Netherlands): Spontaneous Sign Reversals in Self-Organized States of Forced Two-Dimensional Turbulence on a Bounded Square Domain Felix B. Kaplanski (Estonia): A Model for the Formation of ’Optimal’ Vortex Rings with Taking into Account Viscosity Katsuya Ishii (Japan): Numerical Simulation of Vortical Flows Using a Highly Accurate Finite Diﬀerence Scheme Michael D. Patterson (UK): The Development of an Axisymmetric Gravity Current Ewa Tuliszka-Sznitko (Poland): Numerical Investigation of the Laminar-Turbulent Transition of the Flow in a Rotor-Stator Cavity Ramiro Godoy-Diana (France): Viscous Vertical Length Scale Selection in Stratiﬁed Fluids S. Balachandar (USA): On Local Vortex Identiﬁcation Vyacheslav V. Meleshko (Ukraine): The Modelling of The Dynamics of Hairpin Vortex Packets in Wall Turbulence Marcin Kurowski (Poland): Coherent Structure of Point Vortices Inﬂuenced by Uniform Straining Flow Wolfgang Schr¨ ¨ oder (Germany): Strong Shock-Vortex Interaction a Numerical Study Oscar U. Velasco Fuentes (Mexico): Isolated Vortices over Seamounts: Laboratory Experiments and Numerical Simulations Klaus W. Hoyer (Switzerland): Three Dimensional Velocity Field of Vortices Impinging on a Wall Obtained by Scanning Particle Tracking Velocimetry Pawel Regucki (Poland): Study of the Vortex Rings Interaction by 3d Vorticity Particle-In-Cell Method

FM26 – Waves (W.K. Melville, USA & V.I. Shrira, UK) 10245 Lev Shemer (Israel): Unidirectional Steep Waves in Wave Tanks 10746 Tetsu Hara (USA): Wave Breaking and Equilibrium Surface Wave Spectra 10762 T.R. Akylas (USA): Propagation and Interactions of Nonlinear Internal Gravity Wave Beams

Scientiﬁc Program

li

10861 Vladimir E. Zakharov (USA): Weak-Turbulent Theory of Wind-Driven Sea 11126 Hu Huang (China): Shallow-Water Theory for Wave-Current-Bottom Interactions 11134 Gennady El (UK): Unsteady Undular Bore Transition in Fully Nonlinear Dispersive Wave Dynamics 11144 Anatoli Ivanovich Dobrolyubov (Belarus): The Theory of Travelling Deformation Waves and Its Applications in Biomechanics, Engineering, and Geophysics 11179 William R. Phillips (USA): The Spacing of Langmuir Circulation in Strong Wavy Shear 11199 Thomas Peacock (USA): Experiments on Rotating and Reﬂecting Internal Wave Beams 11230 Paul A. Hwang (USA): Spatio-Temporal Measurements of Capillary-Gravity Waves 11290 G´ ´ erard Iooss (France): Standing Gravity Waves in Deep Water 11345 Emilian I. R´ ´ ar´ au (UK): Nonlinear Three-Dimensional Free Surface Flows in Finite and Inﬁnite Depth 11417 Dorian Fructus (Norway): Dynamics of Crescent Wave Patterns in a Channel 11468 Colm Howlin (Ireland): Evolution of Packets of Surface Gravity Waves over Smooth Topography 11548 Walter Craig (Canada): Three Dimensional Gravity Water Waves 11591 Vasyl P. Lukomsky (Ukraine): Sharpening and Breaking of Subharmonic Gravity Waves on Deep Water 11680 Xin Zhang (USA): Short Wind Waves and Surface Wind Drift 11746 Takeshi Kataoka (Japan): Transverse Instability of Surface Solitary Waves 11803 Jan Erik Weber (Norway): A Lagrangian Approach to Wave-Induced Oceanic Mass Transport 12087 Victor I. Shrira (UK): Eﬀect of Horizontal Component of the Coriolis Force on Propagation of Near-Inertial Waves in the Ocean 12089 Fred´ ´ ´ eric Dias (France): Generalized Internal Solitary Waves and Fronts 12268 Igor A. Brovchenko (Ukraine): Intermittent Mixing by Multiscale Breaking of Wind Waves: Implications for Oil Dispersion 12424 Chantal Staquet (France): Focusing of an Inertia-Gravity Wave Packet with a Baroclinic Shear Flow 12715 Jean-Marc Vanden-Broeck (UK): Steep Capillary Waves in Electriﬁed Fluid Sheets 12813 Georgy I. Burde (Israel): Bi- Directional Water Waves and Integrable High Order KDV Equations

SM1 – Computational solid mechanics (T. Belytschko, USA & P. Wriggers, Germany) 10270 Chung-Yue Wang (Taiwan): Elastic-Plastic Large Deformation Analysis of 2D Frame Structure 10274 Terumi Touhei (Japan): Multiscale Analysis of Scattered Elastic Waves Based on the Lippmann-Schwinger Equation ¨ (Germany): Model Updating a Multicriteria Optimization 10423 Hans H. Muller-Slany Process in Mechanics 10559 Yuan Lin (China): Experiment and Quasicontinuum Simulation of Nanoindentation of Single Crystal Copper 10665 Vasilios G. Mokos (Greece): A BEM Solution to Transverse Shear Loading of Beams 10728 Anatoli Stulov (Estonia): Mechanical Features of Piano Hammer Felt 10829 Larry D. Libersky (USA): A Dual Particle Computational Method for Continua 11154 Jaroslaw Knap (USA): Mesh Optimization for the Quasicontinuum Method: A Generalization of VALE 11193 Jiann-Tsair Chang (Taiwan): Derivation of the Higher-Order Stiﬀness Matrix of a Space Frame Element for Geometric Nonlinear Analysis of Structrues 11202 Pavlo A. Steblyanko (Ukraine): The Method of Solving of Non-Stationary Coupled Problems of the Theory Thermal-Plasticity for the Rotation Shells

lii

ICTAM04

11211 Julia Mergheim (Germany): A New Approach for the FE Modelling of Cohesive Cracks 11291 Stanislaw A. Lukasiewicz (Canada): Eﬀective Solution for Finite Element Problems with Nonlinear Constraints 11298 Etienne L.G. Pecquet (Belgium): Lingopti Project: Semi-Continuous Casting Process of Copper-Nickel Alloys 11334 George Mejak (Slovenia): Two Scale Finite Element Method 11389 Grzegorz W. Zboi´ n ´ ski (Poland): Problems of Application of Hierarchical Modelling, Displacement FEM and a Posteriori Residual Error Estimation to Static and Dynamic Adaptive Analysis of Complex Structures 11482 Florian Kovacs (Hungary): Statics And Kinematics of Symmetric Swelling Viruses 11580 Jixin Yang (China): A Numerical Approach for Large-Scale Computation CEM 11720 Zhenhan Yao (China): Some Investigations on FM Bem in Solid Mechanics 11769 Anthony Nouy (France): Radial-Type Approximation Technique for a Space-Time Multiscale Computational Strategy 11797 Julien R´ ´ethor´e (France): An Energy Conserving Scheme for Time Dependent Problems Using the Extended Finite Element Method 11817 Seyoung Im (S. Korea): Development of a Novel ’Crack’ Finite Element for Propagation Simulation 12054 Alexandre V. Vakhrouchev (Russia): Modelling of Static and Dynamic Processes of Nanoparticles Interaction 12101 Andras Lengyel (Hungary): Singularities of the Four-Sided Antiprism Ring 12115 Ercan Guerses (Germany): Analysis of Evolving Deformation Microstructures in Instable Inelastic Solids Based on Energy Relaxation Methods 12131 Rene L.J.M. Ubachs (Netherlands): Microstructural Behaviour of Solder Joints 12167 Sergey N. Medyanik (USA): Molecular Mechanics Simulations of Carbon Nanostructures Using Multi-Scale Boundary Conditions 12177 Hirohisa Noguchi (Japan): Multiscale Buckling Analyses of Corrugated Fiberboard 12237 Pierre Feissel (France): Modiﬁed Error in Constitutive Relation and Its Application to Dynamic Tests with Corrupted Boundary Conditions 12310 Juha A. M¨ ¨ akipelto (Finland): Geometry Based Rational Enrichment Functions for Triangular Plane Elasticity Element 12340 Marek S. Kara´s (Poland): Solving of Indirect Problems Using Treﬀtz Method 12351 Alexey V. Borisov (Russia): Tensor Invariants and Mechanisms of Transition to Chaos in Nonholonomic Dynamical Systems 12359 Mathieu Cloirec (France): Analysis of a Structural Detail Using a Two-Scale Approach 12441 Robert B. Haber (USA): Adaptive Discontinuous Galerkin Method for Elastodynamics on Unstructured Spacetime Grids 12455 Marino Arroyo (USA): Continuum Mechanics and Carbon Nanotubes 12458 Katerina D. Papoulia (USA): Toward Convergence in Initially Rigid Cohesive Fracture Models 12534 Stefan Loehnert (Germany): Computational Homogenisation of Microheterogeneous Materials Including Decohesion at Finite Strains 12562 Antoni John (Poland): The Load Cases in Numerical Model of Pelvic Bone with Artiﬁcial Acetabulum 12579 Andrzej Siemaszko (Poland): Shakedown Safety Criterion in Reliability Analysis 12584 Eiris F.I. Boerner (Germany): A New Finite Element Formulation Based on the Theory of a Cosserat Point 12607 Ilson P. Pasqualino (Brazil): Arc-Length Method for Explicit Dynamic Relaxation 12723 Ekaterina Viatkina (Netherlands): Modelling of Non-Uniform Deformation of Metals with Dislocation Cell Structure 12727 Frederic Grondin (France): The Numerical Homogenization of the Concrete Behavior 12737 A. Amine Benzerga (USA): Discrete Dislocation Calculations of the Stored Energy of Cold Work

Scientiﬁc Program

liii

12844 Huu Nam Tran (Czech Republic): Deformation Analysis of Inﬂated Cylindrical Membrane of Composite Material with Rubber Matrix Reinforced by Cords 12845 Dimitri E. Beskos (Greece): Dynamic Analysis of Gradient Elastic Solids by BEM 12901 Kian-Meng Lim (Singapore): Variable-Order Singular Boundary Element for Calculation of Three-Dimensional Stress Intensity Factors 12924 Ellen Kuhl (Germany): Application of the Material Force Method to Structural Optimization

SM2 – Contact and friction mechanics (A. Klarbring, Sweden & G. Szefer, Poland) 10211 Mykhaylo G. Pantelyat (Ukraine): Thermocontact Interaction of Bodies of Revolution During Induction Heating 10342 Victor M. Musalimov (Russia): Dynamic Characteristics and Monitoring of Rubbing Surfaces Quality 10459 Herman N.V. Parland (Finland): Contact Mechanical Analysis of Elastic Multibody Structures 10686 L.E. Anderson (Sweden): Existence and Uniquness of Steady State Solutions in Thermoelastic Contact With Frictional Heating 10718 Hamid Reza Irannejad (Iran): FE Analysis of Bond for Smooth FRP Rods Embedded in Concrete 10999 Sergey A. Chizhik (Belarus): Modelling of Contact of Structured Materials Based on Data from Scanning Probe Microscopy 11008 Graham J. Weir (new zealand): A Universal Property of Geometrical Hardening 11074 Istvan Paczelt (Hungary): Contact Optimization Problems Associated with the Wear Process 11135 Feodor M. Borodich (UK): Molecular Adhesive Contact for Indenters of Non-Ideal Shapes 11196 Irina G. Goryacheva (Russia): Adhesive Component of the Rolling Friction Force 11215 Leon M. Keer (USA): Fundamental Relations for Frictional and Adhesive Nanoindentation Tests 11365 Denis Elaguine (Sweden): Hertz Contact at Finite Friction and Arbitrary Proﬁles 11452 Jozef ´ Joachim Telega (Poland): Frictional Contact with Wear Diﬀusion 11481 Jan Awrejcewicz (Poland): On the Contact Thermoelastic Problem with Frictional Heating, Wear and Auto-Vibrations 11611 Ganna Shyshkanova (Ukraine): Three-Dimensional Problem of the Contact by Doubly Connected Domain Taking into Account Roughness and Friction 11613 Alfred Zmitrowicz (Poland): Evolutions of Friction Anisotropy and Heterogeneity 11697 Victor A. Shevchuk (Ukraine): Inverse Problems of Thermoelasticity for Frictionally Interacting Layers 11778 Alexey A. Kireenkov (Russia): Multidimensional Model of Combined Dry Friction 11933 Yuriy Letser (Ukraine): Numerical Modeling of Contact Fracture of Elasto-Plastic Cracked Bodies 11987 Demirkan Coker (USA): Crack-Like and Pulse-Like Modes of Frictional Sliding along an Interface Under Dynamic Shear Loading 12082 Leila Abdou (France): Experimental and Numerical Study of the Brick-Mortar Interface 12086 Markus Lindner (Germany): Experimental and Analytical Investigation of Rubber Friction 12382 Marius Cocou (France): A Dynamic Unilateral Contact Problem for a Cracked Body 12412 Larissa Gorbatikh (USA): A Simple Model to Account for the Locking Eﬀect Between Two Rough Surfaces under Cyclic Loading 12553 Stanislaw Stupkiewicz (Poland): Boundary Layers Induced by Contact of Rough Bodies 12559 Alexandr A. Olshevsky (Russia): The Accounting of Surface Roughness in Contact of Arbitrary Shaped Bodies Using FEM

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ICTAM04

12563 Thibaut Putelat (UK): Frictional Sliding of a Multislip System 12646 Antonio Pinto da Costa (Portugal): Second-Order Cone Complementarity Formulation for Quasi-Static Incremental Frictional Contact Problem in Three-Dimensional Space 12773 Yves Gonthier (Canada): A Novel Contact Model Based on Volumetric Information

SM3 – Control of structures (F. Chernousko, Russia & S. Pellegrino, UK) 10098 10457 10458 10879 11036 11163 11219 11243 11265 11287 12756 13004

Anupam S. Ahlawat (India): Coupled Optimal Design of Building with TMD Bartlomiej Blachowski (Poland): Optimal Vibration Control of Guyed Masts Pawel Holobut (Poland): Time-Optimal Control of Hydraulic Manipulators Kazuo Tanizawa (Japan): Surface Accuracy of Inﬂatable Reﬂector Covered with Stretched Cable Daniela G. Marinova (Bulgaria): H-inf Control for Smart Multistory Building Structures Firdaus E. Udwadia (USA): Exact Tracking Control for Nonlinear Structural and Mechanical Systems Nikolai N. Bolotnik (Russia): Pre-Acting Impact Isolation Systems Felix L. Chernousko (Russia): Control of Multibody Systems Moving along a Plane Agnessa S. Kovaleva (Russia): Control of Random Dynamics of a Rigid Rocking Block Jong-Dar Yau (Taiwan): Suppression of Train-Induced Vibrations of Continuous Truss Bridge by Hybrid TMDs Walerian Szyszkowski (Canada): Optimization of Active Control of Structural Vibration by the Beam Analogy Andre Fenili (Brazil): Control of a Nonlinear Slewing Flexible Beam

SM4 – Damage mechanics (M. Chrzanowski, Poland & P. Steinmann, Germany) 10011 Noel ¨ Challamel (France): Stability and Creep Damage of Quasi-Brittle Materials 10456 Akrum Abdul-Latif (France): Modeling of the Damage Evolution at the Granular Scale in Polycrystals under Complex Cyclic Loadings 10865 Robert Svendsen (Germany): Continuum Thermodynamic and Variational Modeling and Simulation of Ductile Failure at Large Deformation with Application to Engineering Structures 10960 Kari J. Santaoja (Finland): Material Models for Hookean Materials with Voids or Cracks 10961 Chi L. Chow (USA): Localized Necking Criterion Based on Acoustic Tensor for Materials with Anisotropic Damage 10974 Ren´ ´ e Billardon (France): An Elasto-Viscoplastic Model Coupled to Damage and Grain Growth to Take Account of Material Variability 11228 Andrzej Litewka (Portugal): Damage and Failure of Brittle Solids 11406 Sabine Ricci (Germany): Numerical Analysis of Nonlocal Anisotropic Continuum Damage 11414 Ekkehard Ramm (Germany): Discrete Models and Their Application in Damagemechanics 11415 Ron H.J. Peerlings (Netherlands): A Nonlocal Plasticity – Damage Formulation Based on the Micromechanics of Defect Growth 11665 Marcin Chrzanowski (Poland): Propagation of Cracks in Terms of Continuum Damage Mechanics 11779 Larisa V. Stepanova (Russia): An Asymptotic Analysis of Mode I Crack in Creeping Damaged Solids 11796 Jacek J. Skrzypek (Poland): Damage Acquired Anisotropy in Elastic-Plastic Materials 11885 Jean-Louis Chaboche (France): A CDM Approach of Ductile Damage with Plastic Volume Changes

Scientiﬁc Program

lv

11886 Issam Doghri (Belgium): Micromechanical Modelling of the Deformation and Damage of Inelastic Brittle Three-Phase Composites: Application to Fiber-Reinforced Concrete 12021 Fran¸cois Hild (France): Damage Field Identiﬁcation using Full-Field Displacement Measurements 12036 Jian-Ying Wu (China): A New Energy-Based Elastoplastic Damage Model for Concrete 12088 Henning Schuette (Germany): Lifetime Prediction with a Damage Model Based on Mixed-Mode Microcrack Propagation 12116 Serdar Goektepe (Germany): A Micromechanically Based Network Model for Rubbery Polymers Incorporating Mullins-Type Stress Softening 12277 Gilles Lubineau (France): Computational Micro-Meso Modeling for Laminates Under Thermomechanical Fatigue and an Oxidizing Atmosphere 12298 Ilaria Monetto (Italy): A Non-Associative Anisotropic Damage Model for Brittle Materials 12333 Andrzej Stachurski (Poland): Robust Identiﬁcation of an Augmented Gurson Model for Elasto-Plastic Porous Media 12335 Artur W. Ganczarski (Poland): Low Cycle Fatigue Based on Unilateral Damage Evolution 12341 Thierry J. Massart (Belgium): Coupled Meso-Macro Simulation of Masonry Cracking 12749 George Chatzigeorgiou (Greece): Coupling Between Progressive Damage and Permeability of Concrete 12913 Vincent P. Godard (France): Anisotropic Damage Model for Concrete Including Unilateral Eﬀects

SM5 – Dynamic plasticity of structures (N. Jones, UK & T. Wierzbicki, USA) 11022 Vikram S. Deshpande (UK): Blast Resistance of Clamped Sandwich Beams 11051 Tongxi Yu (China): Collision Between Two Deformable Structures 11107 D. Karagiozova (Bulgaria): Counterintuitive Response of Long Circular Tubes to Axial Impact 12043 Rami Masri (Israel): Self Similar Dynamic Expansion of a Spherical Cavity in Elastoplastic Media 12949 Piotr Perzyna (Poland): Numerical Investigation of Dynamic Shear Bands in Inelastic Solids as a Problem of Mesomechanics 12982 Narinder K. Gupta (India): On Non-Axisymmetric Collapse of Thin Tubes

SM6 – Elasticity (R. Kienzler, Germany & L. Wheeler, USA) 10442 Katarzyna Kowalczyk-Gajewska (Poland): On Invariants of the Elasticity Tensor for Orthotropic Materials 10972 Tsolo P. Ivanov (Bulgaria): Motion and Stability of an Elastic Heavy Top 11220 David M. Haughton (UK): Stability of Compressible Elastic Blocks 11523 Reuven Segev (Israel): Generalized Stress Concentration Factors 11526 Lenser A. Aghalovyan (Armenia): On Asymptotic Method of Static and Dynamic Boundary Problems Solution 11901 Charles Ruimy (France): Axisymmetric Force Solution for a Semi-Inﬁnite Cubic Solid 11960 Tungyang Chen (Taiwan): An Exactly Solvable Microgeometry in Torsion 11998 Xiaojing Zheng (China): A New Nonlinear Constitutive Relation for Magnetostrictive Materials 12039 Gaza H. Maluleke (South Africa): Nonlinear Radial Oscillations of Anisotropic ThinWalled Cylindrical Tubes 12122 Marcelo Epstein (Canada): Nonlocal Eshelby Entities: a One-Dimensional Example 12189 Iwona M. Jasiuk (USA): Analysis of Trabecular Bone as a Hierarchical Material 12289 Valery P. Matveyenko (Russia): Investigation of Couple-Stress Eﬀects in Elastic Bodies Under Deformation

lvi

ICTAM04

12324 Serge N. Gavrilov (Russia): New Analytical Approach for Investigation of NonStationary Dynamics of Media with Moving Inhomogeneities 12354 Gearoid P. Mac Sithigh (USA): Agmon’s Condition for Incompressible Elasticity: a Variational Formulation 12434 Elena F. Grekova (Russia): Modelling of Complex Elastic Crystals by Means of Micromorphic Gyrocontinua 12575 Vyacheslav V. Lyakh (Ukraine): Truncated Elastic Wedge under Torsional Load 12685 Valeriy A. Buryachenko (USA): Homogenization of Triply Periodic Elastic Media with Random Imperfections 12987 Roberta Sburlati (Italy): On the Impact Law in Elastic Plate-Like Bodies

SM7 – Experimental methods in solid mechanics (I. Emri, Slovenia & J.L. Freire, Brazil) 10492 Keyu Li (USA): An Optical Strain Rosette/Ring – Core Method Applied on Laser Weld 10691 Masashi Sato (Japan): Estimation of Principal Axes of Inertia 11556 Michael N. Osipov (Russia): The Whole Field Non-Destructive Optical Slicing Method in Three-Dimensional Photoelasticity 11617 Sylwester Samborski (Poland): Porous Ceramics – Experimental Research and Modelling 11764 Yilong Bai (China): Critical Sensitivity in Rock Experiments 12272 Vsevolod V. Shchennikov (Russia): Phase Transitions and Mechanical Properties of Ternary Chalcogenides 12551 Saed Mousavi (Sweden): SHPB Technique for Identiﬁcation of Complex Modulus Under Condition of Non-Uniform Stress 12666 Michal A. Miskiewicz (Poland): In-Situ Observation of Fatigue Crack Growth in Carbon Steel 12799 Tadeusz Uhl (Poland): New Solutions in Experimental Modal Analysis of Mechanical Structures

SM8 – Fatigue (J. Dominguez, Spain & K. Reifsnider, USA) 10698 Jaime Dominguez (Spain): Inﬂuence of Contact Conditions on Fretting Fatigue Under Spherical Contact 11371 Krystyna Majorkowska-Knap (Poland): Fatigue Investigations into a Composite Glider Structure 11686 Yi Sun (China): A Microscopic Mechanics Model 11737 Byeongchoon Goo (S. Korea): Fatigue Life Prediction Considering Residual Stress Relaxation 11920 Pavlo Maruschak (Ukraine): Eﬀects of Frequency Temperature and Loading Waveform on Fatigue Crack Growth Rate in Steel 15Kh13MF 11927 Steﬀen Brinckmann (Netherlands): Stress Concentrations Caused by Dislocations at the Free Surface 12034 Ki-Seok Kim (S. Korea): Probabilistic Analysis of Fatigue Crack Growth using Moment Method 12217 Ren´ ´ e C. Alderliesten (Netherlands): Energy Release Rate Approach for Delamination in a Fatigue Crack Conﬁguration in Glare 12228 Marion Risbet (France): Modelling Fatigue Crack Growth with Time-Derivative Equations 12391 Youshi Hong (China): Characteristics of Very-High-Cycle Fatigue for a High Carbon Low Alloy Steel 12914 Dorota I. Koca´ n ´ da (Poland): Modelling of Short Fatigue Crack Growth in a Metal in HCF Range

Scientiﬁc Program

lvii

SM9 – Fracture and crack mechanics (D. Gross, Germany & A. Needleman, USA) 10250 10334 10338 10419 10429 10579 10646 10676 10824 10864 10897 10910 11045 11195 11321 11425 11484 11568 11626 11856 11910 11934 11980 12044 12171 12174 12249 12378 12393 12492 12798

Michael P. Wnuk (USA): A Fractal Cohesive Crack Model Daniel Kujawski (USA): Inﬂuence of Stress State on Crack-Tip Driving Force Mokhtar Adda-Bedia (France): Branching Instability of Brittle Fracture Vidya Sagar Remalli (India): Size Eﬀect in Tensile Fracture of Concrete – A Study Based on Lattice Model Applied to CT-Specimen Octavian Pop (France): Numerical and Experimental Study of the Plastic Zone in the Vicinity of the Crack Tip by the Optical Caustics Method Vera E. Petrova (Russia): Thermoelastic Problems for a Bimaterial with Defects/Singularities Parissa Hosseini-Tehrani (Iran): Dynamic Crack Analysis Under Thermal Shock Huijian Li (China): Experimental Investigation on Concrete Shear Crack Extension Igor Guz (UK): Eﬀect of Inter- and Intralaminar Damage on the Compressive Fracture of Hyperelastic Materials Liviu Marsavina (Romania): Experimental and Numerical Crack Growth in a Special Geometry Nikita F. Morozov (Russia): Elastodynamics Problems in Domains with Edges Alan Needleman (USA): 3D Microstructural Eﬀects on Plane Strain Ductile Crack Growth Andreas Ricoeur (Germany): Weight Functions for Cracks in Piezoelectrics Oleksandr V. Menshykov (Ukraine): Elastodynamic Contact Problem for Two PennyShaped Cracks Jos´ ´e Dominguez (Spain): Numerical Approach for Dynamic Fracture in Piezoelectric Solids Yasuhide Shindo (Japan): Finite Element Analysis of Fracture and Polarization Switching Behavior in Modiﬁed Small Punch Testing of Piezoelectric Ceramics Axel M¨ u ¨ller (Germany): On Crack Assessment at Bimaterial Interfaces Hai-Tao Wang (China): A Quasi-Spherical Coordinate System and Its Application to the Determination of Vertex-Type Singularities Asher A. Rubinstein (USA): Failure Model of Protective Coatings Alla V. Balueva (USA): Modeling of Environment Assisted Delamination I. Quasistatic Case Ewa M. Turska (Poland): The Inﬂuence of Remote Stresses on the Near Crack Tip Stress Field Yu Shouwen (China): The Elasto-Plastic Thin Film/Substrate Via Buckle-Driven Delamination and Crack Growth Andrzej Kaczy´ n ´ ski (Poland): On 3-D Thermoelastic Problems of Interfacial Cracks in a Periodic Stratiﬁed Space Karsten Kolk (Germany): Automatic 3D Crack Growth Simulation Based on Boundary Elements Irene Arias (USA): Massively Parallel Simulations of Dynamic Fracture and Fragmentation of Brittle Solids Ping Wang (China): the Shield Eﬀect of Phase Transformation Stress Field at Crack Tip Volodymyr V. Loboda (Ukraine): Contact Zone Approach to the Analysis of Interface Cracks in Thermomechanically Loaded Piezoelectric Bimaterials Krishnaswamy Ravi-Chandar (USA): Interaction of Propagting Cracks and Shear Waves Jean-Baptiste M. Leblond (France): Disorder of the Front of a Tensile Tunnel-Crack Propagating in Some Inhomogeneous Medium Yichun Zhou (China): Creep Deformation in Thermal Barrier Coatings Due to the Eﬀect of Thermal Growth Oxidation and Temperature Gradient Abdulhamid Al-Abduljabbar (Saudi Arabia): Numerical Analysis of Strain Hardening and Pressure Sensitivity Eﬀects on J-Integral

lviii

ICTAM04

SM10 – Functionally graded materials (R. Batra, USA & G.H. Paulino, USA) 10045 Andrzej Tylikowski (Poland): Dynamic Stability of Functionally Graded Plate Under In-Plane Compression 10135 Lizhi Sun (USA): Micromechanics-Based Elastic Model for Functionally Graded Materials with Particle Interactions 10364 Jiann-Quo Tarn (Taiwan): A State Space Formalism for Piezothermoelasticity of Functionally Graded Materials 10501 Weichen Shi (China): Conservation Laws of Functionally Graded Materials in Elastodynamics 10836 Chuanzeng Zhang (Germany): Transient Dynamic Crack Analysis in FGMs Under Impact Loading 11177 Dhirendra V. Kubair (India): Asymptotic Ananlysis of a Stationary Crack in a Ductile Functionally Graded Material 11551 Linzhi Wu (China): The Plane Crack Problem in a Functionally Graded Orthotropic Strip 11572 Bing-Zheng Gai (China): Frictional Slip Between a Gradient Non-Homogeneous Layer and a Half-Space in Anti-Plane Elastic Wawe Field 11592 Juri Engelbrecht (Estonia): Wave Propagation in Functionally Graded Materials 11671 Eduard Rohan (Czech Republic): Adaptive Modelling of Microscopic Heterogeneous Medium Undergoing Large Deformation 12093 Joel ¨ A. Pouget (France): Actuator and Sensor Modelling for Laminated Piezoelectric Plates 12273 Arthur H. England (UK): Complex-Variable Methods Applied to Functionally-Graded Elastic Plate Problems 12461 Takemasa Seto (Japan): Study of Two-Dimensional Elasticity on FGM 12465 Yoshihiro Ootao (Japan): Three-Dimensional Transient Thermoelastic Analysis of Orthotropic Functionally Graded Rectangular Plate 12594 R.C. Batra (USA): Adiabatic Shear Bands in Functionally Graded Materials 12665 Emilio C.N. Silva (Brazil): Topology Optimization Applied to the Design of Functionally Graded Material (FGM) Structures 12968 Sathyanaraya Hanagud (USA): First Principles-Based Equations of State for Functionally Graded Materials 12978 Minoru Taya (USA): Design of FGM Bimorph Piezo-Actuators 13018 Lavinia S.A. Borges (Brazil): Thermoelastic Analysis of Functionally Graded Materials Submitted to Shocks

SM11 – Impact and wave propagation (A. Norris, USA & K. Wilma´ n ´ ski, Germany) 10072 Bettina Albers (Germany): Surface Waves on an Impermeable Boundary of a Poroelastic Medium 10114 Alexander M. Samsonov (Russia): Bulk Solitons do not Decay in Elastic Wave Guides 10344 Toshiaki Hata (Japan): Stress-Focusing Eﬀect in a Spherical Zirconia Inclusion with Dynamically Transforming Strains 10422 Toshiro Maeda (Japan): Simultaneous Simulation of Dispersion Curves and H/V Spectra 10426 Nikolai N. Myagkov (Russia): Nonlinear Waves in Shock-Loaded Solids 10612 Krzysztof Wilma´ n ´ ski (Germany): Critical Time for Acoustic Waves in Weakly Nonlinear Poroelastic Materials 10891 Semra Ahmetolan (Turkey): Rayleigh-Like Surface Waves on a Nonlinear Layered Elastic Half Space 11085 Istvan A. Veres (Switzerland): Non-Destructive Testing of Wood by Wave Propagation 11099 Ji Wang (China): A Two-Dimensional Analysis of Surface Acoustic Waves in Finite Anisotropic Solids with Electrodes 11242 Karima R. Khusnutdinova (UK): Nonlinear Wawe Processes in a Bi-Layer

Scientiﬁc Program

lix

11271 Gabriel E. Chao (Netherlands): Shock-Induced Surface Waves in Porous Reservoirs 11360 Janusz Klepaczko (France): Trapping of Plastic Waves by Adiabatic Deformation 11377 Kateryna V. Terletska (Ukraine): Modeling of Solitary Impulses in a Composite Material Using Wavelet Analysis 11401 Igor Selezov (Ukraine): Some Degenerated and Extended Wave Models of Elasto- and Hydrodynamics with Finite Velocity Disturbance Propagation 11596 Angelo Morro (Italy): Uniqueness Results for the Reﬂection-Transmission Problem 11620 Andres Braunbrueck (Estonia): Wave Interaction Resonances in Inhomogeneous Elastic Materials 11682 Martin Ostoja-Starzewski (Canada): Acceleration Wavefronts in Random Media 11812 Philippe S. Boulanger (Belgium): Inhomogeneous Circularly Polarized Waves in Orthorhombic Crystals 11813 Andrus Salupere (Estonia): Hidden and Driven Solitons in Microstructured Media 11853 Witold Kosi´ n ´ ski (Poland): Thermo-Mechanical Wave Propagation in Materials with Internal State Variables 11989 Sergey K. Kanaun (Mexico): Self-Consistent Methods in the Problem of Elastic Wave Propagation Through Matrix Composite Matrials 12203 Anton G. Pegushin (Russia): Waves of Deformation Propagation in Nonlinear Viscously Elastic Porous Material 12313 Bernhard Pichler (Austria): Elastoplasticity of Gravel Protecting Rockfall-Endangered Steel Pipelines 12418 Toshihiko Sugiura (Japan): Flaw Identiﬁcation by Angle Beam Electromagnetic Acoustic Transducers 12481 Anton M. Krivtsov (Russia): Impact Fracture of Rock Materials Due to Percussive Drilling Action 12514 Anil C. Wijeyewickrema (Japan): Dispersion and Stability Analysis of Waves in PreStressed Imperfectly Bonded Layered Composites 12568 Olari Ilison (Estonia): On the Propagation of Solitary Waves in Microstructured Solids 12763 Chokri Zammali (France): Level-Sets and Mixed Approaches for Dynamic Contact Problems 12772 Mahmoud I. Hussein (USA): Analysis and Design of Dispersive Materials and Structures

SM12 – Material instabilities (D. Bigoni, Italy & H. Petryk, Poland) 10265 Ahmed Benallal (France): Material Instabilites in Thermo-Mechanical Processes 10314 Peter B. Beda (Hungary): Dynamical Systems Theory in Material Instabilities 10867 Eugene I. Ryzhak (Russia): An Idea and Theory of Hypothetical Device for Investigating the Localization Phenomena 10996 Henryk Petryk (Poland): Incremental Energy Minimization in Material Instability Problems 11254 Davide Bigoni (Italy): Dynamics of Perturbations and Shear Band Instabilities 11277 Paul Steinmann (Germany): On Convexity Conditions in Spatial and Material Settings of Hyperelasticity 11432 Yoshihiro Tomita (Japan): Computational Characterization of Micro- to Macroscopic Mechanical Behavior of Carbon Black-Filled Rubber 11705 Jose Merodio (Spain): Material Instabilities of Fiber-Reinforced Nonlinearly Elastic Solids 12025 Walter J. Drugan (USA): Ultrastiﬀ Elastic Composites via Negative Stiﬀness Inclusions, and Material Stability Implications 12223 Pavel V. Tkachev (Russia): Stability of Ideal Inﬁnite Crystal Under Finite Uniform Deformation 12419 Igor Dobovsek (Slovenia): Explosive Instabilities in a Class of Hyperelastic Materials with Higher-Order Gradients

lx

ICTAM04

12682 Joao A. Martins (Portugal): On the Concept of “Dynamic (In)Stablility of QuasiStatic Paths” 12751 Yves M. Leroy (France): Strain Localization at the Brittle-Ductile Transition of the Earth’s Continental Crust.

SM13 – Mechanics of composites (S. Adali, South Africa & N.A. Fleck, UK) 10189 Mohammad Reza Khoshravan (Iran): Numerical Evaluation of Mixed Mode Delamination in a U.D. Glass/Epoxy Composite in 2D and 3D States 10244 Ryszard Pyrz (Denmark): Interfacial Properties of Nanowire-Polymer Composites 10347 Jozef ´ Ignaczak (Poland): Plane Harmonic Waves in a Microperiodic Layered Thermoelastic Solid Revisited 10412 Tong-Earn Tay (Singapore): Damage Progression by the Element-Failure Method (EFM) and Strain Invariant Failure Theory (SIFT) 10483 Hossein M Shodja (Iran): Eﬀective Properties of Solids Containing Randomly Distributed Multi-Phase Spherical Particles 10555 Lidiya Nazarenko (Ukraine): Porous Anisotropic Composites under Microfructures 10667 Petri J. Kere (Finland): Reissner-Mindlin-Von Karman Type Plate Modle for Postbuckling Analysis of Laminated Composite Structures 10745 Kanmi Aderogba (Nigeria): Three-Dimensional Transmission in Plane Layered Elastic Composites 11014 Y. Jack Weitsman (USA): Aspects of the Mechanical Response of Randomly Reinforced, Chopped Fiber Strand, Polymeric Composites 11080 Javier LLorca (Spain): Computational Modeling of Deformation and Damage in Particle-Reinforced Composites 11162 Martin G. Andrews (USA): Elastic Interaction of Multiple Delaminations in Laminated Structures 11176 Federico J. Sabina (Mexico): Overall Properties of Periodic Biocomposites 11180 Christophe Bouvet (France): Damage Tolerance of Composite Structures with Thermal Shield 11387 Mike J. Cliﬀord (UK): Can it Be Made? Predicting the Formability of Textile Composite Components 11426 Fumio Narita (Japan): Electroelstic Fields Concentrations and Polarization Switching by Circular Electrodes in Piezoelectric Disk Composites 11457 Aleksander Muc (Poland): Fuzzy Set Approach to Modelling Composite Mechanical Properties 11507 Amna Rekik (France): Evaluation of Linearization Procedures Sustaining Nonlinear Homogenisation Theories 11610 Tomo Takeda (Japan): Three-Dimensional Thermoelastic Analalysis of Plain Weave Glass/Epoxy Composities with Cracks at Cryogenic Temperatures 11747 Andras Szekrenyes (Hungary): Advanced Beam Model for Fiber-Bridging in Unidirectional Composite Double-Cantilever Beam Specimens 11782 John R. Willis (UK): Interfacial Jump Conditions in Strain-Gradient Plasticity and Relations of Hall-Petch Type 11794 Brian Nyvang Legarth (Denmark): A Study of Particle Debonding with Anisotropy 11805 Emmanuelle Chabert (France): Nonlinear Aﬃne Extension of the Three-Phase Sphere Model 11895 Barbara Gambin (Poland): H-Convergence and Multilayering in Piezocomposites 12148 Jorn S. Hansen (Canada): A Homogenization Based Laminated Beam Theory 12214 Ulrik Borg (Denmark): Compressive Strength of Fiber Composite with Porosity 12242 Zhong Ling (China): Thermal Residual Stress in Al2O3/SiCnano Ceramic Composites Measured by Nanoindenter 12321 Heinz E. Pettermann (Austria): Composites with Planar Random Fiber Arrangements 12489 Jan Schjødt-Thomsen (Denmark): Inclusion Dispersion: Eﬀects on Stress and Eﬀective Properties

Scientiﬁc Program

lxi

12496 Shiguo Long (China): Thermal Fatigue of MMC Induced by Laser Heating 12531 Lingadahally S. Ramachandra (India): Thermo-Mechanical Stability and Vibration Analysis of Composite Shells 12560 Aleksander Muc (Poland): New Trends in Optimal Design of Composite Materials 12576 Robert Boehm (Germany): An Anisotropic Damage Model for the Prediction of the Degradation Behaviour of Novel Textile Reinforced Composites 12588 Vladyslav Danishevskyy (Ukraine): Asymptotic Study of Imperfect Interfacial Bonding in Periodic Composite Materials 12667 Antoni A. Gaka (Poland): T-Inclusion Regions for the Eﬀective Transport Coeﬃcients of Two–Phase Media 12680 Akke S.J. Suiker (Netherlands): Crack Tunneling in Laminates 12732 Ryszard Wojnar (Poland): Macroscopic Relations for Nonlinear Thermodiﬀusion in Heterogeneous Elastic Medium 12818 Arwen Smits (Belgium): Study of the Usability of Various Cruciform Geometries for Biaxial Testing of Fiber Reinforced Composites 12970 Marek Leﬁk (Poland): Incremental Eﬀective Constitutive Law for Composite Material in the Form of Artiﬁcial Neural Network 13009 Ali Daneshmehr (Iran): Analysis of Thick Laminated Panel With Piezoelectric Sensors Based on Three-Dimensional Theory of Elasticity

SM14 – Mechanics of phase transformations (F.D. Fischer, Austria & A. Molinari, France) 10046 Wojciech K. Nowacki (Poland): Temperature and Strain Rate Eﬀects on TRIP Sheet Steel. Measurement of Temperature by Infrared Thermograph 10088 Thomas Antretter (Austria): A Numerical Approach to Martensitic Phase Transformations 10434 Tatsuo Inoue (Japan): Macro-, Meso- and Micro-Scopic Metallo-Thermo-Mechanics 10648 Sergio R. Turteltaub (Netherlands): Multiscale Modeling of Steels assisted by Transformation-Induced Plasticity 10721 Isaac V. Chenchiah (Germany): The Nature of Stress and Strain Fields in Shape Memory Polycrystals 11308 Elzbieta ˙ Alicja Pieczyska (Poland): Shape Memory Alloy Under Strain and Stress Controlled Conditions – Thermomechanical Aspects of Martensite and Reverse Transformations 11325 Valery I. Levitas (USA): High Pressure Mechanochemistry: Conceptual Multiscale Theory and Interpretation of Experiments 11393 Arkadi Berezovski (Estonia): Stress-Induced Martensitic Phase Transition Front Propagation 11395 Shangping Chen (Netherlands): Modeling Martensite Transformation in the ElastoPlastic Material at Finite Strain 11625 Sabine M. Schl¨ ¨ ogl (Germany): Modeling of the Microstructural Evolution in Cr-Mo Steels During Tempering and Hydrogen Exposure 11837 Alexander B. Freidin (Russia): Equilibrium and Stability of Two-Phase Deformations within the Framework of Phase Transition Zones 11852 Cristian Faciu (Romania): On Modeling the Longitudial Impact of Two Shape Memory Bars 12077 Thorsten Bartel (Germany): A Micromechanical Model for Single-Crystal ShapeMemory-Alloys 12121 Claus Oberste-Brandenburg (Germany): Simulation of Discontinuity Movement by Boundary Elements 12225 Christian Lexcellent (France): Determination of Phase Transformation Yield Surface of Anisotropic Shape Memory Alloys 12421 Fabrice Barbe (France): Numerical Determination of Diﬀusional Transformation Induced Plasticity from Computations of Random Microstructures

lxii

ICTAM04

12747 Salem Meftah (France): Numerical Analyses of the Interaction Classical Plasticity – TRIP 12881 Qingping Sun (China): Nucleation and Motion of Phase Boundary in Shape Memory Alloy Microtubes

SM15 – Mechanics of porous materials (W. Ehlers, Germany & J.M. Huyghe, Netherlands) 10198 Tim Ricken (Germany): Biodegradation in Porous Landﬁll Bodies 10790 Michio Kurashige (Japan): Mandel and Cryer Problems For Fluid-Saturated Foams With Negative Poisson’s Ratio 10792 Wolfgang Ehlers (Germany): Localization and Stability of Unsaturated Soil 11119 Martin Schanz (Germany): Convolution Quadrature Based Boundary Element Method for Quasi-Static Poroelasticity 11194 Bernd Markert (Germany): Theory and Numerics of Multicomponent Mixture Models for Soft Biological Tissues 11463 Jorg ¨ Hohe (Germany): Probabilistic Homogenization of Hyperelastic Solid Foams 11941 Luc Dormieux (France): Coupling Between Permeability and Damage: a Micromechanical Approach 12114 Stefano Dal Pont (France): Thermo-Hydro-Chemical-Mechanical Analysis of Concrete at High Temperatures 12358 Jacek Tejchman (Poland): FE-Investigations on Shear Localizations in Granular Bodies within Hypoplasticity 12518 Ragnar Larsson (Sweden): Modelling of Composites Processing Using a Two-Phase Porous Media Theory 12566 Michal Pakula (Poland): Wave Propagation in High Porosity Bones – a Cellular Model 12567 Mariusz Kaczmarek (Poland): Soft Porous Media Model of Magnetic Fluid 12577 J´ ´ ozef Kubik (Poland): Mechanics of Saturated High Porosity / Soft Materials 12614 Daniela M. Bauer (France): A Three Layer Porous Media Model of Cutaneous Circulation with Application to Mechanical Skin Irritation 12713 Jo¨ ¨el Sarout (France): Identiﬁcation of Some Chemoporoelastic Parameters of a Reactive Shale from Experimental Data 12771 Michael W. Crochet (USA): Mesoscale Predictions for the Thermomechanics of Granular Energetic Composites 12903 Alan C.F. Cocks (UK): The Structure of Constitutive Laws for Powder Metallurgical Components 12917 Csaba I. Sinka (UK): Experimental Characterisation and Numerical Modelling of Density Distribution in Tablets 13006 Jacek Banaszak (Poland): Stresses and Fractures in Capillary – Porous Materials Under Drying

SM16 – Mechatronics (W.O. Schiehlen, Germany & M. Tomizuka, USA) 10510 Ya-Pu Zhao (China): Stability Analyses of Electrostatic Torsional RF MEMS Switch 10553 Dominique de Blaise (France): Improvement of Positioning Accuracy of Delta Parallel Robot 10809 Friedrich G. Pfeiﬀer (Germany): Dynamics and Control of a Hydraulically Driven Boring Plant 10900 Dalius Mazeika (Lithuania): Investigation of Powerful and High Precision Piezoelectric Actuator for Two-Dimensional Positioning 11282 Wim Symens (Belgium): Gain-Scheduling Control of Machine Tools with Varying Structural Flexibilities 12417 Horst Schulte (Germany): A Systematic Load Identiﬁcation Procedure for Parallel Robot Manipulators

Scientiﬁc Program

lxiii

12546 Alfredas Busilas (Lithuania): Development of Positioning of Mechanisms with Piezoelectric Engines 12712 Li-Sheng Wang (Taiwan): Hierarchical Tracking Control of Wheeled Mobile Robot 12758 Matthias Weber (Germany): Rapid Prototyping of Model Based Control Algorithms for Diesel-Engines with Turbocharger

SM17 – Multibody dynamics (M. Geradin, Italy & F. Pfeiﬀer, Germany) 10026 Wojciech Blajer (Poland): A Geometrical Framework for Modeling and Simulation of Nonholonomic Mechanical Systems 10533 Gakhip Ualiyev (Kazakhstan): Research of Movement of the Mechanism Suﬃcient with Elastic Part 10624 Katica Stevan´ ´ ovi´ ´ c Hedrih (Serbia): Homoclinic Orbits Layering in the Coupled Rotor Nonlinear Dynamics and Chaotic Clock Models 10647 Pavel Polach (Czech Republic): Design of the Rear Carriage Stabilizer of a Low-Floor Articulated Trolleybus 10779 Yuriy N. Kononov (Ukraine): Stabilization by Rotating Rigid Bodies for Unstable Rotation of a Rigid Body with Cavities Containing a Fluid 10893 Oleg N. Dmitrochenko (Russia): Simulation of Constrained Rigid and Elastic Bodies Without Constraint Equations 11020 Harry J. Dankowicz (USA): Analysis of Grazing Bifurcations in Impact Microactuators 11223 Nils Guse (Germany): Low Energy Control of Periodic Motions in Manufacturing 11352 Amalia Pielorz (Poland): Selected Problems of Discrete-Continuous Mechanical Systems with Local Nonlinearities 11361 Gilles Saussine (France): Modeling Ballast Behavior Using a Three-Dimensional Polyhedral Discrete Element Method 11375 Arun K. Banerjee (USA): Eﬃcient Generalized Speeds in a Recursive Formulation of Flexible Multibody Dynamics 11640 Dmitry Yu. Pogorelov (Russia): On Approximate Jacobian Matrices in Simulation of Stiﬀ Multibody Systems 11674 Jean Lerbet (France): Intrinsic Formulation of Dynamics of Curvilinear Systems 11688 Dmitry G. Agapov (Russia): Simulation of Track Ballast 11700 Kazuo Tanizawa (Japan): Conﬁguration Control and Dynamic Analysis of Redundant Link-Type Manipulators 12029 Johannes Gerstmayr (Austria): The Absolute Coordinate Formulation with Reduced Strain and Stiﬀening 12071 Juan Valverde (Spain): Stability Analysis of a Tethered System 12133 Beate Muth (Germany): Simulation of Contacting Spatial Polyhedral Particles 12134 Robert Seifried (Germany): Plastic Deformation by Impacts in Multibody Systems 12208 Olivier Bruls (Belgium): A Systematic Model Reduction Method for the Control of Flexible Multibody Systems 12464 Kohichi Miura (Japan): Expression on the Deﬂection of a Flexible Thin Rod and Its Measurement 12537 Naoki Sugano (Japan): Dynamic Analysis and Vibration Control of The Planar Beams Moving Along the Axial Direction 12539 Guy Bessonnet (France): Generating Optimal Motions of Constrained Multibody Systems 12569 Sine Leergaard Pedersen (Denmark): Contact Problems in Roller Chain Drive Systems. 12582 Katsuhisa Fujita (Japan): Motion and Vibration Control of the Lift Mechanism of a Ladder Truck

lxiv

ICTAM04

SM18 – Plasticity and viscoplasticity (E. Van der Giessen, Netherlands & P. Perzyna, Poland) 10064 Ryszard B. P¸echerski (Poland): Metal Forming Processes Conditioned by Cyclic Loading. A New Challenge for the Theory of Plasticity 10383 Milan V. Micunovic (Serbia): Quasi Rate-Independent Viscoplastic FCC-Polycrystals 10393 George Z. Voyiadjis (USA): Physically Based Thermomechanical Modeling of Metals over a Wide Range of Strain Rates and Temperatures 10652 Norimasa Chiba (Japan): Plastic Properties Identiﬁcation With Plural Sharp Indenters 10734 Alexis Rusinek (France): Advanced Thermo-Visco-Plastic Constitutive Relations for Direct Applications in Numerical Analyses 10797 Giulio Maier (Italy): Calibration of Anisotropic Elastic-Plastic Models for Thin Layers and Foils in Microtechnologies: Two Novel Techniques 10955 John D. Clayton (USA): Simulation of Dynamic Polycrystalline Thermoelastoviscoplasticity and Fracture 11005 Ricardo A. Lebensohn (USA): Assessing Diﬀerent Self-Consistent Approximations by Comparison with Full-Field Simulations in Viscoplastic Polycrystals 11110 Andr´ ´ e Dragon (France): Three-Dimensional Modelling of Thermo-Elasto / Viscoplastic Solids Containing Adiabatic Shear Bands 11185 Wiera Oliferuk (Poland): Energy Storage Rate in Non-Homogeneous Deformation 11295 Kazuwo Imai (Japan): Dynamic Behavior of Many-Dislocation Systems in Silicon 11309 Alain L. Molinari (France): The Eshelby Problem for Elastic-Viscoplastic Materials 11344 J.A.W. van Dommelen (Netherlands): Multiscale Modeling of the Structure-Property Relationship for Semicrystalline Materials 11350 Mamoru Mizuno (Japan): Modeling of Viscoplastic Constitutive Equation for Polymers by Taking into Account Strain Recovery 11410 Peter Gudmundson (Sweden): Thickness Dependent Yield Strength of Thin Films 11498 Stephane Andr´ ´e Berbenni (USA): Yield Surfaces Using an Extension of the Regularized Schmid Law to Polycrystalline Materials 11790 Christian F. Niordson (Denmark): Size-Eﬀects in Void Growth 11820 Amit Acharya (USA): On the Accounting of Dislocation Internal Stress in Continuum Plasticity 11930 Nicolaie Dan Cristescu (USA): Steady-Flow of a Non-Homogeneous Bingham Material Over a Wedge 11953 Anguel I. Baltov (Bulgaria): Modelling of Elastic-Plastic or Viscoplastic Materials Sensitive to the Type of Processes – Diﬀerent Approaches 11963 Marc G. Geers (Netherlands): Strain Gradient Crystal plasticity Incorporating Grain Boundary Eﬀects 11975 Wiktor L. Gambin (Poland): Metal Forming and Texture Development Modelling 12058 Dieter Weichert (Germany): Limit and Shakedown Analysis with Decohesive Eﬀects 12339 Fahmi Za¨ri ¨ (France): A Constitutive Law for Glassy Polymers and Blends 12469 Cedric Doudard (France): Development and Identiﬁcation of a Probabilistic TwoScale Model for High Cycle Fatigue Prediction 12549 Thomas B¨ ¨ ohlke (Germany): Modeling the Crystallographic Texture Evolution Based on the Maximum Entropy Method 12634 Absamad El Adb (France): An Elastoplastic Model for Prediction of Permanent Deformations of Unbound Granular Pavement Layers 12697 Maxime Sauzay (France): Intragranular Kinematic Hardening Modelling and Validation 13038 Oliver Pierard (Belgium): Mean Field Homogenization of Elasto-(Visco) Plastic Composites: Formulation for Time-Dependent and Independent Behaviors

Scientiﬁc Program

lxv

SM19 – Plates and shells (H. Mang, Austria & E. Ramm, Germany) 10071 Gennadiy Lvov (Ukraine): The Stress Analysis of the Multilayered Plates and Shells with Defects of the Structure 10233 Parthasarathi Mandal (UK): Some New Thoughts on the Buckling of Thin Cylindrical Shells 10287 Wojciech Pietraszkiewicz (Poland): Continuity Conditions in Elastic Shells with Phase Transformation 10770 Kenzo Sato (Japan): Analytical Solution of Bending of a Clamped Elliptical Plate Under Lateral Load and In-Plane Force 11021 Herbert A. Mang (Austria): Sensitivity Analysis Concerning the Initial Postbuckling Behavior of Elastic Structures 11238 Jan Sladek (Slovakia): Meshless LBIE Formulations for Viscoelastic Thin Plates 11815 Maryna A. Varyanychko (Ukraine): Eﬀect of a ’Static’ Resonance in Elastic ThinWalled Cylinders 11890 Izabela Lubowiecka (Poland): Energy-Conserving Scheme for Nonlinear Dynamics of Shells – Numerical Examples 12224 Takeshi Sakiyama (Japan): Mindlin Cylindrical Panels with Twist and Double Curvature 12328 Zdzislaw Nowak (Poland): Application of the Return Mapping Algorithm to Perzyna Viscoplasticity for Plane Stress 12346 Eelco L. Jansen (Netherlands): A Perturbation Method for Nonlinear Vibrations of Structures 12350 Lidija V. Kurpa (Ukraine): Nonlinear Vibrations of Shallow Shells and Thin Plates of Arbitrary Shape

SM20 – Rock mechanics and geomechanics (Z. Mr´ ´ oz, Poland & I. Vardoulakis, Greece) 10231 Zdzislaw A. Wi¸eckowski (Poland): The Material Point Method in Soil Mechanics Problems 10374 Leopold Kruszka (Poland): Dynamic Behaviours of Soils and Rocks in a Wide Pressure Range 10866 Niels P. Kruyt (Netherlands): Micromechanical Study of Macroscopic Friction and Dissipation in Idealised Granular Materials: The Eﬀect of Interparticle Friction 11584 Marcin Ma´ ´zdziarz (Poland): Inﬂuence of Contact Phenomena on Structure-Subsoil Interaction: Finite Elements Method Analysis 11667 Nathalie Boukpeti (Belgium): Modeling of Static Liquefaction and Evolving Failure Modes 11845 Joseph F. Labuz (USA): Development of Shear Banding in Sandstone 12364 Bojan Guzina (USA): On the Use of Second-Order Topological Information for Subsurface Imaging by Elastic Waves 12462 Yuji Kishino (Japan): Incremental Nonlinearity in Constitutive Relation for Granular Media 12779 Martin J. Schmidt (USA): A High-Pressure Hish Strain Rate Elastic-Viscoplastic Model for Cementitious Materials 12848 Jan Maciejewski (Poland): The Inﬂuence of Teeth on the Earth-Working Processes

SM21 – Solid mechanics in manufacturing (B. Heimann, Germany & T. Inoue, Japan) This session has been cancelled

SM22 – Stability of structures (Z. Gaspar, Hungary & S. Kyriakides, USA) 10378 J. Blachut (UK): Shallow Spherical Caps Under External Pressure 10692 Nobutada Ohno (Japan): Elastoplastic Microscopic Bifurcation and Post-Bifurcation Behavior of Periodic Cellular Solids 10915 Zsolt Gaspar (Hungary): Statical Models to Illustrate Special Instabilities

lxvi

ICTAM04

11097 Sergio Pellegrino (UK): Wrinkles in Square Membranes 11172 Ioannis G. Raftoyiannis (Greece): Nonlinear Dynamic Stability of Multi-Suspended Roof Systems 11174 Alain Combescure (France): Coupling of Axisymmetric and 3D Shell Models for Non Linear Elastoplastic Buckling Prediction of Mainly Axisymmetric Shells 11600 Theodoro A. Netto (Brazil): Dynamic Arrest of Propagating Buckles in Oﬀshore Pipelines 11653 Suresh Shrivastava (Canada): Bifurcation Buckling of Sandwich Plates and Shells in Plastic Range 11816 Hans Troger (Austria): On the Stability of the Sky-Hook 11939 Alexander P. Seyranian (Russia): Stability of Parametrically Excited Structures: New Results 11962 Philippe Le Grognec (France): A Uniﬁed Treatment for the Elastoplastic Bifurcation of Structural Elements 12084 Hans Obrecht (Germany): Buckling and Imperfection – Sensitivity of Axially Compressed Cylindrical Shells with Compliant Cores 12297 Esben Byskov (Denmark): Stability of Shear-Flexible Frames 12420 Djenane C. Pamplona (Brazil): Instabilities of Initially Stressed Hyperelastic Cylindrical Membrane and Shell Under Internal Pressure 12558 Dinar R. Camotim (Portugal): Generalised Beam Theory Formulation to Analyse the Post-Buckling Behaviour of Orthotropic Laminated Plate Thin-Walled Members 12669 Simon D. Guest (UK): The Stiﬀness of Prestressed Frameworks: A Unifying Approach 12688 Giles W. Hunt (UK): Twist Buckling and the Foldable Cylinder: An Exercise in Origami 12931 Szymon Imieowski (Poland): On Stability of Systems Subject to Generalized Follower Force 12934 Jaroslaw Weronko (Poland): Dynamic Instability of a High-Speed Flexible Shaft with a Massive Disc and Follower Load. 12967 Ciprian D. Coman (UK): Secondary Bifurcations and Localisation of Buckle Patterns

SM23 – Stochastic micromechanics (Y. Brechet, France & Y. Shibutani, Japan) 10242 Andrejs Krasnikovs (Latvia): Creep Rupture and Fiber Breaks Accumulation in Unidirectional Composite 12380 Jan Zeman (Czech Republic): Homogenization of Plain Weave Composites with Imperfect Microstructure 12571 Yoji Shibutani (Japan): Collective Prismatic Dislocation Loops Mechanism

SM24 – Structural optimization (K.K. Choi, USA & J. Herskowits, Brazil) 10048 Yoon-Young Kim (S. Korea): Shape Sensitivity Analysis for Fixed-Grid Analysis Based on Oblique Boundary Curve Approximation 10052 Tomasz Lewi´ n ´ ski (Poland): Perturbation of the Compliance Functional Due to the Apperance of a Small Cavity in an Elastic Body 10433 Niels Olhoﬀ (Denmark): Topology Optimization of Vibrating Structures with Hydrodynamic Surface Pressure Loading 10594 Jos´ ´e N. Herskovits (Brazil): A Technique for Nonsmooth Optimization Based on the Interior Point Feasible Directions Algorithm 10621 Timo J. Saksala (Finland): Nash Equilibrium in Bicriteria Structural Optimization 10669 Pauli Pedersen (Denmark): On Shape Optimization for Eigenvalue Problems 11015 Valeri Markine (Netherlands): Shape Optimisation of Railway Wheel Proﬁle 11137 Tadeusz S. Burczy´ n ´ ski (Poland): Shape Optimization of Thermomechanical Structures in the Presence of Convection and Radiation Using Parallel Evolutionary Computation 11316 Jeong Sam Han (Germany): Eﬃcient Optimization of Transient Dynamic Problems for a Micro Accelerometer Using Model Order Reduction

Scientiﬁc Program

lxvii

11349 Krzysztof Dems (Poland): Damage Identiﬁcation in Structures by Means of Thermographic Methods ˙ (Poland): Topological Optimization for Contact Problems 11486 Antoni Zochowski 11532 Yuanxian Gu (China): Coupled Sensitivity and Design Optimization for ThermoStructural Systems 11623 George I.N. Rozvany (Hungary): New Classes of Analytically Derived Optimal Topologies and Their Numerical Conﬁrmation 11634 Pawel Smas (Poland): Optimal Structures for Buckling Forces and Buckling Displacements 11744 Søren Halkjaer (Denmark): Optimization of Beam Properties with Respect to Maximum Band-Gap 11760 Slawomir Czarnecki (Poland): Optimal Layout of Two Materials within the Core Layer of a Sandwich Plate. Relaxed Formulation and Its Computiational Algorithm 11826 Su-Young Chang (S. Korea): Material Cloud Method for Topology Optimization 11858 Doo-Ho Lee (S. Korea): Optimal Design of Unconstrained Damping Layer on Beams 11936 Niels L. Pedersen (Denmark): On Separation of Eigenfrequencies in Two-Material Structures 12074 Gregor Kotucha (Germany): Density Gradient Based Regularization of Topology Optimization Problems ˙ 12075 Michal Zyczkowski (Poland): New Results of Structural Optimization for Post-Buckling Behaviour 12079 Thomas Buhl (Denmark): Compliant Mechanism Design for Adaptive Trailing Edge Flaps 12098 Erik Lund (Denmark): Structural Optimization of Composite Shell Structures Using a Discrete Constitutive Parameterization 12185 Gil Ho Yoon (S. Korea): Why Parameterizing Element Connectivity for Topology Optimization? 12200 Sami Holopainen (Finland): Topology Optimization of the Geometrically Nonlinear Structures Made of Rubber-Like Material 12222 Jakob S. Jensen (Denmark): Optimal Design of Lossy Bandgap Structures 12287 Lijuan Li (China): Non-Gap Design Method and Test for Post-Tensioned Prestressed R.C. Structure 12302 Atsushi Kawamoto (Denmark): Design of Articulated Mechanisms with a Degree of Freedom Constraint Using Global Optimization 12413 Michal Nowak (Poland): Simulation of Trabecular Bone Adaptation – Creating the Optimal Structure 12573 Sandor Kaliszky (Hungary): Optimal Design of Elasto-Plastic Structures Subjected to Normal Loads and Earthquake 12628 Andrzej Garstecki (Poland): Optimal Force Action and Reaction in Structural Design and Identiﬁcation 12679 Piotr Kowalczyk (Poland): DSA for Elastic-Plastic Finite Rotation Shells under Dynamic Loads 12789 Florin Bobaru (USA): Optimization of Functionally Graded Materials with Temperature Dependent Properties. A Meshfree Solution 12870 Vassili Toropov (UK): Optimum Blank Design for Deep Drawing Using Interaction of High and Low Fidelity Simulation 12916 Henrik T. Møller (Denmark): Computational Tricks for Eﬃcient Design Sensitivity Analysis 13013 Gengdong D. Cheng (China): The Concurrent Design of Materials and Structures for Cellular Materials on Eﬃciency of Heat Dissipation

SM25 – Structural vibrations (I. Blekhman, Russia & K. Popp, Germany) 10166 Jianbing Chen (China): Extreme Value Distribution and Dynamic Reliability of Stochastic Structures 10408 Alexander Vakakis (Greece): Experimental Study of Nonlinear Energy Pumping

lxviii

ICTAM04

10470 Marian Wiercigroch (UK): Nonlinear Vibrations of Jeﬀcott Rotor with Preloaded Snubber Ring 10656 Victor Z. Gristchak (Ukraine): Non-Linear Stochastic Vibration Problems for the Plates with Time-Dependent Parameters 10750 Debasish Roy (India): A Multi-Step Transversal Linearization Method in Nonlinear Dynamics 10863 Stephen H. Crandall (USA): Equivalent Stochastic Linearization as an Alternative to Solving the Fokker-Planck Equation 11055 Iliya I. Blekhman (Russia): Nonlinear Eﬀects, Observed in the Process of the Liquid Flowing Out of the Vibrating Vessels: Theory, Experiment and Applications 11066 Akihiko Higashi (Japan): Propagation Analysis of Flexural Waves by Wavelet Transform 11079 Oleg Gendelman (Israel): Bifurcations of Damped Nonlinear Normal Modes: Linear Oscillator with Strongly Nonlinead Attachment 11082 Gamal Mohamed Ashawesh (Libya): Eﬀect of Root Flexibility on the Aeroelastic Analysis of a Composite Wing Box 11246 Herve Riou (France): Reanalysis of an SEA High – Frequency Vibration Calculation Based on the VTCR 11261 Utz von Wagner (Germany): Active Control of Disk Brake Sqeal 11273 Marcin Luczak (Poland): Experimental and Theoretical Modal Analysis of Three Support Rotor Test Rig Using LMS CADA-X and ABAQUS 11338 Jacob P. Meijaard (UK): Stability of a Rotor with Periodically Varying Angular Velocity 11384 Tibor Tarnai (Hungary): Paradoxical Behaviour of Vibrating Systems Challenging Rayleigh’s Theorem 11494 Carlos E.N. Mazzilli (Brazil): Imperfection Sensitivity of Circular Arch’s Non-Linear Modes 11536 Alexei A. Mailybaev (Russia): Optimal Shapes of Parametrically Excited Beams 11679 Alexander J. Fidlin (Germany): Non Trivial Eﬀect of Strong High-Frequency Excitation on a Nonlinear Controlled System 11699 Igor Zeidis (Germany): An Aproach to Worm-Like Motion 11712 Alla D. Firsova (Russia): Dynamics of a Rotor Rolling Along a Circular Surface 11716 Yuri Leonidovich Menshikov (Ukraine): The New Statement of Problem of Unbalance Identiﬁcation 11719 Ekaterina V. Shishkina (Russia): Vibrorheology: Main Results, New Problems 11759 Vladimir Zeman (Czech Republic): Nonlinear Vibrations of Gear Drives 11783 Rob H.B. Fey (Netherlands): Passive Vibration Control of a Piecewise Linear Beam System 11834 Ye Ping Xiong (UK): A power Flow Mode Theory Based on Inherent Characteristics of Damping Distributions in Systems and Its Applications 11867 T.H. Young (Taiwan): Stability of a Spinning Disk Under a Stationary Oscillating Unit 11904 Seyed Saleh Hosseini Yazdi (Iran): High Revolving Speed Spindles Deﬁnition Due to Transient Vibration Conditions 12026 Fadi Dohnal (Austria): Suppressing Self-Excited Vibrations in a Coupled Pendulum System 12322 Ingo Kaiser (Germany): The Running Behaviour of an Elastic Wheelset 12404 Andrew N Norris (USA): Thermoelastic Relaxation in Thin Plates with Applications to MEMS and NEMS Oscillators 12498 Jacek Cie´ ´slik (Poland): Estimation of the Vibration Energy Characteristics for Joints of Constructional Elements 12511 Marek S. Kozien (Poland): Sound Radiation by the White Noise Excited Viscoelastic Shallow Shells 12535 Thomas Marc Richard (Belgium): Self-Excited Stick-Slip Oscillations of Drag Bits

Scientiﬁc Program

lxix

12593 Yoshikazu Sugiura (Japan): Vibration Characteristics of the Main Tower, the Byaon Temple 12632 Viktorija E. Volkova (Ukraine): Application of Extended Phase Space to Investigation of Forced Biharmonic Oscillations 12659 Robert Jankowski (Poland): Non-Linear Modelling of Earthquake Induced Pounding of Buildings 12694 Jon J. Thomsen (Denmark): Discontinuous Transformations and Averaging for VibroImpact Analysis 12759 Pedro M. Ribeiro (Portugal): Experimental Analysis of Modal Interactions in the Non-Linear Vibrations of a Plate 12770 Arkadiusz M¸ezyk ˙ (Poland): Optimum Selection of Design Features of Electromechanical Drive Systems Incorporating a Control Unit 12785 Christian Seidel (Germany): Mode Switching of Rain-Wind Induced Vibrations 12814 Pankaj Wahi (India): Regenerative Tool Chatter Near a Codimension-2 Hopf Point Radkowski (Poland): Characteristics of Vibroacoustic Signals in Diagnosing 12977 Stanisaw Early Stages of Defects 12984 Tadeusz Majewski (Poland): Entering the Excitation into a Mechanical System with Dynamic Eliminators of Vibration 12989 Gayane Manucharyan (Ukraine): Frictional Auto-Oscillations under the Action of Almost Periodic and Periodic Excitations

SM26 – Vehicle dynamics (S. Iwnicki, UK & R. Sharp, UK) 10701 Robin S. Sharp (UK): Optimal Path Following Road Vehicle Steering Control 10909 Simon Iwnicki (UK): Simulation and Testing of a Wheelset with Induction Motor Driven Independent Wheels 11589 Vladislav Yazykov (Russia): Railway Vehicle Simulation Using Non-Elliptical WheelRail Contact Model 11672 Georg Rill (Germany): A Modelling Technique for Fast Computer Simulations of Conﬁgurable Vehicle Systems 11795 Per-Anders J¨ onsson (Sweden): Experimental and Theoretical Aanalysis of Freight Wagon Link Suspension 11832 Zbigniew Lozia (Poland): Mathematical Models and Simulation of Stick-Slip Processes in a Car Steering System

SM27 – Viscoelasticity and creep (F. Cocks, UK & N. Ohno, Japan) 10018 Boris P. Maslov (Ukraine): Nonlinear Overall Viscoelastic Properties of the Random Multicomponent Media 10906 Renata S. Engel (USA): Sintering Simulation of Stainless Steel Powder Compacts 11201 Alan R.S. Ponter (UK): Characterisation of the Cyclic Behaviour of Elastic-PlasticCreeping Bodies 11368 Holm Altenbach (Germany): A Creep Continuum Damage Theory for Beams, Plates and Shells 11506 Jinghong Fan (USA): Multiscale Modeling Schemes Spanning a Large Range of Scales 11827 Ji-Hyun Cho (S. Korea): Constitutive Modeling of Rubber Components Under Small Vibration Superimposed on Large Static Deformation Considering Strain-Dependent Properties 11957 Zbigniew L. Kowalewski (Poland): An Inﬂuence of Cold Work on Creep of Engineering Materials 11981 Roman Lackner (Austria): Multi-Scale Model for Low-Temperature Creep of Asphalt 12450 Fusahito Yoshida (Japan): A Model of Cyclic Viscoplasticity with Special Reference to Yield-Point Phenomena

lxx

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FSM1 – Acoustics (T. Geers, USA & N. Peake, UK) 10294 Nobumasa Sugimoto (Japan): Dissipative Eﬀects on Propagation of the Acoustic Solitary Waves 10907 Jonathan B. Freund (USA): An Empirical ’lower bound’ on Free-Shear-Flow Noise 11249 Jeremy Astley (UK): Special Short Wave Finite Elements for Flow Acoustics 11337 Sergey V. Sorokin (Denmark): Wave Propagation in and Sound Emission from a Sandwich Plate Under Heavy Fluid Loading 11373 David S. Burnett (Italy): 3-D Structural Acoustics Modeling with HP-Adaptive Finite Elements 11508 Dmitry V. Churochkin (Russia): The Low-Temperature Acoustical and Thermal Properties of Materials Due to the Dynamics of Linear Topological Defects 11561 Takao Suzuki (USA): Shock Leakage Through a Vortex-Laden Mixing Layer Causing Jet Screech 11896 Natasha V. Movchan (UK): Transmission of Elastic Waves and Localised Modes in Composite Structures 11973 Murthy N. Guddati (USA): Arbitrarily Wide-Angle Wave Equations and their Applications to Unbounded Domain Modeling and Subsurface Imaging 12149 Tatiana Andreeva (USA): Ultrasonic Travel-Time Technique for Diagnostic of GridGenerated Turbulence 12180 Jeﬀ D. Eldredge (USA): The Acoustics of Two-Dimensional Leapfrogging Vortex Interactions 12477 Edward J. Kerschen (USA): A Theoretical Model for Resonances in Flow Past a Cavity 12497 Paul W. Hammerton (UK): Structure of Sonic Booms in a Medium with Multiple Relaxation Modes 12516 Alexander Alexeev (Germany): Gas Oscillations in a Closed Tube at Resonance 12527 Iain D.J. Dupere (UK): The Eﬀect of Viscosity on the Propagation of Acoustic Waves Through Fine Cylindrical Meshes 12533 Lars V. Hansen (Denmark): Modelling of Hydrophone Based on a DFB Fiber Laser 12555 J´ ´ ozef Lewandowski (Poland): Numerical Analysis of the Texture and Acousto-Elastic Properties of Prestressed Polycrytalline Aggregate 12586 Rossano Stefanelli (Switzerland): Measurements and Calculations Related to Curve Squealing in the Railway System 12643 Fernando Lund (Chile): Acoustic Wave Propagation Through a Random Array of Dislocations 12691 George Biros (USA): Distributed Parameter Control of a 2D Acoustic Helmholtz Problem on a Halfspace 12717 Loukas F. Kallivokas (USA): Frequency- and Directionality- Continuation Schemes for Scatterer Shape Detection in Acoustics

FSM2 – Chaos in ﬂuid and solid mechanics (I. Mezic, USA & G. Rega, Italy) 10252 Gabor Stepan (Hungary): Nonlinear Dynamics of High-Speed Milling 10499 John S. Hogan (UK): The Eﬀect of Smoothing on Bifurcation and Chaos Computations in Non-Smooth Mechanics 11183 Remco I. Leine (Switzerland): A Set-Valued Force Law For Spatial Coulomb-Contensou Friction 11241 Ugo Galvanetto (UK): Chaotic Attractors with Long Regular Sequences 11302 Ekaterina E. Pavlovskaia (UK): Reduction of Multidimensional Flow to Low Dimensional Map for Piecewise Smooth System Experiencing Chaos 11363 P. Piiroinen (UK): Numerical Detection and Continuation of Sliding Bifurcations in a Dry-Friction Oscillator 11416 Elzbieta ˙ Tyrkiel (Poland): On Generating Chaotic Dynamics in Nonlinear Vibrating Systems 11535 Ken Kiyono (Japan): Low-Dimensional Chaotic Dynamics in Dripping Faucets

Scientiﬁc Program

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12014 Kohei Yamasue (Japan): Inﬂuence of Remaining Chaos on Convergence of Solutions in Time Delayed Feedback Controlled Duﬃng System 12129 Giuseppe Rega (Italy): A Dynamical Systems Analysis of the Overturning of Rigid Blocks 12468 Igor Mezic (USA): On the Nonlinear Dynamics of Multicomponent Dynamical Systems 12482 Theodoros Karakasidis (Greece): Short-Time Dynamical Behavior of Fluids at the Atomic Scale 12674 Li-Qun Chen (China): Nonlinear Dynamics of Axially Moving Viscoelastic Strings Based on Translating Eigenfunctions n ´ ski (Poland): Nonlinear Oscillators with Time Delays 12927 Zbigniew Peradzy´ 12972 Radoslaw Iwankiewicz (South Africa): Non-Linear Oscillator Under Random RenewalDriven Trains of Impulses

FSM3 – Continuum mechanics (K.R. Rajagopal, USA & G. Saccomandi, Italy) 11096 11140 11227 11247 11347 11638 11642 11704 11818 11840 11950 12392 12406

James Casey (USA): Pseudo-Rigid Bodies Viewed as Globally Constrained Continua Lev Steinberg (USA): Constitutive Equations of Mesoelastic Deformation Luis A. Dorfmann (Austria): Nonlinear Response of Magnetoelastic Solids Bohuslav Striz (Czech Republic): Application of Continuum Mechanics in the Textile Fabrics Gerard A. Maugin (France): Generalized Continuum Mechanics: Three Paths Victor M. Tigoiu (Romania): Viscoelastic Fluid Flows in a Falling Cylinder Viscometer and the Evaluation of Shear Viscosity Stanisaw Tokarzewski (Poland): Fundamental Inequalities for the Bounds on the Eﬀective Transport Coeﬃcients of Two–Phase Media Glenn B. Sinclair (USA): On the Source of Singularities in Mechanics Michael A. Hayes (Ireland): Extended Polar Decompositions for Finite Plane Strain Wlodzimierz Doma´ nski ´ (Poland): Nonlinear Waves in Elastic Solids Michel Destrade (France): Explicit Secular Equations for Surface and Interface Waves in Anisotropic Solids Andreas Menzel (Germany): Views on Material Forces in Multiplicative Elastoplasticity Jan J. Sawianowski (Poland): Aﬃne Symmetry in Mechanics of Discrete and Continuous Systems

FSM4 – Fluid-structure interaction (J. Grue, Norway & M.P. Paidoussis, Canada) 10520 Elena G. Gavrilova (Bulgaria): Coupled Frequancies of a Fluid-Structure Interaction Cylindrical System 10826 Matej Vesenjak (Slovenia): Fluid Structure Interaction in Multiphase Mixing Vessel 10911 Tatiana Khabakhpasheva (Russia): Piston Impact Onto the Boundary of Two-Layer Fluid 11078 Michael P. Pa¨ ¨ıdoussis (Canada): Nonlinear Dynamics of Pinned-Pinned Cylinders in Axial Flow 11668 Ming-Jyh Chern (Taiwan): Interaction of Oscillating Flow with a Pair of Side-By-Side Square Cylinders 11736 Anthony D. Lucey (Australia): The Hydroelastic Destabilisation of Finite Compliant Panels 11758 Hiroshi Kagemoto (Japan): Water-Surface Dynamics Among a Periodic Array of Floating Bodies Subject to Regular Incident Waves 11801 Charlotte Py (France): The Mixing Layer Instability of Wind Over a Flexible Crop Canopy 11833 Jing Tang Xing (UK): An Updated Arbitrary-Lagrangian-Eulerian Description in Continuum Mechanics and Its Application to Nonlinear Fluid-Structure Interaction Dynamics

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11912 Wolfgang M. Sichermann (Germany): Slender Body Theory Approach to Nonlinear Ship Motions 11986 Andrzej Herczy´ n ´ ski (USA): Inverse Magnus Force in Free Molecular Flow 12521 Yu Zhou (China): Eﬀect of an Oscillating Cylinder on a Neighbouring Cylinder Wake 12557 John R. Chaplin (UK): Disturbed-Laminar Flow Over an Oscillating Cylinder 12956 John Grue (Norway): PIV Experiments on Vortex Induced Vibrating Cylinders at High Reynolds Numbers 13050 Ahcene Bouabdallah (Algeria): Inﬂuence of the Circular Cylinder Cross-Section Variation on the Near Wake Behaviour

FSM5 – Mechanics of foams and cellular materials (S. Hilgenfeldt, Netherlands & D.L. Weaire, Ireland) 10597 Sascha Hilgenfeldt (Netherlands): Bubble Shapes in Foams: The Importance of Being Isotropic 11095 Benjamin Dollet (France): Mechanics of Bidimensional Liquid Foams 11240 Martin E. Glicksman (USA): Average N-hedra as Descriptors of 3D Network Cells 11292 H.X. Zhu (UK): Modelling the Round-oﬀ and the Tensile/Compressive Failure Behaviour of Plant and Vegetable Tissues 11605 Stelios Kyriakides (USA): On the Crushing Response of Open Cell Foams 11663 Alfonso H.W. Ngan (China): A Statistical Mechanics Theory of Random Honeycomb and Open-Cell Foam Structures 12052 Isabelle Cantat (France): Dissipation in 2D Foam Flow 12212 Arnaud Saint-Jalmes (France): Surfactant and Protein Foams: Diﬀerences in Drainage and Rheology 12373 Adrian D. Staicu (Netherlands): Determining Stress During Finger Propagation in 2D Foams 12395 Jing Tian (UK): Thermal Flow through Brazed Woven Screens 12435 James E. Coons (USA): Drainage of Emulsion and Foam Films in Scheludko-Exerowa Cells 12578 Vincent Labiausse (France): Shear-Induced Normal Stress Diﬀerences in Aqueous Foams 12851 Stephen J. Neethling (UK): The Dispersion of Particles within Foams 12986 Pacelli L. Zitha (Netherlands): Iinvestigation of Foam Development in Porous Media Using X-Ray Computed Tomography 13033 Fabian Lipperman (Israel): Nucleation of Cracks in Two-Dimensional Periodic Cellular Material

FSM6 – Multiscale phenomena in mechanics (A. Carpinteri, Italy & C. Miehe, Germany) 10399 Justyna Czerwi´ n ´ ska (Germany): Simulations of Micro- and Nano- Channel Flows by a Dissipative Particle Dynamics Method 11224 Alberto Carpinteri (Italy): Multi-Scaling Approach in the Mechanics of Disordered Materials 11237 Wieslaw Larecki (Poland): Grad-Type Expansion About Nonequilibrium States for the Relaxion-Time Approximation of the Boltzmann-Peierls Equation 11340 Heike Emmerich (Germany): Two-Scale Simulations of Epitaxial Surfaces 11471 Pilar Ariza (Spain): A Geometrical Theory of Discrete Dislocations in Lattices, with Applications to Dislocation Dynamics and Crystal Plasticity 11476 Krishna Garikipati (USA): Stress-Defect Interactions at Molecular / Continuum Scales 11806 Bhushan L. Karihaloo (UK): Deterministic Size Eﬀect in the Strength of Cracked Quasi-Brittle Structures 11830 Varvara Kouznetsova (Netherlands): Multi-Scale Second-Order Computational Homogenization of Heterogeneous Materials 11862 Yutaka Shimomura (Japan): Jumping of a Spinning Spheroid 12040 Luca Placidi (Germany): Characteristics of Orientation and Grain-Size Distributions

Scientiﬁc Program

lxxiii

12078 Antoine Gloria (France): Numerical Homogenization of a Locally Hyperelastic Constitutive Law 12112 Joachim Dettmar (Germany): Multiscale Analyses of Granular Media at Finite Strains Based on Micro-Macro Transitions with Diﬀerent Boundary Constraints 12113 Christian Miehe (Germany): Exploitation of Incremental Energy Minimization Principles in Computational Multiscale Analyses of Inelastic Solids 12118 Martin Becker (Germany): Non-Convex Homogenization of Inelastic Composites with Interaction of Material and Structural Instabilities on Diﬀerent Scales 12334 Luciano Colombo (Italy): Physical Modeling of Fracture Mechanics in Complex Materials 12336 Frederic Legoll (France): Analysis of a Variational Method Coupling Discrete and Continuum Mechanics

FSM7 – Education in mechanics (R. Engel, USA & B. Karihaloo, UK) 10003 Hassan Aref (USA): Toys and Games in Mechanics Education 10016 Keith Moﬀatt (UK): African Institute for Mathematical Sciences: a Capacity Building Initiative in which IUTAM Has an Active Involvement 10226 Carl T. Herakovich (USA): On Mechanics/Engineering Science Education 11724 Aleksandr Kositsyn (Ukraine): Mechanics – a New Internet Tutor 11765 Yilong Bai (China): Teaching Mechanics as an Engineering Science in China 11869 Vasily Yaremchuk (Russia): Education and Tutorial on Fluid Mechanics on the Basis of Computer Laboratory 12259 Vitauts Tamuzs (Latvia): Education in Mechanics in Latvia Higher Schools 12488 Yasuaki Nohguchi (Japan): Simulator, Nohguchi Bottle, of Soil Liquifaction for Education 12603 Anders Bostr¨ om (Sweden): Mechanics Education in Sweden 12642 Donovan L. Evans (USA): Rigid Body Dynamics: Student Misconceptions and Their Diagnosis 12752 Kamal B. Rojiani (USA): Web-Based Instructional Units for Teaching Mechanics

Professor Leen van Wijngaarden delivers the ICTAM04 Opening Lecture

INTERPLAY BETWEEN AIR AND WATER Leen van Wijngaarden J.M.Burgers Centre for Fluid Dynamics University of Twente, Enschede, The Netherlands. [email protected]

Abstract

In the Prologue I recall, among others, the period of the Cold War in which, thanks to Polish colleagues, scientiﬁc contacts between East and West were maintained . After that, several aspects of the ﬂow of mixtures of air and water will be discussed and illustrated by examples. Finally I will give some comments on the diﬀerences and similarities between fundamental and applied science and scientists.

Keywords: Multiphase ﬂow, bubbly ﬂows

1.

Prologue

It is a great honour to be invited to deliver the Opening Lecture at ICTAM 2004, especially now that it is here in Warsaw, a city of great signiﬁcance for Mechanics. It reminds me of the Cold War when East was East and West was West. They could nevertheless meet here in Poland, where Wladek Fiszdon organized once in two years a “Symposium on Advanced Problems and Methods in Fluid Mechanics”. Participation was on invitation and those invited travelled to Warsaw and stayed there one night. The next day they were transported by bus to some place found by magician Wladek where there was food and accommodation, modest but suﬃcient. One could meet in this way with famous Russian scientists as Barenblatt, Zel‘dovich, Ladyshenskaya and others. The ﬂuid dynamics community is greatly indebted to Wladek Fiszdon for organizing these Symposia. Unfortunately, his health condition does not allow him to be here today. From this place I would like to thank him for all he did for Fluid Mechanics in this way. The ﬁrst time that I was invited to participate in such an event was in 1969 in Kazimierz (not named after my friend and colleague Kazimierz Sobczyk who will present the Closing Lecture next Friday). George Bat1 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 1–16. © 2005 Springer. Printed in the Netherlands.

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chelor was a key ﬁgure in these Symposia. He had great authority (he was a Foreign Member of the Polish Academy of Sciences), Wadek Fiszdon asked his advice whom to invite and he was always very relaxed and willing to lecture on everything that he was working on. I remember very well that he gave a lecture on the sedimentation problem on which he was working at the time and what was to become the subject of his celebrated paper “Sedimentation in a dilute dispersion of spheres” [1]. This concerns the velocity with which a cloud of heavy particles sedimentates in a ﬂuid. The, until that time unsolved, diﬃculty in this and similar problems is that the velocity which a small particle induces in its vicinity falls oﬀ very slowly, as the reciprocal distance from its center. The calculation of the average sedimentation speed results, because of this in not uniformly convergent integrals, with which J.M. Burgers struggled already in the 1930‘s. George found a way, a renormalization, to overcome this diﬃculty. His renormalization technique has found wide application in other areas. His presentation in Kazimierz induced me to think about the analogous problem where a cloud of bubbles rises under buoyancy.

2.

Air and Water

The ﬂow around a bubble is, to a good approximation, a potential ﬂow. The velocity which one bubble induces in another falls oﬀ as (distance from centre to centre) −3 . In contrast with the falling particle inertia eﬀects are here dominant, the Reynolds number is large. This (distance)−3 behaviour is faster than that with the falling particle but not fast enough to overcome diﬃculties with not uniformly convergent integrals. When a bubble is accelerated, the surrounding liquid exerts a reaction force on the bubble, which is proportional to the acceleration. The multiplying factor has dimension of mass and is called virtual or added mass, because in calculations it can be treated as a virtual mass of the bubble which is itself of course almost massless. It appears that this mass depends on the presence of nearby bubbles in a manner which gives rise to convergence problems. Consider N bubbles in a conﬁguCN ). When there is always one ration CN with probability density P (C bubble in the point r 0 , such a conﬁguration is indicated with CN −1 | r 0 and the corresponding probability density with P (C CN |r0 ). In the course of the calculation one needs to know the average velocity in the centre of a bubble in the presence of all the others, and with respect to the volume velocity U 0 of the suspension, u − U0 = 1/N !

{u(r0 , CN ) − U0 }P (C CN |r0 )dC CN .

(1)

Interplay between Air and Water

3

For low concentration by volume one considers, just as in the case of dilute gases, the interaction between two particles only or, in this case, bubbles. Then Eq. (1) becomes (2) u − U0 = {u(r0 + r, r0 ) − U0 }P (r0 + r|r0 )d3 r. The quantity in curly brackets in Eq. (2) behaves at large distance r as r−3 and therefore the integral does not converge. The essence of Batchelor‘s renormalisation technique starts in this case with noting that if in Eq. (1) we take just P (C CN ), that means when we consider the average velocity in a point whether in ﬂuid or in gas, then the result is zero, CN )dC CN . (3) 0 = 1/N ! {u(r0 , CN ) − U0 }P (C When reduced to the interaction of two bubbles also this integral does not converge. The only diﬀerence with the right-hand side of Eq. (1) is that there is in the latter always a bubble in r 0 and in Eq. (3) – sometimes. However we know the exact result Eq. (3). Now we subtract Eq. (3) from Eq. (1). Since in the absence of long-range order in the suspension we have at a large distance from r 0 P (C CN |r0 ) = P (C CN ), the diﬀerence of the two integrals converges when the conﬁguration is reduced to two bubbles and this overcomes the problem because we are left with the calculation of the integral {u(r0 , r0 + r) − U0 }{P (r0 + r|r0 ) − P (r0 + r)}d3 r, which is now convergent. Although this problem could be solved, [2], the general problem to understand the dynamics of air-water mixtures is today far from being solved. Particle – liquid ﬂow can either show random conﬁgurations or ﬁxed conﬁgurations. With air and water many more topologies are possible. I mentioned as a ﬁrst example the bubbly suspension, a common device in the chemical industry where it serves as a reactor column. Some more examples are: Niagara Falls (American and Canadian). Air is ﬁrst entrained by river water falling down and mixes with this during its fall. The air leaves eventually together with only a little bit of water, forming with it a spray, or mistﬂow. There are in fact

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Figure 1. The Niagara Falls, in the foreground the horseshoe-shaped Fall on the Canadian side, and the American Fall in the background.

Interplay between Air and Water

5

two falls, one on the Canadian side, with the shape of a horseshoe, and one on the American side. Although the picture is not very good, you can clearly see that with the Canadian fall water droplets are in the upward ﬂow entrained with the air above the original level from where they originated. They derive some energy from the air. This does not happen at the American fall. The reason seems to me that there is, in the restricted space available for the downﬂow of the Canadian one, a pressure built up pushing a strong upward air ﬂow entraining droplets. Breaking wave with trapped air. In the case of a breaking wave air is trapped in the overturning wave. This entrained air plays an important role in the dynamics of the wave, and of its impact on walls. A striking eﬀect of the trapped air is a tremendous change in the compressibility of the mixture. Even an air concentration of a few percent dramatically alters the sound velocity which is directly related to the compressibility. This can be made clear as follows. Denoting the velocity of sound of the mixture with cm , we have from thermodynamics (cm )2 = (dp/dρ)s ,

(4)

Figure 2. A breaking wave at Coogee Beach, Sydney, Australia. Photograph taken by D.H. Peregrine, University of Bristol, and reproduced here with his permission.

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where p, ρ and s denote pressure, density and entropy respectively. The density is made up from that of air, with volume fraction α, and that of water, with volume fraction 1 − α. Referring to air and water with subscripts a and w respectively, we have therefore ρ = α ρa + (1 − α) ρw .

(5)

We neglect a possible velocity diﬀerence between air and water. Then the mass of air in a unit mass of the mixture is constant, whence (6) ρa α/{ρa α + ρw (1 − α)} = constant. We assume further that the pressure in water and air is the same (later we shall see when this ceases to be realistic). Then it follows from Eq. (4)–Eq. (6) that for α not too close to zero or to unity (cm )2 = γp/ρw (1 − α)α,

(7)

γ being the ratio of speciﬁc heats of the air. In Fig. 3 graphs of the complete expression are shown for various ambient pressures. Even with a volume concentration α of air of one percent, the velocity of sound is in an air-water mixture only 100 m/s, far below the sound speed in either air or water. (Air was, of course, also involved in the generation of the wave. That is an old problem in hydrodynamics.

Figure 3. The sound velocity, c, in a mixture of air and water. The air concentration by volume α is indicated along the horizontal axis, the sound velocity along the vertical axis.

Interplay between Air and Water

7

In the last 50, or so, years much has been clariﬁed but it is not completely solved.) The air trapped in water has also a profound eﬀect on the radiation of sound when the ﬂow is turbulent. The late Sir James Lighthill has shown in one of his ﬁnest contributions to ﬂuid mechanics, [3], that turbulence produces, ineﬃciently, quadrupole sound. The presence of air gives a new, by far dominating, monopole contribution, which leads as shown by Crighton and Ffowcs Williams [4], for not too low void fraction, to a sound emission larger by a factor (cw /cm )4 which can be for air and water of the order 106 or an intensity increase of 60 dB. Think about this when you hear these waves speak! Cavitation. Another two-phase situation is encountered in cavitation, for example at a hydrofoil, see Fig. 4. Due to the low pressure in the ﬂow along the hydrofoil, a propeller blade, microscopic nuclei become unstable and grow to macroscopic size. In a region of higher pressure these bubbles, ﬁlled with air and vapour, collapse again and may cause at the ﬁnal stage of the collapse considerable damage to the blade.

Figure 4. Cavitation on a ship‘s model propeller turning in a water tunnel. There is cavitation on the blade but also in the tip vortex. Courtesy of the Maritime Research Institute of the Netherlands (MARIN) at Wageningen.

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In all these cases there is interaction between the gaseous phase and the liquid phase and the title of my lecture refers to this. In a further part of the lecture I shall in a symbolic way imagine the air which is above as the domain of fundamental research and the water, down below,as applied science, which will give opportunity to share some thoughts about science and engineering with you. In the spray above the Niagara waterfall we have a lot of air and some water, in the case of a bubbly suspension we have a lot of water and some air. At both ends there are unsolved problems forming a challenge for ﬂuid mechanics.

3.

Bubbly Flow

Compared with the Niagara fall, the rising suspension looks quiet and peaceful. But lo and behold what happens when we increase, with bubbles of about 1mm radius, the void fraction to about 25 %. A violent transition to slug ﬂow occurs. (During the presentation a video of the transition to slug ﬂow was shown). Both phenomena, the homogeneous rise at low concentration and the transition to slug ﬂow are ill understood. Let us start with the former. The interest in bubbles has always been great. Bubbles smaller than about 0.8 mm radius rise in a straight line. For example bubbles rising in champagne or beer. The application in champagne is due to Dom P´ ´erignon who was a contemporary of Sir Isaac Newton. Their simultaneous occupations are described by Michel Onfray [5] as “while the ﬁrst (P´ ´erignon) prepares beverages with rising bubbles, the second (Newton) derives formulas based on falling fruits“ (my translation from the French text). This quotation from the chapter ´ des bulles” of Onfray‘s book illustrates on a day like “Une petite th´orie this very appropriately the unity of mechanics. Bubbles with a radius above 0.8 mm do not rise linearly in water but perform spiralling or zigzag motion, in contrast to falling particles which fall in a straight line. This was already known to Leonardo da Vinci, who made a sketch of what he saw, Fig. 5, and is therefore called nowadays Leonardo‘s Paradox, see e.g. Ohl, Tijink and Prosperetti [6]. Recently, see e.g. de Vries et al. [7], it has been observed that these spiralling bubbles have a wake trailing behind them consisting of two vorticity bearing threads, see Fig. 6. The relevant bifurcation has been also described numerically, Maugin & Magnaudet [8], but the underlying physical mechanism is not yet understood. The ﬂow around a bubble rising in clean water is well described by potential ﬂow supplemented with thin boundary layers. These, of the

Interplay between Air and Water

9

Figure 5. Sketch by Leonardo da Vinci of a spiralling bubble (Courtesy of Mus´ ´ee du Louvre, Paris). The “ Paradox”, for further details see [6], is in the fact that a falling particle has a straight trajectory but a rising bubble – a spiralling path.

Figure 6. Two mutually perpendicular projections of a bubble spiralling in hyperclean water. The eﬀective bubble radius is about 1 mm. Clearly seen is the double –threaded wake behind the rising bubble. Courtesy of Christian Veldhuis.

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same thickness as with rigid bodies, have now only to overcome the diﬀerence in shear stress between the outer ﬂow and the zero stress at the interface. Just as in the study of dilute gases, it is natural to start with looking at binary interactions. Numerical work and analytical work with this model unfortunately predict clustering of bubbles and not the homogeneously rising bubbles as observed in experiments. This can be made plausible as follows: Imagine two bubbles with their line of centres making, at time t = 0, an angle of more than 36 degrees with the vertical direction which is parallel to gravity. The hydrodynamic forces are such that after a time of order a/V , where a is the bubble radius and V the terminal velocity, the line of centres is horizontal. This is therefore a statistically highly probable situation. But in this conﬁguration there is only an attractive force between the bubbles. They bounce for a while but even when this is a purely elastic collision, the motion gets exhausted by viscous friction, which explains the clustering in the numerical simulation. How in reality the lack of repulsive forces, the basic reason for the clustering, is overcome is another unsolved problem of two-phase ﬂow. I have made recently [9]) the suggestion that the above mentioned twothreaded wakes could play a role here. To continue with unsolved problems in two-phase ﬂow I recall the transition to slug ﬂow. Also this awaits an explanation. There is an analogeous phenomenon in ﬂuid beds. There the homogeneous bed becomes unstable, big gas bubbles are formed, as a result of the instability of concentration waves. It has been thought for some time that transition to slug ﬂow is similarly due to instability of such waves. Evidence, Lammers & Biesheuvel [10] shows this not to be the case. The instability of concentration waves (or void fraction waves) occurs but before slugs appear, there is an intermediate ﬂow regime characterized by a pointed transverse velocity and concentration distribution. You might have the impression by now that in two-phase ﬂow there are mainly riddles and unsolved problems. This is certainly not the case. I will illustrate this with two examples. The ﬁrst is about expansion waves in a two-phase ﬂow. We have seen that the bubbly suspension has a low velocity of sound. So, we can play at low velocities the whole organ of compressible gases. For example the theory and experiments of waves of ﬁnite amplitude. There is, however, an important diﬀerence. If pressure changes become very rapid, the relation Eq. (7) for the speed of sound is no longer valid. My compatriot Marcel Minnaert measured in the 1930‘s the frequency of volume oscillations of small air bubbles in water in an ingenious way, described in [11]. He determined the frequency of the audible popping sound of the bubbles formed in his

Interplay between Air and Water

11

apparatus by means of tuning forks. He also derived a formula for this frequency, f , say, f = 1/(2πa){3γ(p − pv )/ρl }1/2 .

(8)

This is for a bubble with radius of 1mm about 3 kHz. When now in a bubbly suspension pressure changes are not far from this Minnaert frequency, the bubbles do no longer passively follow the pressure changes but pressure diﬀerences between the two phases develop due to the inertia of the liquid. As a result the medium becomes dispersive which expresses itself in various ways. One of these is that the velocity of propagation of a wave of ﬁnite amplitude depends not only on the amplitude, as is the case for “normal” compressible ﬂuids, but also of the frequency, or wavelength. With weak nonlinearity and weak dispersion the famous Korteweg-de Vries equation is valid for the pressure in the wave. Some time ago we did [12] the following experiment: At the entrance of a semi-inﬁnite bubbly ﬂow, a time-dependent pressure was established in the form of a rectangular triangle, a shock wave followed by a rarefaction. For this special initial proﬁle the KdV equation can be solved exactly with help of the so-called inverse scattering theory. The evolution in the mixture of this initial proﬁle is into a train of solitons according to this theory. The nice thing about this is that the associated mathematical equation, a Schrodinger ¨ equation, has for this particular proﬁle an exact solution in the form of an Airy function and that the number of evolving solitons is equal to the zeros of this Airy function in a certain interval. In the experiment that we did, the shape of the evolving waves was not quite that of solitons (they were still evolving) but the number agreed exactly with the predictions, see Fig. 7. Another example is directly connected to Minnaert‘s early ﬁnding. Much later it was discovered that also the sound of rain on a water surface is due to bubbles but in a special way. The falling drop forms a crater in the water which is ﬁlled with air. As the crater closes again the air escapes but sometimes a small air bubble is trapped. This produces noise while oscillating in its Minnaert frequency. Experiments, see Oguz ˇ & Prosperetti [13], with drops of various speeds and sizes show that a bubble is trapped only in a narrow portion of the speed/size plane (see Fig. 8). In nature the speed of the raindrop depends on its size and hence the intersection of this line with the above mentioned area gives the size of the raindrops producing bubbles and accompanying sound. This explains the rather narrow frequency spectrum of rain noise.

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Figure 7. From an initial pressure proﬁle shaped in terms of time as a rectangular triangle by a shock wave followed by an expansion wave, (left-hand side of picture) develops, after the wave has travelled a long distance in the bubbly ﬂow, a series of solitons (right-hand side of picture).

Figure 8. The shaded area bounds that portion out of the diameter/impact plane of falling drops, that marks the occurrence of a trapped air bubble. The broken line represents the relation between diameter and impact velocity of raindrops. Hence the intersection of this line with the shaded area gives the range of raindrops which produce air bubbles and thereby sound. The picture is from [13] and reproduced here with permission of Ann. Rev. Fluid Mech.

Interplay between Air and Water

4.

13

Fundamental and Applied, The Scientist and the Engineer

I will now use the freedom of the lecturer on an occasion as this one to compare in a symbolic sense the interplay just described with the interplay between fundamental and applied science. Fundamental science high up in the air, the applications down below where the water is. Sometimes the interplay is not obvious. Famous are the words spoken by the German mathematician David Hilbert when he was asked to comment on the relation between pure and applied science. He said “Meine Herren, sie haben gar nichts mit einander zu tun” (they have nothing to do with each other). This points at a lack of appreciation. In my case I have been lucky to see a lot of both in my professional life. Both have their peculiarities of which I shall give two examples. First the “∼” and the “=”. What do I mean by this? Suppose you are designing a device which includes ﬂow of water, of kinematic viscosity ν = 10−6 m2 /s, with a velocity U =0.10 m/s in a pipe with diameter D=1 cm. You want to know for the operation of your device, what entry length l is needed for the ﬂow to become fully developed. You turn for advice to a theoretical physicist. He takes his copy of Landau & Lifshitz, Fluid Mechanics [14], from the shelf in his oﬃce and ﬁnds on page 150 l/D ∼ Re. (9) This means that the dimensionless entrance length l is a multiple of order unity times the Reynolds number U D/ν. Landau & Lifshitz point out that the thickness δ of the laminar boundary layer along the pipe wall grows, with x running along the pipe axis, as δ ∼ (νx/U )1/2 , where again ∼ means that the boundary layer thickness is a multiple of order unity times the shown quantity. Putting now the thickness equal to the diameter gives Eq. (9) for the entrance length l. In this particular case Re is 1000. You feel uneasy over it and you ask the physicist, do I really need thousand diameters, which is 10 m? The physicist does not listen anymore. Your problem is now an engineering problem and he does not care. So, you turn to an engineer, for example R.S. Brodkey, who tells you in his book Brodkey 1967 [15] on page 129 that exact calculation gives l/D = 0.06 Re. (10) This is, to your great relief, only 60 diameters or 60 cm. As another example I mention granular materials. This is nowadays a popular subject in physics. It has been, however, widely studied in

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civil engineering and in mechanical engineering since long ago. In the second part of the last century A.W. Jenike (1914-2003) has dominated research in the ﬁeld of bulk solids. The research in this ﬁeld has had results. During the last ICTAM of the last century, Chicago 2000, in one invited lecture, Roux & Radja¨¨ı [16], we read “ The quasi-static behaviour of granular materials is already a mature ﬁeld in which a number of elastico-plastic models reproduce very accurately the available experimental tests. They allow us to design civil engineering structures with conﬁdence”. The authors of these lines come from civil engineering. At the same ICTAM there was another invited lecturer, Goldhirsch [17], a physicist, from whose lecture I quote: “Granular matter is often described as “unpredictable”, “irreproducible” or “erratic”. These and other adjectives used to characterize granular matter are a clear sign that much is still lacking in our understanding of these “materials” (my italics). In my opinion the physicists could have in this case more appreciation for the work done and results obtained by engineers.

5.

Epilogue

The great experience in ICTAM is that both fundamental and applied scientists can listen to each other and talk to each other during and outside the many sessions. And in spite of diﬀerences of approach, illustrated in the previous Section with some examples, there are many aspects in their work that they share. Whereas consultant ﬁrms apply high per-hour rates for every service that they deliver, we all are referees and editors for journals, sit in committees, do work for funding organisations, you name it, without payment or at most a modest compensation for subsistence costs. Why do we do that? There are immaterial rewards in the form of prizes and other signs of recognition. But above all it is out of a sense of duty to the scientiﬁc community. The British writer and philosopher Iris Murdoch [18] points out that our sense of duty stems from the fact that we are not perfect beings, “A totally good being would not experience the call of duty, might be said to lack or not need the concept since all acts and decisions would emerge from virtuous insight and its orderly process”. But just this sense of duty saves us, according to an Editorial in Science [19], from becoming victims of human frailty. I quote from this article entitled “The Roots of Scientiﬁc Integrity”: “The system of rewards and punishments tends to make honest, vigorous, conscientious hardworking scholars out of people who have human tendencies of slothfulness and no more rectitude than the law requires”.

Interplay between Air and Water

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With this quotation Mr Chairman, intended to make us all feel good, I come to the end of my presentation. I wish you all an enjoyable and rewarding ICTAM 2004 and I thank you for your attention. I thank my colleagues of the Physics of Fluids Group of the University of Twente for their helpful comments, Michel Versluis for teaching me Power Point and Raymond Bergmann, Peter Eshuis, and Christian Veldhuis for their expert help in preparing the Power Point version of this lecture.

References [1] G.K. Batchelor, Sedimentation in a dilute dispersion of spheres, J.Fluid Mech., Vol.52, pp.245-268, 1972. [2] L. van Wijngaarden, Hydrodynamic interaction between gas bubbles and liquid, J.Fluid Mech., Vol.77, pp.27-44, 1976. [3] M.J. Lighthill, On sound generated aerodynamically 1. General theory, Proc.R.Soc.London, Vol.A211, pp.564-587, 1952. [4] D.G. Crighton and J.E. Ffowcs Williams, Sound generation by turbulent twophase ﬂow, J. Fluid. Mech., Vol.36, pp.585-603, 1969. [5] M. Onfray, Gourmande La Raison, Grasset et Fasquelle, 1995. [6] C.D. Ohl, A. Tijink, A. Prosperetti, The added mass of an expanding bubble, J.Fluid Mech., Vol.482, pp.271-291, 2003. [7] A.W.G. de Vries, A. Biesheuvel, L. van Wijngaarden, Notes on the path and wake of a gas bubble rising in pure water, Intn‘l J.Multiphase Flow, Vol.28, pp.1823-1834, 2002. [8] G. Mougin, J. Magnaudet, Path instability of a rising bubble, Phys.Rev. Lett., Vol.88, 014502- 1-014502-3, 2002. [9] L.van Wijngaarden, Bubble velocities induced by trailing vortices behind neighbours, Submitted to J.Fluid Mech., 2004. [10] J.H Lammers, A. Biesheuvel,, Concentration waves and the instability of bubbly ﬂows, J.Fluid Mech., Vol.328, pp.67-93, 1996. [11] M. Minnaert, On musical air bubbles and the sound of running water, Phil. Mag., Vol.16, pp.235-245, 1933. [12] L.van Wijngaarden, Evolving Solitons in Bubbly Flows, Acta Applicanda Mathematicae, Vol.39, pp.507-516, 1995. [13] A. Prosperetti, H.N. Oˇ ˇ guz, The impact of drops on liquid surfaces and the underwater noise of rain, Annu. Rev. Fluid Mech. Vol.25, pp.486-537, 1993. [14] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, 1959. [15] R.S. Brodkey, The Phenomena of Fluid Motion, Addison-Wesley, 1967. [16] S. Roux, R. Radja¨, ¨ Statistical Approach to the mechanical behaviour of granular media, In Mechanics for a New Millenium [Eds.] H. Aref, and J.W. Phillpps, pp.181-197, Kluwer Academic Publishers, 2001.

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[17] I. Goldhirsch, Kinetic and continuum descriptions of granular ﬂows, Mechanics for a New Millenium, Eds. H. Aref and J.W. Phillips, pp.345-359, Kluwer Academic Publishers, 2001. [18] I. Murdoch, Metaphysics as a guide to morals, Penguin Books, 1993. [19] Editorial Essay “The roots of scientiﬁc integrity”, Science, Vol.139, pp.3161, 1963.

Professor Sobczyk delivers the ICTAM04 Closing Lecture

STOCHASTIC DYNAMICS OF ENGINEERING SYSTEMS Origins, challenges and results Kazimierz Sobczyk Institute of Fundamental Technological Research Polish Academy of Sciences ´ ¸etokrzyska 21, 00-049 Warszawa Swi¸ [email protected]

Abstract

This lecture presents a concise exposition of the basic features of contemporary stochastic dynamics of physical/engineering systems with emphasis on its methodological principles, applicatory power and recent challenges.

Keywords: Stochastic systems, random vibration, stochastic degradation models, random loads, reliability assessment, failure models, information dynamics, noise-induced phenomena

1.

Introduction

Historical Origins: Hundred Years from the Beginnings – W. Gibbs (1903), A. Einstein, M. Smoluchowski (1905/06), P. Langevin (1908) Stochastic dynamics is today a greatly advanced ﬁeld of science investigating real dynamical systems with use of stochastic process theory. It develops the models and methods for investigation of various dynamical systems subjected to parametric and external random excitations. The genesis of stochastic dynamics is connected with problems in physics. Although the ﬁrst probabilistic/statistical concepts were introduced to physics already in the 19-th century (kinetic theories of gases; the Maxwell distribution of velocities of the molecules of a gas, Boltzman H-theorem), the ﬁrst years of the 20-th century brought systematic formulations of statistical mechanics/dynamics, including the stochastic description of the phenomenon of the Brownian motion.

19 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 19–60. © 2005 Springer. Printed in the Netherlands.

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One should mention ﬁrst the Gibbsian construction of statistical mechanics (described by Gibbs [1] in 1903) where the problem of time evolution of a large number of material particles (described by Hamiltonian diﬀerential equations) is treated probabilistically. Because of diﬃculties in the exact determination of the dynamical variables and the fact that the systems considered in statistical physics include a very large number of particles, a statistical description of the motion of the system was introduced. The basic role is played by the so-called phase probability density f (x, t), where x denotes a point in the phase space (of generalized coordinates and velocities): x = (x1 , x2 , . . . x6N ), and N denotes the number of particles. Since the motion of the system is governed by the deterministic Hamiltonian equations, the principle of conservation of probability leads to the well-known Liouville equation for f (x, t) ∂f ∂ + {x˙ k f (x, t) } = 0, ∂t ∂xk 6N

f (x, t0 ) = f0 (x0 ),

(1)

k=1

where f0 (x0 ) denotes the probability density of the initial state of the system. The initial value problem described by Eq. (1) can be regarded as the ﬁrst connection between probability and diﬀerential equations. Another phenomenon belonging to physics, which inﬂuenced tremendously the development of probabilistic thinking in natural sciences, is the Brownian motion – an extremely irregular movement of a small particle suspended in a ﬂuid. This phenomenon discovered experimentally by R. Brown in 1827 is one of the most interesting examples of random physical processes. During many years after the Brown’s discovery, various experiments were performed attempting to measure the properties of the Brownian particles (e.g. experiments of F. Exner and R.A. Zsigmondy). Although some qualitative hypotheses were formulated, as well as some results of quantitative nature were provided (e.g. dependence of the particle displacement on its size or the temperature of the medium), none of these ﬁndings was able to shed brighter light on the true nature of the phenomenon of Brownian motion. As S.G. Brush [2] writes in his historical work: “three quarters of a century of experimentation produced almost no useful results, simply because no theorist had told the experimentalists what quantity should be measured!”. The ﬁrst fruitful and breakthrough results came from mathematical model of the Brownian motion by Einstein [3,4] and Smoluchowski [5,6] – (1905/06). These great physicists, proposed - independently and via diﬀerent approaches – the theoretical description and explanation of the Brownian motion. Einstein’s reasoning was inspired by the ideas of the

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21

Figure 1. Illustration of the trajectory of the Brownian particle on a plane. (cf. www.math.yale.edu)

diﬀusion theory, whereas Smoluchowski’s arguments were based on combinatorics and the mean-free-path approximation of the kinetic theory. It is worth noting that Einstein and Smoluchowski pointed clearly that the basic measurable characteristic of the phenomenon of Brownian motion should be – not, as their predecessors believed, the velocity of particles but – the mean square of their displacements per second. Einstein and Smoluchowski obtained for the mean square displacement x2 in time interval t the same formula (with a slight discrepancy in the numerical factor – due to various approximations used; in his later papers Smoluchowski accepted the Einsten’s numerical factor). This formula is as follows: kT 1 , (2) x2 = t N 3πηr where k is the Boltzmann constant, N is the Avogadro number (a number of point molecules in unit volume), T is the absolute temperature, r is the radius of the spherical particle and η is the viscosity coeﬃcient of the medium. Equation (2) shows that the mean square of the displacement of the Brownian particle grows linearly in time – the result which (as we know today) has had very profound implications in the mathematical theory

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of Brownian motion and stochastic dynamics in general (e.g. the Wiener process/model of Brownian motion). The exact experimental conﬁrmation of the Einstein and Smoluchowski theory was provided by Perrin [7] (using the Brownian motion experiments Perrin determined also the Avogadro number N ). In this context, the value of experimental work of T. Svedberg should also be underlined. Although the scientists mentioned above worked in diﬀerent places, there was among them quite a strong, natural interaction by correspondence. For example, as Pais [8] writes (pp.101) “Six letters between Einstein and Smoluchowski have survived. All show cordiality and great ` natural respect”; cf. also Sredniawa [9]. Nearly in the same time (exactly, in 1908) Langevin [10] formulated a “phenomenological” description of the erratic motion of a “heavy Brownian particle” of mass m immersed in a liquid, using the Newtonian equation for the particle. The interaction of the surrounding ﬂuid with the Brownian particle gives rise to two distinct forces: a dissipative force (due to dynamic friction in the course of a motion of a particle in viscous ﬂuid), and a ﬂuctuation force (arising from the molecular collisions). So, Langevin wrote down a diﬀerential equation of the motion of the particle m

dv = f (t), dt

f (t) = Fr (t) + Firr (t),

(3)

where v is the component of velocity of particle along the x axis and f (t) – the total force caused by a surrounding medium consists of two parts: Fr (t) = −βv(t) being a regular component, and Firr (t) representing irregular or random force acting on the particle by collisions. Denoting: β/m = α,

Firr (t)/m = ξ(t)

one obtains

dv + αv = ξ(t). (4) dt If one assumes that the particle is spherical with radius r and the liquid has a viscosity parameter η then the constant α = 6πrη/m. Symbol ξ(t) represents the unknown force due to the molecular impacts; this force is random in nature and can only by described probabilistically (the Langevin force). Langevin assumed that its mean value (over the ensemble) should be zero and the correlation of each two ﬂuctuating forces at diﬀerent times should be negligibly small when time diﬀerence t2 − t1 is meaningful. The above hypotheses concerning the features of molecular collisions are usually formalized in physics as follows: ξ(t) = 0,

ξ(t1 )ξ(t2 ) = Dδ(t2 − t1 ),

D > 0,

(5)

Stochastic Dynamics of Engineering Systems

23

where δ(t2 −t1 ) is the Dirac function, D is a constant and · denotes averaging over the ensamble of collision process. Therefore, ξ(t) in Eq. (4) is a very peculiar process; today, in stochastic theory it is called a “white noise”. Such a process does not exist in the conventional sense. For this reason, during a long time period it was not clear what should be the appropriate rigorous interpretation of the Eq. (4). Equation (4) is commonly known as the Langevin equation. It can be regarded historically as the ﬁrst stochastic diﬀerential equation. Although the original Langevin equation is linear (and in a scalar form), nowadays (especially in physics) more general, nonlinear equations (also – in vectorial form) with white noise additional excitation are often termed the Langevin equations; various forms of this equation serve as models of real systems in contemporary stochastic dynamics. It is worth noting, that the value for the mean square of the Brownian particle displacement x2 obtained from the Langevin equation (4) agrees exactly with the Einstein-Smoluchowski formula (2) – cf. Lindsay [11].

Origins of Stochastic Dynamics of Real Engineering Systems Uncertainty and random ﬂuctuations are a very common feature of a variety of real dynamical engineering systems. Most of engineering systems (control systems, mechanical, structural, etc.) are subjected to complicated external and internal (time-varying) inﬂuences. These complex excitations and the associated responses can, most rationally, be described in terms of stochastic processes. Among the examples are: structural response due to earthquake, wind load, sea waves etc., random vibration of road vehicles (response to random road roughness), response/and reliability of aerospace structures to random turbulent ﬁeld, response (and stability) of suspension bridges. In the case of such systems as above, the ultimate objective of stochastic dynamics is to provide a new tools for the reliability estimation. In other situations, the qualitative characterization and eﬀects of random excitation are of interest (e.g. stochastic bifurcations, stochastic resonance, eﬀect of random noise on deterministic chaos). Stochastic dynamics of engineering systems emerged nearly exactly ﬁfty years ago. First – in the context of automatic control theory (cf. Booton [12], Kazakov [13]) and, a little later, in the analysis of dynamics of aerospace, mechanical and structural systems. The primary reasons for stochastic analysis was the need to assure a satisfactory performance of engineering systems in the presence of real random noises/excitations.

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For a proper assessment of reliability of a wide class of systems the stochastic analysis of their response turned out to be necessary. In the context of engineering systems of mechanical nature, the pioneering work belongs to the following great scientists (alphabetically): Ariaratnam [14] (1960), Bolotin [15] (1959), Caughey [16] (1959), Crandall [17] (1958), Kozin [18] (1961), Lin [19] (1963), Lyon [20] (1956), Shinozuka [21] (1964). The books by Bolotin [22], Crandall and Mark [23], Lin [24], Robson [25] give an excellent exposition of the early eﬀorts and results. Today, stochastic dynamics of physical/engineering systems has a very extensive literature dealing both with mathematical bases as well as with speciﬁc applications. This literature includes also a number of books-monographs. The books of Bolotin [26], Lin and Cai [27], Roberts and Spanos [28], Sobczyk [29], and Soong and Grigoriu [30] provide an adequate presentation of the existing results.

General Unifying Scheme of Stochastic Dynamics A general methodical scheme of stochastic dynamics of mechanical systems can be illustrated as it is shown in the Fig. 2. The random excitation acting on the system is described by a random/stochastic process (in general vectorial) X(t, γ), where time t belongs to a prescribed time interval [t0 , ∞] or [t0 , T ], whereas γ symbolizes randomness, γ ∈ Γ where Γ is a space of elementary events. More exactly, we have the basic probability space (Γ, F, P ) where F is a family of subsets of Γ (σ–algebra) on elements of which the probability P is deﬁned; 0 ≤ P (A) ≤ 1, A ∈ F . A random variable X(γ) is a measurable function which maps Γ into Rn . A stochastic process X(t, γ) is a function which for each t gives a random variable. This function

Figure 2.

General scheme of stochastic dynamics.

Stochastic Dynamics of Engineering Systems

25

is characterized partially by its mean X(t, γ) = mX (t), by its correlation function KX (t1 , t2 ) = X(t1 , γ)X(t2 , γ) and by other – more complicated – averages. A complete characterization of X(t, γ) is given by probability distributions for various subsets of t-values. Having the probability distribution of X(t, γ) we can calculate the probability of various events associated with X(t, γ), e.g. the probability that values of X(t, γ) belong to a given set D. A stochastic process can be, for example, stationary or non-stationary, Gaussian or non-Gaussian, Markovian etc. For a systematic presentation, a reader is referred to the books on stochastic processes (e.g. see the references in Sobczyk [29]). In stochastic dynamics problems the excitation process X(t, γ) is assumed to be given; most often it must be inferred from the empirical data on real processes. A dynamical system transforms X(t, γ) into another process Y (t, γ); this response process is unknown and should be characterized via the mathematical/stochastic analysis.

Major Challenges The following problems constitute the major tasks of stochastic dynamics of engineering systems. System modeling and characterization of real random excitations. This is a problem of formulation of the governing equations adequately to the speciﬁc system under consideration and selection of the appropriate stochastic processes characterizing external and/or parametric excitations. Although the system modeling follows, to a great extent, the basic principles of model building in deterministic theory, here in stochastic dynamics there are some speciﬁc factors which should be taken into account. For example, in modeling of the system dynamics (under random excitations) with simultaneous degradation taking place in it, the coupled responsedegradation model has to be consistent with the nature of randomness; also, the initial and boundary conditions (which are posed for random functions) should be deﬁned according to stochastic nature of the problem. The characterization of random loads acting on speciﬁc engineering systems is an involved problem itself (we will discuss it in the next section). Characterization of the response; eﬀective solution methods. This is a problem of solving the adequate systems of stochastic diﬀerential equations (mostly nonlinear) with speciﬁed real random excitations. This means that we are looking for a stochastic process, which satisﬁes (in the appropriate sense) a given system

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of equations along with the eﬀective method of quantifying the probabilistic characteristics of this process (moments, probability distribution, etc.). The most notable methods will be discussed in Sec. 5. Failures of stochastic dynamical systems; reliability assessment. The ultimate purpose of stochastic analysis of engineering systems is characterization of safety or reliability of systems of practical importance. Having obtained the probabilistic characteristics of the response (e.g. displacement, stress, etc.) from the dynamic analysis, we can use them to assess the conditions of a system performance for various failure mechanisms. In most cases the reliability of systems in question can be deﬁned and quantiﬁed in terms of some random variables associated with the response process. However, for a wide class of problems (where the Markov process theory can be used) the reliability can be assessed more directly via differential equations for the reliability function (derivable from the governing stochastic diﬀerential equations). Section 6 expounds this problem. Qualitative phenomena / eﬀects. In addition to the problems indicated above and primarily important for engineering practice, there exists a class of interesting questions which are qualitative in nature and are associated with the basic dynamical features of a system in the presence of random noise. Is a random excitation acting on / in a system just an annoying factor - which makes our life more diﬃcult or – maybe – it can generate some new and interesting physical phenomena? It turns out that random perturbations, when combined with nonlinearity, can induce multifarious speciﬁc “noise-induced” phenomena and eﬀects; they have a potential to change some internal features of system dynamics (stability, bifurcations, resonances, etc.). The last part of this lecture will shed some light on these problems.

2.

General Mathematical Model of Stochastic Dynamics

Stochastic Diﬀerential Systems. Basic Interpretations A general model for a wide class of physical and engineering systems subjected to time-varying random disturbances can be represented in

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the vectorial form as follows: dY/dt = F(Y, t) + G(Y, t)X(t, γ),

Y(t0 ) = Y0 ,

t ∈ [t0 , T ],

(6)

where Y(t, γ) = [Y Y1 (t, γ), . . . , Yn (t, γ))] is an unknown vectorial response process, X(t, γ) = [X1 (t, γ), . . . , Xn (t, γ))] is m-dimensional stochastic process (characterizing random excitations acting on the system); F(y, t) = [F F1 (y, t), . . . , Fn (y, t)] is a given vector-valued function of indicated variables describing the regular (deterministic) component of the motion, G(y, t) = [Gij (y, t)]j=1,...,m i=1,...,n is a given matrix-valued function characterizing the state-dependent intensity of the random excitation X(t, γ), Y0 is an initial state of the system (deterministic or random). If G in Eq. (6) does not depend on Y, the model Eq. (6) describes dynamics with an external random excitation. It is seen that the classical Langevin equation (4) is a special case of Eq. (6) when F = −αv,

GX = X/m = ξ.

It is clear that functions F(y, t) and G(y, t) – taking on speciﬁc mathematical forms in modeling real systems – should belong to the class of functions which satisfy the appropriate conditions assuring the existence and uniqueness of a solution of Eq. (6). If the stochastic process X(t, γ) is suﬃciently regular (e.g. continuous and diﬀerentiable) then system Eq. (6) can be called a regular stochastic diﬀerential system. The majority of problems for such systems can be analyzed by use of the methods which are analogous to those in deterministic theory of ordinary diﬀerential equations; in spite of this fact, such stochastic equations give rise to serious solution problems (cf. Sobczyk [29]). If the stochastic process X(t, γ) is very irregular (e.g. white noise, Brownian motion process, jump process) then the system Eq. (6) requires more sophisticated probabilistic analysis. Let us assume that X(t, γ) = ξ(t, γ) where ξ(t, γ)is a vectorial white noise i.e. ξ(t, γ) = [ξ1 (t, γ), . . . , ξm (t, γ)]. In this case the stochastic system Eq. (6) is commonly represented in the form of the Langevin-type equation dY/dt = F(Y, t) + G(Y, t)ξ(t, γ),

Y(t0 ) = Y0 ,

t ∈ [t0 , T ].

(7)

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As we have already mentioned, Eq. (7) – as it stands – should be regarded as a “pre-equation” which needs an appropriate interpretation. The generally accepted interpretation is associated with the equation dY/dt = F(Y, t) + G(Y, t)dW(t, γ),

Y(t0 ) = Y0 ,

t ∈ [t0 , T ], (8)

or, equivalently, with the following stochastic integral equation: t t Y(t) = Y0 + F Y(s), s ds + G Y(s), s dW(s), t0

(9)

t0

where W(t, γ) = [W W1 (t, γ), . . . , Wm (t, γ)] is the m-dimensional Wiener (or Brownian motion) process. Because of the peculiar properties of the Wiener process (e.g. trajectories of W(t, γ) are continuous but nowhere diﬀerentiable and have unbounded variation on each ﬁnite interval), the second integral in Eq. (9) has to be deﬁned in a special way. Two basic deﬁnitions are associated with the names of Ito ˆ and Stratonovich (cf. Arnold [31], Sobczyk [29]). Depending on the choice of the definition of the integral with respect to dW(t), we obtain two diﬀerent interpretations of Langevin-type Eq. (7) via Eq. (8); those are the Itˆ o and Stratonovich interpretations. When G(y, t) occurring in Eq. (7) depends explicitly on y, the Ito ˆ and Stratonovich interpretations lead to diﬀerent solution processes Y(t). There exists, however, simple relationship between the Itoˆ solution and Stratonovich solution. Namely, the Stratonovich solution coincides with the Itoˆ solution of Eq. (8) if the components of the drift term F(y, t) in Eq. (8) are replaced by the following ones 1 ∂Gik (Y, t) + Gjk (Y, t) , 2 ∂Y Yj n

Fi (Y, t) =

Fi∗ (Y, t)

m

(10)

j=1 i=1

where Fi∗ (Y, t), i = 1, . . . , n are the drift components in the Stratonovich equation. The diﬀusion term is the same in both interpretations. In what follows we will adopt the Itoˆ interpretation of the nonlinear Langevin “pre-equation” (7).

Main Theorem; Relation to F-P-K Equation Let us concentrate our attention on the Itˆ oˆ stochastic diﬀerential model Eq. (8). Suppose the following conditions are satisﬁed: (a) the vector-valued function F(y, t) and the (n × m)–matrix valued function G(y, t) are deﬁned and continuous for t ∈ [t0 , T ], y ∈ Rn ,

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(b) the functions F (y, t) and G(y, t) satisfy the Lipschitz condition (with respect to y), (c) the functions F(y, t) and G(y, t) grow (with respect to y) at most linearly, (d) the random variable Y(t0 ) = Y0 is independent of W(t) − W(t0 ) for each t > t0 . Then Eq. (8) has on t ∈ [t0 , T ] a unique solution satisfying the initial condition, almost all realizations of the solution process Y(t) are continuous, and the solution Y(t) is a Markov diﬀusion process on [t0 , T ] with the following drift vector A(y, t) and diﬀusion matrix B(y, t): A(y, t) = F(y, t),

B(y, t) = G(y, t)GT (y, t).

(11)

This means that the transition probability density p(y, t|y0 , t) of process Y(t) satisﬁes the following Fokker-Planck-Kolmogorov (F-P-K) equation n n 1 ∂ ∂p ∂ + [F Fi (y, t)p(y, t)] − [bij (y, t)p(y, t)] = 0, (12) ∂t ∂yi 2 ∂yi ∂yyj i=1

i,j=1

with the appropriate initial and boundary conditions. In Eq. (12) bij (y, t) = {G(y, t)GT (y, t)}ij =

n

Gir (y, t)Gjr (y, t).

(13)

r=1

The theorem stated above shows that a wide class of diﬀusion Markov processes can be constructed – via the stochastic Eq. (8) – on the basis of increments of the Wiener process. It also indicates that a wide class of real dynamical systems modeled by the Langevin-type equation (7) can be characterized by solving the partial diﬀerential Eq. (12). The methods for obtaining solutions of the F-P-K equation (12) have been a subject of a great research eﬀort (cf. Soize [32], Langtangen [33], Spencer and Bergman [34]). Although the progress is signiﬁcant, the effective characterization of the transition probability density for systems of higher dimension (e.g. n > 5) still constitutes a serious problem. The situation is simpler if one is interested in stationary solutions of stochastic systems (8), i.e. when t → ∞, and the F-P-K equation becomes “time-independent”.

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3.

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Engineering Stochastic Vibratory Systems: Response and Degradation Models

Basic Model A general model for dynamics of mechanical/structural systems with random excitation (both – external and/or parametric) can be formulated in the following form ¨ ˙ ˙ MY(t) + CY(t) + R[Y(t), Y(t), X1 (t, γ)] = X2 (t, γ),

(14)

where M and C represent the constant mass and damping matrices, respectively, Y(t) is an unknown response vector process, R characterizes the nonlinear restoring force, and X1 (t, γ), X2 (t, γ) are the random processes characterizing parametric and external excitations, respectively. When the original system is of a continuous type (e.g. beam, plate, shell), the Eq. (14) is a spatially discretized version (e.g. via Galerkin of ﬁnite-element methods) of the original equations and it describes the system response (as a function of time) in ﬁxed spatial points. The stochastic system of Eq. (14), which can be easily represented in the form of a system of the ﬁrst order equations, characterizes (when appropriately speciﬁed) adequately a variety of real systems of engineering practice, e.g. complicated multibody vibrating vehicle systems (cf. Schiehlen [35]), structural/ mechanical vibrating components in bridges, oﬀshore structures as well as aerospace systems (cf. examples in Roberts and Spanos [28], Lin and Cai [27]). In modeling of real physical / engineering problems, the stochastic processes X1 (t, γ) and X2 (t, γ) describing the parametric and external exci-

Figure 3. Exemplary multibody vehicle system with random excitation (cf. W. Schiehlen [35]).

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tations may have quite diverse probabilistic properties. From the point of view of analytical treatment and computational eﬃciency it is most convenient if these random processes can be assumed to be random white noises (i.e. uncorrelated for diﬀerent instants of time); as we know from Sec. 2, in such a case one can make use of the advantages of the theory of Itoˆ stochastic equations (and Markov process theory). It turns out, however, that these theories can also be used for a wide class of random excitations with ﬁnite correlation time; especially for excitations which can be represented as a response of dynamical systems to a white noise excitation ξ(t, γ). An arbitrary Gaussian and stationary random process with rational spectral density can be obtained as an output of a linear system (ﬁlter) with white noise as the input. Therefore, in order to represent the model Eq. (14), with random excitations X1 (t, γ), X2 (t, γ) being real (or, “colored”) random processes, we extend the state space of the system by deﬁning the extended state ˙ The vibratory sysvector [Y1 , Y2 , X1 , X2 ], where Y1 = Y, Y2 = Y. tem governed by Eq. (14) can in this way be represented as the following system of ﬁrst-order Itˆˆo stochastic equations: dY1 (t) = Y2 (t)dt, dY2 (t) = −M−1 [CY2 + R(Y1 , Y2 , X1 (t)) − X2 ]dt, dX1 (t) = −A1 X1 (t)dt + φ1 (t)dW1 (t, γ), dX2 (t) = −A2 X2 (t)dt + φ2 (t)dW2 (t, γ),

(15)

where X1 (t, γ) and X2 (t, γ) are the outputs of the ﬁlter driven by white noises ξ1 (t, γ) and ξ1 (t, γ), respectively; A1 and A2 are the ﬁlters operators (matrices) associated with real excitations X1 (t, γ) and X2 (t, γ), whereas φ1 (t, γ) and φ2 (t, γ) are the time-dependent intensities of white noises ξ1 (t, γ) and ξ2 (t, γ), respectively; W1 (t, γ) and W2 (t, γ) are the Wiener (or Brownian motions) processes in the Itˆ oˆ representation of “pre-equations” of Langevin – type with white noises ξ1 (t, γ) and ξ2 (t, γ), respectively. Therefore, adding the additional ﬁlter equations (for real excitations X1 (t, γ) and X2 (t, γ)), to the original system Eq. (14) enables us to use the Markov process theory, including the analytical and numerical methods developed for Itoˆ stochastic equations (cf. Sobczyk [29]). Of course, the ﬁlter can also be governed by higher order diﬀerential equations. Another possibility of treating system Eq. (14) with real random noises gives the Khasminskii averaging method (cf. Lin and Cai [27], Sobczyk [29], Soong and Grigoriu [30]).

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Stochastic Dynamics of Degrading Systems In fact, stochastic governing equations for many mechanical / structural systems should be represented in a more general form which accounts for inelastic behaviour and simultaneous degradating processes taking place in the system. Above all, these are elastic-plastic vibratory systems (under severe random loadings) in which the restoring force has a hereditary nature (cf. Casciati [36], Irshik and Zigler [37], Wen [38]. Therefore, instead of model (14) we construct the coupled system of equations for the response problem in which the restoring force, in general, depends on the time history of the response. The model is as follows: ¨ ˙ ˙ MY(t) + CY(t) + R[Y(t), Y(t), Z(t), X1 (t, γ)] = X2 (t, γ),

(16)

where Z(t) characterizes various speciﬁc processes responsible for degradation phenomena. Process Z(t) is governed by its own equation (coupled with Y(t)). In general, it has a form ˙ ˙ Z(t) = H[Z(t), Y(t), Y(t)],

Z(t0 ) = Z0 ,

(17)

where function H[z, y, y] ˙ should be constructed for speciﬁc situations. Its mathematical form is inferred from the elaboration of empirical data, or it is derived from the analysis of the physics of the process. It seems that a need for the coupled models dynamics Eqs. (16,17) arose for the ﬁrst time in the analysis of structural response to earthquake. Indeed, structures under strong earthquake excitation become inelastic with restoring forces being nonlinear and depending on the time history of the response. In this situation process Z(t) describes a hysteretic loop and is most often represented by model (17) in which Z(t) = Z(t) is a scalar process, and function H depends only on (z, y) ˙ and has the form (the Bouc-Wen [39] model) n−1 − δ y|z| ˙ n, H[z, y] ˙ = αy˙ − β|y||z||z| ˙

(18)

where α, β, δ, n characterize the amplitude and shape of the hysteretic loop. In the situation considered, a degradation D(t) taking place in the system has been deﬁned in terms of the total hysteretic energy dissipation characterizing the cumulative eﬀect of severe response and expressed by the state variables (z, y). ˙ Therefore, the coupled responsedegradation problem for randomly excited dynamic hysteretic systems is governed by Eqs. (16,17). These equations have been the subject of detailed analysis for many speciﬁc situations under various hypotheses concerning random external excitation (the restoring force R does not depend on X1 (t, γ); cf Wen [39]).

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There is a class of practically important problems governed generally by the coupled Eqs. (16,17) in which Z(t) characterizes directly the degradation of the system. A general model can be formulated in the following form ¨ ˙ ˙ MY(t) + CY(t) + R[Y(t), Y(t), D(t)] = X(t, γ),

(19)

˙ ˙ Q[D(t), D(t), Y(t), Y(t)] = 0,

(20)

where Q[˙] symbolizes a relationship between degradation and response processes. In Eq. (20) dependence on the response can be regarded in some relaxed sense, i.e. degradation rate may depend not on the actual values of Y(t) but – on some functionals of Y(t). In the fatigue degradation problem (more exactly – in the analysis of response of vibrating system with the stiﬀness degradation due to fatigue accumulation), it is natural to quantify the degradation D(t) = D(t) by a “normalized” crack size and adopt as an evolution Eq. (20) one of the “kinetic” equations for fatigue crack growth. These equations, however, ˙ contain the stress intensity factor range, so the degradation rate D(t) depends on the quantity related to Ymax − Ymin . In this situation, Eq. (20) has the form ˙ D(t) = H[D(t), Ymax − Ymin ].

(21)

Another version of an equation for D(t) in the coupled response– degradation problem is obtained if the functional relationship (20) does ˙ not include D(t), and the degradation D(t) depends on some functionals ˙ deﬁned on the response process [Y(t), Y(t)], i.e. (20) takes the form (F denotes here the appropriate functional) ˙ D(t) = F Y(t), Y(t) . (22) Important examples include randomly vibrating systems in which a degradation process depends on the time length which the response Y(t) spends above some critical level y∗ (or, D(t) depends on the number of crossings of the level y∗ by the trajectories of the process Y(t) within a given interval [0, T ]). This is the case of an elastic-plastic oscillatory system with D(t) interpreted as an accumulated plastic deformation generated by the “excursion” of the response process Y(t) into plastic domain (in this situation y∗ = y ∗ may be regarded as the yield limit of the material component in question, cf. Grossmayer [40]. This is also a situation of randomly vibrating plate with fatigue – induced stiﬀness degradation; in this case D(t) is interpreted as accumulated fatigue damage due to exceeding the fatigue limit by the response process.

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The analysis of the stochastic response-degradation problem for elastic - plastic vibratory systems and for the system with fatigue-induced degradation can also be analyzed by the more explicit cumulative model for degradation D(t). We mean the situation in which relationship (20) is represented as follows (in scalar form):

N (t)

D(t) = D0 +

∆i (Y, γ),

(23)

i=1

where ∆i = ∆i (Y, γ) are random variables characterizing the elementary degradations taking place in the system; the magnitude of ∆i depends on the characteristics of the process Y (t) above a ﬁxed (critical) level y ∗ . Process N (t) is a stochastic counting process characterizing the number of degrading events in the internal [t0 , t]. In the case of elastic-plastic oscillator (cf. Grossmayer [40]) ∆i (Y, γ) are the yielding increments taking place in a single yielding duration τY which is related to the time interval which the response process spends above the yield level during a single excursion or during a single clump of excursions. In the case of fatigue ∆i , i = 1, 2, . . . , N (t) can be regarded as the magnitudes of elementary (e.g. within one cycle) crack increments (cf. Sobczyk and Trebicki ¸ [41,42] as well as Sobczyk and Spencer [43]).

4.

Characterization of Real Random Dynamic Loads

General Remarks It is clear that a key factor aﬀecting the system behaviour is the excitation (load) to which a system is exposed. So, the appropriate (adequate to the physical / mechanical situation) modelling of real random loads is a crucial task within the whole methodical eﬀort of applied stochastic dynamics. This is a problem of statistical inference from the empirical data (and from the basic physical mechanisms of the excitation in question) about the most informative features of a random excitation under consideration. Depending on the type of engineering structure and its operational task, we meet various kinds of random load processes. For example: gust wind loads (acting on tall slender, tower-shaped structures such as TV masts, chimneys, some bridges, etc. as well as on various aerospace structures) induced by atmospheric turbulence; sea wave loads (acting on oﬀshore platforms, ships, storage tanks etc.) dominated by gravity forces; earthquake excitations (acting on all structural systems) caused by tectonic phenomena and complex interaction of seismic waves; ground-

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induced excitations (acting on road vehicles) generated by a roughness of real road surfaces; traﬃc loads (acting on long-span bridges) caused by moving vehicles. The character and intensity of the load ﬂuctuations in the examples given above depend on the shape of a structure and its orientation with respect to the load direction. Of course, in the case of turbulent wind its probabilistic characteristics depend additionally on meteorological conditions, the geographical position, the height over the Earth’s surface, etc. So, the statistical inference on the real random excitations acting on engineering systems has to make use of various hypotheses and simpliﬁcations. Let us consider here two important types of random excitation: the sea wave excitation acting on the steel oﬀshore platform and the earthquake excitation.

Characterization of Sea Wave Excitation This type of excitation is generated by wind and sea waves. Waves, in turn, occur as a result of complicated interaction between wind and water. This leads to a loading process that is often described by a series of continuously varying sea states. The nature of oﬀshore loading and the complex interactions likely in the seawater environment make establishment of standard load characteristics (e.g. spectra) for oﬀshore structures much more diﬃcult than for aircraft structures. Although sea motion (or sea states) can be partially characterized by some parameters (e.g., the wave height hs the mean wave period Ts , the wave direction θ), an underlying quantity in stochastic theory is the sea elevation η(x, y, t), which is regarded as a random function of position and time. Probabilistic properties of η(x, y, t) are derived partially from the measurements and partially from hydrodynamic wave theory. In almost all studies in ocean engineering, it is assumed that the sea wave process is a stationary stochastic process. Under such a hypothesis, the process η(x, y, t), for ﬁxed (x, y), is characterized by the spectral density gη (ω). Various forms of the spectral density of sea surface elevation η(t), for ﬁxed (x, y), have been proposed in the literature. The most popular form used in practice is the Pierson-Moskowitz spectrum gη (ω) = Aa2g ω −5 exp[−B(ag /ωv0 )4 ],

ω > 0,

(24)

where A and B are dimensionless constants taken to be A = 8.1 × 10−3 , B = 0.74; v0 is the mean wind velocity at a height of 19.5 m above the still sea surface; and ag is the acceleration of gravity. Using a linearized wave theory, one obtains a relationship between the ﬂuid particle velocity u(x, y, z, t) and the surface elevation η(x, y, t). This relationship allows

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to determine the statistics of particle velocity; e.g. the spectral density of the horizontal component of u and acceleration u˙ (cf. Madsen [44] et al.) If the characteristic dimension of a structure is small compared with the wave length, then the load consists of two basic components: a drag force proportional to the square of the normal component of the incident particle velocity and an inertia or mass force associated with the normal component of the particle acceleration. These forces are combined in the Morison formula for the force per unit length of a ﬁxed cylinder: P = kd u|u| + km u˙ ≡ X(t, γ),

(25)

where u is the incident particle velocity normal to the cylinder, and kd and km are given in terms of the drag and mass coeﬃcients. The total Morison force on a ﬁxed vertical cylinder is obtained by an integration of Eq. (25) with respect to z over the interval [−d, 0]. It should be noticed that even when the sea surface elevation is Gaussian, the non-linearity of the Morison formula yields a force P(t) which, in general, is a non-Gaussian process. The departure from the Gaussian distribution (at a given cross-section) depends on the coeﬃcients kd and km . The non-Gaussian character of the forces acting on oﬀshore structures causes additional problems with their proper characterization. The spectral density, in this situation, provides only a partial characterization of the process. Higher order statistics should be estimated from the data. The local extremes of a random wave force P(t) were investigated by Grigoriu [45]. The application to the response analysis of tension-leg platform can be found e.g. in Spanos and Agarwal [46].

Description of Earthquake Excitation An earthquake action, i.e. a complicated ground motion caused by tectonic phenomena, is a result of complex interaction of seismic waves propagating from the source through inhomogeneous layered media. Multiple scattering of waves at randomly distributed inhomogeneities makes the surface displacement ﬁelds highly unpredictable (cf. Sobczyk [47]). Various stochastic models for a strong ground motion have been proposed. Modelling started from uncorrelated impulses (Housner G.W.) and white-noise representations (Bycroft G.N.) and has been developed to account for non-uniform spectra (Kanai K., Tajimi H.) as well as for the temporal non-stationarity of a random seismic action (Bolotin V.V., Amin M., Ang A.H.S.). These investigations lead to a commonly accepted model for (horizontal) ground acceleration having the form of

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a non-stationary modulated stochastic process ¨ γ) = A(t)X1 (t, γ), X(t,

(26)

where A(t) is a deterministic envelope function imposed on stationary process X1 (t, γ). More general model has the form ¨ γ) = Ak (t)Xk (t, γ)I(t), (27) X(t, k

where Xk (t, γ) are stationary random processes, Ak (t) – deterministic functions (envelopes) and I(t) is the Heaviside function. One of the methods of obtaining a tractable model for the reliability predictions consists in treating the system transmitting the motion from the source to the ground surface, as a suitable ﬁlter characterized by a frequency transfer function. This transfer function characterizes approximately the averaged eﬀects of wave propagation through the earth strata. The required frequency transfer function is approximated on the basis of the analytical theory of wave propagation and system identiﬁcation techniques. A speciﬁc common formula for the spectral density of the ground displacement was identiﬁed in: Kanai [48], Ruiz and Penzien [49]. Recently, Suzuki and Minai [50] elaborated the model: ¨ γ) = a1 (t)V (t, γ) + a2 (t)ζ2 (t, γ), X(t,

(28)

where V (t, γ) is the output of a time-dependent linear ﬁlter driven by Gaussian white noise ζ1 (t, γ); ζ2 (t, γ) is another white noise independent of ζ1 (t, γ). There is also possible another way of characterizing the earthquake excitation acting on the structures. It consists in representation of the earthquake process as a series of random impulses. The original idea is attributed to Housner (1947), but the model which we have in mind has the general form N (t) Zk (γ)s(t, τk ), (29) X(t, γ) = k=1

where function s(t, τk ) describes the shape of a pulse at random time instant τk , Zk (γ) is a random amplitude of the k-th pulse and N (t) is a stochastic counting process characterizing the number of impulses in the interval [t0 , t]. Lin [24] indicated the conditions under which there may be valid reasons for modelling an earthquake excitation by uncorrelated or correlated random impulses. The response analysis of

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many speciﬁc systems subjected to excitation in the form (29) has been performed by many authors (cf. book of Iwankiewicz [51] and references therein). The processes of the form (29) can also be used for modelling vehicular traﬃc ﬂow and the corresponding loading of highway bridges.

5.

Characterization of Response: Eﬀective Solution Methods

Existing Approaches The most notable approaches used in engineering stochastic dynamics are as follows. Perturbation method. If the nonlinearity in the system is weak, a small parameter ε 1 is introduced into the governing equations, and the solution process Y(t) is looked for in the form of a series expansion with respect to this small parameter. Introducing this expansion into the equations of dynamics and equating terms of the same order in ε, we obtain a recurrent sequence of linear diﬀerential equations for the successive terms Y0 , Y1 , Y2 , . . . of the expansion. These equations, especially when the excitation is Gaussian, can serve to obtain two ﬁrst statistical moments of the solution. Statistical linearization. A nonlinear stochastic system is replaced by “equivalent” linear equations whose coeﬃcients are determined from the condition of minimum of the diﬀerence between nonlinear and the “equivalent” linear part of the system equations (e.g. mean-square criterion); to evaluate the coeﬃcients of the “equivalent” linear equations one has to assume a speciﬁc form of the probability distribution of the unknown solution. Most often the Gaussian approximation is used (for systems with Gaussian external excitation and very weak nonlinearity). The procedure has been used mostly to compute the second order statistical moments of the stationary response; cf. Spanos [52], Socha and Soong [53]. Equivalent nonlinear systems. A given nonlinear system (with random excitation) which is too complicated for eﬀective quantitative analysis is replaced “equivalently” by another nonlinear system which is simpler to handle mathematically and computationally; e.g. by a nonlinear system

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for which a stationary solution exists and its probability density is available (cf. Caughey [54], Cai and Lin [55]). Equations for moments and closure hypotheses. From a given nonlinear stochastic system the equations for the statistical moments of the response are derived, which are diﬀerential for a nonstationary solution process and algebraic for a stationary solution. Because of nonlinearity of the original equations, these moment equations constitute an inﬁnite hierarchy of equations, so the appropriate closure assumptions have been proposed and used to obtain a ﬁnite system of equations (these are various hypothetical relationships between higher- and lower-order moments); cf. Crandall [56], Wu and Lin [57], Sobczyk and Tr¸e¸bicki [58]. Once the equations were “closed” they can be solved and, therefore, give the approximate moments of the response. Stochastic averaging method. This method is associated with the question: can the system with real random excitations be treated by the use of the Itˆ oˆ stochastic diﬀerential equations (without extension of the state space of an unknown process)? A positive answer to this question is due to Stratonovich and Khasminskii. Stratonovich [59] noticed that for a wide class of excitation processes X(t) acting on nonlinear systems, the stochastic eﬀects become truly important for the time intervals of order 1/ε or 1/ε2 . He also enunciated heuristically a theorem assuring that on time intervals of length of order 1/ε2 , process Yε (t) approaches a Markov diﬀusion process. A rigorous mathematical formulation and proof of this theorem was provided by Khasminskii [60] along with explicit formulae for the drift vector and diﬀusion matrix of a limiting diﬀusion process. This theorem constitutes a ground for the eﬃcient method known in stochastic dynamics as the stochastic averaging method; its numerous applications to practical problems can be found in the books cited above. Numerical schemes for stochastic diﬀerential systems. In the last decades various numerical schemes for approximation of the solutions of stochastic diﬀerential and integral equations have been elaborated. Although they can be viewed as being an extension of the corresponding schemes of deterministic numerical analysis (e.g. stochastic Euter scheme, stochastic Runge-Kutta scheme, etc.), in the stochastic case one meets speciﬁc and complex problems. In contrast to the deterministic situation, in the

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case of stochastic Ito ˆ equation diﬀerent schemes can converge to diﬀerent solutions for the same noise sample and initial conditions. In addition we can consider various types of approximation; the most common are: (i) mean-square approximation, (ii) pathwise (or, sample function) approximation, (iii) approximation of moments [f (Y Yt )]. Additional diﬃculties occur in multidimensional case; in this case the solution process can not be, in general, expressed as a continuous functional of the Wiener process alone; the detailed presentation can be found in the book by Kloeden and Platen [61], cf. also Chapter 5 of the book by Sobczyk [29]. Numerical methods for the Fokker-Planck-Kolmogorov equation. Since a wide class of physical engineering stochastic systems can be analysed via the diﬀusion Markov processes (generated by the Itoˆ stochastic diﬀerential equations), the numerical solutions of the appropriate F-P-K equations are of a great interest. The F-P-K equation(12) associated with the basic stochastic model Eq. (8) is a partial diﬀerential equation of parabolic type and, as such, is – in principle – accessible to the existing numerical methods. However, one should keep in mind that the F-P-K equation for the transition probability density p(y, t|y0 , t0 ) is for practical problems an equation in multi-dimensional space with variable coeﬃcients and with speciﬁc conditions associated with the probabilistic nature of unknown function (initial, boundary conditions and global normalization condition for the probability density). So, the applicability of the numerical methods (including FEM) is still limited to systems of lower dimensions, that is to the cases when unknown probability density depends on at most four spatial variables. Extension to higher dimensions, as concluded by the authors of paper by Spencer and Bergman [34] “ . . . while posing no logical problems . . . is beyond the capability of the current computer hardware”. An interested reader is referred to the book by Soize [32], and the paper of Spencer and Bergman [34]. An approach to approximate solving the Fokker-Planck-Kolmogorov equation which has attracted some interest has its roots in R. Feyman’s work on a space-time approach to quantum mechanics and is known as the path integration method; it consists in using the ChapmanSmoluchowski equation for discretized time variable (cf. Wehner and Wolfer [62], Naess and Johsen [63]). But, like in the previous techniques, its application to higher-dimensional systems seems to be diﬃcult. Each of the above approaches to quantitative characterization of the solution (or response) of stochastic systems has its own methodical draw-

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backs and restrictions. As far as numerical methods are concerned, the computational eﬃciency (especially, for real multi-dimensional systems) is still not satisfactory. Undoubtedly, elaboration of the eﬃcient computational methods for multidimensional stochastic systems constitutes today one of the greatest challenges within the stochastic dynamics research. In what follows we will report brieﬂy on the approach elaborated recently for evaluation of the probability distribution of the solution of stochastic equations and based on the moment equations and the informational entropy of the system.

Maximum Entropy Method for Stochastic Systems The maximum entropy principle (MEP), originating in statistical physics, states that of all the probability distributions that satisfy the appropriate moment constraints (given information) one should choose the distribution having the largest informational Shannon entropy. Since the entropy characterizes a global uncertainty of a random quantity in question, the principle of maximum entropy means that maximum entropy distribution is maximally noncommittal with regard to missing information. The above idea has been widely used in statistics and in variety of other applications. In its classical formulation MEP deals with random variable with unknown probability density, the partial information about which is given by a ﬁnite number of moments. It has been tempting to try to adopt this principle to determining the unknown probability distribution of the solution of a stochastic system. The ﬁrst attempt toward such a goal has been made by Sobczyk and Tr¸ebicki [64] where the general scheme of the method and some illustrative examples have been presented. In the papers by Sobczyk and Tr¸ebicki [65,66], Tr¸e¸bicki and Sobczyk [67], the idea has been extended to more complicated situations. Let the system of interest be governed by the following stochastic Itˆ o equation for the vector process Y(t) = [Y Y1 (t), . . . , Yn (t)]: dY(t) = F[Y(t)]dt + G[Y(t)]dW(t, γ),

(30)

where W(t, γ) = [W W1 (t, γ), . . . , Wm (t, γ)] is the m-dimensional Wiener process. Under known conditions speciﬁed in Sec. 2 the solution of Eq. (30) is a diﬀusion Markov process with the drift vector A(y) and diﬀusion matrix B(y) deﬁned in Sec. 2 (see Eq. (11)). The equations for moments are derived easily by use of the Itˆ oˆ formula k1 (or, Itˆ oˆ diﬀerentiation rule) to the function hk = Y1 , . . . , Ynkn of the solution and taking the average. The symbol k denotes here the multiindex, i.e. k = (k1 , . . . , kn ); we will denote: |k| = k1 + . . . + kn and

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|k| = 1, 2 . . . , K. The moments of process Y(t) at time t are deﬁned as usual by Y1k1 . . . Ynkn f = hk (Y(t))f , (31) mk = Y where f denotes the mean value of the quantity indicated, i.e., f is the integral of the product function hk (y) = y1k1 . . . ynkn with respect to the true probability density f (y; t) = f (y1 , . . . , yn ; t) of the solution process. The general form of the moment equations is ∂hk 1 ∂ 2 hk dmk (t) Fi Gil Gjl = + . (32) dt ∂Y Yi f 2 ∂Y Yi ∂Y Yj f i

i,j

l

The initial conditions mk (t0 ) are speciﬁed from the given probability density f (y; t0 ) of the initial condition Y0 (γ). According to the spirit of the maximum entropy principle, the approximate probability density p(y; t) of the stochastic process Y(t) governed by the general system Eq. (30) is determined as a result of maximization of the information entropy functional H[p [ ] = − p(y; t)lnp n (y; t)dy (33) under constraints Eq. (32) and the normalization condition p(y; t)dy = 1.

(34)

The integration in Eqs. (33,34) is extended over the range of the possible values Y(t) for each t. Let us notice that constraints Eqs. (32,34) in the maximum entropy scheme can be represented as p(y; t) − 1 = 0, (35) dmk (t)/dt = Qk (y)p , where Qk (y) =

i

Fi

∂hk (y) 1 ∂ 2 hk (y) + Gil Gjl . ∂yi 2 ∂yi ∂yyj l

(36)

i,j

It has been shown (cf. Tr¸e¸bicki and Sobczyk [67]) that the probability density which maximizes the entropy functional Eq. (33) under constraints Eq. (35) has the form K λk (t)Qk (y) , p(y; t) = C(t)exp − |k|=1

(37)

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where |k| = k1 + . . . + kn = 1, 2, . . . K < n, ki = 0, 1, . . . , n and C(t) is the normalizing factor equal to e−λ0 (t)−1 . Functions λk (t) being the unknown Lagrange multipliers are determined by substituting density Eq. (37) into constraints Eq. (35). This means that all moments mr , for r > |k| occurring in the set of moment Eq. (35), which is not closed, are calculated with the use of probability density Eq. (37); this is just the maximum entropy closure. In the stationary case when the probability density of the solution process Y(t) does not depend on t, multipliers are constant, i.e., λk (t) = λk and the moment equations are algebraic (cf. Sobczyk and Tr¸ebicki [65]). Hence, instead of Eq. (37) we have K p(y) = C exp − λk Qk (y) . (38) |k|=1

In such a way the problem of determining a probability distribution of the solution of a general stochastic system is reduced to the solution of a system of deterministic equations (diﬀerential or algebraic) for the Lagrange multipliers; the approximate probability density itself is represented in analytical form. A reader interested in details of the method is referred to the papers cited at the beginning of this subsection.

Empirical Characterization of Random Response, Optimal Experiment Design, Remarks Although the theoretical analysis dominates the research in stochastic dynamics, in many situations the empirical information acquired from measurements on randomly vibrating structures is necessary. In addition, the nature of the state (response) variables often does not allow much ﬂexibility as to which states can be measured. The problem consists in estimation of the probabilistic characteristics of random ﬁelds (e.g. displacements, stresses of randomly vibrating beams, plates, shells) on the basis of a statistical sample obtained in a ﬁnite number of points. It is clear that the informational content of the data depends on the number of measurement points and their locations. The experiments should be designed optimally. In distributed parameter systems an important optimal experiment design variable is the spatial location of the measurement sensors. In the paper by Papadimitriou, Haralampidis and Sobczyk [68] a general method was given for optimising the number and positions of sensors on randomly vibrating structures for the purpose of the response prediction at the unmeasured locations. The dynamics of the structure is governed by a linear partial diﬀerential equation subjected to space-time

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random excitation with given mean and correlation function. The response characteristics, when obtained from the random vibration analysis, are used in the kriging method to obtain the response predictions (and the corresponding mean-square errors) at unmeasured locations. The optimal sensor locations are chosen to minimize the total averaged (over all prediction points) mean-square error of the response prediction in unmeasured points. The dependence of the optimal sensor locations on the type of response variable (displacement, strain), the characteristics of the random excitation and number of sensors have been determined and illustrated via numerical calculations. The interested reader will ﬁnd the details of the analysis in the paper cited above, along with the appropriate references.

Information Dynamics, Remarks Analysis of many problems of stochastic dynamics leads naturally to the concepts and tools of the information theory. This is not only due to the fact that the basic notions of mathematical information theory are based on the probability theory, but also because the apparatus of information theory is applicable to any probabilistic system of inference (in which we seek information). When the language and tools of information theory (e.g. Shannon entropy, mutual information between random events and processes, information ﬂow) are used in system dynamics, we come to the notion of information dynamics. The premises, challenges and results of this emerging ﬁeld are presented in Sobczyk [69].

Eﬀects of Spatial Randomness, Remarks In the present lecture I have restricted my attention to dynamical systems in which randomness is time-dependent. The systems material parameters remain deterministic and constant. It is clear however that in reality, an engineering system can never fulﬁll strictly such ideal requirements; manufacturing processes of structural/mechanical components introduce some imperfections and inhomogeneity into the material structure. Therefore, in many situations there is a need for taking into account the spatial randomness of the system properties (e.g. random bending stiﬀness of beams, plates, etc., or span length in an N-span beam). This randomness, when characterized by random variables, is usually called a random disorder (cf. Lin and Cai [27] – Sec. 9). In many situations one should assume that the material property randomly varies in space; this leads to the governing partial diﬀerential equations with spatial randomness in coeﬃcients, and subsequently – to

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random eigenvalue problems (cf. early papers by Shinozuka and Astill [70], Sobczyk [71]) and stochastic ﬁnite elements (Shinozuka and Deodatis [72], Ghanem and Spanos [73]). In a sense, the analysis of wave propagation in stochastic media belongs to this category (cf. the book by Sobczyk [47] and references therein) and it constitutes today a fairly advanced ﬁeld.

6.

Failures of Stochastic Dynamical Systems – Reliability Assessment

General Formulation Next important step in stochastic analysis of dynamical engineering systems – strongly connected with the response characterization – is the evaluation of the system performance and its reliability. The character of systems being a subject of stochastic dynamics (complexity of interaction between the system constituents and external excitation, material property degradation due to dynamics, etc.) requires much more sophisticated formulations and methods than the traditional safety assessment. Not only the probability concepts have to play a key role in the analysis but also (and above all) the fact that system performance changes in time. So, today there is a common agreement between the researchers that failures (of various modes) of dynamical systems should be deﬁned as outcrossings or exits of the appropriate stochastic processes (or, random ﬁelds) out of an acceptable (safety) domain. Let us assume that the system states in each time instant are characterized by a random response process Y(t, γ), i.e. for each t ∈ [t0 , ∞] – by a random vector Y belonging to the state space of the system. In order to describe the system performance and its reliability it is convenient to deﬁne quality states of the system and the quality space. A quality is ˆ For each state yt in the state space there characterized by a vector Y. exists a corresponding quality state y ˆt in the quality space. A set of system states and consequently, a set of corresponding quality parameters admissible from the point of view of quality (or reliability) requirements deﬁnes in the quality space a set Dr which can be interpreted as a safety or reliability domain. A boundary of Dr corresponds to the limit (or critical) states. Reliability function R(t) of a system under consideration is deﬁned as the probability of its admissible performance during the time interval [0, t], i.e.

ˆ ) ∈ Dr ; R(t) = P {Y(τ

τ ∈ [0, t]}.

(39)

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Figure 4.

Illustration of the system reliability problem.

The complement of R(t) to unity deﬁnes the probability of failure, Pf (t) = 1 − R(t)

(40)

i.e. the probability that at least one outcrossing of set Dr by process ˆ Y(t), in the direction normal to the boundary of Dr , will occur within the time interval [0, t]. The time of satisfactory system performance TDr (i.e. the time duraˆ t ∈ Dr , is called a life-time; it is a random variable. tion within which Y Reliability is related to TDr by the formula R(t) = P {T TDr > t}.

(41)

Analysis of the reliability problems depends crucially on the speciﬁc failure mechanisms. Engineering systems (e.g. machines, structures, etc.) subjected to random dynamic load may fail due to various failure modes that can occur during the designed lifetime. They depend on the material properties, system characteristics and excitations. The basic failure modes are: (1) the motion of a system becomes unstable, (2) the system response (or, the appropriate function of it) at a critical location exceeds, for the ﬁrst time, the prescribed safety boundary, (3) the accumulated damage (due to Y(t) – ﬂuctuations) exceeds the ﬁxed critical limit (e.g. fatigue failures). It is clear that evaluation of the reliability function R(t) is a basic problem in the reliability assessment of stochastic dynamical systems. Solving such a problem in practical situations (i.e. obtaining exact values of the probability Eq. (39)) meets serious diﬃculties. So, various approximations and bounds on reliability and failure probability have been proposed (e.g. cf. Bolotin [74], Lin and Cai [27] – Chapter 8). In general, the boundary of safety domain Dr can be random. Therefore, the problem consists in evaluation of the probability of passing of the ˆ process Y(t) outside the safety domain Dr with random boundary; this introduces additional diﬃculties.

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47

Stochastic Stability/Instability As far as stability or instability is concerned, it has always been of a great concern in the analysis of dynamical systems. In the stochastic case, when the system response is generally an n-dimensional stochastic vector process Y – measured from a referenced solution (regarded as the trivial solution) – stability is deﬁned in terms of boundedness and convergence (for t → ∞) of the norm ||Y(t)||. But, these properties can be characterized in diﬀerent probabilistic meanings. So, various stochastic stability notions have been introduced and a variety of stochastic stability criteria obtained. Nowadays the existing literature concerning stability of stochastic dynamical systems is very extensive; it is concerned both with a beautiful mathematical analysis and with speciﬁc applications. The essential advances are concerned with elaboration of the stochastic Lapunov function method and with the analysis of the asymptotic behaviour (when t → ∞) of stochastic systems, including stability, by using the Lapunov exponents; cf. for example: Khasminskii [75], Arnold and Wihstutz [76], Ariaratnam and Xie [77], Wedig [78], Bucher and Lin [79], Tylikowski [80,81]. It is worth noticing that the system stability/instability can be characterized via the basic formula (39). Indeed, the system stability can be deﬁned in terms of probability that the response process Y does not leave a spherical domain, say Dst of radius ε > 0 centered at the equilibrium point y0 , where Dst is a set of points y such that ||y − y0 || < ε.

First–excursion Failures: Stochastic Diﬀusion Markov Systems As we have already stated in Section 7, the transition probability oˆ stochastic dynamical density p(y, t; y0 , t0 ) of the solution process of Itˆ system (let us assume here that its drift and diﬀusion coeﬃcients do not depend explicitly on time) satisﬁes the Fokker-Planck-Kolmogorov Eq. (12) with respect to y and t. It turns out that p(y, t; y0 , t0 ) satisﬁes also – with respect to the “backward” variables y0 , t0 - the backward Kolmogorov equation. Using this equation it can be shown that the reliability function R(t|y0 , t0 ) satisﬁes the equation n n ∂R 1 ∂2R ∂ R+ mi (y0 ) 0 + bij (y0 ) 0 0 = 0. ∂t0 2 ∂yi ∂yi ∂yyj i=1

(42)

i,j=1

This equation is supplemented by the appropriate initial and boundary conditions.

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Equation (42) along with the imposed conditions constitutes a general mathematical basis for characterization of the reliability of a wide class of stochastic dynamical systems. As one could expect, however, closed form solutions are known only for simple cases (cf. the book by Lin and Cai [27]). To make the reliability problem more tractable, instead of evaluation of the (conditional) reliability function we restrict ourselves to the calculation of statistical moments of the probability distribution of the (ﬁrst) passage time TD of the response process Y(t) across the boundary of the safety domain Dr . For example, the mean value of the ﬁrst passage time satisﬁes the equation (known as the Pontryagin equation) 1 d2 dT dT b(y0 ) + m(y0 ) = −1. 2 dy0 dy0

(43)

The boundary conditions for this equation at the ends of the safety interval considered, say [α, β] are: T = 0 for y0 = α, y0 = β, since the ﬁrst passage time is zero when Y (t) starts on the boundary of the safety set. The moments of the ﬁrst passage time have been evaluated for various practical situations. In general, the Galerkin ﬁnite element method can be used (cf. Spencer [82]). However, numerical diﬃculties are usually encountered when we are dealing with higher-dimensional random response processes.

Cumulative Failures: Fatigue in Randomly Vibrating Systems An important deterioration or failure mechanism in structural and mechanical components subjected to time-dependent (deterministic or stochastic) loading is fatigue. According to fracture mechanics, the fatigue is due to nucleation and growth of cracks. In engineering, a measurable characteristic of fatigue is usually the size of a dominant crack, and ultimate failure occurs when this crack reaches the critical size. To capture the basic features of random fatigue crack growth, various stochastic models have been proposed in the literature (cf. Sobczyk and Spencer [43] and references therein). A model which takes into account the empirical information and randomness inherent in the fatigue crack growth consists in randomization of the empirical crack growth equation (e.g. Paris-Erdogan equation) by introducing to the equation an appropriate stochastic process X(t, γ). The resulting equation has the general form dA(t) = F [A(t), ∆S, constants]X(t, γ), (44) dt

Stochastic Dynamics of Engineering Systems

49

where A(t) is the crack size at time t, ∆S characterizes the stress range, and F is the empirical (nonlinear) √ function of the indicated variables. Assuming that X(t, γ) = mx + 2Dξ(t, γ) and using the empirical Paris equation, the following probability density of fatigue life-time TF was derived in Sobczyk [83]

α 1 (α − βt)2 √ , (45) fTF (t) = exp − 2 t 2πt3/2 which is the inverse Gaussian distribution (α, β are constant parameters including Paris constants, A0 , mx , D and ∆S). Such distribution has been earlier hypothesized as a possible lifetime model. The inverse Gaussian distribution for a fatigue life-time has also been derived by Ditlevsen [84] by an alternate approach based on an incremental version of Paris-Erdogan equation. Other approaches to the characterization of the fatigue life-time can be found in some recent publications (cf. Doli´ n ´ski and Colombi [85]).

7.

Qualitative Phenomena: Noise – Induced Eﬀects. Examples

Introductory Remarks In the previous four sections I focused my attention on the quantitative aspects of stochastic dynamics, which are of interest in applied (engineering) problems. One could therefore get an impression that stochastic dynamics is primarily concerned with numerical eﬀects of random noises and it does not deal with the phenomena which might be generated in the system solely by random excitations and which have a power to change the system dynamics qualitatively. Such a view would not be correct. The questions which naturally come to mind are, for example: (i) Does a random excitation (acting on the system) have any inﬂuence on the most essential, internal features of a system dynamics? (ii) Is a random noise just an annoying factor we have to live with or is there any interesting physics induced that is not present when the random ﬂuctuations are absent? All macroscopic systems of interest in physics, biology, engineering, chemistry, etc. are subjected to irregular perturbations, internal and external, which – when combined with nonlinearity – can display a rich variety of speciﬁc “noise-induced” phenomena and eﬀects. For example, stochastic instability, bifurcations, the also called – noise-induced transitions of the equilibrium states of the system, etc. It has to be kept in

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mind that in the presence of random (internal/external) excitations the state of the system is no longer characterized by a simple number (or vector) but by a probability distribution. In order to shed some light on the phenomena and eﬀects which random excitations may generate in the dynamical system we will discuss a few “simple” examples.

Stabilization and Destabilization by Random Noise Since some time it has been observed by engineers (ﬁrst, in radioelectronic systems) that noise can aﬀect a system in two opposite ways. It can destabilize a system as well as stabilize it (cf. the survey article of Roberts and Spanos [86] and the paper by Bucher and Lin [79]). As it is well known, a linear oscillatory system (cf. Arnold [87] et al.) Y¨ (t) + 2β Y˙ (t) + Y (t) = 0

(46)

is stable for β > 0. Let us perturb the constant stiﬀness term by a random noise ξ(t), where ξ(t) is a stationary random process with a given spectral density g(ω) and with intensity σ > 0. So, we have the damped linear oscillator with random restoring force Y¨ (t) + 2β Y˙ (t) + [1 + σξ(t)]Y (t) = 0.

(47)

The Lapunov exponent λ = λ(β, σ), a counterpart of the real parts of the eigenvalues, is the indicator of stability (λ < 0) or instability (λ > 0). For small noise (σ → 0) and underdamped case (β 2 < 1): πg(2 1 − β 2 ) 2 σ + O(σ 3 ). λ = −β + 4(1 − β 2 )

(48)

For overdamped case (β 2 > 1): 2 −1 2 λ = −β + β − 1 − 4[(β − 1)]

∞

e−2τ

√

β 2 −1

K(τ )dτ.

(49)

0

Therefore, in the underdamped case, a small noise destabilizes the motion since a positive term is added to λ = −β. However, in the overdamped case, a small noise stabilizes the motion since a positive term is subtracted from the value λ = −β + (β 2 − 1)1/2 . For a more general analysis of the stabilization by noise cf. Arnold [88] and the references therein, and the book by Khasminskii [75].

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Noise-Induced Bifurcations Stochastic bifurcation analysis deals with the qualitative changes in parameterised families of stochastic systems ˙ Y(t) = F[Y(t), X(t), λ],

(50)

where λ is a parameter. It seems that the bifurcation problems for stochastic equations were ﬁrst studied by physicists (cf. Horsthemke and Lefever [89] and references therein). In these studies the qualitative change of stationary solutions of the Fokker-Planck-Kolmogorov equation is used as an indicator of bifurcation. More explicitly, the extremes of the stationary density pst (y; λ) are regarded as indicators of qualitative changes in the system dynamics; these changes are called the phase transitions. The number and positions of the extrema of pst in the stochastic case and the extrema of the potential (of the system considered) in the deterministic case are the characteristic features of the steady-state behaviour of the system. Let us write down the deterministic system corresponding to Eq. (50) in the form (for n = 1) Y˙ (t) = Fλ (Y (t)).

(51)

It is convenient to represent this equation as d Vλ (Y ), Y˙ (t) = − dY

Vλ (Y ) = −

Y

Fλ (z)dz,

(52)

0

where Vλ (y) is called the potential of Eq. (50). The stable steady states of the system Eq. (51) correspond to the minima of Vλ (y) and the unstable steady states to the maxima. The maxima of pst (y) are the states in whose neighbourhood the system spends relatively much time, and they are most likely to be observed in an experiment; the minima of pst (y) correspond to the valleys of a potential (stable steady states). The minima of pst (y) are the states that the system leaves rather quickly (the unstable steady states). Depending on the value of the bifurcation parameter λ, density pst (y) may exhibit one-peak to two-peak or crater-like density. The analysis of the extrema of stationary probability density in multidimensional cases is, of course, much more involved. The analysis of a nonlinear Duﬃng-Van der Pol oscillators along this line was presented by Wiesenfeld and Knobloch [90]. Let us take the deterministic Duﬃng-Van der Pol oscillator Y¨ = αY (t) + β Y˙ (t) − Y 3 (t) − Y 2 (t)Y˙ (t),

α, β ∈ R1 .

(53)

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For this system in a deterministic case, trajectories do not explode in ﬁnite time, and there exists a (parameter-dependent) attracting compact set. When β is ﬁxed and is less than zero, whereas α varies, the pitchfork bifurcation occurs for α = 0. When α < 0 is ﬁxed and β varies, the Hopf bifurcation can be observed at β = 0 (cf. the bifurcation diagrams in Schenk-Hoppe´ [91]). Let the noisy version of Eq. (53) have the form Y¨ (t) = [α + σξ(t)]Y (t) + β Y˙ (t) − Y 3 (t) − Y 2 (t)Y˙ (t).

(54)

The analysis of a stationary F-P-K equation corresponding to Eq. (54) shows that for β suﬃciently negative, the Dirac delta shape of probability density is the only possible. Increasing β, we observe the birth of a bellshaped density and when β increases further (crossing β = 0) this density undergoes a “P-bifurcation” (at β = βp ) and it becomes crater-like. The above eﬀects of noise on “P-bifurcation” are inferred from the stationary probability density pst (y) which characterizes a system for long times and therefore it does not carry information on the transient states. However, recently a serious eﬀort has also been made to understand the bifurcations of stochastic systems on the dynamic level. Extensive simulations of stochastic dynamic systems, their random attractors and invariant measures provided interesting results. For example, it has been found (cf. Arnold et al. [92]) that random noise splits deterministic multiple eigenvalues. For β 2 < −4α (where α is ﬁxed and negative) and σ = 0, the deterministic linear system has two complex-conjugate eigenvalues 0.5β ± iωd what amounts to just one Lapunov exponent λ1 (β) = β/2 with multiplicity 2; for σ = 0, however, a linearized system has two diﬀerent simple Lapunov exponents.

Chaotic Dynamics Subjected to Random Noise An important inner property of deterministic nonlinear systems is chaos. It causes unpredictability of the long-term behaviour of the system. As one may expect, there has to exist the inﬂuence of external random noise on various characteristics of a chaotic dynamics. The eﬀects of noise on chaos, and – more generally – the interplay between chaos and externally introduced randomness has been a topic of research in nonlinear dynamics and statistical physics (cf. Grasman and Roerdink [93], Kapitaniak [94], Ying Cheng Lai [95] et al.). Among various reported results are the following: in the common route to chaos (the period-doubling bifurcations) the random noise tends to smooth out the transition and induces chaos in the parameter regime where there is no chaos otherwise,

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some authors found (cf. Grasman and Roedrik [93]) that noise may regularize chaotic dynamics causing a decrease of the Lapunov exponent λ (“noise-induced order”), for multidimensional systems (in chaotic regime) represented in the form of a system of ﬁrst-order diﬀerential equations, diﬀerent Lapunov exponents λi , (i = 1, 2, . . . , n) may react diﬀerently to the changes of the noise intensity. To obtain a measure of the averaged eﬀect of noise, some authors compute a “global” exponent deﬁned as

λg = m +

m i=1

λi , |λm+1 |

(55)

where λ1 ≥ λ2 ≥ . . . and m is the largest integer such that λ1 + . . . + λm > 0. For the Van der Pol oscillator (represented by a system of three diﬀerential equations of the ﬁrst order) λg decreases as the noise intensity σ increases. Although the maximum exponent λ1 slightly increases with σ, the system is “regularized” by noise in a global sense.

Stochastic Resonance. Remarks Another phenomenon which makes stochastic dynamics fascinating is stochastic resonance. It occurs as a result of interplay of nonlinearity, periodicity and randomness. Intuition suggests that when noise is added to a signal prior to or during transmission through a system/communication channel, the received signal will be more corrupted (deteriorated) than if the uncorrupted signal had been transmitted. The amount of corruption is usually characterized by the so-called signal – to noise ratio (SNR) of the output. For linear systems, the output SNR decreases monotonically with increasing noise intensity. The peculiarity of stochastic resonance lies in the fact that (for a large class of nonlinear systems) there occurs an increase in the SNR up to a maximum, with added random noise. This phenomenon has attracted much attention in the recent years. Besides numerous theoretical studies also numerical simulations and experimental work are today in the progress (cf. Benzi et al. [96], McNamara and Wiesenfeld [97], and references therein).

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Korteweg de Vries Solitons in Randomly Varying Medium. Remarks Phenomenon which occurs in spatially extended material media (ﬂuids, solids) and results due to the interplay between nonlinearity and dispersion of the medium is known as solitary waves or solitons. One of the best known equations describing solitons is the Korteweg de Vries equation ∂u ∂ 3 u ∂u +u + 3 = 0, ∂s ∂ξ ∂ξ

(56)

where u = u(s, ξ) characterizes the medium disturbance as a function of time and one-dimensional spatial coordinate. A characteristic feature of this equation is the existence of the so-called stationary waves (solitons) which do not change their form during propagation (usually, nonlinearity distorts the wave proﬁle). For example, waves in shallow water, ion-acoustic waves in plasma are governed by the above equation. If, however, the medium is perturbed by a random inhomogeneity (e.g. random roughness of the bottom of the water channel, random impurity of plasma density), the KdV solitary waves are attenuated and the amount of attenuation depends on the magnitude of randomness of the medium (cf. Sobczyk [99] and references therein).

8.

Closing

Stochastic dynamics is still at the stage of its development. Nowadays, the methodology presented in this lecture extends its models and methods to new ﬁelds of human endeavours. For example, a great intellectual eﬀort is concentrated today on stochastic dynamics of economic (and ﬁnancial) systems as well as on the atmospheric processes (weather and climate forecasting). Are there any other, fundamental, expectations concerning probabilistic / stochastic methods in science (and . . . in everyday life)? David Munford, the past president of the International Mathematical Union, in his article “The dawning of the age of stochasticity”, published in the distinguished volume [100] writes: “My overall conclusion is that I believe stochastic methods will transform pure and applied mathematics in the beginning of the third millennium. Probability and statistics will come to be viewed as the natural tools to use in mathematical and scientiﬁc modelling”.

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Acknowledgments I wish to express my sincere thanks to the Congress Committee of IUTAM for selecting stochastic dynamics for a very prestigious, plenary presentation at the XXI-st IUTAM Congress. I was very pleased and honoured to deliver this exceptional lecture. Also I owe a great debt of gratitude to many colleagues of my Institute who have contributed, by their friendly advices, to the clarity of my lecture. In particular, I wish to express my appreciation to dr. Jerzy Trebicki ¸ for his painstaking work on the visualization of my presentation, as well as – for bringing the typescript to its ﬁnal form.

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[14] S.T. Ariaratnam, Random vibrations of nonlinear suspensions, J. Mech. Eng. Sci., Vol.2, 195-201, 1960. [15] V.V. Bolotin, Statistical theory of seismic resistance of structures (in Russian), Izv. Acad. Nauk SSSR, Mekkanika I Mashinostrojenije, No 4, 123-129, 1959. [16] T.K. Caughey, Response of nonlinear string to random loading, J. Appl. Mech., Vol.26, 341-348, 1959. [17] S. H. Crandall (Ed.), Random Vibration, Vol.I, Technology Press, Cambridge, Mass, 1958. [18] F. Kozin, On the probability densities of the output of some random systems, J. Appl. Mech., Vol.28, 161-164, 1961. [19] Y. K. Lin, Nonstationary response of continuous structures to random loading, J. Acoust. Soc. Amer., Vol.35, 222-227, 1963. [20] R. H. Lyon, Response of strings to random noise ﬁelds, J. Acoust. Soc. Amer., Vol.28, 391-398, 1956. [21] M. Shinozuka, Probability of structural failure under random loading, J. Eng. Mech. Div., Amer. Soc. Civil Engrs., Vol.90, (EM 5), 147-170, 1964. [22] V.V. Bolotin, Statistical Methods in Structural Mechanics (in Russian: Moscow 1961; English transl., Holden-Day, San Francisco, 1965). [23] S.H. Crandall, W. D. Mark, Random Vibration of Mechanical Systems, Academic Press, N. York, 1963. [24] Y.K. Lin, Probabilistic Theory of Structural Dynamics, Mc Graw Hill, N. York, 1967. [25] J.D. Robson, An Introduction to Random Vibrations, Elsevier, Amsterdam, 1964. [26] V.V. Bolotin, Random Vibrations of Elastic Systems (in Russian), Izd. Nauka, Moscow, 1979; English language translation: Martinus Nijhoﬀ Publ., Hague, 1984. [27] Y.K. Lin, G. Q. Cai, Probabilistic Structural Dynamics: Advanced Theory and Applications, Mc Graw Hill, N. York, 1995. [28] J.B. Roberts, P.T.D. Spanos, Random Vibration and Statistical Linearization, Chichester, Wiley, 1990 [29] K. Sobczyk, Stochastic Diﬀerential Equations with Applications to Physics and Engineering, Kluwer Acad. Publ., Dordrecht, 1991. [30] T.T. Soong, M. Grigoriu, Random Vibration of Mechanical and Structural Systems, Prentice Hall, Englewood Cliﬀs, New Jersey, 1993. [31] L. Arnold, Stochastic Diﬀerential Equations: Theory and Applications, Wiley, N. York, 1974. [32] C. Soize, The Fokker-Planck equation for stochastic dynamic systems and its explicit steady state solutions, World Scientiﬁc, Singapore, 1994. [33] H. P. Langtangen, A general numerical solution method for Fokker-Planck equations with application to structural reliability, Probab. Eng. Mech., Vol.6, 1, 33-48, 1991. [34] B.D. Spencer, L. A. Bergman, On the numerical solutions of Fokker-Planck equation for nonlinear stochastic systems, Nonlinear Dynamics, Vol.4, 357372, 1993.

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[35] W. Schiehlen, Probabilistic analysis of vehicle vibrations, Probab. Eng. Mech., Vol.1, 2, 99-104, 1986. [36] F. Casciati, Stochastic dynamics of hysteretic systems, Structural Safety, Vol.6, 2-4, 1987. [37] H. Irschik, F. Ziegler, Nonstationary random vibrations of yielding frames, Nucl. Eng. Design, Vol.90, 357-364, 1985. [38] Y.K. Wen, Methods of random vibration for inelastic structures, Appl. Mech. Rev., Vol.42, 2, 39-52, 1989. [39] Y.K. Wen, Stochastic response and damage analysis of inelastic structures, Probab. Eng. Mech., Vol.1, 49-57, 1986. [40] R.I. Grossmayer, Elastic-plastic oscillators under random excitations, J. Sound and Vibrations, Vol.65, 3, 353-379, 1979. [41] K. Sobczyk, J. Tr¸¸ebicki, Modelling of random fatigue by cumulative jump processes, Eng. Fracture Mechanics, Vol.34, 477-493, 1989. [42] K. Sobczyk, J. Tr¸¸ebicki, Stochastic dynamics with stiﬀness degradation, Probab. Eng. Mech., 15, 91-99, 2000. [43] K. Sobczyk, B. D. Spencer, Random Fatigue: From Data to Theory, Academic Press, Boston, 1992. [44] H. O. Madsen, S. Krenk, N. C. Lind, Methods of Structural Safety, PrenticeHall, N. Jersey, 1986. [45] M. Grigoriu, Extremes of wave forces, J. Eng. Mech., ASCE, Vol.110, EM12, 1731-1742, 1984. [46] P.T.D. Spanos, V.K. Agarwal, Response of a simple tension-leg platform model to wave forces, J. Energy Res. Techn., Vol.103, 243-249, 1981. [47] K. Sobczyk, Stochastic Wave Propagation, Elsevier, Amsterdam, 1985. [48] K. Kanai, Some empirical formulas for the seismic characteristics of the ground, Bull. Earthquake Res. Institute, Univ. Tokyo, Vol.35, 309-325, 1957. [49] P. Ruiz, J. Penzien, Stochastic seismic response of structures, J. Eng. Mech. Div., ASCE, 441-456, April 1971. [50] Y. Suzuki, R. Minai, Application of stochastic diﬀerential equations to seismic reliability analysis of hysteretic structures, Probabilistic Eng. Mech., Vol.3, 1, 1988. [51] R. Iwankiewicz, Dynamical Mechanical Systems under Random Impulses, World Scientiﬁc, Ser. on Advances in Math. and Appl. Sci. (36), World Scientiﬁc, Singapore, 1995. [52] P.T.D. Spanos, Statistical linearization in structural dynamics, Appl. Mech. Rev., Vol.34, 1, 1-8, 1981. [53] L. Socha, T.T. Soong, Linearization in analysis of nonlinear stochastic systems, Appl. Mech. Rev., Vol.44, 10, 399-422, 1991. [54] T.K. Caughey, On the response of non-linear oscillators to stochastic excitation, Probab. Eng. Mech., Vol.1, 2-4, 1986. [55] G.Q. Cai, Y. K. Lin, A new approximate solution technique for randomly excited nonlinear oscillators, Intern. J. of Nonlinear Mech., Vol.23, 409-420, 1988.

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[56] S.H. Crandall, Non-Gaussian Closure for random vibration of nonlinear oscillators, Int. J. Nonlinear Mech., Vol.15, 303-313, 1980. [57] W.F. Wu, Y.K. Lin, Cumulant-neglect closure for nonlinear oscillators under parametric and external excitations, Int. J. Nonlinear Mech., Vol.19, 349-362, 1984. [58] K. Sobczyk, J. Tr¸¸ebicki, Maximum entropy closure for nonlinear stochastic systems, in: Vibration of Nonlinear, Random, and Time-Varying Systems, Proc. of 1995 Design Eng. Conf., DE-Vol.84-1, ASME, 1995. [59] R.L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach, N. York, 1963 (translation from Russian). [60] R.Z. Khasminskii, A limit theorem for the solution of diﬀerential equations with random right-hand side (in Russian), Tieoria Vieroyatn. Prim., Vol.11, 3, 1966. [61] P.E. Kloeden, E. Platen, The Numerical Solutions of Stochastic Diﬀerential Equations, Springer, Berlin, 1992. [62] M.F. Wehner, W. G. Wolfer, Numerical evaluation of path integral solutions to F-P-K equations, Phys. Rev. Vol.A27, 2663-70, 1983. [63] A. Naess, J. M. Johsen, Response statistics of nonlinear, compliant oﬀshore structures by the path integral solution method, Probab. Eng. Mech., Vol.8, 91-106, 1993. [64] K. Sobczyk, J. Tr¸¸ebicki, Maximum entropy principle in stochastic dynamics, Probab. Eng. Mech., Vol.5, 3, 102-110, 1990. [65] K. Sobczyk, J. Tr¸¸ebicki, Maximum entropy principle and nonlinear stochastic oscillators, Physica A, Vol.193, 448-468, 1993. [66] K. Sobczyk, J. Tr¸¸ebicki, Approximate probability distributions for stochastic systems: maximum entropy method, Comput. Methods Appl. Mech. Eng., Vol.168, 91-111, 1999. [67] J. Tr¸¸ebicki, K. Sobczyk, Maximum entropy principle and nonstationary distributions of stochastic systems, Probab. Eng. Mech., Vol.11 (3), 169-178, 1996. [68] C. Papadimitriou, Y. Haralampidis, K. Sobczyk, Optimal experiment design in stochastic structural dynamics, Probab. Eng. Mech., 2004. [69] K. Sobczyk, Information dynamics: Premises, challenges and results, Mech. Systems and Signal Processing, Vol.15(3), 475-498, 2001. [70] M. Shinozuka, C. J. Astill, Random eigenvalue problems in structural analysis, AIAA Journal, Vol.10, 4, 456-462, 1972. [71] K. Sobczyk, Free vibrations of elastic plate with random properties – the eigenvalue problem, J. Sound and Vibration, Vol.21, 4, 1972. [72] M. Shinozuka, G. Deodatis, Response variability of stochastic ﬁnite element systems, ASCE Jour. Eng. Mech., Vol.114, 39, 499-519, 1988. [73] R.G. Ghanem, P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, Berlin, 1991. [74] V.V. Bolotin, Prediction of Service Life of Machines and Structures (in Russian: Mashinostroienje, Moskov, 1984; English edition: ASME Press, N. York, 1989).

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[75] R.Z. Khasminskii, Stability of Diﬀerential Equations with Random Perturbation of Parameters (in Russian) Nauka, Moskov, 1969; English transl. Stochastic Stability of Diﬀerential Equations, Sijthoﬀ and Noordhoﬀ, ﬀ Alphen, 1980. [76] L. Arnold, V. Wihstutz, Lapunov exponents: a survey, in: L. Arnold, V. Wihstutz (Eds.): Lapunov exponents, Lecture Notes in Math. 1186, Springer, 1-26, Berlin, 1986. [77] S.T. Ariaratnam, W.C. Xie, Lapunov exponents and stochastic stability of coupled linear systems under real noise excitation, ASME J. Appl. Mech., Vol.59, 3, 664-673, 1992. [78] W. Wedig, Stability of nonlinear stochastic systems, in: C. Dafermos, G. Ladas, G. Papanicolau (Eds.) Lecture Notes in Pure and Appl. Math., Dekker, N. York, 1988. [79] C.G. Bucher, Y. K. Lin, Eﬀect of spanwise correlation of turbulence ﬁeld on the stability of long-span bridges, J. of Fluids and Structures, Vol.2, 437-451, 1988. [80] A. Tylikowski, Dynamic stability of nonlinear antisymetrically laminated crossply rectangular plates, J. Appl. Mech., ASME, Vol.56, 375-381, 1989. [81] A. Tylikowski, Stabilization of parametric vibrations of a nonlinear continuous system, Meccanica, Vol.38, 6, 659-668, 2003. [82] B.F. Spencer, Reliability of randomly excited hysteretic structures, Lecture Notes in Engineering (C. A. Brebbia , S. H. Orszag, eds.) Springer, N. York, 1986. [83] K. Sobczyk, Modelling of fatigue crack growth, Eng. Fracture Mech., Vol.24, 609-623, 1986. [84] O. Ditlevsen, Random fatigue crack growth – a ﬁrst passage problem, Eng. Fracture Mechanics, Vol.23,2, 467-477, 1986. [85] K. Doli´ n ´ ski, P. Colombi, Fatigue life time under stochastic loading with random overloading pulse trains, Comp. Meth. Appl. Mech. Eng., 168, 1999. [86] J. B. Roberts, P. D. Spanos, Stochastic averaging: An approximate method of solving random vibration problems, Int. J. Nonlinear Mechanics, Vol.21, 111-134, 1986. [87] L. Arnold, G. Papanicolau, V. Wihstutz, Asymptotic analysis of the Lapunov exponent and rotation number of the random oscillator and applications, SIAM J. Appl. Math., Vol.46, 3, 427-450, 1986. [88] L. Arnold, Stabilization by noise, ZAMM, M Vol.70, 7, 235-246, 1990. [89] W. Horsthemke, R. Lefever, Noise-Induced Transitions, Springer, Berlin, 1984. [90] K.A. Wiesenfeld, E. Knobloch, Eﬀect of noise on the dynamics of a nonlinear oscillator, Phys. Rev. Vol.A26, 5, 2946-2953, 1982. [91] K.R. Schenk-Hopp´ ´e, Bifurcation scenarios of the noisy Duﬃng-Van der Pol oscillator, Nonlinear Dynamics, Vol.11, 255-274, 1996. [92] L. Arnold, N. Sri Namachchivaya, K. R. Schenk-Hopp´ ´e, Toward an understanding of stochastic Hopf bifurcation: A case study, Int. J. Bifurc. and Chaos, Vol.6, 11, 1947-1975. [93] J. Grasman, J. B.T.M. Roerdink, Stochastic and chaotic relaxation oscillations, J. Statist. Physics, Vol.54, 3/4, 949-970, 1989.

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[94] T. Kapitaniak, Chaos in Systems with Noise, World Scientiﬁc, Singapore, 1988. [95] Y-Cheng Lai, Z. Liu, L. Billings, I.B. Schwartz, Noise-induced unstable variability and transition to chaos in random dynamical systems, Phys. Rev., Vol.E 67, 026210, 2003. [96] R. Benzi, A. Sutera, A. Vulpiani, The mechanism of stochastic resonance, Journ. of Physics A, Vol.141, L453-L457, 1981. [97] B. McNamara. K. Wiesenfeld, Theory of stochastic resonance, Phys. Rev. Vol.A39, 4854-4869, 1989. [98] L. Schimansky-Geier et al., Noise induced order: Stochastic resonance, Int. J. Bifurc. and Chaos, Vol.8, 5, 869-879, 1998. [99] K. Sobczyk, Korteweg-de Vries solitons in a randomly varying medium, Intern. Journ. Nonlin. Mech., Vol.27, 1, 1-8, 1992. [100] D. Munford, The Dawning of the Age of Stochasticity, in: V. Arnold, M. Atiyah, P. Lax, B. Mazur (Eds.) Mathematics: Frontiers and Perspectives, Amer. Math. Soc., 2000.

MULTIBODY DYNAMICS: BRIDGING FOR MULTIDISCIPLINARY APPLICATIONS Jorge A.C. Ambr´ o´sio IDMEC Instituto Superior Tecnico, ´ Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Abstract

Simple or complex systems characterized by large relative motions between their components ﬁnd in the multibody dynamics formalisms the most general and eﬃcient computational tools for their analysis. Initially restricted to the treatment of rigid bodies, the multibody methods are now widely used to describe the system components deformations, regardless of their linear or nonlinear nature. The ease of including in the multibody models diﬀerent descriptions of the contact problems, control paradigms or equations of equilibrium of other disciplines is demonstrated here to show the suitability of these approaches to be used in multidisciplinary applications

Keywords: Flexible multibody dynamics, contact, biomechanics, vehicle dynamics, railway dynamics, crashworthiness.

1.

Introduction

The design requirements of advanced mechanical and structural systems and the real-time simulation of complex systems exploit the ease of use of the powerful computational resources available today to create virtual prototyping environments. These advanced simulation facilities play a fundamental role in the study of systems that undergo large rigid body motion while their components experience material or geometric nonlinear deformations, such as vehicles, deployable structures, space satellites, machines operating at high speeds or robot manipulators. Some examples of engineering and biological systems for which the large overall motion is of fundamental importance are exempliﬁed in Fig. 1. If on the one hand the nonlinear ﬁnite element method is the most powerful and versatile procedure to describe the ﬂexibility of the system components, on the other hand the multibody dynamic formulations are the basis for the most eﬃcient computational techniques that 61 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 61–88. © 2005 Springer. Printed in the Netherlands.

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Figure 1. Natural biological and artiﬁcial engineering systems for which multibody dynamics provides irreplaceable modeling methodologies.

deal with large overall motion. Therefore, it is no surprise that many of the most recent formulations on ﬂexible multibody dynamics and on ﬁnite element methods with large rotations share some common features. In multibody dynamics methods, the body-ﬁxed coordinate frames are generally adopted to position each one of the system components and to allow for the speciﬁcation of the kinematic constraints that represent the restrictions on the relative motion between the bodies. Several formalisms are published suggesting the use of diﬀerent sets of coordinates, such as Cartesian [1], natural [2] and relative coordinates [3]. Depending on the type of applications, each of these types of coordinates has advantages and disadvantages. Due to the ease of the computational implementation, their physical meaning and the widespread knowledge of their features, all the formalisms presented in this work are based on the use of Cartesian coordinates. The methodological structure of the equations of motion of the multibody system obtained allows the incorporation of the equilibrium equations of a large number of disciplines and their solution in a combined form. The description of the structural deformations exhibited by the system components by using linear [5] or non-linear ﬁnite elements [6] in the framework of multibody dynamics is an example of the integration of the equations of equilibrium of diﬀerent specialties. Of particular importance for the applications pursued with the methodologies proposed is the treatment of contact and impact, which is introduced in the multibody systems equations by using either unilateral constraints

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[7] or a continuous contact force model [8]. The availability of the state variables in the multibody formulation allows for the use of diﬀerent control paradigms in the framework of vehicle dynamics, biomechanics or robotics and their integration with the multibody equations [9]. The coupling between the ﬂuid and structural dynamics equations allows for the development of applications, where the ﬂuid-structure interaction is analyzed, especially for cases with large absolute or relative rotations in the system components, are of importance [10, 11]. The research carried at IDMEC provides the examples oﬀered in this work. Application cases involving the modeling of realistic mechanisms, passive safety of road and rail vehicles, impact and human locomotion biomechanics, automotive and railway dynamics are used to demonstrate the developments reviewed here.

2.

Rigid Multibody Dynamics

A multibody system is deﬁned as a collection of rigid and/or ﬂexible bodies constrained by kinematic joints and eventually acted upon by a set of internal and/or external forces. The position and orientation of each body i in the space is described by a position vector ri and a set of rotational coordinates pi , which are organized in a vector as [1]: qi = [rT , pT ]Ti .

(1)

According to this deﬁnition, a multibody system with nb bodies is described by a set of coordinates in the form: q = [qT1 , qT2 , . . . , qTnb ]T .

(2)

The dependencies among system coordinates, which result from the existence of mechanical joints interconnecting several bodies, are deﬁned through the introduction of kinematic relationships written as [1]: Φ (q, t) = 0,

(3)

where t is the time variable, which is used only for the driving constraints. The second time-derivative of Eq. (3) with respect to time yields: ¨ (q, q, Φ ˙ q ¨, t) = 0 ≡ D¨ q = γ, (4) where D is the Jacobian matrix of the constraints, q ¨ is the acceleration vector and γ is the vector that depends on the velocities and time. The system kinematic constraints are added to the equations of motion using the Lagrange multipliers technique [1]. Denoting by λ the

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vector of the unknown Lagrange multipliers, the equations of motion for a mechanical system are written as

q ¨ f M DT = (5) λ γ D 0 where M is the global mass matrix, containing the mass and moments of inertia of all bodies, and f is the force vector, containing all forces and moments applied to the system bodies plus the gyroscopic forces. The Lagrange multipliers, associated with the kinematic constraints, are physically related with the reaction forces generated between the bodies interconnected by kinematic joints, given by [1] f (c) = −ΦTq λ,

(6)

The usual procedures to handle the integration of sets of diﬀerentialalgebraic equations must still be applied in this case in order to eliminate the constraint drift or to maintain it under control [1, 2].

Forward Dynamics The computational strategy used to solve the forward dynamics of the system, represented by Eq. (5), is outlined in Fig. 2. The solution procedure starts by the determination of the initial positions and velocities of the system components. Next, the system inertia, the Jacobian matrices, the forces and the right-hand-side of the kinematic acceleration constraint equations vectors, are calculated and assembled in the equations of motion. Equation (5) is then solved to ﬁnd the system accelerations, and in the process the Lagrange multipliers. By integrating the current velocities and the system accelerations, at time t, the new positions and velocities for time t + ∆t are calculated by using a variable

Figure 2.

Solution of the forward dynamic analysis of a multibody system.

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65

order, variable time-step integration procedure [1, 2, 4]. The forward dynamics simulation proceeds until the previously set ﬁnal time is reached. The procedure outlined in Fig. 2 is used in general purpose multibody dynamics codes, such as DAP-3D [1]. Throughout this work it is demonstrated that all engineering applications foreseen here are implemented, either by developing speciﬁc kinematic constraints or by implementing force models in Eq. (5).

Application Example of a Roller Coaster. When a body travels along a guide, not only its path has to be followed, but also its spatial orientation has to be prescribed, according to spatial characteristics of the curve. The formulation adopted to implement these kinematic constraints, using the moving Frenet frame associated with the track centerline based on the work by Pombo and Ambrosio ´ [12], is outlined next. Prescribed Motion Constraint. The objective here is to deﬁne the constraint equations that enforce that a point of a rigid body follows the reference path [12]. Consider a point R, located on a rigid body i, that has to follow the speciﬁed path depicted in Fig. 3. The path is deﬁned by a parametric curve g(L), which is controlled by a global parameter L that represents the length travelled along the curve until the current location of point R. The kinematic constraint is Φ(pmc,3) = 0 ≡ rR i − g(L) = 0,

(7)

R where rR i = ri +Ai s i represents the coordinates of point R with respect to the global coordinate system (x, y, z), ri is the vector that deﬁnes the location of the body-ﬁxed coordinate system (ξ, η, ζ)i , Ai is the transformation matrix from the body i ﬁxed coordinates to the global reference frame, and s R i represents the coordinates of point R with re-

x Figure 3.

Local frame alignment constraint.

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spect to the body-ﬁxed reference frame. For notational purposes (·) means that (·) is expressed in body-ﬁxed coordinates. The second part of the constraint ensures that the spatial orientation of body i remains unchanged with respect to the moving frame of the reference path represented in Fig. 3. Consider that (uξ , uη , uζ )i represent the unit vectors associated with the axis of the body-ﬁxed coordinate system (ξ, η, ζ)i . Let the Frenet frame of the general parametric curve g(L) be deﬁned by the principal unit vectors (t, n, b)L . At the initial time of analysis, the relative orientation between the body vectors (uξ , uη , uζ )i and the local frame (t, n, b)L leads to ⎫ ⎧ ⎫ ⎧ T ⎨n · uξ ⎬ ⎨a⎬ Φ(lf ac,3) = 0 ≡ bT · uξ − b = 0. (8) ⎭ ⎩ ⎭ ⎩ T c n · uζ This kinematic constraint ensures that the alignment remains constant throughout the analysis. The transformation matrix from the body i ﬁxed coordinates to the global coordinate system is: Ai = [uξ

uη

uζ ]i

(9)

deﬁning the following unit vectors as: u1 = {1

0

0}T ;

u2 = {0

1

0}T ;

u3 = {0 0 1}T . (10)

Equation (8) is now rewritten in a more usable form as: ⎫ ⎧ ⎫ ⎧ T ⎨n Ai u1 ⎬ ⎨a⎬ Φ(lf ac,3) = 0 ≡ bT Ai u1 − b = 0, ⎭ ⎩ ⎭ ⎩ T c n Ai u3

(11)

which constitutes the second part of the path following constraint.

Roller-Coaster Dynamics. Let the roaller-coaster rail be deﬁned with the spatial geometry described in Fig. 4. The path-following constraint is used to enforce the vehicles to follow the rail for the prescribed geometry. The roller coaster vehicle consists of a train with three cars that are interconnected by linking bars, represented in Fig. 5. The multibody model of the vehicle is assembled using eleven rigid bodies, corresponding to 3 car bodies, 6 wheelsets and 2 connection bars. The complete vehicle model only has 1 d.o.f., which is the longitudinal motion of the cars. The motion of the vehicle is guided by the dynamics described by Eq. (5). A view of the motion of the roller coaster is displayed in Fig. 6 and the details of the analysis are found in reference [13].

Multibody Dynamics: Bridging for Multidisciplinary Applications

Figure 4.

67

View of the roller coaster as used in the simulation.

z 3

6

10

7

11 x

1

Figure 5.

Figure 6.

2

4

5

8

9

Multibody model of the roller coaster vehicle.

Snap shots of the roller coaster motion as observed from the second car.

Note that the study of these vehicles only requires the use of the pathfollowing constraint. The contact forces are not explicitly used but they can be calculated from the Lagrange multipliers associated to the path constraint.

Inverse Dynamics In many applications all external forces are known and the motion of the system is also known. Therefore, the only unknowns are the internal forces. Let the ﬁrst row of Eq. (5) be re-written as M¨ q + DT λ = g

(12)

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which is q. DT λ = g − M¨

(13)

Equation (13) emphasizes that the only unknowns of the system are the Lagrange multipliers. The reaction forces at the joints are given by: g(c) = −DT λ.

(14)

The solution of the equations of motion in inverse dynamics can be used to solve for the internal forces of the human body, i.e., muscle and anatomical joint reaction forces, that develop for known motions.

Application to Biomechanics: Gait Analysis For biomechanical applications in gait a three-dimensional model, presented in Fig. 7, is used [14]. It has a kinematic structure made of thirtythree rigid bodies, interconnected by revolute and universal joints, in such a way that sixteen anatomical segments are represented. 16

3 v20

111

12 3

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v12

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v17

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v4

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Figure 7. The biomechanical model, its kinematic structure and a detail of the ankle joint.

Joint Moments-of-Force: A Determinate Problem. To drive the biomechanical model in the inverse dynamic analysis, joint actuators such as the one represented in Fig. 8 for the knee joint, are speciﬁed. The actuators are the kinematic constraints in which the angle between two adjacent segments is a known function of time. These additional equations are added to the system kinematic equation so that the number of

Multibody Dynamics: Bridging for Multidisciplinary Applications

69

O m1 m1

Im1 =

O m2 m2

Im22 = I m3

Figure 8.

O m3

m3

Joint actuator associated with the knee joint and muscle actuator.

non-redundant constraint equations becomes equal to the number of coordinates. Equation (13) is solved to obtain the Lagrange multipliers associated with the joint actuators, representing the net moments-of-force of the muscles that cross those joints. The inverse dynamics problem, as stated here, is totally determined.

Muscle Forces: A Redundant Problem. The solution of the inverse dynamics problem with muscle actuators introduces indeterminacy in the biomechanical system, since it involves more unknowns than equations of motion. By using optimization techniques to ﬁnd the muscle forces that minimize a prescribed objective function, a solution for the problem is obtained. The optimization problem is stated as: minimize F0 (ui ) ⎧ ⎪ ⎨ fj (ui ) = 0, subject to fj (ui ) 0, ⎪ ⎩ lower ui uupper ui i

j = 1, ..., nec , j = (nec + 1) , . . . , ntc , i = 1, . . . , nsv

(15)

where ui are the state variables bounded respectively by ulower and uupper , F0 (ui ) is the objective or cost function to minimize and fi (ui ) are constraint equations that restrain the state variables. The minimization of the cost functions simulate the physiological criteria adopted by the central nervous system when deciding which muscles to recruit and what level of activation to obtain the adequate motion. Several cost functions have been proposed for the study of the redundant problem in biomechanics [15]. The minimization of the sum of the cube of the muscle stresses [16] is often used in applications involving human

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Figure 9.

Figure 10.

Lower extremity muscle apparatus.

Muscle forces for the hamstrings and triceps surae.

Multibody Dynamics: Bridging for Multidisciplinary Applications

locomotion

m m 3 Fl Fl˙ m σ ¯ a F0 = F0m2 m=1 n ma

71

(16)

¯ is the speciﬁc where nma are the number of muscle actuators and σ muscle strength with a constant value of 31.39 N/cm2 [17]. The human locomotion apparatus, represented in Fig. 9, is modeled having the muscles with the physiological data described in Yamaguchi [17]. The state variables associated with muscle actuators represent muscle activations that can only assume values between 0 and 1. To illustrate the type of results obtained for the muscle forces in a case of normal cadence gait of a 50%ile male, the muscle forces for the hamstrings and triceps surae are presented in Fig. 10.

3.

Contact and Impact

Let a triangular patch, where point k of the body shown in Fig. 11 will impact, be deﬁned by points i, j and l. The normal to the outside surface of the contact patch is deﬁned as n = rij × rjl . The position of the point k with respect to point i of the surface is rik = rk − ri

(17)

which is decomposed in a tangential and a normal component, given by (18) rtik = rik − rTik n n; rnik = rTik n n. The necessary conditions for contact are that node k penetrates the ‘front’ surface of the patch, but not through its ‘back’ surface, with which a thickness e is associated. These conditions are written as 0 rTik n e.

k

riik

rikn = ( ik

rlli

l k*

i

Figure 11.

rikt = rik − ( ik

rij

(19)

)n n

)n

r jjl j

Contact detection between a ﬁnite element node and a surface.

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The remaining necessary conditions for contact result from the need for the node to be inside of the triangular patch. These three extra conditions are t T ˜ rik rij n 0;

T T t t ˜ rik rjl n 0 and ˜ rik rki n 0.

(20)

Equations (19) and (20) are necessary conditions for contact. However, depending on the contact force model actually used, they may not be suﬃcient to ensure eﬀective contact.

Unilateral Constraints If contact between a node and a surface is detected, a kinematic constraint is imposed. For ﬂexible bodies let us assume a fully plastic nodal contact, i.e., the normal components of the node k velocity and acceleration, with respect to the surface, are null during contact (−) (−)T (−) (−)T ¨k = q ¨k − q ¨k n n (21) q˙ k = q˙ k − q˙ k n n; q (−)

(−)

where q˙ k and q ¨k represent the nodal velocity and acceleration immediately before impact respectively . The kinematic constraint implied by Eqs. (21) is removed when the normal reaction force between the node and the surface becomes opposite to the surface normal, i.e., fkn = −ffkT n > 0.

(22)

It should be noted that the contact force is related to the Lagrange multiplier associated by the kinematic constraint deﬁned by Eqs. (22). Therefore, the change of sign of the force is in fact the change of sign of the multiplier. This contact model is not suitable to be used directly with rigid body contact. The sudden change of the rigid body velocity and acceleration would imply that the velocity and acceleration equations resulting from the kinematic constraints would not be fulﬁlled. Other forms of this contact model can be found in the work by Pfeiﬀer and Glocker [7].

Continuous Contact Force Model An alternative description of contact considers this to be a continuous event where the contact force is a function of the penetration between the surfaces. This leads to the continuous force contact model, proposed by Lankarani and Nikravesh [8], and brieﬂy described here. Let the contact force between two bodies be written as (23) fs,i = Kδ n + Dδ˙ u

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where δ is the pseudo-penetration, δ˙ is the pseudo-velocity of penetration, K is the equivalent stiﬀness, D is a damping coeﬃcient and u is a unit vector normal to the impacting surfaces. Using the hysteresis dissipation model and the equivalent stiﬀness, calculated for instance for Hertzian elastic contact [18], the nonlinear contact force is 3 1 − e2 δ˙ n u (24) fs,i = Kδ 1 + 4 δ˙ (−) where δ˙ (−) is the initial contact velocity and e is the restitution coeﬃcient. Note that K is a function of the geometry and material properties of the impacting surfaces.

Application to Railway Dynamics – The Wheel-Rail Contact Problem One of the interesting applications of multibody dynamics with contact mechanics is the description of the wheel-rail contact in railway dynamics, represented in Fig. 12. The stability of the running vehicle depends ultimately on the rail-wheel contact and on the vehicle primary suspension. Therefore, methodologies that provide accurate models to represent the phenomena are of particular importance. In a general case of a railway vehicle one or two points of each wheel are in contact with the rail, as shown in Fig. 12. The diametric section that contains the wheel ﬂange contact point makes an angle sfRw with the diametric section that contains the wheel tread contact point. The possibility of detecting contact in diﬀerent diametric sections allows predicting derailment and it is, therefore, of utmost importance. Let the generalized geometry of the rail and wheel be described by generalized surfaces resulting from sweeping the rail proﬁle along the rail Fla (Le

ct ct) e Tre Tr con

Figure 12.

Two points of contact in the rail and wheel surfaces: lead contact.

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centerlines and the wheel proﬁles around the base circle of the wheel. In order to ensure that the search for the contact points is between convex surfaces, the wheel proﬁle is divided in treat and ﬂange proﬁles. The contact between the rail and one of the wheel surfaces is described generically in Fig. 13, where the mating surfaces are represented as free surfaces.

x Figure 13.

Candidates to contact points between two parametric surfaces.

The geometric conditions for contact between the convex surfaces are deﬁned by vector products deﬁned between the surfaces. The ﬁrst condition is that the surfaces normals ni and nj at the candidates to contact points have to be parallel. This condition means that nj has null projections over the tangent vectors tui and tw i : ⎧ ⎨nTj tui = 0, (25) nj × ni = 0 ⇔ ⎩nT tw = 0. j i The second condition is that the vector d, which represents the distance between the candidates to contact points, has to be parallel to the normal vector ni . This condition is mathematically written as: ⎧ ⎨dT tui = 0, (26) d × ni = 0 ⇔ ⎩dT tw = 0. i The geometric conditions (25) and (26) provide four nonlinear equations with four unknowns, the four parameters u, w, s and t that deﬁne the two surfaces. This system of equations provides solutions for the location of the candidates to contact points that have to be sorted out.

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75

The coordinates of the candidates to contact points are determined by solving an optimal problem and the distance between such points is calculated in the process. The points are in contact if dT nj 0.

(27)

When contact is detected, the normal force is calculated using Eq. (24) and the tangential forces are evaluated using the Kalker theory, the Polach formulation or the Heuristic nonlinear creep model. It has been found that the Polach formulation provides the best approach for the tangent forces, and it is used hereafter [13]. The wheel-rail contact model outlined here is used to model the ML95 trainset, shown in Fig. 14, which is used by the Lisbon subway company (ML) for passengers’ traﬃc. The multibody model of the trailer vehicle of the train, developed in the work by Pombo [13], is composed of the car shell suspended by a set of springs, dampers and other rigid connecting elements on the bogies. This assembly of connective elements constitutes the secondary suspension, sketched in Fig. 15, which is the main one responsible for the passenger’s comfort. The connections between the bogies chassis and the wheelsets, also achieved by another set of springs, dampers and rigid connecting ele-

Figure 14.

Figure 15.

Schematic representation of the ML95 trainset.

Secondary suspension model of the ML95 trailer vehicle.

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Axlebox

Three-dimensional spring-damper elements a)

Wheelset b)

Figure 16. Primary suspension model of the ML95 trailer bogie: a) Threedimensional spring-damper elements; b) Suspension model with springs and dampers.

Figure 17. Lift of the right wheel of the leading wheelset for vehicle forward velocities of 10 and 20 m/s, using the Kalker linear theory.

ments, constitute the primary suspension represented in Fig. 16. The primary suspension is the main suspension responsible for the vehicle running stability. The simulation results of the vehicle, running in a circular track with a radius of 200 m with velocities of 10 and 20 m/s, show that the prediction of ﬂange contact is of fundamental importance. Fig. 17 shows that contact forces obtained with the Kalker linear theory originate the lift of the outer wheel of the front wheelset at the entrance of the curve. Despite this wheel lift, derailment does not occur and the analysis proceeds up to end. Nevertheless, such results are not realistic since the existence of ﬂange contact involves high creepages, which makes the Kalker linear theory inappropriate to compute the creep forces. Therefore only the Heuristic and the Polach creep force models must be considered. Another aspect to note is that ﬂange contact is detected with all creep force models. Even when running at the speed of 10 m/s, where the centrifugal forces eﬀect is balanced by the track cant, ﬂange contact occurs. Lateral ﬂange forces develop on the wheels of both wheelsets of the front bogie as presented in Fig. 18 for a vehicle forward velocity of 10 m/s and

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using the Polach creep force model. During curve negotiation, the outer wheel of the leading wheelset and the inner wheel of the rear wheelset have permanent ﬂange contact. Referring to Fig. 19, for the velocity of 10 m/s, the ﬂange contact occurs on the outer and in the inner wheels of the vehicle. For the velocity of 20 m/s, only the outer wheels have ﬂange contact. This is explained by the fact that, when running at 20 m/s, the vehicle negotiates the curve with a velocity higher than the balanced speed.

4.

Flexible Multibody Dynamics with Plastic Hinges

Many applications of multibody dynamics require the description of the ﬂexibility of its components. For structural crashworthiness it is 25 000

Le ft W s 3 ( P ol ach) R ig ht W s 3 ( P olach )

20 000

La te ra l Flang e F orc e [N ]

Le ft W s 4 ( P ol ach) R ig ht W s 4 ( P olach )

15 000

10 000

5 000

0

-5 0 0 0 0

3

6

9

12

15

18

21

24

27

30

T im e [s ]

Figure 18. Lateral ﬂange forces on the wheels of both wheelsets of the front bogie for a vehicle forward velocity of 10 m/s, using the Polach creep force model. Flange contact Front wheelset (Ws 4)

Rear wheelset (Ws 3)

Flange contact

Figure 19.

Contact conﬁguration during curve negotiation.

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often unfeasible to use large nonlinear ﬁnite element models. The use of multibody dynamics with plastic hinges is an alternative formulation that allows building insightful models for crashworthiness.

Formulation of Plastic Hinges In many impact situations, the individual structural members are overloaded giving rise to plastic deformations in highly localized regions, called plastic hinges. These deformations, presented in Fig. 20 for structural bending, develop at points where maximum bending moments occur, load application points, joints or locally weak areas [19]. Multibody models obtained with this method are relatively simple, which makes the procedure adequate for the early phases of vehicle design. The methodology described herein is known in industry as conceptual modeling [20].

Figure 20.

Localized deformations on a beam and a plastic hinge.

The plastic hinge concept has been developed by using generalized spring elements to represent constitutive characteristics of localized plastic deformation of beams and kinematic joints to control the deformation kinematics [21], as illustrated in Fig. 21. The characteristics of the spring-damper that describes the properties of the plastic hinge are obtained by experimental component testing, ﬁnite element nonlinear analysis or simpliﬁed analytical methods. The plastic hinge constitutive equation can be modiﬁed to account for the strain rate sensitivity of some materials. A dynamic correction factor is used to account for the strain rate sensitivity given by [21]. Ps = 1 + 0.07V V00.82 , Pd /P

(28)

where Pd and Ps are the dynamic and static forces, respectively, and V0 is the relative velocity between the adjacent bodies. The force or moment

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Multibody Dynamics: Bridging for Multidisciplinary Applications

Analytical Test

Moment ( kNm) 15 30 45 0 Figure 21.

0

79

.05

.10

.15 .20 .25 Rotation ( Rad )

Plastic hinge bending moment and its constitutive relationship.

to apply due to the plastic hinge is multiplied by the ratio calculated in Eq. (28) before it is used in the force vector of the multibody equations of motion.

Application of the Plastic Hinge Approach to Crashworthiness of Surface Vehicles The multibody of an oﬀ-road vehicle with three occupants, shown in Fig. 22, is used to demonstrate the plastic hinge approach to complex crash events. The model includes all moving components of the vehicle, suspension systems and wheels, and a tire model [16]. The biomechanical models for the occupants are similar to those described in Fig. 7. The three occupants, with a 50%tile, integrated in the vehicle are seated. Two occupants in the front of the vehicle have shoulder and lap

Figure 22.

Initial position of the vehicle and occupants for the rollover.

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seat belts while the occupant seated in the back of the vehicle has no seatbelt. The rollover situation for the simulation is such that the initial conditions correspond to experimental conditions where the vehicle moves on a cart with a lateral velocity of 13.41 m/s until the impact with a waterﬁlled decelerator system occurs. The vehicle is then ejected with a roll angle of 23 degrees. The results of the simulation are pictured in Fig. 23 by several frames of the animation. The vehicle ﬁrst impacts the ground with its left tires. At this point the rear occupant is ejected. The rollover motion of the vehicle proceeds with an increasing angular velocity, mainly due to the ground – tire contact friction forces. The occupants in the front of the vehicle are held in place by the seat belts. Upon continuing its roll motion, the vehicle impacts the ground with its rollbar cage, while the ejection of the rear occupant is complete. Bouncing from the inverted position, the vehicle completes another half turn and impacts the ground with the tires again. The HICs for all occupants largely exceed 1000, which indicates a very high probability of fatal injuries for the occupants under the conditions simulated. An experimental test of the vehicle was carried out at the Transportation Research Center of Ohio [22], being an overview of the footage obtained shown in Fig. 24. The outcome of the experimental test, which is rather similar to the outcome of the simulation, is further used to validate the vehicle model [21].

Figure 23. occupants.

Views of the outcome of the rollover simulation of a vehicle with three

Figure 24.

View of the experimental test for the truck rollover.

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5.

Flexible Multibody Dynamics with Finite Elements

General Equations of Motion of a Single Body Let the principle of the virtual works be used to express the equilibrium of the ﬂexible body in the current conﬁguration t+∆t and let an updated Lagrangean formulation be used to obtain the equations of motion of the ﬂexible body [23]. Let the ﬁnite element method be used to represent the equations of motion of the ﬂexible body. Referring to Fig. 25, the assembly of all ﬁnite elements used in the discretization of a single ﬂexible body results in its equations of motion written as [6] ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ gr ¨ r sr 0 Mrr Mrf Mrf ⎣Mφr Mφφ Mφf ⎦ ⎣ω ˙ ⎦ = ⎣g φ ⎦ − ⎣s φ ⎦ − ⎣0⎦ Mf r Mf φ Mf f u ¨ g f s f f ⎤⎡ ⎤ ⎡ 0 0 0 0 ⎦ ⎣ 0 ⎦ (29) 0 − ⎣0 0 u 0 0 KL + KN L

⎡

where ¨ r and ω ˙ are respectively the translational and angular accelerations of the body-ﬁxed reference frame and u ¨ denotes the nodal accelerations measured in body ﬁxed coordinates. The local coordinate frame ξηζ attached to the ﬂexible body, is used to represent the gross motion of the body and its deformation.

t

ζ

t

η t

b ∆t

ζ

ted updated ation

t ∆

ζ

0 0

η

t ′

t

Figure 25.

t

h

∆ b

General motion of a ﬂexible body.

b

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Linear Deformations of Flexible Bodies In many situations it is enough to consider that the components of the multibody system experience only linear elastic deformations. Furthermore, assume that the mode superposition technique can be used. Then, the ﬂexible part of the body is described by a sum of selected modes of vibration as (30) u = Xw where the vector w represents the contributions of the vibration modes towards the nodal displacements and X is the modal matrix. Due to the reference conditions, the modes of vibration used here are constrained modes. Due to the assumption of linear elastic deformations the modal matrix is invariant. The reduced equations of motion for a linear ﬂexible body are [5] gr sr Mrf X Mr 0 q ¨r = (31) − − Λw w ¨ I X T gf XT sf XT Mf r where Λ is a diagonal matrix with the squares of the natural frequencies associated with the modes of vibration selected. For a more detailed discussion on the selection of the modes used the interested reader is referred to [5]. The methodology is demonstrated through the application to the simulation of the unfolding of a satellite antenna, the Synthetic Aperture Radar (SAR) antenna that is a part of the European research satellite

Figure 26. The European satellite with the folded and unfolded conﬁgurations of the antenna.

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ERS-1, represented in Fig. 26. During the transportation the antenna is folded, in order to occupy as little space as possible. After unfolding, the mechanical components take the conﬁguration shown in Fig. 26(a). The SAR antenna consists in two identical subsystems, each with three coupled four-bar links that unfold two panels on each side. The central panel is attached to the main body of the satellite. Each unfolding system has two degrees of freedom, driven individually by actuators located in joints A and B. In the ﬁrst phase of the unfolding process the panel 3 is rolled out, around an axis normal to the main body, by a rotational spring-damper-actuator in joint A, while the panel 2 is held down by blocking the joints D and E. The second phase begins with the joint A blocked, next the panels 2 and 3 are swung out to the ﬁnal position by a rotational spring-damped-actuator. The model used for one half of the folding antenna, schematically depicted Fig. 27, is composed of 12 bodies, 16 spherical joints and 3 revolute joints. The central panel is attached to the satellite, deﬁned as body 1, which has much higher mass and inertia. The data for this antenna is reported in the work of Anantharamann [24]. Panel 3 (B3) Panel 2 (B2 (B2))

1.3 Actuator (1) (1)

a)

Figure 27. model.

Panel 1 (B1)

b)

c)

The SAR antenna: a) half unfolded state b) folded antenna; c) multibody

In the ﬁrst phase of the unfolding antenna, the rotational springdamped-actuator is applied in the revolute joint R3 . For the second phase, the revolute joint R3 is blocked and the system is moved to the next equilibrium position by a spring-actuator-damped positioned in joint R1 . The unfolding processes for rigid and ﬂexible models are shown in Fig. 28, only for its ﬁrst phase. The diﬀerent behavior between the rigid and the ﬂexible models is noticeable in Fig. 28. Though not shown here, the rotational actuator moment responsible for the start of the unfolding is not correctly predicted by the rigid multibody model. Being a very light and ﬂexible structure, the discrepancies, if not identiﬁed during the design stage, would lead to the failure of the unfolding process.

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Figure 28. models).

First phase of the unfolding of the SAR antenna (rigid and ﬂexible

Nonlinear Deformations in Multibody Systems For ﬂexible multibody systems experiencing nonlinear geometric and material deformations, the equations of motion for a ﬂexible body are given by Eq. (29). However, due to the time variance of all its coeﬃcients, Eq. (29) is not eﬃcient for computational implementation. Instead, by considering a lumped mass formulation for the mass matrix and referring the nodal accelerations to the inertial frame, the equations of motion for a single ﬂexible body take the form of [6] ⎡ ¯T ¯ ∗A mI + AM ⎣ − AM ¯ ∗S T 0

⎤⎡ ⎤ ¯ ∗S −AM 0 ¨ r ˙ ⎦ J + ST M∗ S 0 ⎦ ⎣ω q ¨f 0 Mff ⎤ ⎡ ¯ δ fr + AC = ⎣n − ω˜J ω − ST C δ − ¯ IT C θ ⎦ (32) g f − f − (KL + KN L ) u

where the absolute nodal displacements are written as

T − x ¨ ˙ ¨ r d ˜ + δ A k k =u ¨ k + q ¨kf ≡ α ¨ k ω˙ 0 I ˜ (xk + δ k ) + 2˜ ω δ˙ k ω ˜ ω (33) + ω ˜ θ˙ k with xk being the position of node k in the reference conﬁguration. In Eq. (32) M∗ is a diagonal mass matrix containing the mass of the n boundary nodes, T T T ¯ T = [A . . . A]T , S = x A ... x ˜n + δ˜n ˜1 + δ˜1

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Multibody Dynamics: Bridging for Multidisciplinary Applications

and ¯ I = [I . . . I]T where A is the transformation matrix from the body ﬁxed to global coordinate coordinates and xk denotes the position of node k. Vectors C δ and C θ represent respectively the reaction force and moment of the ﬂexible part of the body over the rigid part, given by C δ = g δ − Fδ − (KL + KN L )δδ δ − (KL + KN L )δθ θ ,

(34)

C δ = g θ − Fθ − (KL + KN L )θδ δ − (KL + KN L )θθ θ .

The coupling between the rigid body motion and the system deformations is fully preserved. For a more detailed description of the formulation, and the notation, the interested reader is referred to reference [6]. As an application example of the nonlinear formulation for ﬂexible multibody systems, a sports vehicle with a front crash-box is analyzed for various impact scenarios, represented in Fig. 29, where the angle of Angle 10º no friction

Angle 20º no friction

Angle 20º friction = 0.5

Angle 10º friction = 0.5

10 cm ramp

(a)

(d)

Figure 29.

(e)

Diﬀerent impact scenarios for the sports vehicle.

Figure 30. Motion of the vehicle for a 20◦ oblique impact without contact friction and for impact with an oblique surface for a vehicle traveling over a ramp.

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impact and the topology of the road are diﬀerent. The simulations are carried until the vehicle reaches a full stop. The vehicle motion, for the oblique impact scenario presented in Fig. 29, is characterized by a slight rotation of the vehicle during impact. This rotation is more visible in the case of frictionless impact. At the simulated impact speed the inﬂuence of the car suspension elements on the deformation mechanism is minimal.

6.

Conclusions

The multibody dynamics formalisms provide an extremely eﬃcient framework to incorporate diﬀerent disciplines. The behavior of a good number of phenomena in diﬀerent problems can be represented by kinematic constraints (e.g., contact, muscle action, guidance) or by contact forces (e.g, impact phenomena, control, general interactions). However, diﬀerent disciplines use diﬀerent preferred numerical methods to solve their equilibrium equations which lead to diﬃculties in the co-simulation of diﬀerent systems. The use of multibody formalisms in biomechanics presents a strong increase due to the suitability to model contacts, muscles, anatomical joints, data processing, etc. The treatment of structural components with large rotations or of rotating bodies with structural deformations ﬁnds in the ﬂexible multibody dynamics eﬃcient methods to deal with the problem. A continued eﬀort to close the gap between the ﬂexible multibody dynamics and the nonlinear ﬁnite element method is required. The need for more robust and eﬃcient numerical methods to handle the speciﬁc forms of the MBS equations and the discontinuities associated to intermittent and ‘fast’ behaviors are still required.

Acknowledgements The contents of this work result from a team eﬀort and collaborations with many co-workers among which the contribution by Miguel Silva, Joao ˜ Gon¸calves, ¸ Jo˜˜ao Pombo, Manuel Seabra Pereira, Jo˜ ao ˜ Abrantes, Augusta Neto and Rog´ ´erio Leal are gratefully acknowledged.

References [1] P. Nikravesh, Computer-Aided Analysis of Mechanical Systems, Prentice-Hall, Englewood Cliﬀs, New Jersey 1988. [2] J. Garcia de Jalon, E. Bayo, Kinematic and Dynamic Simulation of Mechanical Systems – The Real-Time Challenge, Springer-Verlag, Berlin 1994. [3] P. Nikravesh and G. Gim, Systematic construction of the equations of motion for multibody systems containing closed kinematic loops, Journal of Mechanical Design, Vol. 115, No.1, pp.143–149, 1993.

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[4] C.W. Gear, Numerical solution of diﬀerential-algebraic equations, IEEE Transactions on Circuit Theory, Vol. CT-18, pp.89–95, 1981. [5] J. Gon¸calves and J. Ambr´ ´ osio, Complex ﬂexible multibody systems with application to vehicle dynamics, Multibody System Dynamics, Vol. 6, No.2, pp.163–182, 2001. [6] J. Ambr´ ´ osio and P. Nikravesh, Elastic-plastic deformation in multibody dynamics, Nonlinear Dynamics, Vol. 3, pp.85–104, 1992. [7] F. Pfeiﬀer and C. Glocker, Multibody Dynamics with Unilateral Contacts, John Wiley and Sons, New York 1996. [8] H. Lankarani and P. Nikravesh, Continuous contact force models for impact analysis in multibody systems, Nonlinear Dynamics, Vol. 5, pp.193–207, 1994. ˇ [9] M. Valasek, Z. Sika, Evaluation of dynamic capabilities of machines and robots, Multibody System Dynamics, Vol. 5, pp.183–202, 2001. [10] H. Møller and E. Lund, Shape Sensitivity Analysis of Strongly Coupled FluidStructure Interaction Problems, [in:] Proc. 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA. AIAA Paper No.2000–4823, 2000. [11] H. Møller, E. Lund, J. Ambr´ ´ osio and J. Gon¸¸calves, Simulation of ﬂuid loaded ﬂexible multiple bodies, Multibody Systems Dynamics, Vol. 13, No.1, 2005. [12] J. Pombo and J.Ambr´ ´ osio, General spatial curve joint for rail guided vehicles: kinematics and dynamics, Multibody Systems Dynamics, Vol. 9, pp.237–264, 2003. [13] J. Pombo and J. Ambr´ ´ osio, A multibody methodology for railway dynamics applications, Technical Report IDMEC/CPM-04/002, IDMEC, Instituto Superior T´ ´ecnico, Lisboa, Portugal, 2004. [14] M. Silva and J. Ambr´ ´ osio, Kinematic data consistency in the inverse dynamic analysis of biomechanical systems, Multibody System Dynamics, Vol. 8, pp.219– 239, 2002. [15] D. Tsirakos, V. Baltzopoulos and R. Bartlett, Inverse Optimization: Functional and Physiological Considerations Related to the Force-Sharing Problem, Critical Reviews in Biomedical Engineering, Vol. 25, Nos.4&5, pp.371–407, 1997. [16] M. Silva and J. Ambr´ ´ osio, Human Motion Analysis Using Multibody Dynamics and Optimization Tools, Technical Report IDMEC/CPM-04/001, IDMEC, Instituto Superior T´ ´ecnico, Lisboa, Portugal, 2004. [17] G.T.Yamaguchi, Dynamic Modeling of Musculoskeletal Motion, Kluwer Academic Publishers, Boston 2001. [18] H. Hertz Gesammelte Werke , Leipzig, Germany 1895. [19] N.W. Murray, The static approach to plastic collapse and energy dissipation in some thin-walled steel structures, [in:] Structural Crashworthiness, N. Jones and T. Wierzbicki [eds.], pp.44–65, Butterworths, London 1983. [20] C.M. Kindervater, Aircraft and helicopter crashworthiness: design and simulation, [in:] Crashworthiness Of Transportation Systems: Structural Impact And Occupant Protection, J.A.C. Ambrosio, ´ M.S. Pereira and F.P Silva [eds.], NATO ASI Series E. Vol. 332, pp.525–577, Kluwer Academic Publishers, Dordrecht 1997.

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[21] P.E. Nikravesh, I.S. Chung, and R.L. Benedict, Plastic hinge approach to vehicle simulation using a plastic hinge technique, J. Comp. Struct. Vol. 16, pp.385–400, 1983. [22] 30 mph Rollover Test of an AM General Model M151-A2 1/4 Ton Jeep, The Transportation Research Center of Ohio, Test Report, 1985. [23] K.-J. Bathe and S. Bolourchi, Large displacement analysis of three-dimensional beam structures, Int. J. Num. Methods in Engng., Vol. 14, pp.961–986, 1979. [24] M. Anantharaman and M. Hiller, Numerical simulation of mechanical systems using methods for diﬀerential-algebraic equations, Int. J. Num. Meth. Eng., Vol. 32, pp.1531–1542, 1991.

RAPID FORMATION OF STRONG GRADIENTS AND DIFFUSION IN THE TRANSPORT OF SCALAR AND VECTOR FIELDS Konrad Bajer Institute of Geophysics Warsaw University [email protected]

Abstract

An important issue in the theory of transport by moving ﬂuids is the role of dissipation when the medium is nearly ideal. The central problem of this nature is understanding of the viscous dissipation at very large Reynolds numbers. We will discuss a few problems in the same category but linear and therefore more promising although, as it turns out, surprisingly rich and far from being solved. Their common denominator is the interplay between diﬀusion and advection. In a typical ﬂow the latter tends to decrease the characteristic length scales of the spatial variations of the transported quantity, thus increasing the rate of diﬀusion. Depending on a particular conﬁguration, either this rapid diﬀusion prevails and eﬃciently annihilates all gradients, or a kind of balance is reached and a quasi-steady dissipative structure emerges.

Keywords: Accelerated diﬀusion, shear dispersion, current sheets, passive scalar, vortex dynamics, Poiseuille ﬂow.

Introduction Dissipative processes are, in general, strongly aﬀected by the motion of the medium. In a stationary ﬂuid, a passive quantity like temperature or concentration of a contaminant would change its distribution only through molecular processes that, in a wide range of physical problems, may be well modelled as ‘simple’ Fickian diﬀusion. When the ﬂuid is in motion, the diﬀusive process in the Lagrangian frame of every material element may result in great complexity of the Eulerian distribution of the transported quantity. In particular, the interplay of advection and diﬀusion typically increases the rate at which the latter smoothes out the spatial variations of that quantity’s distribution. The archetypal 89 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 89–101. © 2005 Springer. Printed in the Netherlands.

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formula describing this interplay is the advection-diﬀusion equation for an intensive quantity, say temperature, ∂T + u · ∇T = κ∇2 T. ∂t

(1)

The dynamics of various physical processes, other than advectiondiﬀusion of a passive scalar, often reduces to the Eq. (1) or to another linear equation of similar type that exhibits the same salient feature of enhanced rate of diﬀusion associated with non-uniform ﬂuid motion. In simple geometries the evolution of magnetic ﬁelds in electrically conducting liquids and plasmas can sometimes be reduced exactly to an advection-diﬀusion problem with important implications in all branches of magnetohydrodynamics whether cosmic, solar, geophysical or that applied in the theory of nuclear fusion devices and in metallurgy. The evolution of weak background vorticity in presence of a strong coherent structure – an important mechanism in two-dimensional turbulence – is described by a linearised vorticity equation that, in spite of some extra complications, is a linear equation (scalar when the ﬂow is twodimensional) containing both advection and diﬀusion ingredients. The same equation also describes the evolution of small perturbations induced in a strong vortex by a weak external irrotational ﬂow. We shall illustrate the phenomenon, called accelerated diﬀusion, with simple solutions of the Eq. (1) for two basic steady ﬂows u(x).

1.

Accelerated Diﬀusion

Let us consider an initial value problem for the advection-diﬀusion equation (1) in an unbounded two-dimensional domain. The problem is linear, so every intial distribution of the scalar, T (x, t = 0), can be decomposed into independently evolving Fourier modes. For simplicity we take the initial distribution to be harmonic in the x direction (see Fig. 1b), (2) T (x, t = 0) = T0 eik0 x , i.e. the initial wave vector is k0 = (k0 , 0). We shall consider two steady ﬂows u(x) (see Fig. 1a) - a stagnation point ﬂow (irrotational linear strain), u(x) = (−αx, αy, 0) ,

(3)

u(x) = (αy, 0, 0) .

(4)

and a linear shear ﬂow

91

Strong Gradients & Diﬀusion in the Transport . . .

(a)

(b)

Figure 1. (a) Two types of the ﬂow considered — a stagnation point ﬂow (irrotational linear strain, top) and linear shear (bottom); (b) Initial distribution of the scalar ﬁeld subject to advection and diﬀusion.

Both ﬂows impose a characteristic advection time scale determined by the value of strain/shear coeﬃcient α, tA = α−1 .

(5)

The initial scalar distribution has a characteristic length scale, k0−1 , and the associate diﬀusion time scale −1 . (6) tD = κk02 The ratio of the two time scales is called Schmidt number when the transported scalar ﬁeld T is the mass concentration of some material admixture (an extensive quantity) and P´ ´eclet number in the case of temperature (an intensive quantity), Pe =

α tD = 2 . tA k0 κ

(7)

In both cases the solution has the general form of a single Fourier mode but with time-dependent wave vector and amplitude, T (x, t) = T0 F (t)eik(t)·x .

(8)

For the stagnation point ﬂow (3) we obtain ∂T ∂T − αx = κ∇2 T ∂t ∂x

(9)

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and the solution is 1

k(t) = k0 eαt , 0 , −1 e2αt

F (t) = e− 2 Pe

(10a)

= exp (− exp(2αt − ln Pe − ln 2)) .

(10b)

The wavelength in the x-direction decreases exponentially. That in the y-direction would, in general, increase exponentially but with the initial condition (2) it started being already inﬁnite (cf. Fig. 2a). The decay of the amplitude F (t) is super-exponential, i.e. extremely fast. The initial variations of the scalar ﬁeld distribution are therefore eliminated by a very rapid process of strain diﬀusion on the time scale tstrain = (ln Pe) tA

(11)

which is much shorter than that of ordinary diﬀusion, in fact only slightly longer than the advection time scale, even when the P´eclet number is very large. For the linear shear (4) we have ∂T ∂T + αy = κ∇2 T ∂t ∂x

(12)

and the corresponding solution is k(t) = k0 (1, −αt) , F (t) = exp −

(a)

1 t − tD 3

t

(13a) 3

.

−2/3 tD Pe

(13b)

(b)

Figure 2. The distribution of the scalar subject to advection and diﬀusion in a stagnation point ﬂow (a) and in a linear shear ﬂow (b) (cf. Fig. 1a). The initial distribution is shown in Fig. 1b.

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Here the wavenumber in the x-direction remains unchanged while that in the y-direction increases linearly in time. This linear growth, considerably slower than the exponential one in the straining ﬂow, accounts for slower decay of the amplitude F (t). For times shorter than tA we have F (t) ∼ e−t/tD ,

t tA = Pe−1 tD .

(14)

Therefore, initially, for a short time, the dissipative process is simple diﬀusion. Later we have 1

3

F (t) ∼ e− 3 (t/tS ) ,

t tA ,

(15)

which means that the process considerably accelerates. Firstly, the exponent changes from t1 to t3 . Secondly, the time scale of the decay becomes tS = Pe−2/3 tD = Pe1/3 tA . (16) This intermediate shear diﬀusion time scale, considerably shorter than that of the ordinary diﬀusive process, but longer than tA , was identiﬁed early in the context of scalar diﬀusion (Rhines & Young 1983) and in the context of magnetic ﬁeld transport (Moﬀatt & Kamkar 1983). The two special linear ﬂows chosen here, i.e. a stagnation point ﬂow and linear shear, may seem to be too simple and lacking all the complications encountered in real transport problems. However, there are at least two reasons to single them out. Firstly, they are often good local approximations of the large-scale, time-dependent ﬂows when the initial spatial scale of the transported scalar is small compared with that of the ﬂow itself and the local time-scale, say tS , is short compared with that of the ﬂow variations. Secondly, these ﬂow conﬁgurations, possibly with a modiﬁed geometry, are often found in diﬀerent physical problems of special interest. We will discuss the transport in a stagnation point ﬂow in the context of ﬂux- and current-sheet formation, common in magnetohydrodynamics, where a ﬂow of this kind is essential. Further, we will consider shear that is always associated with a strong, concentrated vortex, both in its core and outside and aﬀects any passive scalar or weak vorticity that may be present.

2.

Current Sheets and Flux Sheets

The evolution of the magnetic ﬁeld B(x, t) penetrating an electrically conducting ﬂuid moving with velocity u(x, t) is governed by the induction equation, ∂B = ∇ × (u × B) + η∇2 B, (17) ∂t

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where the magnetic diﬀusivity of the ﬂuid, η, is inversly proportional to its electrical conductivity. This approximation of the Maxwell’s equation is used within the framework of magnetohydrodynamics (MHD) to describe processes much slower than electromagnetic waves. This is a kind of advection-diﬀusion equation for a vector quantity B(x, t). In general it describes a much richer variety of phenomena than its scalar counterpart, the kinematic dynamo problem possibly being the most spectacular one (Childress & Gilbert 1995), but in some special symmetric conﬁgurations the two are actually equivalent1 . We will now consider two simple steady solutions of the induction equation (17), both with u(x, t) corresponding to the stagnation point ﬂow (3), that illustrate an important MHD phenomenon, namely the formation of large, localised gradients of the magnetic ﬁeld (Bajer 2004). Strong electric currents, j(x, t) = ∇ × B, are associated with such gradients implying intense Ohmic heating. The structures of this type, called current sheets, play an important role in various MHD processes like, for example, the heating of the solar corona. The ﬁrst solution describes the magnetic ﬁeld lying in the same plane as the ﬂow, B(x) = (Bx (x, y), By (x, y), 0), in which case (17) reduces to the advection-diﬀusion equation (1) for the only component of the vector potential, ∂A ∂A − αx = η∇2 A, ∂t ∂x

B = ∇ × A,

A = (0, 0, A(x, y, t)) .

(18)

As we could see in the previous section, individual Fourier modes ‘travel’ across the Fourier space towards smaller scales (larger k). However, an imposed boundary condition may act as a ‘source’ of one or more Fourier modes and a steady state may be established. In particular, forcing the k = 0 mode, A(x, t) = A(x)ˆ ez ,

A(x) −−−−→ ±A0 ,

we obtain A(x) = A0 erf

α 2η x

,

(19)

x→±∞

B(x) = −A (x) = −A0

2α πη

α 2 − 2η x

e

.

(20)

This solution represents a ﬂux sheet, i.e., a layer of strong magnetic ﬁeld with Gaussian proﬁle, similar in nature to a Burgers vortex layer in which stretching is in balance with diﬀusion. This is also a double current 1 With

linear stretching in the direction of the ﬁeld B the Eq. (17) reduces to a scalar advection-diﬀusion equation with an extra ‘source term’ (Bajer & Moﬀatt 1997).

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sheet with two adjacent layers of strong current in opposite directions. The Ohmic heating Dm is strong, in fact for a given ﬂux Dm −−−→ ∞, η→0

but the magnetic energy Em stored in such sheets also grows without bounds as η → 0, so they cannot be easily created in ﬂows with ﬁnite energy supply (Bajer 2004). They are, therefore, more relevant to the solar dynamo than to the coronal heating problem. When the ﬁeld B = B(x)ˆ ey is perpendicular to the stagnation point ﬂow u = (−αx, 0, αy), we also obtain an advection-diﬀusion equation but this time for a single componet of the ﬁeld, B(x), rather than its vector potential. Imposing B(x) → ±B0 for x → ±∞ we now obtain a steady solution describing a current sheet, B(x) = B0 erf

α 2η x

,

j(x) = B (x) = B0

2α πη

α

2

e− 2η x .

(21)

Ohmic dissipation in such current sheets is small, Dm −−−→ 0, so the η→0

direct heating eﬀect of an individual sheet is weak. However, in a highly conducting medium the magnetic energy stored in such current sheets is small, Em → 0 as η → 0, so they can be easily created even in weak ﬂows. The cumulative direct heating eﬀect may be augmented when many such sheets are created (Parker 1963a). This is likely to be the case when a weakly perturbed integrable ﬁeld relaxes towards a new equilibrium and current sheets appear on the whole range of spatial scales with their distribution, at least in some regions, becoming dense in the limit of inﬁnite electric conductivity of the medium (Bajer 2004). Besides their energetics in a steady or quasi-steady state, the details of the dynamic formation process of strong ﬁeld gradients are also important. The collapse of an X-type neutral point is probably the generic mechanism (Parker 1957, 1963a; Moﬀatt 1990; Linardatos 1993). In spite of the relatively low heat release, the associated magnetic reconnection process locally lifts the topological constraints of ideal MHD thus enabling further relaxation towards yet lower energy states. The role of topological constraints and their violation is a challenging open problem from the physical and from the mathematical point of view, both in linked and in open-ended, braided ﬂux systems (Moﬀatt 1985; Moﬀatt & Ricca 1992). In this section we have discussed some fundamental, ubiquitous MHD processes involving stagnation point ﬂows. Now we will consider a category of archetypal transport processes ruled by the shear ﬂows.

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Figure 3. The formation of a current sheet as a result of the collapse of an X-type neutral point. The cylindrical cavity, surrounded by perfectly conducting solid is ﬁlled with viscous, perfectly conducting ﬂuid penetrated by non-equilibrium magnetic ﬁeld initially containing an X-type neutral point (left panel). Under the evolution governed by the full, nonlinear MHD equations, the neutral point collapses and the system relaxes towards an ideal magnetostatic equilibrium containing a tangential discontinuity of the ﬁeld across which the ﬁeld direction reverses. In a perfectly conducting ﬂuid this corresponds to a singular, delta-like current while in a ﬂuid of ﬁnite conductivity the singular layer would be replaced by a quasi-steady current sheet of ﬁnite length and with ﬁnite current.

3.

Passive Scalar Outside and Inside a Vortex

The dynamics of many ﬂows is best understood in terms of the evolution of coherent vortices and their interactions. For some time this is known to be the case with two-dimensional turbulence (McWilliams 1984) where vortices are sizeable compared with the length scale of the forcing (driven ﬂow) or with the size of the domain (decaying ﬂow). In three-dimensional turbulence the vortices are prominent at small scales (Bajer & Moﬀatt 2002), their diameters seem to be of the order of the Kolmogorov scale (Moﬀatt, Kida & Ohkitani 1994). It is therefore of interest to understand the advection-diﬀusion processes in the presence of concentrated vortices. Locally, the ﬂow near a vortex is simple shear. An initial blob of passive scalar which is small compared with the streamline curvature experiences shearing by a ﬂow similar to (4) (Flohr & Vassilicos 1997) and the initial-value problem for its evolution can be decomposed into Fourier modes evolving according to (8). However, if the initial spatial variations of the scalar are much larger than the size of the vortex, then, on the scale of the vortex, the initial scalar isolines are, to a good approxmation, straight. They are then wrapped around the vortex in the manner shown in Fig. 4. Should the

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Figure 4. The evolution of two stretches of diﬀerent material lines, AB and CD, in the ﬂow due to a point vortex located at the centre. Dashed lines in the left panel mark the streamlines of the ﬂow. Each of the two intervals, originally rectilinear, is wound into a double spiral (right panel). As time progresses, the spirals are increasingly tight and the separation of the two material lines decreases like t−1 .

scalar be non-diﬀusive, this spiral wind-up would continue forever. In Fig. 4 we see that the distance between given isolines decreases with time. It is proportional to t−1 , just like for a single Fourier mode in linear shear (cf. eqn. 13). Such decrease in the radial length scale boosts diﬀusion which eventually prevails. The details of this process inside and outside the vortex are somewhat diﬀerent, but the essential ingredient, i.e. shear, is present in both regions. Looking at the outer region we can approximate the ﬂow by that of a point vortex and obtain both a steady solution (when one mode is forced by a distant boundary condition) or an unsteady, similarity solution in an unbounded domain (Bajer 1998). The steady solution describes the balance inside a cylindrical cavity of radius R ﬁlled with inviscid liquid of ﬁnite thermal diﬀusivity. The motion is irrotational but for a line vortex of circulation Γ on the axis. Uniform temperature gradient, T = T0 y = T0 r sin θ, is imposed on the solid surface of the cylinder. This surprisingly simple solution of Eq. (1) with u = Γ/(2πr)ˆ eθ , T = T0 Re f (r, t)eiθ , (22a) r √1+iP e

Γ , Pe = , (22b) R κ describes, in particular, a thermal boundary layer on the cylinder surface with isotherms wound into a double logarithmic spiral and, also, the f (r, t) = f (r) = −i

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Concentration, Pe = 100

0.09 0.8 1

1

0.08 0.6 0.07 0.4 0.5

0.5

0.06 0.2 0.05 0

0

0 0.04

−0.2

−0.4

−0.5

0.03 −0.5 0.02

−0.6

−0.8

−1

−1

−0.5

0

0.5

1

0.01 −1

−1

−0.5

0

0.5

1

0

Figure 5. Point vortex in the initialy uniform temperature gradient. Boundary condition ﬁxes one mode which makes a steady state possible. The left panel shows isotherms being wound into double logarithmic spirals in a thermal boundary layer on the surface of the cylinder (lighter colours correspond to warmer ﬂuid). The right panel shows the heat ﬂux expelled from the central region (solid lines - integral curves of the heat ﬂux; grey scale - magnitude of the heat ﬂux).

expulsion of the heat ﬂux from the central region (Fig. 5). A straightforward MHD analogue of this solution describes the magnetic ﬂux expulsion, an important phenomenon discovered some fourty years ago (Parker 1966; Weiss 1966). Corresponding similarity solutions, f (r, t) = f (r2 /t), for a point-vortex ﬂow (Bajer 1998) and for the Lamb vortex (Pearson & Abernathy 1984; Moore 1985) describe spiral structures propagating outwards in an unbounded domain. Spiral structures appear in diﬀerent contexts in the theory of turbulence and vortex dynamics (Gilbert 1988; Moﬀatt 1993; Kimura & Herring 2001; Pullin & Lundgren 2001). A gigantic magnetic structure of this kind encompassing the entire solar system is called Parker’s heliospheric spiral (Parker 1963b). The wind-up and accelerated diﬀusion are direct consequences of shear in a vortical ﬂow. This is present both outside a vortex and in its core. However, the radial proﬁles of the shear in the two regions are diﬀerent. The shear, deﬁned as the angular velocity gradient, is proportional to r−3 in the outer region and to r near the centre of the vortex. The details of the gradient annihilation will thus be diﬀerent. In particular, the shear vanishes at r = 0. Interestingly, there is another exact solution eθ (where of (1) with parabolic core vorticity proﬁle, i.e. u = (α0 r+α1 r3 )ˆ α0 , α1 = 0 are arbitrary constants) of the same general form (22a) with (Lighthill 1966; Brunet & Haynes 1995; Bajer, Bassom & Gilbert 2001) 2

f (r, t) = g(t)e−iα0 t−ih(t)r ,

(23a)

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µ = (1 + i) 2α1 /P e. (23b) This solution describes the three phases of annihilation of the gradient of a passive scalar inside a vortex: spiral wind-up, accelerated (shear) diﬀusion moving in from the region of stronger shear towards the centre and the survival of longer-living remnant around the origin where the shear vanishes. The latter is an ‘eigenmode’ decaying without change of shape on a time scale tI = tD P e−1/2 , intermediate between tS and tD (cf. Eq. (16)) (Bajer, Bassom & Gilbert 2001). g(t) = (cosh µt)−2 ,

4.

h(t) = α1 µ−1 tanh µt,

Vortex-Background Interaction

The interplay between advection and diﬀusion resulting in the formation of strong gradients and in accelerated diﬀusion is also a feature of transport of physical quantities that cannot be regarded as passive. Vorticity in a two-dimensional ﬂow is, for example, an ‘active’ scalar ﬁeld governed by the vorticity equation. How important is its coupling to the transporting velocity ﬁeld depends on particular circumstances. When vorticity ﬁeld can be naturally separated into a strong coherent part, Ω and weak background ω (Kevlahan & Farge 1997; Farge, Holschneider & Colona 1990; Beta et al. 2003), the vorticity equation for the background can be linearised and becomes, in some sense, an extension of the advection diﬀusion equation for ω in the ﬂow U associated with Ω, ∂ω + U · ∇ω = ∂(ψ, Ω) + F + ν∇2 ω, ∇2 ψ = −ω. (24) ∂t Compared with the Eq. (1) there is the source term, F , that corresponds to a possible external forcing of the ﬂow. There is also an extra term, ∂(ψ, Ω), corresponding to the diplacement of the coherent vorticity Ω by ˆz associated with the weak background vorticity ω. the ﬂow u = ∇ψ × e This is an Eulerian eﬀect which needs to be separated from the ‘genuine’ evolution of the background. If we choose for Ω the Lamb vortex (diﬀusing Gaussian vorticity dis˙ tribution) which is moving with prescribed velocity (X(t), Y˙ (t)) and for the initial ω – a distribution with locally uniform gradient, we ﬁnd that there is only one choice of (X(t), Y (t) consistent with the far-ﬁeld boundary conditions. We then obtain both the evolution of the background and the motion of the strong vortex. An important conclusion is that a single, strong coherent vortex immersed in a non-uniform weak background vorticity is set in motion by the ﬂow associated with a perturbation of the background that the strong vortex creates (Bajer, Bassom & Gilbert 2004). The vortex also homogenises the background around itself, thus making a ‘hole’ in the

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background vorticity gradient. Therefore, the viscous solution of the linearised vorticity equation conﬁrms Batchelor’s general prediction about uniformisation of vorticity in ﬂows with closed streamlines (Batchelor 1956).

Acknowledgement Much of the material presented in this review relates to the collaborative work with Andrew Bassom and Andrew Gilbert that I enjoyed over several years. I am indebted to Keith Moﬀatt for drawing my attention to a range of focal physical problems and for sharing his insights. The support from the Centre of Excellence in Small Scale Atmospheric Research, CESSAR (EU grant no EVK2-CT-2002-80010), is gratefully acknowledged.

References K. Bajer, Flux expulsion by a point vortex, Eur. J. Mech. B/Fluids 17(4):653–664, 1998. K. Bajer, Abundant singularities, Fluid Dyn. Res. Submitted, 2004. K. Bajer, A.P. Bassom, and A.D. Gilbert, Accelerated diﬀusion in the centre of a vortex, J. Fluid Mech. 437:395–411, 2001. K. Bajer, A.P. Bassom, and A.D. Gilbert, Vortex motion in a weak background shear ﬂow. J. Fluid Mech. 509:281–304, 2004. K. Bajer, H.K. Moﬀatt, On the eﬀect of a central vortex on a stretched magnetic ﬂux tube, J. Fluid Mech. 339:121–142, 1997. K. Bajer, and H.K. Moﬀatt, Tubes, Sheets and Singularities in Fluid Dynamics, Kluwer, 2002. G.K. Batchelor, On steady laminar ﬂow with closed streamlines at large Reynolds number, J. Fluid Mech. 1:177–190, 1956. C. Beta, K. Schneider, M.Farge, and H. Bockhorn, Numerical study of mixing of passive and reactive scalars in two-dimensional turbulent ﬂows using orthogonal wavelet ﬁltering, Chem. Eng. Sci. 58 (8):1463–1477, 2003. G. Brunet, and P.H. Haynes, The non-linear evolution of disturbances to a parabolic jet, J. Atmos. Sci. 52:464–477, 1995. S. Childress and A.D. Gilbert, Stretch, Twist, Fold: The Fast Dynamo, Springer, 1995. M. Farge, M. Holschneider, J.F. Colona, Wavelet analysis of coherent structures in two-dimensional turbulent ﬂows, [In:] Topological Fluid Mechanics (ed. H.K. Moffatt & A. Tsinober), Cambridge University Press, 1990. P. Flohr, J.C. Vassilicos, Accelerated scalar dissipation in a vortex, J. Fluid Mech. 348:295–317, 1997. A.D. Gilbert, Spiral structures and spectra in two-dimensional turbulence, J. Fluid Mech. 193:475–497, 1988. N.K.R. Kevlahan, M. Farge, Vorticity ﬁlaments in two-dimensional turbulence: creation, stability and eﬀect, J. Fluid Mech. 346:49–76, 1997.

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Y. Kimura and J.R. Herring, Gradient enhancement and ﬁlament ejection for a nonuniform elliptic vortex in two-dimensional turbulence, J. Fluid Mech. 439:43–56, 2001. M.J. Lighthill, Initial development of diﬀusion in a Poiseuille ﬂow, J. Inst. Maths. Applics. 2:87–108, 1966. D. Linardatos, Determination of 2-dimensional magnetostatic equilibria and analogous Euler ﬂows, J. Fluid Mech., 246:569–591, 1993. J.C. McWilliams, The emergence of isolated coherent vortices in turbulent ﬂow, J. Fluid Mech. 146:21–434, 1984. H.K. Moﬀatt, Magnetostatic equilibria and analogous Euler ﬂows with arbitrary complex topology, J. Fluid Mech. 159:359–378, 1985. H.K. Moﬀatt, Structure and stability of solutions of the Euler equations: a Lagrangian approach, Phil. Trans. Roy. Soc. A 333:321–342, 1990. H.K. Moﬀatt, Spiral structures in turbulent ﬂows, [in:] New Approaches and Concepts in Turbulence (Monte Verit` ` a), Birkhauser 1993. H.K. Moﬀatt and H. Kamkar, On the time-scale associated with ﬂux expulsion, [in:] Stellar and Planetary Magnetism, pp.91–97. Gordon & Breach, 1983. H.K. Moﬀatt, S. Kida, and K. Ohkitani, Stretched vortices – the sinews of turbulence: high Reynolds number asymptotics, J. Fluid Mech. 259:241–264, 1994. H.K. Moﬀatt and R. Ricca, Helicity and the Cˇ ˇ alugˇ ˇ areanu invariant, J.Proc. Roy. Soc. A, 439:411–429, 1992. D.W. Moore, The interaction of a diﬀusing vortex and an aligned shear ﬂow, Proc. R. Soc. Lond. A 399:367–375, 1985. E.N. Parker, Sweet’s mechanism for merging magnetic ﬁelds in conducting ﬂuids, J. Geophys. Res. 62:509–520, 1957. E.N. Parker, Dynamics of the interplanetary gas and magnetic ﬁelds, Astrophys. J. 128:664–676, 1958. E.N. Parker, The solar-ﬂare phenomenon and the theory of reconnection and annihilation of magnetic ﬁelds. Astrophys. J. Suppl. 8:177–211, 1963a. E.N. Parker, Interplanetary Dynamical Processes, Wiley-Interscience, New York 1963b. R.L. Parker, Reconnection of lines of force in rotating spheres and cylinders, Proc. Roy. Soc. A, 291:60–72, 1966. C.F. Pearson and F.H. Abernathy, Evolution of the ﬂow ﬁeld associated with a streamwise diﬀusing vortex, J. Fluid Mech. 146:271–283, 1984. D.I. Pullin and T.S. Lundgren, Axial motion and scalar transport in stretched spiral vortices, Phys. Fluids, 13(9):2553–2563, 2001. P.B. Rhines, W.R. Young, How rapidly is a passive scalar mixed within closed streamlines? J. Fluid Mech. 133:133–145, 1983. N.O. Weiss, The expulsion of magnetic ﬂux by eddies, Proc. Roy. Soc. Lond. A 293:310–328, 1966.

WAVE-VORTEX INTERACTIONS IN THE ATMOSPHERE, AND CLIMATE PREDICTION Onno Bokhove Numerical Analysis and Computational Mechanics, Department of Applied Mathematics IMPACT, University of Twente, P.O. Box 217, Enschede, The Netherlands [email protected]

Abstract

Can we construct an accurate atmospheric climate model with a balanced model representing its ﬂuid mechanics, and with dissipative as well as non-dissipative parameterization schemes for the gravity-wave activity? To address this question, we focus our attention on a 1 21 -layer atmospheric model with an isentropic troposphere and isothermal stratosphere. We investigate parcel dynamics in a hybrid Eulerian-Lagrangian formulation, potential vorticity conservation, static stability, linear modes and the concept of balanced ﬂow; and brieﬂy discuss wave-vortex interactions and recent advances in numerical solution techniques.

Keywords: hybrid Eulerian-Lagrangian ﬂuid parcel dynamics, linear modes, balanced models, gravity waves, weather and climate prediction

Introduction Numerical weather and climate prediction is complicated because only the ﬂow scales larger than at least ∼ 10 × 10 × 1 km3 can be resolved to date. When we use the (inviscid) primitive Navier-Stokes equations on these scales, the commonly used (semi-Lagrangian) numerical schemes implicitly ﬁlter all acoustic waves and some of the gravity-wave (GW) motion. The rapid small-scale three-dimensional turbulence is then certainly not resolved. Consequently, also the feedback of the unresolved wave and (quasi-two-dimensional) turbulent motions on the large-scale dynamics requires parameterization. A lot of attention has been paid to simpliﬁed or balanced versions of the primitive equations, in which preservation of the conservation laws (of the inviscid dynamics) such as mass, energy and potential vorticity (PV) has been advocated to enhance the stability of these so-called ba103 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 103–116. © 2005 Springer. Printed in the Netherlands.

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lanced models. The large-scale ﬂow is generally close to hydrostatic and geostrophic balance; the former due to the anisotropy of horizontal and vertical scales and the latter due to the rapid rotation of the Earth, both in these balanced models (by default) and the numerical integration schemes used. Consequently, one has a choice to use either the primitive equations or balanced models. Birner et al. (2002) observed that vertical temperature proﬁles are nearly constant in the stratosphere with a distinct kink at the tropopause between the troposphere and stratosphere. To analyze several properties of atmospheric ﬂows, we therefore derive a conceptual model of the atmosphere with an isentropic troposphere and isothermal stratosphere, where the entropy or potential temperature θ and temperature T are constant, respectively. Subsequently, we illustrate the concept of balance by deriving a balanced model describing only the vortical motion from this so-called “θ-T -model”. A novel derivation of this θ-T -model from the three-dimensional Euler equations, using a combination of asymptotic methods and physical simpliﬁcations, is given in the framework of a hybrid Eulerian-Lagrangian description of a ﬂuid parcel (Section 1). This hybrid formulation of the Euler equations (Dixon and Reich, 2004) describes the Hamiltonian dynamics of each parcel as a dynamical system with six degrees of freedom with the internal and potential energy as function of space and time. The formulation is passive when this function is given. In contrast, an integral equation for the density using the Jacobian between Eulerian and Lagrangian space links the dynamics of all ﬂuid parcels into a dynamically consistent continuum. In the linearized θ-T -model, three time derivatives (or four in the parcel framework) in the model give rise to a pair of fast GW modes and one slow geostrophic mode, whose eigen-periods are separated in time on the f -plane. However, the dynamics are nonlinear and there may be a conversion of energy and momentum between these slow and fast modes. In Section 2 (i.e., Fig. 4), this is illustrated in simulations of the nonlinear dynamics initialized by a linear mode at ﬁnite amplitude, in which a simple hydraulic, dissipative wave-breaking parameterization is used. In the nonlinear dynamics, the slow modes survive approximately on a slow manifold of lower dimension. Balanced models of vortical dynamics describe the slow motion on a slow manifold (Section 3), on which the dimension of phase space is reduced by two thirds (or half in the parcel framework) due to the removal of the pair of (fast) GW modes. Within the Eulerian-Lagrangian framework of parcel dynamics, we illustrate the derivation of (Hamiltonian) balanced models using two velo-

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city constraints which arise in asymptotic expansions in a relevant small parameter (such as the Rossby or Froude number). These constraints deﬁne the reduction of the phase space to the slow or slaving manifold. We compare balanced and unbalanced trajectories in simpliﬁed simulations in Fig. 1(b). The generation of gravity waves by instabilities of a balanced ﬂow, or the absorption of gravity waves by the nearly balanced mean ﬂow through wave-vortex interactions indicate, however, that the slow manifold is not an exact manifold (B¨ u ¨hler and McIntyre, 2003; Vanneste and Yavneh, 2004). In balanced models, these unbalanced GW eﬀects can only be included by explicitly parameterizing the gravity waves. Likewise, in numerical weather and climate prediction small-scale, unresolved gravity waves require parameterization. We ﬁnish by brieﬂy discussing idealized wavevortex interactions and some recently developed numerical schemes for geophysical ﬂows (Sections 4 and 5), and their relevance to General Circulation Models (GCMs).

1.

Eulerian-Lagrangian Dynamics of Fluid Parcels

Three-dimensional compressible Euler equations Consider Newton’s equations of motion for a ﬂuid parcel with position x = (x, y, z)T and velocity u = (u, v, w)T [(·)T denotes the transpose] in a rotating reference frame with rotation vector Ω ∂H H3D dx =u= , dt ∂u

dθ = 0 and dt

∂H H 3D ∂H H3D du = −θ∇Π − ∇φ − 2Ω × u = − − 2Ω × dt ∂x ∂u

(1) (2)

with the parcel energy (extending Frank and Reich, 2003) H3D (x, u, θ, t) = |u|2 /2 + θ Π(x, t) + φ(x),

(3)

three-dimensional gradient ∇; external potential φ, e.g., φ = gz; potential temperature θ = T (p/pr )−κ ; temperature T (x, t); pressure p(x, t) and reference pressure pr ; and Exner function Π = cp (p/pr )κ for an ideal gas p = ρRT with density ρ(x, t), gas constant R, speciﬁc heat at constant pressure cp , and κ = R/ccp . Note that θ∇Π = (1/ρ)∇p, so that (2) attains its usual form (Dixon and Reich, 2004). We can write (1)–(3) as a non-canonical Hamiltonian system dq/dt = J∂H H3D /∂q with state vector q = (x, y, z, u, v, w, θ)T and a skew-symmetric tensor J. The state vector q is a function of time and ﬂuid labels a = (a, b, c)T =

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(a1 , a2 , a3 )T , so q = q(a, t). If the parcel energy is a function of q and t, then (1)–(3) is non-autonomous. The continuum character of the ﬂuid, albeit hidden in the representation (1)–(3), appears through the density ρ(x, t) =

ρ(x , t)δ(x − x )dx dy dz D = ρ0 (a)δ(x − x (a, t))dadbdc (4) D0

with x = x (a, t) in a domain D or D0 , since the Jacobian between label and position space is proportional to the inverse density ρ0 (a)/ρ(x, t) = det |∂x/∂a|.

(5)

Hence, an element of mass dm relates to the density as follows: dm = ρ(x, t)dxdydz = ρ0 (a)dadbdc.

(6)

A common choice is ρ(x, 0) = ρ0 (a) and x(a, 0) = a. The system (1)–(4) is closed and represents the ﬂuid as a continuum. From (1), (4) and (5), we can derive the continuity equation dρ(x, t)/dt = ∂t ρ(x, t) + u · ∇ρ(x, t) = −ρ(x, t)∇ · u(x, t).

(7)

Similarity to 2D vorticity dynamics. We note that this hybrid description is akin to the (more familiar) situation in inviscid, incompressible, two-dimensional vorticity dynamics, where the passive or kinematic advection of each ﬂuid parcel is described by a given stream function ψ(x, y, t) as Hamiltonian with horizontal coordinates xh = (x, y)T and time t. Thus, dx/dt = u = −∂ψ/∂y and dy/dt = v = ∂ψ/∂x. In contrast, a dynamically consistent formulation appears when the vorticity ω = ∇2h ψ is conserved on each ﬂuid parcel and linked to the continuum of parcels using ω(x, y, t) = ω0 (a, b)δ(xh − xh (a, b, t))dadb D0

with domain D0 , delta function δ(·), parcel position xh (a, b, t), and ω0 (a, b) denoting the initial distribution of vorticity on parcels identiﬁed by labels a and b. Given ω on each parcel, we calculate ψ. Hence, the dynamical description is closed, since incompressibility yields dxdy = dadb. With v = (u, v)T and gradient ∇h in the horizontal direction, we ﬁnd dω/dt = ∂ω/∂t + v · ∇h ω = 0.

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Static stability. To illustrate the formulation of the hybrid parcel dynamics, parcel oscillations in a static atmosphere with given parcel energy H3D = |u|2 /2 + θΠ(z) + g z are shown in Fig. 1(a). We choose Π = Π(z) with potential temperature θ = θg (z) in its thermodynamics to satisfy hydrostatic balance θg (z)∂Π/∂z = −g. Hence, we ﬁnd d2 z /dt2 = −N 2 z for small amplitude oscillations with z = z − zr and a reference a¨is¨¨ala¨ frequency N , level zr . Oscillations are then stable with Brunt–V¨ 2 when N = [g(dθg /dz)/θg ]z=zr > 0, neutral when N 2 = 0, and unstable when N 2 < 0. Note that θ is conserved on each parcel of air and generally diﬀerent from θg (z).

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Figure 1. (a) Three trajectories are shown of parcel oscillations in the atmosphere for a given parcel energy. The stratiﬁcation of the atmosphere is statically stable (dθ/dz = cst. > 0) for x < −10 km, unstable (dθ/dz = cst. < 0) for −10 km< x < 10 km, and neutral (constant θ) for x > 10 km and z < 10 km, and isothermal and stable for z > 10 km. The stable oscillations have a period of 10.84 min. When the atmosphere is hydrostatic, these oscillations disappear as the thin lines at zr = 5 km illustrate. (b) 41.7 days of (dimensionless) geostrophically balanced and unbalanced Hamiltonian motion of a particle in a simple, given Montgomery potential M2 (x, y) starting at (x, y) = (1, 1). The predictability horizon lies around 14 days whereafter the balanced (dashed lines) and unbalanced (solid lines) trajectories depart from one another signiﬁcantly.

Hydrostatic primitive equations The atmosphere is shallow for larger scales, and the aspect ratio δ between vertical and horizontal length and velocity scales (D, W and L, U ) arises as a small parameter δ = W/U 1. At leading order in δ, we ﬁnd from the scaled version of system (1)–(4) the dynamics and

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hydrostatic balance ∂H Hp dxh =v= , dt ∂v

(8)

∂H Hp ∂H Hp dv ˆ×v =− ˆ× = −θ∇h Π − f z , − fz dt ∂xh ∂v dθ = 0 and dt

(9)

0 = −θ∂Π/∂z − g

(10)

ˆ the unit vector in the vertical direction, and the hydrowith f = 2 Ω3 , z static parcel energy Hp (x, y, z, u, v, θ, t) = (u2 + v 2 )/2 + M (x, y, z, t)

(11)

with Montgomery potential M = θΠ(x, y, z, t)+g z. The vertical velocity dz/dt follows by insisting hydrostatic balance persists in time. We use these interim results next in the derivation of the conceptual 1 12 - and 2-layer models.

A 1 12 - and 2-layer atmosphere Birner et al. (2002) measured the vertical temperature proﬁles which suggest a conceptual model with an isentropic troposphere and an isentropic or isothermal stratosphere. We therefore simplify the stratiﬁcation of the atmosphere into an isentropic tropospheric layer and an isentropic or isothermal stratospheric layer, see Fig. 2(a). In this ﬁgure, we deﬁne the variables and constants used subsequently and denote their dependence, if any, on x, y and t. (a)

(b)

45 km x

constraining

real dynamics

fast f

w fold

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slow

Figure 2. (a) Sketch of a simpliﬁed atmosphere with an isentropic troposphere and isentropic or isothermal stratosphere. p0 is a passive and constant pressure, and p1 and p2 are active pressures. (b) The slow manifold sketched has a third (or half) of the dimension of the entire Eulerian (or Lagrangian) phase space, with fast and slow variables f and s. Constraining forces, “the hand”, place the full dynamics on the manifold. When s and f are small, the dynamics is linear and separated in time such that f → 0, as it is sketched.

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Integrating hydrostatic balance θ∂Π/∂z + g = 0 in the tropospheric layer from z = z2 = hb to z with z2 < z < z1 , we obtain the Montgomery potential M2 (p2 ) = θ Π + g z = cp θ(p/pr )κ + g z = cp θ2 (p2 /pr )κ + g hb .

(12)

In an isentropic stratospheric layer one ﬁnds likewise, by integrating from z1 to z with z1 < z < z0 and using (12) at z = z1 with p(x, y, z1 , t) = p1 , that M1 (p1 , p2 ) = θΠ + gz = g(z0 − Z0 ) = cp θ(p/pr )κ + gz = cp (θ1 − θ2 )(p1 /pr )κ + cp θ2 (p2 /pr )κ + g(hb − Z0 ), (13) while in an isothermal stratospheric layer, one obtains similarly M1 (p1 , p2 ) = θΠ + gz = g(z0 − Z0 ) = cp θ(p/pr )κ + gz = RT T1 ln(p1 /p0 ) + cp θ2 ((p2 /pr )κ − (p1 /pr )κ ) + g(hb − Z0 ). (14) Note that, without any loss of generality, we have added a constant reference level Z0 to which we can ﬁx the top of the stratospheric layer z0 at a later stage. Any initial z-independence in each layer remains intact, so only two parcels in a vertical column of ﬂuid suﬃce for closure. Hence, the two-layer tropospheric-stratospheric model is the hydrostatic model (8)–(9) applied in each layer ∂H Hα dxα = vα = , dt ∂xα

(15)

dvα ∂H Hα ∂H Hα ˆ × vα − ∇h Mα = −f z ˆ× = −f z − dt ∂vα ∂xα

(16)

with α = 1, 2; x = (x, y)Tα and parcel energy Hα (xα , yα , uα , vα , t) = (u2α + vα2 )/2 + Mα (xα , yα , t).

(17)

Closure of these two-layer equations is reached via the layer pseudodensity σ0 (a, b)δ(x − xα (a, b, t))dadb (18) σα (x, y, t) = DH0

relating the horizontal label and position spaces with (using ∂p/∂z = −ρg) dm = ρdxdydz = −dxdydp/g d = dadbdc ∆m2 = σ2 dxdy = [(p2 − p1 )/g]dxdy = σ20 (a, b)dadb ∆m1 = σ1 dxdy = [(p1 − p0 )/g]dxdy = σ10 (a, b)dadb ∂y b − ∂y a∂ ∂x b. σα0 /σα = ∂x a∂

(19) (20) (21) (22)

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We emphasize that in each layer the Eulerian velocity is independent of depth, so vα = vα (x, y, t). Hence ∂z vα remains zero once it was initially so. Again, we can derive continuity equations ∂t σα + ∇h · (σα vα ) = 0 and a materially conserved PV: dQα /dt = 0 with ˆ · ∇ × vα )/σα . Qα = (f + z 1 12 -layer models. When the stratospheric layer is much deeper than the tropospheric layer, e.g., 45 km versus 15 km, we approximate the top z0 to Z0 and neglect the motion in the stratospheric layer. Thus, from (13) or (14) one ﬁnds M1 (p1 , p2 ) = 0. The stratospheric pressure p1 remains active, but the dynamics [(15) and (16) for α = 2] is evolved in the tropospheric layer. The bottom pressure p2 used to deﬁne M2 [(12)] in the tropospheric momentum equations (16) is then determined from σ2 = (p2 − p1 )/g and M1 (p1 , p2 ) = 0.

Static stability. Static stability means that a ﬂuid parcel perturbed in the vertical oscillates around a certain height with the Brunt-V¨ a¨is¨¨al¨ a frequency N rather than taking oﬀ. The eigen-values a of the 1 12 -layer equations [(15) and (16) for α = 2 with σ2 = (p2 −p1 )/g and M1 (p1 , p2 ) = 0] are (θ2 –θ1 -model) cp κ(θ1 − θ2 )(p1 /pr )κ−1 (p2 − p1 )/pr 2 a ∝ (23) κ−1 T1 /p1 − cp θ2 κ(p1 /pr ) ) (θ2 –T T1 -model). (p2 − p1 )(pr RT These eigen-values are real when the atmosphere is statically stable or dθ/dz > 0: this occurs when θ1 > θ2 in the θ2 -θ1 -model, and when T1 model. While the θ2 -θ1 model remains T1 > θ2 (p1 /pr )κ in the θ2 -T T1 statically neutral or stable if it is initially so, the stability of the θ2 -T model thus depends on p1 (x, y, t).

2.

Linear Modes

Linearized around a “rest depth” H with σ = σ2 = H(x, y) + η, the 1 12 -layer models [(15) and (16) for α = 2] become ∂t v = −f zˆ × v − g ∇h η

and ∂t η + ∇h · (Hv) = 0

(24)

with velocity v = v2 and eﬀective gravity g . This linearized system is akin to the classical, linearized shallow water equations with (for hb = 0) ⎧ (θ1 −θ2 )P P1κ−1 ⎪ ⎨ (θ2 –θ1 -model) κ−1 (θ −θ )P P +θ2 P2κ−1 1 2 1 (25) g ∝ (P P1 /pr )κ−1 ⎪ ⎩ pr RTT1 /PP1 −cp κθ2κ−1 (θ –T T -model) 2 1 κ−1 RT T /P P +c κθ (P P −P P ) 1

1

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18 16 14

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Figure 3. model.

Dispersion relation of linear modes in a periodic channel for the θ2 -θ1 -

and the rest state H = (P P2 − P1 )/g, which is consistent with (23). In a periodic channel with constant H, the linear modes using η ∝ ei(kx+ωt) (i2 = −1) consist of: a vortical or geostrophic mode with frequency ω = 0, Poincar´ ´e GW modes with ω 2 = f 2 + g H (k 2 + l2 ), and counterclockwise propagating boundary-trapped Kelvin modes with ω 2 = g H k 2 (for constant f > 0). We observe in the dispersion diagram in Fig. 3 that there is a time-scale separation between the vortical and GW modes, except perhaps for the lowest-order Kelvin modes and the geostrophic solution. As usual, a linear mode analysis is limited in scope. First, the dynamics is nonlinear, so there is no clear notion of a time scale separation anymore. Nonlinear “slow” dynamics can have high-frequency overtones triggering resonances or interactions with “fast” dynamics. Second, approaching the equator, the eﬀective Coriolis parameter f → 0 , giving rise to equatorial Kelvin waves and mixed Rossby-gravity waves or mixed slow-fast linear modes. Consequently, linear Kelvin or gravity mode solutions of larger amplitude used as initial condition, can develop vortex motion and, vice versa, linear geostrophic or Rossby modes can develop GW motion from instabilities. Mixed fast-slow motion emerges in simulations, see Fig. 4, of the nonlinear evolution of a linear Kelvin mode solution in a zonally periodic channel used as initial condition. In particular PV is constant (in time and/or space) for a Kelvin or Poincar´e mode, ˆ · ∇ × v)/σ = f /H Q2 = Q = (f + z

and ∂t Q + (v · ∇h )Q = 0, (26)

before the occurrence and parameterization of wave breaking. These constant PV regions are then distinguished, ideally, from regions where

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Figure 4. Top: contour plots of σ in simulations of the 1.5-layer θ2 -θ1 -model with shock-capturing numerics which, however, do not conserve absolute vorticity σQ. Bottom: simulations displaying σ (left) and 100× PV anomaly (Q − f /H) (right) at the ﬁnal time. Non-dimensional quantities are displayed, for example in a domain of 4000 × 2000 km with z1 ≈ 15 km and Z0 ≈ 60 km.

a wave breaking parameterization generates non-constant PV anomalies Q − f /H (cf., Peregrine and Bokhove, 1998).

3.

Balanced Dynamics

The concept of balanced large-scale ﬂow arises from the observation that at mid-latitudes the atmosphere and oceans are in approximate geostrophic balance, and near the equator the Earth’s rotation remains inﬂuential. Locally — due to topography, strong (tropical) convection, dissipative and non-dissipative (GW) instabilities — balance often fails. The notion of balance may be formalized in various ways: small Rossby and Froude numbers are identiﬁed from measurements, observations or simulations, and then used in scaling arguments. Subsequently, a perturbative or iterative approach is applied to approximate the full or parent

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model. The resulting dynamics evolves on a slow or slaving manifold of reduced dimensionality, see the sketch in Fig. 2(b). The preservation of certain conservation laws, or the variational or Hamiltonian structure, may be imposed heuristically in these balanced approximations. Whether the conservative or non-conservative approach to balanced dynamics is better, remains undecided and depends on the, perhaps subjective, value placed on (point-wise) accuracy, and long-term stability. Geostrophic balance denotes the alignment of the wind vectors along the pressure or Montgomery potential isobars. To derive this leading order balance, we rewrite the 1 12 -layer equations [(16) for α = 2] and drop the layer subscripts f 1 f ∂H 1 ∂H dui = ij uj − ∂xi M = ij − dt R R R ∂ui R ∂xi

(27)

with the permutation symbol ij , v = (u1 , u2 )T and i, j = 1, 2. The Rossby number R = U/(f L) 1 is placed in (27) at the relevant locations, as the ratio of the GW time scale 1/f and the vortical time scale L/U with typical length and velocity scales L and U . At leading order in R, we ﬁnd geostrophic balance from (27) as a constraint on the velocity with M/f being a stream function in the balance relations u = −∂ ∂y M/f and v = ∂x M/f . In general, (higher-order) velocity constraints obtain the form e.g.

φi = ui − uC i [σ(x)] = ui +

1 ij ∂xj M, f

(28)

in which uC [σ] operates (non-locally) on σ and, hence, through σ on the parcel coordinates x and y. Next, we use these constraints to derive balanced models.

Conservative balanced models: slaved Hamiltonian approach We illustrate the derivation of Hamiltonian balanced models in the hybrid parcel framework. The variables (xi , ui ) are transformed to (xi , φi ) using (28), and a constrained variational derivative is introduced $ ∂H ∂H ∂uC ∂H $$C j = + , $ ∂xi ∂xi ∂uj ∂xi

(29)

where (·)|C denotes that φi = 0 in derivatives of x and y. The evolution on the slow manifold of reduced dimensionality becomes, using (15)

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and (16), dxi ∂H = = ui dt ∂ui

and

0=

∂H C dφi =− + ij σQC uj dt ∂xi

(30)

with σ QC = f + ∂v C /∂x − ∂uC /∂y. The slaved Hamiltonian dynamics on the slow manifold is concisely written as ∂H C dxi = (L−1 )ij dt ∂xj

or

dF C ∂F C −1 ∂H C = (L )ij dt ∂xi ∂xj

(31)

[cf. Dirac (1958)] with skew-symmetric matrix Lij = ij σQC and arbitrary function F C = F C (x, y) and H C = H(x, y, uC , v C ). Simpliﬁed numerical integrations are explained in Fig. 1(b). It is unclear whether the parcel balanced dynamics (31) presented is a didactic simpliﬁcation, or equivalent to the results for the Eulerian balanced equations in Vanneste and Bokhove (2002).

4.

Wave-Vortex Interactions and Numerical Schemes

The parameterization of unresolved gravity waves is a critical component in numerical GCMs. Gravity waves can inﬂuence the large-scale dynamics in various ways: (i) breaking gravity waves dissipate energy to small scales and deposit momentum to drive the mean, large-scale ﬂow (McFarlane, 1987); (ii) instabilities of balanced vortical ﬂows locally excite gravity waves, which can transport energy and momentum away (Vanneste and Yavneh, 2004); and (iii) non-dissipative wave-vortex interactions, such as remote recoil, can lead to a cumulative forcing of the mean vortical ﬂow (B¨ u ¨hler and McIntyre, 2003). The crucial question is how to parameterize these unresolved GW-eﬀects, studied hitherto in isolation, in numerical models for large-scale ﬂows on advective time scales, given the resolved large-scale ﬂow. New numerical schemes have emerged with a focus on improved meshes without pole problem, conservation properties and advection-dominated time integration. Based on gas dynamics and novel ﬁnite-element discretizations (Bokhove, 2005; Fig. 4), an impulse formulation of the θT -model with 3 prognostic equations can be used, which are shockcapturing but with explicit time stepping limited by the largest GW speed. In atmospheric dynamics, the velocity formulation with 3 prognostic equations is often preferred (Ringler and Randall, 2002). The GW speed is then still the limiting factor. Hamiltonian Particle Mesh methods (HPM) involve (15)–(18) with 4 prognostic equations and 1 integral equation (Frank and Reich, 2003). By smoothing the pseudo-density,

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time step restrictions can be lifted. To emphasize the vortical dynamics, a mass-divergence-vorticity formulation is used by Thuburn (1997) and Ringler and Randall (2002) resulting in 3 prognostic and 2 elliptic equations. The advective time step is then used after some numerical stabilization. The approaches by Ringler and Randall conserve mass, energy, potential enstrophy and vorticity. Mass or PV conserving balanced models consist of a prognostic equation and 2–4 elliptic equations of the ﬁrst and second order. These elliptic inversions are time consuming and require special (multi-grid) techniques.

5.

Conclusions

The HPM and related semi-Lagrangian numerical schemes, as well as the ones using vorticity-divergence variables (Frank and Reich, 2003; Thuburn, 1997; and Ringler and Randall, 2002) seem to be most advantageous as they use the larger advective time step, at the expense of introducing an artiﬁcial numerical GW-vortex parameterization. It may be a good strategy to test GW-vortex parameterizations in both the balanced models and high-resolution (in space and time) primitive equations. Otherwise, it is unclear to what extent the (artiﬁcial) numerical GW parameterizations in the numerical schemes jeopardize the physical ones. Clearly, the potential interplay between physical and (hidden) numerical parameterizations of gravity waves is a research question with important implications for GCMs. Finally, a thorough answer to the initial question whether a balanced model can provide accurate climate predictions needs to be postponed, although Olaguer’s (2002) results seem to be encouraging.

Acknowledgments The criticism of J. Frank and B.J. Geurts has been much appreciated. The θ-T -model originates from an unpublished work with W.T.M. Verkley.

References T. Birner, A. D¨ ¨ ornbrack and U. Schumann. How sharp is the tropopause at midlatitudes? Geophys. Res. Lett. 29: 10.1029, 2002. O. Bokhove. Flooding and drying in ﬁnite-element discretizations of shallow-water equations. Part 1: One dimension. J. Sci. Comput. 22, To be published, 2005. O. Bokhove and W.T.M. Verkley. Constrained isentropic models of tropospheric dynamics. Submitted to Quart. J. Roy. Meteor. Soc., 2004. ¨ and M.E. McIntyre. Remote recoil: a new wave-mean interaction eﬀect. J. O. Buhler Fluid Mech. 492, 207–230, 2003.

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P.A.M. Dirac. Generalized Hamiltonian dynamics. Proc. Roy. Soc. Lond. A 246, 326– 332, 1958. M. Dixon and S. Reich. Symplectic time-stepping for particle methods. GAMM, M to appear, 2004. J. Frank and S. Reich. Conservation properties of smoothed particle hydrodynamics applied to shallow water equations. BIT 43, 40-54, 2003. N.A. McFarlane. The eﬀect of orographically excited gravity wave drag on the general circulation of the lower stratosphere and troposphere. J. Atmos. Sci. 44, 1775– 1800, 1987. E.P. Olaguer. An eﬃcient 3-D model for global circulation, transport and chemistry. IMA Vol. Math. Appl. 130, 205–276, 2002. D.H. Peregrine and O. Bokhove. Vorticity and surf zone currents. Proceedings of the 26th International Conference on Coastal Engineering 1998, ASCE, Copenhagen. 745–758, 1998. T.D. Ringler and D.A. Randall. A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations on a geodesic grid. Mon. Wea. Rev. 130, 1397–1410, 2002. J. Thuburn. A PV-based shallow-water model on a hexagonal-icosahedral grid. Mon. Wea. Rev. 125, 2328–2347, 1997. J. Vanneste and O. Bokhove. Dirac-bracket approach to nearly-geostrophic Hamiltonian balanced models. Physica D 164, 152–167, 2002. J. Vanneste and I. Yavneh. Exponentially small inertia-gravity waves and the breakdown of quasi-geostrophic balance. J. Atmos. Sci. 61, 211–223, 2004.

NEAR-CRITICAL POINT HYDRODYNAMICS AND MICROGRAVITY Daniel A. Beysens CEA, Service des Basses Temp´ ´eratures, Grenoble & ESPCI, PMMH, 10, rue Vauquelin, 75015, Paris Cedex 05, France [email protected]

Abstract

Near their critical point, ﬂuids exhibit anomalous behavior of thermodynamic parameters (divergence of speciﬁc heat, compressibility and expansion coeﬃcients) and transport coeﬃcients (heat conductivity, thermal diﬀusivity). Weightlessness (”microgravity”) environment permits to go very close to the critical point, thus allowing key tests of the Renormalization Group theory to be made. It also results in a very particular hydrodynamics of dense and hyper-compressible gases, where weightlessness experiments play a key role. For instance, a very fast thermalization eﬀect (”Piston eﬀect”) is evidenced, where a thermal boundary layer expands and adiabatically heats the whole ﬂuid, leading in some cases to an (apparent) violation of the laws of thermodynamics. Another one is concerned with the use of critical slowing down and microgravity to investigate the dynamics of phase separation with no gravity-induced sedimentation. The key role of the coalescence of domains makes valid only two simple growth laws; they can be successfully applied to a quite diﬀerent situation, the evolution laws in the well-known biological problem of sorting of the embryonic cells. Other situations are concerned with the eﬀect of vibrations. The investigation of the above thermal and phase transition problems suggest that a periodic excitation can act as a kind of artiﬁcial gravity, which induces thermal convection, speeds up phase transition and localizes the liquid and vapor phases perpendicular to it. Some of these phenomena still persist at higher temperature and pressure. Fluids in such supercritical conditions are very appealing to the industry as non-polluting solvents or hosts of chemical reactions with high yield.

Keywords: Critical point, supercritical ﬂuids, phase transition, thermalization, cell sorting, Piston eﬀect, microgravity, vibrations, cell sorting

117 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 117–130. © 2005 Springer. Printed in the Netherlands.

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Introduction

It is widely believed that a ﬂuid can exist only as a gas or a liquid. However, there is another state, the ”supercritical state”, that ﬂuids can exhibit (Fig. 1a). Since Baron Charles Cagniard de La Tour discovered in 1821 that the liquid and gas phases of a carbon dioxide sample became undistinguishable after crossing a ”critical” temperature of 31◦ C and a ”critical” pressure of 72 bar, the intriguing properties of this very particular state has motivated a great number of studies. The critical point co-ordinates vary according to the particular ﬂuid under study. For instance, the critical point of carbon dioxide (CO2 ) is observed at 31◦ C and 72 bar, that of water (H2 O) at 375◦ C and 225 bar and that of hydrogen (H2 ) at 33 K and 13 bar. Above the critical temperature and pressure, supercritical ﬂuids exhibit a number of speciﬁc properties (large density, low viscosity, large diﬀusivity) which make them intermediate between liquids and gases [1]. In addition, their isothermal compressibility and thermal expansion can become very large, especially when they approach the critical point. The highly variable properties of near-critical ﬂuids make them very attractive for studying many phenomena that hold for all ﬂuids because of the critical universality. Supercritical ﬂuids are increasingly used by the food and waste management industry [3] for their solubilization properties (e.g. supercritical CO2 ), as host of ”cold” combustion (e.g. supercritical water), in energetics (supercritical thermal or nuclear plants), and in astronautics (e.g. storage of cryogenic ﬂuids).

Figure 1. (a): Phase diagram of a pure substance. The supercritical ”state” corresponds to a compressed gas that exhibits the density of a liquid. (b): Critical anomaly of the speciﬁc heat at constant volume (C Cv ) measured under zero-gravity in SF6 (Spacelab D2, 1993). (From [2]).

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Near-critical Fluids and Microgravity Fluids in their near-critical or supercritical state are strongly aﬀected by gravity. As they are highly compressible, gravity compresses them under their own weight. The density varies in the sample, thus preventing a close approach to the critical point to be made, cf. [4]. They exhibit anomalies in the transport of heat so that convection and buoyancy phenomena, often turbulent, appear for even minute temperature gradients. We show in the following that space experiment have enabled new phenomena to be discovered thanks to a close approach to the critical point and the removal of convection and buoyancy. The eﬀects of gravity can be removed in space thanks to spacecrafts and satellites. On earth, microgravity conditions can be obtained during a short time period in a free fall tower (a few s) and in parabolic ﬂights of planes (20 s) and sounding rockets (2–12 min.). Some other means can be used with liquid mixtures. Liquid mixtures near their consolute critical point exhibit a number of common features with pure ﬂuids, when concentration is replaced by density as an order parameter of the transition. Some aspects can be then studied with binary liquid mixtures that have been made density-matched by partial deuteration [5]. The compensation of gravity forces in simple ﬂuids is more diﬃcult. The large diamagnetism susceptibility of H2 was used to compensate gravity by a magnetic force, proportional to its density. By using a superconducting coil it became possible to study, without gravity eﬀects, the process of solidiﬁcation and gas-liquid phase transition over a wide range, from the triple point (13.8 K) to above the critical point (33 K) [6]. A vibration device has also been implemented using a special sapphire sample.

2.

Universality and Scaling Laws

An important aspect of the critical region is that most of the anomalies of the thermodynamic and transport properties can be set in the form of scaled, universal (power law) functions with respect to the critical point (CP) coordinates. Then, any results obtained with one single ﬂuid can be immediately generalized to a whole class of systems, the ”class of ﬂuids”, to which belong also liquid mixtures, including polymer melts and solutions, microemulsions, molten salts, monotectic liquid metals, etc. [1]. This scaling is of fundamental nature and stems from the universal behavior that the free energy must asymptotically obey to fulﬁll the conditions of a 2nd order phase transition – the CP. In a sense, scaling is generic to CP phenomena. By permitting the measurements to be extremely close to the critical point, zero-g experiments

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have made possible the precise measurements of important, weak powerlaw divergence, such as that of the speciﬁc heat at constant volume Cv (Fig. 1-b). From space experiments, the temperature divergence of the speciﬁc heat has been determined with a very high precision. With the Tc (T is temperature, Tc is the abreduced temperature τ = (T − Tc )/T solute critical temperature), the speciﬁc heat diverges as Cv ∼ τ −α near the critical point. The ”critical” exponent α is universal. Its precise determination was a key test of the ’Renormalization Group’ theory, which has been developed in order to try to improve the classical macroscopic description of ﬂuid behavior close to the critical point [7]. The value deduced from the space experiments, α = 0.1105 ± 0.027 [2], indeed appears to be very close to the result of the Renormalization Group theory, α = 0.110 ± 0.005.

3.

Phase Transition

Kinetics and Morphology Let us describe a typical phase separation experiment (Fig. 2a) and what has been learned from the space experiments [9]–[10]. The supercritical ﬂuid is thermally quenched from a region of the phase diagram where it is homogeneous (at temperature Ti ) to a region where it is thermodynamically stable as two phases (at temperature Tf ). Droplets nucleate and their development is limited by coalescence events. When the volume fraction of the new phase that has nucleated is low (Fig. 2b), the droplets collide by Brownian motion and coalesce. The average radius of the drops R or the average distance Lm between them evolves as Rφ−1/3 ∼ Lm = 2π(kB T /6πη)t1/3 ,

(1)

where t is time, η is the shear viscosity, T is absolute temperature and kB is the Boltzmann constant. It is worth noting that this law is practically independent of the distance (T − Tc ) from the critical point. To a given φ corresponds a typical interaction length (≈ R/3) between domains. When φ > 0.3, it is found [10] that the ﬂow generated by a coalescence event is able to move a neighboring drop and thus induces another coalescence. Such a process therefore creates a chain reaction of coalescence. In the viscous limit, the pattern looks to be interconnected. Growth is limited by the balance between the capillary pressure gradient σ/R (σ is the gas-liquid surface tension) and the friction due to the shear viscosity, so that (late times) Lm = b(σ/η)t.

(2)

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Figure 2. (a): Schematic phase diagram for simple ﬂuids and liquid mixtures in the plane T − M . CP: critical point. T : temperature. M : order parameter; M = (ρ/ρc ) − 1 for simple ﬂuids and M = c − cc for liquid mixtures. ρ(ρc ) is density (critical density). c (cc ) is concentration (critical concentration). The coexistence curve is described by M = Bτ β with B being a system-dependent amplitude and β = 0.325 an universal exponent. (b): Growth laws when gravity eﬀects are absent. Fluids (SF F6 , CO2 ): all data points except open squares. Liquid mixtures (partially deuterated Cyclohexane and Methanol): letters and open squares. The evolution of the average distance between the domains (Lm ) is expressed in the scaled units ∗ = 2πξ/Lm and t∗ = t/tξ (see text). Lines are theoretical predictions. The Km lower curve (φ > 0.3) corresponds to a ” fast ” growth law and an interconnected morphology (pattern in insert b1) and the reduced upper curve ( φ < 0.3) refers to a ” slow ” growth and a disconnected morphology (pattern in insert b2). (From Ref. [9]).

Here b ≈ 0.03 is a universal constant. All experiments can be rescaled by the natural lengthscale and timescale: the correlation length ξ of density ﬂuctuations (ξ diverges as τ −ν ), and the associated diﬀusion time tξ (tξ diverges as τ −3ν ) .

Application to Biological Tissues The development of domains by coalescence events, as reported above, is very general. The universality of behavior, which is observed in ﬂuids and liquid mixtures, can be extended to other areas of science. In particular, it can be applied to developmental biology where tissues can be considered as very viscous liquids (viscosity η ≈ 106 Po), with a surface tension arising from the balance of adhesion sites between the tissue cells (eﬀective interfacial tension σ ≈ 10 dyn.cm−1 ). When analyzing both the kinetics and morphology of cell sorting in embryonic chicken tissues, [11](Fig. 3), the development of the pattern can be interpreted as

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Figure 3. Upper sequence: gas and liquid phase ordering in SF6 under reduced gravity, after a thermal quench to Tc − 0.7 mK. Cell diameter: 12 mm. Gas and liquid eventually order with the liquid phase wetting the container wall and surrounding the gas phase, corresponding to σwl < σwg . Here σwl and σwg are the wall-liquid and wallgas interfacial tensions, respectively. The three pictures from the left correspond to states at 120 s, 275 s and 3960 s after quench, respectively. Lower sequence: sorting out of chicken embryonic pigmented epithelial cells (dark) from chicken embryonic neural retinal cells (light). Aggregate size: 200 µm. At the end of sorting, neural retinal cells preferentially wet the external tissue culture medium surrounding the aggregates. Here σtn (=1.6 dyn/cm)< σtp (=12.6 dyn/cm), where σtn and σtp are the tissue culture medium-neural retina and the tissue culture medium-pigmented epithelium interfacial tensions, respectively. The three pictures from the left correspond to 17 h, 42 h and 73 h after initiation of sorting, respectively. (From [11]).

a result of the coalescence of domains that rearrange like droplets. The domains continuously coalesce and form a network, showing the same linear evolution as the fast growth in liquids. The pseudo-period between domains can indeed be ﬁtted to a linear growth law Lm = bt. The value of parameter b turns out to be comparable to that extracted from Eq. (2), although surface tension and viscosity values diﬀer by factors as large as 108 .

4.

Thermalization

The ”Piston” Eﬀect The thermal diﬀusivity of ﬂuids vanishes near the critical point and a simple calculation [12] shows that it would need more than one month to reach thermal equilibration in a sample of 1 cm3 at T − Tc = 1 mK.

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At the beginning of the microgravity experiments, it thus seemed hopeless to the scientists to try to homogenize in temperature and density a sample of ﬂuid close to the critical point in a reasonable experiment time. However, in 1986, a preliminary experiment by Straub in a sounding rocket [8] indicated that thermalization might be much faster than expected. In a workshop in 1989 on thermal equilibration near the critical point, Onuki pointed out the importance of ”adiabatic heating”. We proposed a hydrodynamic mechanism of thermalization that was based on hydrodynamics where, at the sample wall the hot diﬀuse boundary layer expands and compresses adiabatically the whole ﬂuid. Thermalization proceeds at the velocity of sound. As a result, a spatially uniform heating of the bulk ﬂuid should be observed. There should be a real ﬂow at the boarder between the bulk ﬂuid and the expanding diﬀuse layer, later observed under microgravity as real ”jet” ﬂows. Onuki and Ferrell and the Gammon team proposed a similar process, an ”adiabatic heating”. In the Physical Review A issue of December, 1990 [12]-[14] Onuki and Ferrell, the Gammon team and our group exposed their views on the subject and all concluded a fast thermalization by this eﬀect. Other reports (Meyer in [15]) came out nearly at the same time. We coined in [12] this adiabatic heating phenomenon the ”Piston Eﬀect” (PE), a name which has been accepted by the scientiﬁc community.

Can Heat Flow Backwards? This eﬀect is at the origin of a very particular behavior [17] when the vapor is in equilibrium with liquid below the critical point (Fig. 4b). While heating the cell, the temperature of the vapor becomes greater than that of the wall. That heat ﬂow could seemingly ﬂow from cold to hot contradicts the laws of thermodynamics. However, as we are here in presence of a thermo-mechanical conversion where the hot boundary layer compresses more the gas than the liquid, the violation is only apparent. These results concerning the PE have been adapted and modiﬁed to the earth’s environment, where gravity couples to the PE-induced ﬂows and the geometry of the phases. Paradoxical phenomena, such as the cooling of the bulk ﬂuid after a heat pulse, have also been obtained [18]. Also, the accelerations of the shuttle have been used (rotation and maneuvering the shuttle) to investigate the eﬀect of density destratiﬁcation, a study performed with Air Liquide Company to validate the codes that are now used in the pressurization of the reservoirs of the Ariane 5 rocket.

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(a) L

δ

ρ

T

T

Fast & homogeneous F The «Piston»:

ρ

Figure 4. (a): The Piston Eﬀect mechanism [12]-[16]: a thin hot boundary layer expands and compresses the bulk ﬂuid. The corresponding temperature proﬁle shows a thin zone of strong gradients near the heated boundary (thermal boundary layer δ) and a homogeneous rise in the rest of the ﬂuid, that settles at the speed of sound. (b): overheating of nearly 20% obtained in the gas phase of a SF F6 sample at 10 K below the critical point (ALICE in MIR, 1999). A temperature rise of Tw = 0.1K is TG ), that of the imposed at the cell wall. The temperature evolution of the gas ( δT liquid ( δT TL , at two locations) are shown. In the insert is reported the sample with the thermistors. The fact that heat can ﬂow from “cold” to “hot” apparently violates thermodynamics. It is a spectacular demonstration of the reality of the “piston” in the “Piston Eﬀect”. (From Ref.[17]). 0.06

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Figure 5. Yield Y = transmitted power/ incident power vs time (s) at various Tc = 33 K). At time t0 , power (7.5 mW) is sent for 300 s at one T − Tc . Fluid is H2 (T end of the cell. The transmitted power is measured at the other end. The conduction Ywall by the wall of the sample cell (curve ”wall”) has been subtracted.

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Since temperature outside the thermal boundary layers is homogeneous, the bulk ﬂuid acts as a thermal short circuit. A question arises whether it is possible to use this Piston eﬀect as a kind of heat pipe to carry heat on long distances. Experiments and simulation have been performed with H2 [19] in a magnetic gravity compensation set-up [6]. The results (Fig.5) show that the heat transfer is very fast, in contrast to the usual conduction process. In particular, the yield Y (see Fig. 5 caption) shows a ﬁnite slope at initial time, in contrast with conduction in the cell walls where the slope is zero. The maximum yield increases when T approaches Tc . Its value corresponds to the heat transfer in the steady conduction state (at inﬁnite time).

5.

High Frequency Vibrations

At high frequency excitation - i.e. frequencies larger than the inverse typical hydrodynamics times - the time average of the Bernoulli pressure, which is proportional to the ﬂuid square velocity, is non-zero. The pressure gradient that appears in a non-homogeneous ﬂuid can thus induce ﬂows perpendicular to the vibration direction. However, at low frequency, vibration acts by its instantaneous acceleration and can induce ﬂows parallel to the vibration (as usually gravity does). In the following, a will denote the vibration amplitude, f the frequency and ω = 2πf the angular frequency.

Vibrational Thermal Eﬀects When a ﬂuid is submitted to a vibrational acceleration in a thermal gradient in the Rayleigh-B´ ´enard conﬁguration, convection is able to start at conditions corresponding to a vibrational Rayleigh number [20] Rav =

[aω(∂ρ/∂T )p ∆T e]2 2ηD

(3)

larger than a few thousands. Here ∆T is the temperature diﬀerence between two ﬂuid layers separated by the distance e and D is the thermal diﬀusivity coeﬃcient. As the ﬂuid temperature becomes closer to the critical temperature, Rav diverges as (T − Tc )−1.9 . The ﬂuid then becomes extremely sensitive to vibration as the critical point is approached. Measurements of ﬂow velocities performed in the MIR station in CO2 and in SF6 conﬁrm this expectation [22]-[23] (Fig. 6). A heat ﬂux was sent into the ﬂuid from a point-like source (thermistor). Depending on the oscillation velocity, two regimes of heat propagation are observed: (i) at low frequency, heat is convected during one oscillation period to form plumes parallel to vibration (Fig. 6b); (ii) at high frequency, heat

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c

ω

ω

Figure 6. Interferometer image of the hot boundary layer around a heating thermistor T h1, supported by a thread. Fluid: SF F6 at Tc +0.5 K. (a) no vibration (hot region underlined in white); (b) under low frequency; (c) under high frequency vibration. Here convection rolls form. (ALICE in MIR, 1999, from [27]).

is convected by convection rolls perpendicularly to the direction of oscillation (Fig. 6c). A numerical simulation and analysis of the convection has been performed by Jounet in [21], emphasizing the role of vortices.

Vibrational Phase Ordering A plane liquid-vapor layer vertically vibrating parallel to gravity displays two diﬀerent regimes [24]. Far from the critical point, a square wave-pattern deformation arises (the usual Faraday instability). At Tc − T0 ≈ 20 mK a temperature T0 close to the critical temperature (T for CO2 ), a transition to a new pattern comprised of lines occurs. This transition is due to the increase of dissipation near the critical point. This is a rather unique example of a strong coupling between two diﬀerent critical point phenomena: the critical point of interface instability and the thermal critical point of the liquid-vapor phase transition. When acceleration is perpendicular to gravity, a Kelvin-Helmholtzlike instability is observed [25], with the interface modulated as a ”frozen” roll wave pattern (Fig. 7a). The mechanism of the instability results from the relative motion of the two ﬂuids induced by vibration. A perturbation becomes unstable if the cell velocity (aω) is larger than the threshold velocity [26] (ρL + ρG )3 σg . (4) (aω)0 = ρL ρG ∆ρ ∆ρ Here ρL (ρV ) is the liquid (vapor) density and ∆ρ = ρL −ρG is the liquidvapor density diﬀerence. This destabilization is due to the increasing eﬀect of the Bernoulli - type pressure arising from the velocity diﬀerence between gas and liquid.

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Figure 7. (a-b): CO2 gas-liquid phases show up under 1-g as gas phase (G) above the liquid phase (L) separated by a ﬂat meniscus. When submitted to vibration, the phases can order in a diﬀerent way: (a) under 1 − g, the interface exhibits ”frozen waves”; (b) under zero-g, gas-liquid phases order in layers. Cell diameter: 10 mm (sounding rocket Maxus 5, April, 2003). (c): Typical phase separation in H2 at T c − 1.06 mK with a = 0.3 mm and f = 20s−1 . From [28].

Three samples of diﬀerent gas volume fractions and various T −T Tc were vibrated at several amplitudes and frequencies ranging from 0.1 mm to 5 mm and 0.1 Hz to 60 Hz, respectively [27]. Although the initial state of the sample was either an emulsion of vapor drops, or a single drop, the ﬁnal state remains the same: vapor and gas phases are forming alternate layers perpendicular to the direction of acceleration. An instability, similar to Kelvin-Helmotz’ with inviscid, zero surface tension, develops as liquid ﬁngers from the cell walls. The ﬁngers coalesce with the droplets in the bulk and/or with the ﬁngers that have grown from the opposite side. Here the viscous boundary layer λ = (2η/ρω)1/2 is the natural lengthscale of the phenomenon.

Phase Transition under Vibration A study of phase separation was performed in H2 under magnetic compensation of gravity (Fig. 7c), for volume fraction φ > 0.3 [28]. The domains are interconnected. When Lm is lower than the viscous boundary layer λ, liquid and gas domains have the same velocity and their growth is unaﬀected. From Eq. (2), the growth velocity U = dLm /dt = bσ/η. When Lm > λ, the domains exhibit diﬀerent velocities whose diﬀerence is proportional to the gas-liquid diﬀerence. The corresponding shear ﬂow

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between the domains speeds up the growth (see Fig. 7c where the crossover is denoted L0 ) as U /U = 1 + ∆U/U , with ∆U/U ∼ (∆ρ)(a/λ). These observations suggest that a periodic excitation can act as a kind of artiﬁcial gravity, which speeds up phase transition and localizes the liquid and vapor phases perpendicular to it.

6.

Conclusion

Supercritical ﬂuids are of both the fundamental interest (universality of phase transition, supercritical hydrodynamics) and industrial interest (supercritical solubilization, nucleation of nanomaterials, oxidation, thermalization, storage). The ﬁeld of Critical Point Phenomena has achieved some major breakthroughs during the last 15 years thanks to microgravity research; in particular, a new thermalization process has been discovered, the ”Piston Eﬀect”, that reveals novel hydrodynamics in such near-critical ﬂuids. The study has strongly modiﬁed our vision of critical point phenomena and even of hydrodynamics: the very unusual hydrodynamics of these supercritical ﬂuids, compressible, dense, and weakly viscous, makes their behavior quite particular when compared to gas or liquids. Future experiments will certainly lead to the discovery of new and unexpected phenomena that will be of interest for both the fundamental and applied science.

Acknowledgments This review has been made possible thanks to the friendly help and contribution of Y. Garrabos, B. Zappoli, J. Hegseth, P. Evesque and V. Nikolayev. The ﬁnancial support of CNES is gratefully acknowledged.

References [1] See e.g. H.E. Stanley, Introduction to phase transitions and critical phenomena, Clarendon Press, Oxford, New York, 1971; D. Beysens, J. Straub, D. Turner, [in:] ”Fluid Sciences and Materials Science in Space”, H.U. Walter [Ed.], pp.221–256, Springer, Berlin, 1987. [2] A. Haupt, J. Straub, Phys. Rev. E 59, pp.1795–1802, 1999. [3] F. Cansell, P. Beslin, B. Berdeu, Environmental Progress 17, pp.258–263, 1998; S. Yesodharan, Current Science 82, pp.1112–1122, 2002. [4] M.R. Moldover, J.V. Sengers, R.W. Gammon, R.J. Hocken, Rev. Mod. Phys. 51, pp.79–99, 1979. [5] D. Beysens, Acta Astron. 12 525-530, 1985; C. Houessou, P. Guenoun, R. Gastaud, F. Perrot, D. Beysens, Phys. Rev. A 32, pp.1818–1833, 1985. [6] R. Wunenburger, D. Chatain, Y. Garrabos, D. Beysens, Phys. Rev.E 62, pp.469– 476, 2000; see http://www.spaceflight.esa.int.

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[7] K.G. Wilson, J. Kogut, Phys. Reports C 12, 75, 1974. [8] K. Nitsche, J. Straub, Naturwissenschaften 73, 370, 1986; J. Straub, L. Eicher, A. Haupt, Phys. Rev. E 51, pp.5556–5563, 1995. [9] D. Beysens, Y. Garrabos, Physica A 281, 361-380, 2000 and refs. therein. [10] V. Nikolayev, D. Beysens, P. Guenoun, Phys. Rev. Lett. 76, pp.3144–3147, 1996; V. Nikolayev, D. Beysens, Physics of Fluids 9, pp.3227-3234, 1997. [11] D.A. Beysens, G. Forgacs, J.A. Glazier, P.N.A.S. 97, pp.9467-71, 2000; ibid., Networks of droplets induced by coalescence: application to cell sorting, in: Dynamical Networks in Physics and Biology, D. Beysens and G. Forgacs [Eds.], pp.161–169, Springer & EDP Sciences, Berlin & Les Ulis, 1998. [12] B. Zappoli, D. Bailly, Y. Garrabos, B. Le Neindre, P. Guenoun, D. Beysens, Phys. Rev. A 41, pp.2264–2267, 1990. [13] A. Onuki, H. Hao, R.A. Ferrell, Phys. Rev.A 41, pp.2255-2259, 1990; A. Onuki, R.A. Ferrell, Physica A 164, pp.245–264, 1990. [14] H. Boukari, J.N. Shaumeyer, M.E. Briggs, R.W. Gammon, Phys. Rev. A 41, pp.2260–2263, 1990. [15] R.P. Behringer, A. Onuki, H. Meyer, J. Low Temp. Phys. 81, pp.71–102, 1990. [16] See e.g. Y. Garrabos, M. Bonetti, D. Beysens, F. Perrot, T. Fr¨¨ohlich, P. Carl´es, B. Zappoli, Phys. Rev. E. 57, pp.5665-5681, 1998. [17] R. Wunenburger, Y. Garrabos, C. Chabot, D. Beysens and J. Hegseth, Phys. Rev. Lett. 84, pp.4100–4103, 2000; M. Sincell, Science 288, pp.789–791, 2000. [18] T. Fr¨ ¨ olich, D. Beysens, Y. Garrabos, V. Nikolayev, A heat ﬂux can cool a nearcritical ﬂuid, preprint, 2004. [19] D. Beysens, D. Chatain, V. Nikolayev, Y. Garrabos, 4th International Conference on Launcher Technology Space Launcher Liquid Propulsion, 3-6 December 2002 – Liege, Belgium. [20] G.Z. Gershuni, D.V. Lyubimov, Thermal Vibrational Convection, John Wiley & Sons, New-York, 1998. [21] A. Jounet, Phys. Rev. E, 65, pp.37301–37304, 2002 [22] S.V. Avdeev, A.I. Ivanov, A.V. Kalmykov, A.A. Gorbunov, S.A. Nikitin, V.I. Polezhae, G.F. Putin, A.V. Zuzgin A.V., V.V. Sazonov, D. Beysens, Y. Garrabos, T. Frohlich, ¨ B. Zappoli, Experiments in the far and near critical ﬂuid aboard the MIR station with the use of the ’Alice-1’ instrument, in Proceedings of the Joint Xth European and VIth Russian Symposium on Physical Science in Microgravity, St. Petersburg, Russia, 15-21 June 1997, Vol.1, pp.333–340, V.S. Avduyevsky and V.I. Polezhaev [Eds.], Institute for Problems in Mechanics, RAS, Moscow. [23] Y. Garrabos, D. Beysens, C. Chabot, R. Wunenburger, V. Polezhaev, V. Emelianov, A. Ivanov, A. Kalmykov, Thermo-Convectional Phenomena Induced by Vibrations in Supercritical SF6 Under Low Gravity, preprint, 2004. [24] S. Fauve, K. Kumar, C. Laroche, D. Beysens, Y. Garrabos, Phys. Rev. Lett. 68, pp.3160–3163, 1992. [25] R. Wunenburger, P. Evesque, C. Chabot, Y. Garrabos, S. Fauve, D. Beysens, Phys. Rev. E 59, pp.5440–5445, 1999.

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[26] D.V. Lyubimov, A. Cherepanov, Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza 6, 8-13, translated in Fluid Dynamics 86, pp.849–854, 1987. [27] D. Beysens, C. Chabot, Y. Garrabos, Microgravity Sci. Technol. 11, pp.113–118, 1998. [28] D. Beysens, D. Chatain, P. Evesque, Y. Garrabos, Phase separation under vibrations in near-critical hydrogen free of gravity eﬀects, submitted, 2004.

FLAW TOLERANT NANOSTRUCTURES OF BIOLOGICAL MATERIALS Huajian Gao∗ , Baohua Ji, Markus J. Buehler, and Haimin Yao Max Planck Institute for Metals Research, Heisenbergstrasse 3, 70569 Stuttgart, Germany

Abstract

1.

Bone-like biological materials have achieved superior mechanical properties through hierarchical composite structures of mineral and protein. Gecko and many insects have evolved hierarchical surface structures to achieve superior adhesion capabilities. We show that the nanometer scale plays a key role in allowing these biological systems to achieve such properties, and suggest that the principle of ﬂaw tolerance may have had an overarching inﬂuence on the evolution of the bulk nanostructure of bone-like materials and the surface nanostructure of gecko-like animal species. We demonstrate that the nanoscale sizes allow the mineral nanoparticles in bone to achieve optimum fracture strength and the spatula nanoprotrusions in Gecko to achieve optimum adhesion strength. Strength optimization is achieved by restricting the relevant dimension to nanometer scale so that crack-like ﬂaws do not propagate to break the desired structural link. Continuum and atomistic modeling have been conducted to verify this concept.

Introduction

New challenges in materials science in the 21st century will include the development of multi-functional and hierarchical materials systems. Nanotechnology promises to enable mankind to design materials using a bottom-up approach, that is, to construct multi-functional and hierarchical material systems by tailor-designing structures from atomic scale and up. However, to this date, there is almost no theoretical basis on how to design a hierarchical material system to achieve a particular set of functions. One strategy is to look among solutions in nature for hints on advanced materials design.

∗ [email protected]

131 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 131–138. © 2005 Springer. Printed in the Netherlands.

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(a)

(b)

Figure 1. Nanostructure of bones (a) and the toe of geckos (b) that consists of a terminal nanostructure called spatula of about 200–500 nm in diameter.

Biological materials, such as bone [1] exhibit many levels of hierarchical structures from macroscopic to microscopic length scales. The smallest building blocks in such materials are generally on the nanometer length scale. For instance, the nanostructure of bone (Fig. 1a) consists of mineral crystal platelets with thickness around a few nanometers embedded in a collagen matrix [1,2]. Interesting nanostructures of biological systems for superior mechanical properties are not just limited to the nanocomposite structure of bone. Gecko and many insects have evolved elaborate hierarchical surface structures in their foot hair to achieve extraordinary adhesion capabilities. These animals possess ability to adhere to vertical surfaces and ceilings. A gecko is found to have hundreds of thousands of keratinous hairs or setae on its foot; each seta is 30 ∼ 130µm long and contains hundreds of protruding nanoscale structures called spatula (Fig. 1b). We attempt to address the following questions. Why is nanoscale is so important to biological systems? What are the basic mechanisms and principles behind biological nanostructures?

2.

The Protein-mineral Bulk Nanostructure of Bone-like Biocomposites

Experimental observations (e.g. [1,3] and further references in [4]) have shown that, at the most elementary structure level, biological materials exhibit a generic structure consisting of staggered mineral platelets embedded in a soft matrix (Fig. 2a). Under an applied tensile stress, the path of load transfer in the mineral-protein biocomposites can be represented by a tension-shear chain model [4] where the mineral platelets carry tensile load and the protein transfers load between mineral crystals via shear (Fig. 2b). In this tension-shear chain model, the mineralprotein composite is simpliﬁed to a one-dimensional chain consisting of tensile springs (mineral) interlinked by shear springs (protein). The integrity of the composite chain structure is hinged upon the strength of mineral platelets since breaking of the platelets would destroy the cri-

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(a)

(b)

Figure 2. A simple tension-shear chain model of biocomposites. (a) Schematic of staggered mineral crystals embedded in a soft (protein) matrix. (b) Tension-shear chain model showing the path of load transfer in the mineral-protein composites.

tical structural links in the composite, leading to disintegration of the protein-mineral network. The strength of mineral platelets plays a crucial role in the fracture energy of the composite. In order to achieve high fracture energy, the mineral platelets must be able to sustain large tensile stress without fracture. How to optimize the strength of the mineral platelets? The Griﬃth theory of fracture [5] and common engineering experiences have shown that the strength of brittle solids is determined by pre-existing ﬂaws. It was pointed out that the nanometer scale is the key to optimizing mineral strength [4]. At the simplest level, this can be understood from the following consideration. A perfect, defect-free mineral particle should be able to sustain mechanical stress near the theoretical strength σth of the material. However, we assume that the particle contains crack-like ﬂaws. For example, protein molecules trapped within the mineral crystals during the biomineralization process are mechanically equivalent to embedded microcracks. Considering all potentially existing cracks in a thin strip, the largest crack, and hence the most dangerous one, will be a crack about half the strip width. The key idea of ﬂaw tolerance [4,6] is that cracks conﬁned in a small structure do not propagate until the material around the crack uniformly reaches the theoretical strength. This can also be demonstrated with the crack conﬁguration shown in Fig. 3. In this conﬁguration, the strength of the material can be calculated from the Griﬃth theory as σf = 4γE ∗ /h for a mineral platelet ∗ 2 width h and fracture surface energy γ, where E = E/ 1 − ν , E being the Young’s modulus and ν the Poisson ratio. According to this expression, the strength of the material approaches inﬁnity when h goes to zero. This is physically impossible since the largest stress a material can sustain is limited by an upper bound (theoretical strength) σth . This suggests that there exists a transition between crack propagation governed by the Griﬃth criterion and uniform rupture of atomic bonds

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at theoretical strength at a critical length scale [4] hcr =

4γE ∗ 2 . σth

(1)

Taking a rough estimate γ = 1 J/m2 , Em = 100 GPa, and σth = Em /30, we found hcr to be around 30 nm for a half-cracked platelet [4]. The nanometer scale not only allows the strength of mineral particles to be optimized near theoretical strength but also renders these particles insensitive to crack-like defects (ﬂaw tolerance). This concept has so far also been studied by atomistic simulations (details see [6]). Figure 3(a) plots the critical failure stress normalized by the theoretical strength, indicating a smooth transition between crack propagation governed by the Griﬃth condition for thick layers (hcr /h < 1) to uniform rupture at theoretical strength for thin layers ( hcr /h > 1). This result is fully consistent with the continuum mechanics analysis [4]. Figure 3(b) plots the distribution of normal stress ahead of the crack. As the strip width is decreased, stress concentration at crack tip disappears and the stress distribution becomes uniform near the crack tip, and thus the solid has become insensitive to ﬂaws. Further analysis of the protein-mineral bulk nanostructure of bone on stiﬀness (discussion of the interplay of the soft protein matrix and the stiﬀ mineral platelet material, and the impact of the aspect ratio of mineral platelet) and fracture energy (including a discussion on sacriﬁcial Ca++ bonds) can be found in [6]. The interested reader is referred to references [4,6-9] for further details of our group on this topic.

(a)

(b)

Figure 3. (a) Fracture strength as a function of layer width h, and (b) stress distribution ahead of the crack for diﬀerent layer widths h.

Flaw Tolerant Nanostructures of Biological Materials

3.

135

Flaw Tolerant Surface Nanostructure of Gecko for Adhesion

The concept of nanoscale ﬂaw tolerance can be discussed in a more general context to include the surface nanostructure of gecko. Among the hairy biological attachment systems, the density of surface hairs (setae) increases with the body weight of animal, and gecko has the highest density among all animal species that have been studied [10]. The most terminal (smallest) structure of gecko’s attachment mechanism is called spatula (Fig. 1b) which is about 200–500 nanometers in diameter. Why is the spatula size in the nanometer range? To understand this, we have modeled the spatula as an elastic ﬂat-ended cylindrical hair in adhesive contact with a rigid substrate [11]. The radius of the cylinder is R. To test the ability of the ﬂat cylinder to adhere in the presence of adhesive ﬂaws, imperfect contact between the spatula and substrate is assumed such that the radius of the actual contact area is a = αR, and 0< α 0), and an additional input of energy is required to stabilize the protein cluster. For this thermodynamically unfavorable growth process, increasing the force per unit length, f , leads to smaller adhesions. FA can only grow when the overall free energy of Eq. (10) is negative, corresponding to e > 0 – i.e. favorable aggregation energy and an exothermic, local chemical interaction. However immobilization of the extracellular matrix (k → 0) or very large forces can lead to positive value for ∆E, and, if large enough, may arrest the growth process, even if e > 0. This analysis can be generalized to predict the kinetics of growth of FA [30].

Acknowledgments The authors acknowledge very fruitful experimental collaborations with L. Addadi, N. Balaban, A. Bershadsky, B. Geiger, D. Riveline, and theoretical discussions with I. Bischofs and M. Kozlov. This work has been supported by the Israel Science Foundation, the U.S. – Israel Binational Science Foundation, the Minerva Foundation, the German – Israel Foundation and an EU Network Grant. USS is grateful for the support of the Emmy Noether Program of the German Science Foundation.

References [1] B. Alberts, D. Bray, J. Lewis, M. Raﬀ, K. Roberts, J. Watson, Molecular Biology of the Cell, Garland Publishing, New York, 1994. [2] D. Boal, Mechanics of the Cell, Cambridge University Press, Cambridge, 2002. [3] B. Geiger, A.D. Bershadsky, Cell, Vol. 110, pp.139–142, 2002. [4] N.Q. Balaban, U.S. Schwarz, D. Riveline, P. Goichberg, G. Tzur, I. Sabanay, D. Mahalu, S.A. Safran, A.D. Bershadsky, L. Addadi, B. Geiger, Nature Cell Biol., Vol. 3, pp.466–472, 2001.

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[5] U.S. Schwarz, S.A. Safran, Phys. Rev. Lett., Vol. 88, 048102, 2002. [6] I.B. Bischofs, U.S. Schwarz, Proc. Natl. Acad. Sci. USA, Vol. 100, pp.9274–9279, 2003; I.B. Bischofs, S.A. Safran, U.S. Schwarz, Phys. Rev. E, Vol. 69, 021911, 2004. [7] P.C. Dartsch, H. H¨ ¨ ammerle, E. Betz, Acta Anat., Vol. 125, pp.108, 1986. J.H.-C. Wang and E.S. Grood, Connect.Tissue Res., Vol. 41, 29, 2000. [8] K. Jakab, A. Neagu, V. Mironov, R.R. Markwald, G. Forgacs, PNAS, Vol. 101, pp.2804, 2004. [9] B. Geiger, Science Vol. 294, pp.1661–1663, 2001. [10] F.G. Giancotti, E. Ruoslahti, Science, Vol. 285, pp.1028–1032, 1999. [11] R. Zaidel-Bar, C. Ballestrem, Z. Kam, B. Geiger, J. Cell Sci., Vol. 116, pp.4605– 4613, 2003. [12] R.J. Jr Pelham, Y.-L. Wang, Proc. Natl. Acad. Sci. USA, Vol. 94, pp.13661– 13665, 1997. [13] C.-M. Lo, H.-B. Wang, M. Dembo, Y.-L. Wang, Biophys. J., Vol. 79, pp.144–152, 2000. [14] D. Riveline, E. Zamir, N. Q. Balaban, U. S. Schwarz, T. Ishizaki, S. Narumiya, Z. Kam, B. Geiger, A.D. Bershadsky, J. Cell Biol., Vol. 153, pp.1175–1185, 2001. [15] K.A. Beningo, Y.-L. Wang, Trends Cell Biol., Vol. 12, 79, 2002. [16] A.K. Harris, P. Wild, D. Stopak, Science, Vol. 208, pp.177–179, 1980; A.K. Harris, D.Stopak, P.Wild, Nature, Vol. 290, pp.249–251, 1981. [17] M. Dembo, T. Oliver, A. Ishihara, K. Jacobson, Biophys.J., Vol. 70, pp.20082022, 1996; M. Dembo, Y.-L. Wang, Biophys.J., Vol. 76, pp.2307-2316, 1999. [18] R. Merkel, P. Nassoy, A. Leung, K. Ritchie, E. Evans, Nature, Vol. 397, 50, 1999. [19] L. D. Landau, E. M. Lifshitz, Theory of Elasticity, Pergamon, Oxford, 1970. [20] U.S. Schwarz, N.Q. Balaban, D. Riveline, B. Geiger, S.A. Safran, Biophysical Journal, Vol. 83, pp.1380, 2002. [21] R. Siems, Phys. Stat. Sol., Vol. 30, pp.645, 1968. [22] H. Wagner, H. Horner, Adv. Phys., Vol. 23, pp.587, 1974. [23] J. P. Butler, I. M. Tolic-Norrelykke, B. Fabry, J. J. Fredberg, Am. J. Physiol. Cell Physiol., Vol. 282, pp.C595, 2002. [24] J. Y. Wong, A. Velasco, P. Rajagopalan, Q. Pham, Langmuir, Vol. 19, pp.1908, 2003. [25] F. Grinnell, Trends in Cell Biol., Vol. 10, pp.362, 2000. [26] P.F. Davies, , A. Robotewskyj, M.L. Griem, J. Clin. Invest., Vol. 93, pp.20312038, 1994. [27] A. Nicolas, S.A. Safran, Phys. Rev. E, Vol. 69 pp.051902-1–051902-7, 2004. [28] B.Z. Katz, E. Zamir, A.D. Bershadsky, Z. Kam, K.M. Yamada, B. Geiger, Mol. Cell Biol., Vol. 11, pp.1047–1060, 2000. [29] D. Choquet, D.P. Felsenfeld, M.P. Sheetz, Cell, Vol. 88, pp.39–48, 1997. [30] A. Nicolas, B. Geiger, S. A. Safran, PNAS, Vol. 101, pp.12520–12525, 2004.

ELECTROKINETIC FLOW INSTABILITIES IN MICROFLUIDIC SYSTEMS Hao Lin, Michael H. Oddy and Juan G. Santiago Mechanical Engineering Department, Stanford University Stanford, CA 94305, USA [email protected]

Abstract

The stability of electrokinetic ﬂow in a rectangular cross-section microﬂuidic channel with transverse conductivity gradients and driven by streamwise electric ﬁelds was explored. Such a system exhibits a critical electric ﬁeld above which the ﬂow is highly unstable, resulting in ﬂuctuating velocities and rapid stirring. The problem was studied using theoretical and numerical analyses, as well as experimental observations. It was found that the internally generated electric body force was responsible for the instability, whereas the diﬀusion of ion species provided a stabilizing mechanism. Various models including two-dimensional and depth-averaged formulations were studied; modeling results compare well with experimental observations. These results have application to the design and control of on-chip assays that require high conductivity gradients, and provide a rapid mixing mechanism for low Reynolds number ﬂow in microchannels.

Keywords: Electrokinetics, electrokinetic instability, critical electric ﬁeld, electric conductivity gradient, microﬂuidics, microchannel, micromixing

1.

Introduction

Over the past decade there has been an extensive research into the design of devices that perform chemical analysis in micro-fabricated ﬂuidic channel structures. Often referred to as Micro Total Analysis Systems (µTAS), these systems exhibit a mass transport regime that is often diﬀerent from that of macro-scale ﬂow devices. Many of these devices apply electrokinetic liquid-phase, bioanalytical techniques including capillary electrophoresis and isoelectric focusing, and often manipulate the samples having poorly characterized or unknown electrical conductivi343 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 343–354. © 2005 Springer. Printed in the Netherlands.

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ties. As a result, conductivity mismatches often occur between the sample/reagent species and the background electrolyte. In the presence of applied electric ﬁelds, conductivity gradients can induce electrohydrodynamic coupling, which can in turn generate complex, unstable ﬂowﬁelds. Flows exhibiting these physics have been reported in the classical electrohydrodynamics literature (see for example, the seminal paper by Hoburg and Melcher, 1976, and a later work by Baygents and Baldessari, 1998). In this paper, we review ﬂow instabilities due to electric ﬁeld and conductivity gradients coupling in electrokinetic systems. Electrokinetic ﬂows are a subset of electrohydrodynamics characterized by the presence of an electrical double layer and regimes, where transport due to molecular diﬀusion is important. Although desirable for rapid-mixing applications, the electrokinetic instabilities are unwanted in microﬂuidics applications such as sample injection, separation, and controlled diﬀusion-limited reaction processes where minimal sample dispersion is required. This motivates research toward a better understanding of the conditions necessary for the onset of electrokinetic ﬂow instability. In 2001 Oddy et al. ﬁrst reported observation of electrokinetic instability (EKI) in a microchannel system. These experiments were performed in 4-mm-long glass capillaries with rectangular cross-sections, and the instabilities were in general of temporal nature (Oddy et al., 2001). In a slightly diﬀerent geometry (microﬂuidic T-junction), Chen and Santiago also reported spatial ampliﬁcation of disturbances which was later identiﬁed as convective instability (Chen and Santiago, 2002, Chen et al., 2004). In all of these experiments, conductivity gradients were in the spanwise direction (perpendicular to the electric ﬁeld), and there existed critical values of the applied streamwise electric ﬁeld above which instabilities and signiﬁcant stirring occurred. Following these initial experimental observations, there has been a development of models for electrokinetic ﬂow instabilities. Models are useful in predicting threshold conditions for instability onset as well as other ﬂow features including coherent wave structures and mixing rate. Lin et al. (2004) and Chen et al. (2004) showed that a generalized EHD modeling framework (derived from the so-called “leaky dielectric” model ﬁrst developed by Melcher and Taylor, 1969) can be used to describe both the low-conductivity, non-diﬀuse charge dynamics of classical EHD, and the more recently reported ﬂow instabilities of high-conductivity electrolyte in electrokinetic microsystems. Lin et al. (2004) analyzed the temporal stability properties based on a two-ion model, comprising the conductivity transport equation along with the conservation of momentum and electromigration current. They showed that the model provided good qualitative and fair quantitative agreement with regard to

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the threshold electric ﬁelds and critical wavenumbers. Lin et al. (2004) also presented non-linear simulations of their set of governing equations that capture the high Peclet (or the so-called electric Rayleigh) number stirring events observed in experiments. Using a convective framework, Chen et al. (2004) showed that EKI could manifest itself convectively in the presence of a strong electroosmotic ﬂow. In the latter analysis, EKI is modeled using a linearized, thin-layer limit of the Navier-Stokes equations coupled with conservation equations for electrical conductivity and current. The model reveals both convectively and absolutely unstable eigenmodes. More recently, Storey et al. (2004) presented a depth-averaged version of the governing equations used by the Lin et al. (2004) model. Their depth-averaged model compared favorably with a complete three-dimensional model for thin channel geometries. In this paper we present our experimental, analytical, and computational results and some progress in the pursuit of modeling and understanding of EKI in electrokinetic microchannels. We shall brieﬂy introduce the experimental results, followed by a general theoretical formulation developed in Lin et al. (2004). Using these equations we show the results from various linear analyses as well as nonlinear simulations, and assess their qualitative and quantitative agreements with experimental data. We conclude the paper by introducing our latest development in a depth-averaged framework suitable for the study of generalized electrokinetic ﬂows in microchannels with thin channel geometry.

2.

Experimental Observations

Here we show a few exemplary results from our experiments. The microchannel consisted of a borosilicate glass capillary (Wilmad-Labglass, NJ) with a rectangular cross-section. The channel was 4 mm long (in the x or streamwise direction), 1 mm wide (in the y or spanwise direction), and 0.1 mm deep (in the z or depth direction). Two buﬀer streams initially occupied the upper and lower halves of the microchannel, resulting in a diﬀuse conductivity gradient along the spanwise, y-direction; an electric ﬁeld was subsequently applied along the streamwise, x-direction. The conductivity of the buﬀer streams were 50 and 5 µS/cm, respectively, resulting in a conductivity ratio of γ = 10. For ﬂow visualization, an electrically neutral, heavy-molecular-weight dye (70 kDalton) composed of a dextran-rhodamine B conjugate (Molecular Probes, OR) was added to the high-conductivity buﬀer stream. The imposed electric potential initiated an electroosmotic ﬂow in the channel and, for electric ﬁelds above a threshold value, electrokinetic instabilities.

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A representative set of images from experiments conducted at 1, 2, and 3 kV applied potentials are shown in Fig. 1. The potentials were applied over a distance of 40 mm, such that they were equivalent to applied ﬁelds of 25,000, 50,000 and 75,000 V/m, respectively. In each case, the top ﬁgure of each series shows the initial, undisturbed interface between the dyed and undyed buﬀer streams in the channel (t = 0). The successive images in each column show the temporal evolution of the imaged dye under a constant, DC potential. In this color scheme, blue corresponds to the undyed, low-conductivity stream, and red to the dyed high-conductivity stream. For an applied ﬁeld of 25,000 V/m, the interface was only slightly perturbed and only slight ﬂuctuations are apparent in the images captured at 4.0 s and 5.0 s. At the two higher applied voltages, the interface exhibited a rapidly-growing wave pattern within the ﬁrst 1 s. The unstable ﬂuid motion in the channel buckled the interface and proceeded to stretch and fold the material lines. The transverse

Figure 1. Sample images from the experiment, shown for applied ﬁelds of 25,000, 50,000, and 75,000 V/m, corresponding to the ﬁrst, second, and third column. Images obtained at various times are shown for each column. The electric ﬁeld and bulk ﬂow directions were from left to right. High voltage was applied as a Heaviside function at t = 0 s. Each image corresponds to a physical area 1 mm wide (y) and 3.6 mm long (x). The depth of the channel is 100 µm along the z-direction (into the page). Small amplitude waves at t = 1 s grew and led to rapid stirring of the initially distinct buﬀer streams. The instability stretched and folded material lines and, after about 4 s for the 75,000 V/m applied ﬁeld, resulted in a well-stirred, relatively homogeneous dye concentration ﬁeld. The time of the images in each row are shown in the ﬁgure.

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and ﬂuctuating velocities associated with this unstable motion resulted in rapid mixing of the two streams. At the 75,000 V/m applied ﬁeld, the channel reached a well-stirred state and with nearly-homogeneous concentration ﬁelds observable within 5 s.

3.

Formulation

The description of experiments given above serves as an introduction to the problem and describes the observed features of electrokinetic ﬂow instability. We now turn to a theoretical formulation of the ﬂow. In this section, we summarize the governing equations and discuss the parameters of interest in our experiment. The equations of our model are derived from the conservation laws for a dilute, two-species electrolyte solution (Probstein, 1994), and we have adopted (with justiﬁcation) a relaxation assumption to simplify the equations. The scaling analysis and derivations are discussed in detail in Lin et al. (2004) and should not be repeated here. The (dimensionless) equations read 1 ∂σ + v · ∇σ = ∇2 σ, ∂t Rae ∇ · (σ∇Φ) = 0, ∇2 Φ = −ρE , Re

∇v = 0, ∂v + v · ∇v = −∇p + ∇2 v − ρE ∇Φ, ∂t

(1) (2) (3) (4) (5)

where σ is the conductivity, v is the bulk ﬂuid velocity, Φ is the electric ﬁeld (which includes both the applied and generated components), and p is pressure. The electric Rayleigh number (similar to the Peclet number) and the Reynolds number that arise from the nondimensionalization are deﬁned as Uev H Uev H , Re ≡ . Rae ≡ D ν Here H is half-channel width, D is an eﬀective diﬀusivity of the conductivity, and ν is the kinematic viscosity of the electrolyte solution (usually aqueous). The velocity Uev , the so-called electroviscous velocity, is velocity scale derived by setting equal the electric body force and the viscous force in the momentum equation. (See Hoburg and Melcher, 1976, Lin et al., 2004 and Chen et al., 2004; more discussions on these parameters can also be found in these references.)

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Two-dimensional Model

We ﬁrst present a model where we assume that the ﬂow exists only in the x − y plane, and that there is no dynamics in the z-direction. This analysis will capture the basic physics of the instability mechanisms due to the conductivity gradient. We ﬁrst use a linear stability analysis to predict the regimes where we would expect rapid mixing to occur. The base states are a diﬀused, spanwise conductivity proﬁle σ0 = σ0 (y) with a (high-to-low) conductivity ratio of 10, and a shear electroosmotic ﬂow u0 = u0 (y). Note that the spanwise dependence of the latter was induced by that of the former via the dependence of zeta potential on conductivity. We assume that disturbance is periodic in the x direction, and the growth of amplitude is exponential in time. We have obtained, for each streamwise wave number k and applied ﬁeld E0 , a set of eigenvalues (the exponential growth rates), together with their respective eigenfunctions. In Fig. 2 we show a contour plot of the growth rates of the most unstable eigenfunction in the wave number-Rayleigh number (electric ﬁeld) parameter space. Symbol s denotes the real and dimensional growth rate. The neutral stability curve is obtained by setting s = 0. A threshold electric ﬁeld is successfully captured from the minimal value of E0 on the neutral stability curve. As originally reported by Baygents and Baldessari (1998), we found that the inclusion of the diﬀusive term ∇2 σ/Rae in Eq. (1) is crucial for the existence of the neutral stability curve. 5

10

s = 40 sec−1

Rae

s = 4 sec−1 4

10

Eo ((V/m)

s = 20 sec−1

5

10

4

s = 1 sec−1

10

3

10

Neutral 0

10

1

k

10

Figure 2. Contour plot of growth rates (s) versus wave number and Rayleigh number. Dimensional applied electric ﬁeld is provided on the right-hand axis. For the case plotted here, the ratio of the conductivity between the two streams is 10.

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Figure 3. Snapshots of the simulated dye ﬁeld at various instances in time for diﬀerent driving electric ﬁelds. In this color scheme red corresponds to the high conductivity buﬀer, and blue to the low one. Each column indicates a diﬀerent applied ﬁeld and the rows within each column present the selected snapshots in time. The image corresponds to a physical domain of 3.6 mm×1 mm. The time for noticeable waves to develop is decreased as the ﬁeld is increased.

In addition to the two-dimensional linear stability analysis presented above, we have also solved the full nonlinear governing Eqs. (1-5) numerically to capture the nonlinear evolution of the instability observed in the experiments. The initial conditions are the base states plus a small white noise perturbation. The solution details are documented in Lin et al. (2004). Figure 3 shows the nonlinear evolution of the simulated dye at various instances in time and for three diﬀerent electric ﬁelds. The model reproduces many of the essential features observed in the experiments, including the shape and initial break-up dynamics of the interface, the transverse growth of a wave pattern in the interface, and the roll-up of scalar structures observed at later times. Note the similarity in the most unstable (and most apparent) wave number at later times between the simulation and experiments. Despite the similarities between the wave number and dynamics of the interface breakup, the threshold imposed ﬁelds from both the linear and nonlinear predictions are lower than those shown for the experiment in Fig. 1. For example, compare the evolution of the dye at 25,000 V/m from the experiments (Fig. 1, column 1) and the simulation (Fig. 3, column 3). We see that the simulation at 25,000 V/m predicts

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a well-stirred ﬂow ﬁeld in less than three seconds, while experiments show that the ﬂow is stable on the time-scale of the experiments. The simulation of 25,000 V/m is qualitatively similar to the experimental ﬂow at 75,000 V/m (Fig. 1, column 3). Despite the discrepancy in the magnitude of the applied ﬁeld, our simulation captures a threshold ﬁeld and scalar features qualitatively similar to the experiment. In the following section we address possible causes for the under-prediction of the threshold electric ﬁeld by including three-dimensional eﬀects. In comparison with the temporal instability analysis presented here (which is consistent with the experimentally observed instability in the previous section), Chen et al. (2004) analyzed the instability in a convective framework which is consistent with the spatial growth that was observed in T-junction microﬂuidic channels. Among other contributions, the authors found that an important dimensionless group Rν , deﬁned as the ratio of the electroviscous to electroosmotic velocity, is critical in demarcating the absolute/convective instability boundary. Interested readers are referred to Chen et al. (2004).

5.

Depth-averaged Model

In the previous sections we have provided a two-dimensional framework which appears to capture the primary physics of our ﬂow. However, the primary ﬂaw in that model is the two-dimensional assumption, for a channel with an aspect ratio of δ ≡ d/H = 0.1, where d denotes the channel half-width. Such a thin channel geometry was also used in the experimental work of Chen et al. (Chen et al., 2002; Chen et al., 2004). In three dimensions, an EDL forms not only on the top and bottom walls of y = ±1, but also along the side walls (z = ±1), and strongly drives the ﬂow due to the small depth of the channel. The three-dimensional nature of a thin channel has the added dynamics that the ﬂuid motion in the interior of the channel is directly coupled to the top and bottom wall boundary condition (determined in part by the local value of ion density). In Lin et al. (2004) we presented a 3D linear analysis and found that the predicted threshold ﬁeld was one order-of-magnitude higher than that from the 2D linear analysis, in much closer agreement with the experimental observations. However, a depth-averaging approach is preferred since in general, the 3D analysis (simulations) are computationally more expensive, and considering that the microchannels of our interest are “shallow” in the depth (z) direction. In Chen et al. (2004) a depth-averaged analysis was performed on a set of linearized governing equations and the resulted linear equation system was used for convective

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instability analysis. In Storey et al. (2004) a more complete Hele-Shaw type of integrated momentum equation was used. Both models yielded favorable quantitative results when compared with experimental data. Here we extend and complete the ideas that were ﬁrst developed by Chen et al. (2004) and Storey et al. (2004). We develop a generalized, nonlinear depth-averaged model suitable for the study of electrokinetic microchannel ﬂows with thin channel geometries. We accomplish this through a complete asymptotic analysis of the full threedimensional equations based on a smallness parameter which is the channel cross-sectional aspect ratio δ. Our general methodology follows a combined lubrication (for the momentum equations) and Taylor-Aris (for the convective-diﬀusion of the conductivity ﬁeld) approach. Without presenting the details of the derivation, we list the ﬁnal equations as 1 2 ∂σ ¯ 2 2 2 ¯ ¯ +u ¯ · ∇H σ ∇H σ Ra δ ∇H · [U (U · ∇H σ ¯= ¯+ ¯ )] , (6) ∂t Rae 105 e

Reδ 2

∂u ¯ +u ¯ · ∇H u ¯ ∂t

¯ = 0, σ ∇H Φ) ∇H · (¯

(7)

∇H · u ¯ = 0,

(8)

¯ HΦ ¯ − 3U ¯ + δ 2 ∇2H u = −∇H p¯ + ∇2H Φ∇ ¯.

(9)

Here the overbar denotes depth-averaged quantity, the operator ∇H denotes the in-plane gradient (to distinguish from the full three-dimen¯ ≡ u sional gradient), and U ¯ − u∞ is the diﬀerence between the total depth-averaged velocity and the electroosmotic velocity. The main contributions of this new equation set are the Taylor-dispersion-type term in the conductivity equation, and the Darcy-BrinkmanForchheimer (DBF) type of momentum equation which is of second-order consistency in δ (Chen et al., 2004; Liu and Masliyah, 1996). We present preliminary results in the assessment of the validity and accuracy of the model. Figure 4 compares the linear stability results for growth rate of disturbances versus wave number at a single Rayleigh number of Rae = 5.000. We perform linear analyses using the following three models: 1. The linear integrated momentum equations used by Storey et al. (2004). 2. The linearized three-dimensional equations (Lin et al., 2004). 3. The depth-averaged DBF formulation presented here.

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Integrated momentum

0.35

3D linear

0.3

s

0.25 0.2

DBF Formulation

0.15 0.1 0.05 0

2

4

6

8

k

10

12

14

16

18

Figure 4. Comparison of growth rates of disturbances as predicted by three models. Shown here are the real part of the growth rates versus the wave number for Rae = 5.000 as computed with the DBF momentum equation presented here, the integrated momentum equation (δ 0 approximation, Storey et al., 2004), and the three-dimensional equation set. The DBF formulation represents in-plane viscous stresses that quench the unphysically high wave number growth and is in agreement with the three-dimensional result.

Note that at the linear regime, the Taylor dispersion in Eq. (6) drops as a higher-order term, and the only diﬀerence between the models are within the momentum equations. We ﬁnd that all three models are in good agreement for wave numbers below about 4. However, when compared with the more accurate three-dimensional analysis, the DBF momentum equation provides signiﬁcantly better results at higher wave numbers than the lower-order, integrated momentum approximation. The characteristics of the DBF momentum equations also make it more advantageous to use in nonlinear simulations when compared with the integrated momentum equation. In particular, the inclusion of in¯ preserves a mathematical structure similar to the plane diﬀusion δ 2 ∇2H u original Navier-Stokes equations, and enables reproduction of the boundary eﬀects (e.g., at y = ±1 walls) that are not captured by lower-order approximations. This will be discussed further in a future work, and here we simply show some sample results of the full nonlinear, depth-averaged simulations with Eqs. (6–9). That is, a model with the combined eﬀects of Taylor dispersion and the DBF momentum equation. We try to reproduce the experimental image presented in Fig. 1 at the two lower voltages (25,000 and 50,000 V/m); the result is shown in Fig. 5. Again the model reproduces essential features observed in the experiments such

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Figure 5. Nonlinear simulation of the depth-averaged equation system. This model includes the combined eﬀects of the Taylor dispersion and the DBF momentum equation. In the strongly nonlinear regime the Taylor dispersion acts as an extra smoothing mechanism. The results compare favorably with the experimental data presented in Fig. 1.

as the fastest growing wave numbers and the growth rates of the interface disturbance amplitude. However, note that the computations are now at exactly the same ﬁeld strength as those applied in the experiments (as opposed to the unnaturally low ﬁelds used for comparison with the simple 2D model results of Fig. 3). Future work will also include the application of the model to diﬀerent ﬂow conﬁgurations such as those used in ﬁeld ampliﬁed sample stacking (FASS).

6.

Summary

In this work we have presented experimental, numerical, and analytical results that explain the basic mechanisms behind an electrokinetic mixing phenomenon observed in microﬂuidic channels. We have presented analysis and computations based on various sets of assumptions for electrokinetic ﬂows in a long, thin channel with a transverse conduc-

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tivity gradient. Our models are able to predict general trends in the data, as well as many of the qualitative and quantitative aspects of the observed ﬂow ﬁeld. Ongoing work includes the development of a generalized, depth-averaged model for a wide class of electrokinetic ﬂows (such as FASS) in thin microﬂuidic channels. The models presented in this work are useful in optimization studies, as parameter space can be spanned in simulations more quickly than in the laboratory. The results described by Oddy et al. (2001) has demonstrated that oscillatory electric ﬁeld can potentially drive even more vigorous mixing. The models presented here can be used to optimize the form of the forcing function, to design the shape of a micro-mixer, and to develop optimal control strategies for both the micro-mixing and the suppression of instabilities.

Acknowledgments This work was sponsored by DARPA (Contract Number F30602-002-0609) with Dr. Anantha Krishnan as contract monitor and by an NSF CAREER Award (J.G.S.) with Dr. Michael W. Plesniak as contract monitor.

References [1] J. Baygents, F. Baldessari, Electrohydrodynamic instability in a thin ﬂuid layer with an electrical conductivity gradient, Phys. Fluids, Vol.10, 1, 301–311, 1998. [2] C.-H. Chen, J.G. Santiago, Electrokinetic instability in high concentration gradient microﬂows, Proceedings of IMECE-2002, CD Vol.1, #33563, 2002. [3] C.-H. Chen, H. Lin, S.K. Lele, J.G. Santiago, Convective and absolute electrokinetic instability with conductivity gradients, J. Fluid Mech., in press, 2004. [4] J.F. Hoburg, J.R. Melcher, Internal electrohydrodynamic instability and mixing of ﬂuids with orthogonal ﬁeld and conductivity gradients, J. Fluid Mech., Vol.73, 333, 1976. [5] H. Lin, B.D. Storey, M.H. Oddy, C.-H. Chen, J.G. Santiago, Instability of electrokinetic microchannel ﬂows with conductivity gradients, Phys. Fluids, Vol.16(6), 1922–1935, 2004. [6] S. Liu, S. Masliyah, Single ﬂuid ﬂow in porous media, Chem. Eng. Comm. Vol.148–150, 653-732, 1996. [7] J.R. Melcher, G.I. Taylor, Electrohydrodynamics: a review of the role of interfacial stresses, Annu. Rev. Fluid. Mech., Vol.1, 111–146, 1969. [8] M.H. Oddy, J.G. Santiago, J.C. Mikkelson, Electrokinetic instability micromixing, Anal. Chem., Vol.73, 5822–5832, 2001. [9] R.F. Probstein, Physicochemical Hydrodynamics, John Willey & Sons, New York, 1994. [10] B.D. Storey, B.S. Tilley, H. Lin, J.G. Santiago, Electrokinetic instabilities in thin microchannels, Phys. Fluids, in review, 2004.

MOLECULAR MECHANICS OF CYTOSKELETAL COMPONENTS M. Atakhorrami Vrije Universiteit Amsterdam, Dept. Phys. Amsterdam, The Netherlands

K.M. Addas Rowland Institute at Harvard Cambridge, MA, USA

M. Buchanan, G.H. Koenderink, F.C. MacKintosh Vrije Universiteit Amsterdam, Dept. Phys. Amsterdam, The Netherlands

J.X. Tang Brown University, Dept. Phys. Providence, RI, USA

Christoph F. Schmidt Vrije Universiteit Amsterdam, Dept. Phys., Amsterdam, The Netherlands [email protected]

Abstract

Semiﬂexible polymers are of great biological importance in determining the mechanical properties of cells. We have used optical tweezers to trap pairs of micron-sized silica spheres in solutions of semiﬂexible polymers, and laser interferometry to detect their thermal motions with high bandwidth. Frequency-dependent complex shear moduli were extracted from the auto- and cross-correlated bead motions, with the response functions being derived from position-ﬂuctuation data using dispersion relations from linear response theory.

Keywords: Cytoskeleton; semiﬂexible polymers; microrheology; optical tweezers; laser interferometry

355 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 355–364. © 2005 Springer. Printed in the Netherlands.

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Introduction

One of the characteristic lengths that describes a polymer in a solution is the persistence length Lp , which is the length traversed along the ﬁlament contour before the tangent vector thermally randomizes its orientation. Lp is proportional to the bending stiﬀness of the polymer. Flexible polymers are characterized by Lp > L. Semiﬂexible polymers are in the intermediate regime, characterized by Lp ≈ L >> d, with ﬁlament diameter d. The dynamics of polymer solutions and networks greatly depends on the degree of ﬂexibility of the individual polymers. Recent theoretical treatments have addressed semiﬂexible polymer networks [1-4]. Such polymers are found particularly in biology where they form networks that determine the mechanical properties of cells. We report here on rheology experiments on such polymer systems, and in particular on a new microrheology method. There are clear advantages to the miniaturization of a rheology experiment: only small amounts of material are necessary, spatial inhomogeneities can be studied, and the bandwidth of the measurement can be high. We have developed a microrheology technique using laser interferometric tracking of the Brownian motion of micron-sized beads embedded in viscoelastic materials [5]. The simplest implementation of the technique consists of tracking one probe particle at a time (1-bead microrheology) [5]. A further development, avoiding local artefacts due to the insertion of the probe particles into the system, consists of measuring the correlated ﬂuctuations of pairs of probe particles (2-bead microrheology). We have studied the dynamics of several systems and have compared 1-bead and 2-bead results for wormlike micelle, semiﬂexible fd virus and actin solutions. Particularly in the actin systems, which is a ﬁrst approximation to the cytoskeleton of cells, we explore the rich multitude of length and time scales in the dynamic behavior of these networks, which are not accessible to conventional macrorheology.

2.

Principle of the Technique

The microrheology (MR) technique used here is a passive one in which the thermal ﬂuctuations of pairs of micron-sized beads are observed. Complex auto- and cross-correlation response functions of the beads are calculated using the ﬂuctuation-dissipation theorem [5,6]. The complex (n,m) (n) response function αij (f ) relates the Fourier transform ri (f ) of the (n)

(m)

displacement of the nth bead ri (t) to the Fourier transform Fj the force

(m) Fj (t)

(f ) of

acting on the mth bead (i and j are either 1 or 2 for

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the parallel and perpendicular directions with respect to the line joining the bead centers, respectively), (n)

(n,m)

ri (f ) = αij

(m)

(r, f )F Fj

(f ).

(1)

The measured response function must ﬁrst be corrected for the trapping eﬀect of the optical tweezers [7]. The ﬂuctuation-dissipation theorem [5] provides the link between the single-sided power spectral density (PSD) [8] (n,m) (n) (m)∗ (f ) = lim (2/t)rt i (f )rt j (f ) (2) Sij t →∞

(n)

and the imaginary part of the response function, where rt i (f ) is the (n) Fourier transform of the position ri (t), deﬁned over a ﬁnite time t . A Kramers-Kronig integral can then be used to calculate the real part of the response function, provided that the imaginary part is known over a large enough frequency range. The connection between the auto(n,m) correlation response αii (f ) of a bead and the corresponding complex shear modulus (n,m)

Gii

(n,m)

(f ) = Gii

(n,m)

(f ) + iGii

(f )

(3)

of the viscoelastic medium surrounding the bead is assumed to be provided by the generalized Stokes-Einstein relation (GSER) [5], (n,m)

Gii

(f ) =

1 (n,m) [6πaαii (f )]

,

(4)

where G and G are the elastic and loss modulus respectively, and a is the radius of the bead. The cross-correlation shear modulus of the (1,2) solution is derived from the parallel a ≡ α11 response by [9] Gcross (f ) =

1 , [4πrα (f )]

(5)

where r is the distance between the centers of the beads.

3.

Experimental Setup

Two focused laser beams were used to form optical tweezers and trap pairs of probe particles imbedded in solutions of semiﬂexible polymers. Position ﬂuctuation data was recorded with a 200 kHz sampling rate, using laser interferometry and quadrant photodiode detection as described in detail elsewhere [5,6,10]. A near-infrared (1064 nm) laser was used for one trap and a 830 nm laser for the second trap. The beams

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Figure 1. Schematic illustration of the 2-bead microrheology experiment. Two spheres of radius a, are held in two optical tweezers, created by focusing lasers of diﬀerent wave lengths λ = 830 nm and λ = 1064 nm at a separation distance r. The displacements of both particles are detected separately in x and y direction with interferometry.

are brought to a focus in the sample chamber by a high numerical aperture objective. The laser light emerging from the condenser lens, after passing through the sample, is projected onto two quadrant photodiodes in such a way that the back-focal plane of the condenser is imaged [10]. The signals from the four quadrants of each photodiode are combined to obtain the X- and Y -voltages corresponding to the displacements of each bead in these directions in the plane normal to the propagation direction of the laser. The output voltages are, after analog ampliﬁcation and pre-processing, recorded using an A/D interface and digital data acquisition.

4.

Results

Worm-like Micelle Solutions Figure 2 presents the loss and the viscous shear modulus of a worm-like micelle solutions for 1wt % concentration. The worm-like micelle solutions were made of binary mixtures of cetylpyridium chloride (CPyCL) and sodium salicylate (NaSal) dissolved in brine (0.5 M NaCl). A characteristic feature of worm-like micelle solutions is that they relax lowfrequency applied stress with a single dominant relaxation time, reﬂecting breakage and reptation of the ﬁlaments [11]. The system we used

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G', G'' [Pa]

100

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0.1 0.1

1

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1000

10000

100000

f [HZ] Figure 2. Elastic modulus and loss modulus for a 1wt% worm-like micelle solution. Figure shows the comparison of 1-bead and 2-bead microrheology. The diameter of the beads was D = 2.1 µm.

produces micelles with typical diameters of 2-3 nm, a persistence length Lp 10 nm, and an average length between 100 nm and 1 µm. We have performed 1-bead MR and 2-bead MR for diﬀerent concentrations of micelles. Bead diameters were 1.16 µm and 2.1 µm. For this system 1-bead and 2-bead MR results show excellent agreement (Fig. 2). More experimental noise is usually found in 2-bead results since the cross-correlation signal between the displacement of two beads is always weaker than the auto-correlation signal of one bead. The abrupt decrease in both G and G at high frequncies is artefactual and due to anti-alias ﬁltering at the Nyquist frequency and to the ﬁnite frequency cut-oﬀ eﬀects on the result of the Kramers-Kronig integral. The eﬀect on G is stronger than on G , reﬂected in about a decade less usable data for G [5]. It is expected that 1-bead measurements could be sensitive to the local medium properties in the direct vicinity of the probe bead, which could be diﬀerent from bulk properties due to surface chemistry or steric (entropic) eﬀects. The unmodiﬁed bulk viscoelastic properties of the solutions should, on the other hand, be reported by the 2-bead results [9]. Even in the absence of speciﬁc chemical or electrostatic interactions, steric depletion of polymer in the vicinity of the probe particles is expected to cause diﬀerences in the 1- and 2-bead results, provided that probe size and network length scales are comparable. In the case of

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the worm-like micelle solutions we studied here, network length scales are at least a decade smaller than bead radius and we therefore did not expect diﬀerences due to surface artefacts. The data nicely conﬁrms this expectation and conﬁrms the validity of both methods. It should be noted that independently measured parameters enter the data evaluation for the two cases, bead diameter for 1-bead results and bead distance (only) for 2-bead results. These results further establish microrheology, both 1-bead and 2-bead methods, as a highly sensitive and high-bandwidth technique to measure shear elastic moduli of viscoelastic materials. Good agreement between more conventional macrorheology and microrheology on similar samples is reported elsewhere [12].

Fd Virus Solutions Fd virus is a ﬁlamentous DNA bacteriophage [6], which is here used as a monodisperse model system for semiﬂexible polymers. Filaments have a diameter of 7 nm, persistence length Lp = 2.2 µm and contour length L = 0.9 µm. Fd solutions were probed with a bead diameter of D = 1.16 µm. Fig. 3(a) shows the concentration dependence of the 1bead (circles) and 2-bead (triangles) elastic moduli. The loss modulus is shown in Fig. 3(b). Data are displayed for 2, 5, 10, and 14 mg/ml fd concentrations. The high-frequency slope of the loss modulus is consistent with single ﬁlament dynamics predictions [1,4]. At such high frequencies, the entanglement of the polymer in its network does not contribute to the modulus. The modulus is proportional to concentration and expected to scale with a power law exponent of 3/4. The elastic modulus is expected to exhibit the same behavior but due to the cut-oﬀ in the Kramers-Kronig integral, the reliably measured elastic modulus extends to about 10 kHz, a frequency apparently still in the transition regime between collective dynamics and high-frequency single ﬁlament dynamics because a true scaling regime would require both moduli to scale with the same power law. The high-frequency behavior is attributed to tension in the ﬁlaments due to the shear-induced extension. At lower frequencies, the tension contributes less, and the dominating contribution to the stress is that of bending and orientational dynamics [1,2]. The data demonstrate that for this system 1-bead MR and 2-bead MR are in good agreement. This is remarkable since the length scales of the system, contour length and persistence length, are comparable with the probe bead diameter. If there was a steric depletion eﬀect, it has evidently led to a negligible eﬀect on the 1-bead results.

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Shear Elastic Modulus,G' [Pa]

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0

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-2

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Shear Loss Modulus,G" [Pa]

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1000 10000 100000

Frequency [HZ]

Figure 3. (a) The concentration dependence of 1-bead and 2-bead elastic modulus data of 1.16 µm diameter beads in 2, 5, 10 and 14 mg/ml fd solutions. (b) The same for the loss modulus. Pairs of 1-bead (circles) and 2-bead (triangles) moduli are higher for higher concentrations.

Entangled Actin Solutions Actin is a protein of 42 kD molecular weight and a major component of the cytoskeleton of most eukaryotic cells. It forms double-helical ﬁlaments of about 7 nm diameter, with a persistence length Lp = 17 µm, and with a polydisperse length distribution with an average length of approximately L = 17 µm in our in vitro reconstituted model systems.

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Actin stress relaxation is essential for cytoskeleton dynamics and is in vivo strictly regulated by a large number of associated regulatory proteins. We study here simpliﬁed model systems of just pure actin without additional proteins to explore the fundamental properties and dynamic characteristics of the networks it forms. In contrast to the two systems presented above, characteristic length scales in actin networks are large compared to bead size and we expect interesting diﬀerences between 1bead and 2-bead results, which would report on local network structure near the probes. Figures 4(a-b) show the results of 1-bead and 2-bead MR for entangled actin networks at 1 mg/ml actin concentration as a function of bead size. Figure 4(a) shows the elastic shear modulus for 1-bead MR, recorded with various probe bead sizes in comparison with the 2bead results. Both the 1- and 2-bead curves show the 3/4 power-law scaling at high frequencies, which is expected for semiﬂexible polymer networks [1,2,4,5]. This regime connects to more complex dynamics with no clear power law characteristics at lower frequencies. There are also clear diﬀerences among the results on identical samples, with 1-bead MR systematically underreporting the bulk moduli of the sample in the high frequency regime. These diﬀerences between 1-bead and 2-bead MR as well as the systematic trend with probe bead size in the 1-bead results are consistent with the formation of an entropic depletion layer around the beads. The extent of the depletion layer is, roughly speaking, dominated by the shortest of the three characteristic lengths: bead radius, entanglement length and persistence length [1,2,5]. As long as bead size is the shortest length, this depletion layer will dominate the bead response and thereby the 1-bead results. This appears to be the case in the actin networks, and the trend towards less of an inﬂuence of local environment with increasing bead size reﬂects the fact that the other length scales, particularly entanglement length which is estimated to be on the order of 5-10 µm, begin to compete with bead radius. The depletion eﬀect nevertheless does not entirely disappear even at the largest probe size, D = 5 µm. This behavior is visible in both the elastic and the loss modulus (Figs. 4(a-b)). A diﬀerent type of discrepancy is observed at low frequencies within the 1-bead results for the elastic modulus with varying bead size, as well as between 1-bead results with smaller beads and 2-bead results. 2-bead results were independent of bead size and bead distance (after conversion to shear moduli) and the curve shown is the result of averaging data taken at the diﬀerent bead distances (data for D = 1.16 µm bead diameter is shown). We hypothesize that this phenomenon is due to local non-aﬃne deformations of the actin ﬁlaments forming the network and will publish further research on these dyna-

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G' [Pa]

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-2

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-1

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f [HZ]

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-1

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f [HZ]

Figure 4. (a) Elastic modulus and (b) loss modulus for 1 mg/ml entangled actin solutions. Comparison of the moduli from 1-bead MR for bead diameters D = 0.5µ m, D = 1.16µ m and D = 5µ m. 2-bead MR data taken at diﬀerent distances all collapse after scaling by bead distance. Data for bead diameter D = 1.16µ m is shown.

mics elsewhere. It is important to note that video-based microrheology methods that have also been applied to actin networks will mainly test this regime and may therefore be diﬃcult to interpret.

5.

Conclusions

We have applied a passive microrheology technique, evaluating both single-probe particle ﬂuctuations and correlated two-particle ﬂuctuations to measure the complex shear moduli of solutions of various semiﬂexible

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polymers. Results with fd solutions and worm-like micelle solutions show good agreement between 1-bead and 2-bead MR. This strongly conﬁrms that one can measure bulk viscoelastic properties of these polymer systems with 1-bead MR, which is the simpler and less noisy technique compared to 2-bead MR. Such a conﬁrmation has been lacking up to now. More complex behavior was observed for actin solutions, providing strong evidence for local depletion eﬀects near the bead surface as well as indications of non-aﬃne ﬁlament dynamics visible at low frequencies. The results together advocate 1-bead MR as a reliable high resolution and high bandwidth technique to study simple systems far surpassing conventional rheology both in sensitivity and bandwidth. A combination of 1-bead and 2-bead MR, on the other hand, provides a way to explore more complex scale-dependent and local dynamic properties in various polymers, biological and colloidal systems.

Acknowledgments We thank David Morse, Alex Levine and Matteo Pasquali for extensive discussions. Erwin Peterman helped with technical advice. Joost van Mameren, Fredrick Gittes, Mark Buchanan and Joanna Kwiecinska helped at various stages with software for data processing. Karen Vermeulen puriﬁed actin. This project was supported by NSF DMR 9988389 (J.X.T.) and the Dutch Foundation for Fundamental Research on Matter (FOM).

References [1] D.C. Morse, Macromolecules, Vol.31, pp.7030, 1998. [2] D.C. Morse, Macromolecules, Vol.31, pp.7044, 1998. [3] A.C. Maggs, Phys. Rev. E, Vol.55, pp.7396, 1996. [4] F. Gittes, F.C. MacKintosh, Phys. Rev. E, Vol.58, pp.1241, 1998. [5] B. Schnurr, F. Gittes, F.C. MacKintosh, et al., Macromolecules, Vol.30, pp.7781, 1997. [6] K. Addas, J.X. Tang, C.F. Schmidt, Phys. Rev. E, Vol.70, 021503, 2004. [7] A. J. Levine and F. Mackintosh, pp.manuscript in preparation. [8] F. Gittes, C.F. Schmidt, Method. Cell. Biol., Vol.55, pp.129, 1998. [9] A. J. Levine, T.C. Lubensky, Phys. Rev. E, Vol.63(4), 041510, 2001. [10] M. W. Allersma, F. Gittes, M. J. deCastro, et al., Biophys. J., Vol.74, pp.1074, 1998. [11] J. F. Berret, J. Appell, G. Porte, Langmuir, Vol.9, pp.2851, 1993. [12] M. Buchanan, M. Atakhorrami, J.F. Palierne, F.C. Mackintosh, C. F. Schmidt, (manuscript in preparation).

TOPICS IN ASTROPHYSICAL FLUID DYNAMICS Edward A. Spiegel Astronomy Department Columbia University, New York, NY, USA [email protected]

Abstract

This brief description of some ﬂuid dynamical problems of astrophysical interest focuses on two eﬀects that are characteristic of the subject: selfgravity and radiative forces. Self-gravity is important in determining the basic structures of cosmic bodies as well as producing some intriguing instabilities. Radiation, by which we observe these bodies, produces forces that may be disruptive to their basic structures and be a source of vigorous ﬂuid dynamical activity in the form of photon bubbles and radiatively driven vortices.

Keywords: Gravitational instability, solitary waves, astrophysical vortices, photon bubbles

This sampler of problems in AFD begins with the classical problem of gravitational instability and continues with some current problems that I ﬁnd very intriguing. Given the imposed space limitations, I can only touch on these and I need to be very stingy with references (not of my own, of course) and will assume that I need not provide many ﬂuid references for the expected readership of this volume. (But feel free to use the email address above.)

1.

Gravitational Instability

Massaged Models The observable part of our universe looks homogeneous in the large, but it is clearly rather lumpy on smaller scales. There are stars, clusters of stars, galaxies and clusters of galaxies. Possibly there are even larger structures but here we come to a less clearly deﬁned topic where people may yet argue about how (or whether) to study such issues as the dimension of the point set that approximates the distribution of galaxies. But it is undoubtedly interesting to inquire into the origin of the inho365 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 365–377. © 2005 Springer. Printed in the Netherlands.

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mogeneities of the universe. Our current understanding of cosmology is that the early universe was dense and hot and very dissipative. Hence it is thought to have been rather homogeneous in the earliest times that we can reasonably think about and the lumpiness is considered to have been caused by gravity. The breakup of a homogenous ﬂuid into reasonably discrete structures is believed to be caused by an instability whose qualitative nature was already imagined by Newton. Jeans [13] was apparently the ﬁrst to formulate this problem. He assumed a perfect barotropic ﬂuid with selfgravity and used these simple equations of motion: ∂t (ρu) + ∇ · (ρuu) = −∇p − ρ∇V, ∂t ρ + ∇ · (ρu) = 0, ∇2 V = 4πGρ, p = KρΓ .

(1) (2) (3) (4)

Jeans posited a static homogenous solution to these equations (though one presumes that he knew better) and studied perturbations on this “solution”. But there is no inﬁnite, homogenous, static, self-gravitating medium in cosmology, not even in Einstein’s original theory of gravity [11]. Einstein added a term to his equation for the curvature of spacetime that allowed him to ﬁnd a static solution, unstable though it is. Similarly, we may add a term to the right side of the Euler equation to provide our Newtonian universe with a static solution. This term, ρλr, represents a repulsive force, as Eddington [10] pointed out. Now we may introduce V˜ = V − 12 λr2 and replace (1) and (3) by ∂t (ρu) + ∇ · (ρuu) = −∇p − ρ∇V˜ ,

(5)

∇2 V˜ = 4πG(ρ − ρλ )

(6)

where ρλ = λ/G. This recalls the device used in plasma physics to maintain charge neutrality and we may now ﬁnd a homogenous static solution and study its stability to clumping. Einstein’s extra term, “the cosmological term”, intrigued people from its beginning, but its status was uncertain until recently. The recent discovery that the expansion of the universe is accelerating has brought this extra term into vogue, though its physical meaning remains unclear. Does the cosmological term mean what the simple statement given here suggests — the existence of matter with negative gravitational mass [21]? If so, we should perhaps add a ﬂuid equation for this odd material

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as in the two-ﬂuid plasma model. Many cosmologists look on the added term as a negative pressure that can drive the expansion velocity ever upward. Here it merely serves Einstein’s original purpose for it: to produce a static solution.

The Dispersion Relation To study the stability of the homogenous medium, we let ρ = ρ0 + δρ, V˜ = V˜0 + δ V˜ and so on. With the extra term in place, we may assume that the basic ﬁelds are constant. If we then linearize the equations about the static state, we are led by standard manipulations to a KleinGordon equation for δρ: 2 (7) ∂t − ∇2 δρ = (4πGρ0 )δρ. For plane waves with δρ ∝ exp(iωt − ik · x), we obtain the dispersion relation (8) ω 2 = c2 (k 2 − kJ2 ) with kJ2 = (4πGρ0 )/c2 , and c2 = Γp0 /ρ0 . The quantity kJ is called the Jeans wavenumber and, when the wavenumber of the perturbation is much larger than this, we recover ordinary sound waves. But for k < kJ , ω 2 becomes negative and we have instability. Because of the diﬀerence in sign from the electrostatic case, we get instability rather than oscillation. The Jeans length 1/kJ tells us the scale on which the gravity just balances the pressure gradient much as the balance of surface tension and pressure dictates the size of a liquid drop. When the length scale of a perturbation is larger than this critical size, collapse occurs. For a perfect gas with temperature T , we ﬁnd from −1 the number these formulae that kJ ≈ 100 (T /n) light years where n is √ density and the collapse time for very small k is ≈ 5 × 107 / n yr.

Cosmology’s Fictitious Forces To study the formation of inhomogeneities in the universe properly, we need to take account of its expansion. This was ﬁrst done for the linear stability problem by Lifshitz in 1946 in the context of relativistic cosmology, but the Newtonian case conveys the idea [5]. Even that story is on the long side so, to indicate how the expansion reduces the degree of instability, a brief look at the kinematics in an expanding medium may suﬃce. For a universe that is iniﬁnite and homogeneous in the large, it does not matter where we put the origin of coordinates, so let us presume that one has been chosen. The position with respect to the arbitrary origin

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of a ﬂuid element is r. We then transform to isotropically expanding coordinates such that r = R(t)x (9) where R(t) is a nondimensional function that tells us how the global scale of the universe changes in time. For some suitable origin of time we let t0 be the present and take R(t0 ) = 1. We ﬁnd ˙ + Rx˙ = Hr + Rx˙ r˙ = Rx

(10)

where v = Hr is the Hubble ﬂow or global expansion and H0 = H(t0 ) is called the Hubble constant. The acceleration of the ﬂuid element is then ˙ + H 2x . ¨r = R x ¨ + 2Hx + Hx (11) In the expanding coordinates, we acquire three additional terms or ﬁctitious forces reminding us of those gained in going into a rotating frame. The scalar H plays the role of the rotation rate in that comparison, hence the analogue of the Coriolis force is a drag term. Material particles moving through an expanding medium are slowed down with respect to the background. This is an eﬀect analogous to the cosmological redshift of photons. (You may think of this as a stretching of the de Broglie wave˙ corresponding lengths.) Since H is not a constant, we get a force Hx to the Euler force. Finally there is the analogue of the centrifugal force, H 2 x . These extra terms appear in the Euler equation when we go to expanding coordinates. The main point is that the cosmological drag term inhibits the development of gravitational instability but it does not kill it completely in standard cosmological models. Rather, it converts the exponential growth to an algebraic growth. It appears that this feeble instability may suﬃce to produce the structures we see in the universe around us according to many simulations. Still, if you want to get your hands analytically on the way gravitational instability develops, as one may do for weak instabilities, it is best to consider another static conﬁguration of the mass.

Polytropic Slabs If, in equation (4), you treat Γ as a parameter and not necessarily the ratio of speciﬁc heats, you have what is called the polytropic gas law. The spherically symmetric, static, self-gravitating solutions of Eqs. (1)– (4), served as models of stars in the nineteenth century and they remain qualitatively useful even now. The disks in spiral galaxies may also be modeled as polytropes to good eﬀect and these are useful for studying their gravitational instability.

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As a prelude to studying ﬂuid dynamics in disks, one may study simpler ﬂattened objects such as polytropic layers or slabs. The simplest case is a static conﬁguration with the layer extending inﬁnitely in two directions that we may call horizontal. The third, or vertical, direction is given the designation z. Just as for a stratiﬁed atmosphere, we may write the hydrostatic condition, the only diﬀerence being that the gravity is not speciﬁed but is governed by the Poisson equation. Static solutions have ρ = ρ0 (z) with ρ0 (z) = ρ0 (0)CΓ (z)

(12)

where CΓ (z) is given by a simple integral that, for general Γ, is a beta function. Several special cases may identiﬁed, notably C1 = sech2 (kJ z),

Here kJ2

4πGρ0 = c2

C2 = cos(kJ z).

= z=0

4πG [ρ0 (0)]2−Γ . ΓK

(13)

(14)

The characteristic thickness of the slab is ∼ 1/kJ as before but now we have a static solution with no artiﬁce. The model is admittedly simpliﬁed but it has scope for interesting dynamics. For instance, the distribution of density on the midplane controls the layer thickness and this may vary in the rotating case so that we may ﬁnd Rossby waves propagating through disks. Ledoux [14] derived the marginal stability condition for linearized perturbations on the isothermal slab (Γ = 1). He found two horizontal wavelengths that are marginal, with khor = 2kJ and 0. Here is a situation in which we can make use of asymptotic approaches for the modes of long wavelength where the instability ﬁrst arises weakly. This is unlike the homogenous case, for which the maximum growth rate occurs at k = 0. The real case is even more favorable to this approach since disks of galaxies are embedded in very massive halos that make the disk thicknesses even less than the Jeans lengths of the disks. The halos themselves, though not visible, are generally believed to exist on the basis of their gravitational eﬀects. Their inﬂuence strengthens the validity of the the thin-layer approximation. Linear theory reveals that there are acoustic modes and gravity modes, just as in models of standard atmospheres. The surprise is that the instability occurs in the gravity modes and not in the acoustic modes as in the original Jeans problem. The gravity waves combine both thickness variations and true density variations, so have all that is needed. Since the largest scales are nearly marginal, one can assume slow times

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and long lengths to develop shallow layer theories [18]. With the additional restriction to small amplitudes, as in weakly nonlinear theories, one may derive nonlinear wave equations in the manners of Boussinesq and Korteweg and de Vries [19]. The familiar form of the theory is modiﬁed by a term representing the eﬀect of self-gravity that comes in through the Poisson equation. For instance, in the case of nonlinear waves of small amplitude in a thin layer with Γ = 2 (the easiest case) one ﬁnds for the surface deformation that: 1 (15) ηT − 3ηηX + ηXXX = µH[η] 2 where the Hilbert transform is 1 H[η] = P π

∞ −∞

η(Y ) dY Y −X

(16)

and X and T are suitably stretched variables. This equation has pole solutions, but we cannot say whether it is completely integrable. However, there are certainly solutions resembling solitons. The interest in such a result is that it has sometimes been thought that the highly dispersive nature of the waves in this kind of problem would not allow the formation of coherent structures. In fact, the nonlinearity in this problem leads to the formation of long-lived nonlinear waves. What happens in the more realistic case of a rotating disk? This is an issue that is unresolved as yet in the nonlinear case. The ﬁniteness of the disk brings discrete modes into play and they behave chaotically [4]. Still, some global order may emerge, perhaps with the help of other eﬀects.

2.

Astrophysical Vortices

The masses of stars range from 60-80 times the mass of the sun (4 × 1033 gm) down to a few tenths of a percent of the solar mass, that is, down to the masses of the giant planets. The massive stars have very hot atmospheres, (tens of thousands of Kelvins), while the low mass ones have much cooler atmospheres as a rule (6000 K for the sun). At both ends of this spectrum, atmospheric turbulence is observed. At the cool end, this is caused by thermal convection resulting from the lowering of the thermal conductivity through the raising of opacity by partially ionized hydrogen. The ionization of hydrogen also favors convection by raising the speciﬁc heat. In the atmospheres of the hottest stars, turbulence is detected through broadening of the spectral lines and it is often supersonic. The origin of this turbulence is not agreed upon, though

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there is no shortage of possible sources. Hot atmospheres are fully ionized and are not subject to thermal convection in the usual way, but they are rapid rotators and they pulsate. When the pulsation is vigorous, thermal convection may be driven parametrically. Moreover, hot stars have high radiation pressure that provokes instability and complicates the dynamics. Already at the qualitative level there are some interesting ﬂuid dynamical issues. We see spots on the sun and these are caused by magnetic ﬂux tubes that protrude from the solar surface. The ﬁelds are locally strong enough to inhibit the convective transport of heat outward and so relatively cool (but still quite warm) spots are produced. By contrast, in Jupiter’s atmosphere, a much cooler place, we ﬁnd evidence for vortex tubes at the surface. Since there is a full range of masses between the two limits (sun and Jupiter) we may someday be able to observe the transition between the two kinds of coherent structures, but there is no reason why this transition could not be studied theoretically at present, perhaps numerically. This is a potentially revealing instance of the transition between the purely ﬂuid and the magnetoﬂuid regimes. Related questions arise in the study of the accretion disks that form around condensed objects on many scales. These are rather diﬀerent from the disks in spiral galaxies. Accretion disks represent inﬂows of mass from various sources such as companion stars in the case of binary stars, to the ambient stars around massive black holes in the centers of galaxies. The primitive nebula that might have preceded the formation of our solar system, as Kant and Laplace ﬁrst suggested, are cooler examples of this kind of structure. As the matter ﬂows toward the central object, its net angular momentum makes itself felt and a disk is formed. Before it can settle into the central object, the inﬂowing matter must get rid of its angular momentum. Various mechanisms have been proposed for expelling the angular momentum, mostly calling on some form of turbulence though waves and magnetic ﬁelds have been considered. Vortices could also play a part in the process. There are some similarities here to the ﬂow around the polar vortex on earth that is central to the ozone problem, though accretion disks frequently are magnetized. The large scale ﬂow in accretion disks is governed mainly by Kepler’s laws so that the circular velocity √ around the central object in an axially symmetric disk varies like 1/ r. This represents a linearly stable shear and it is not yet decided whether nonlinear instability can occur in such a ﬂow. (I ﬁrst heard people arguing about that some thirty-ﬁve years ago.) However, as Chandrasekhar and others showed, magnetic ﬁelds can catalyze the conversion of the energy in the rotational ﬂow of disks

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into turbulence, but it was some time till the importance of this result for disks was appreciated [3]. Can the resulting disorder lead to the formation of vortices? Twenty-ﬁve years ago I put a drawing of a disk with whorls in it in an article on turbulence for an encyclopaedia aimed at twelve-year olds. The hope was that in ten years one of them would appear in my ofﬁce with a fully completed simulation that revealed what the disk really looked like. This did not happen, so the next message in a bottle was sent as a remark at the end of a paper on vortices in stars and planets [9]. This got a response from P.A. Yecko whose thesis revealed that largescale spiral vortices formed. (At the suggestion of A. Ingersoll, Yecko adapted a code written by E. Chassignet for simulating the oceanic thermocline.) Several astronomers objected that the Keplerian shear in disks would shred a vortex. The shredding is avoided by anticyclonic vortices which shield themselves with protective cocoons of reduced shear. Several subsequent simulations with higher resolution have shown their robustness [6, 12, 15]. A recent, 2-D, compressible simulation at high resolution by G. Murante and colleagues in Torino strikingly shows how a single anticyclonic vortex survives in the Keplerian shear, at least for the ten disk rotations they followed. That vortex also generated largescale spiral extensions in line with Yecko’s results. The issue of what magnetic ﬁelds do to these processes needs clariﬁcation as do theoretical questions about how vortices form. But their existence would play a role in forming observable inhomogeneities on disks [1]. An example of a disk simulation in a two-dimensional Keplerian ﬂow with slowly decaying turbulence is shown after [7] in Fig. 1.

Figure 1.

Vortices on a Keplerian Disk

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Photoﬂuiddynamics

The radiation that permits us to observe cosmic bodies also plays a role in their formation, structure and evolution. The thermal aspects of the radiation are familiar to ﬂuid dynamicists, at least qualitatively. What may be of more interest in an introduction to AFD is the way that the force of radiation on matter may inﬂuence the dynamics of the ambient material medium. The phenomena arising in this subject seem suﬃciently diﬀerent from ordinary ﬂuid dynamics that I have followed the advice of M. E. McIntyre and sought a suitably distinctive terminology. The name used here for the subject is inspired by Lighthill’s “Bioﬂuiddynamics.” What follows is a very brief introduction to some aspects of this subject. For some background on the equations see [16].

The Radiative Fluid The most direct derivation of the basic equations of ﬂuid dynamics subject to stresses from a coexisting radiation ﬁeld is by way of the transport equations for radiation and matter. These are kinetic equations that contain terms representing the interaction of the matter with the radiation. This somewhat technical problem is best formulated by treating the radiation ﬂuid as a gas of photons. Nevertheless, there are some real complications depend on the frequencies of the photons and the state of the matter. Such details are out of place here, so we simply assume that the matter is grey, that is, indiﬀerent to the frequencies of the photons passing through it. We shall also omit the details of the state of the matter such as the degree of ionization, which we shall take to be complete for very hot objects. So we may go straight to the ﬁrst two moments of the transfer equation which, for a gas of photons, are equations for the moments of the distribution function or speciﬁc intensity. Let E, F and IP be the energy density, energy ﬂux and pressure tensor of the radiation ﬂuid. These satisfy the two moment equations ∂t E + ∇ · F = interactions with matter

(17)

and (18) ∂t F + c2 ∇ · IP = −ρκc F −1 where (ρκ) is the mean free path of a photon. We see that the latter equation may be written as ρκ (19) ∇ · IP = − F + O(c−2 ). c Naturally, we face the usual problem of closing oﬀ the moment hierarchy and here we take the simplest closure, IP = 13 EIII . Thus we ignore

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the (sometimes important) radiative contribution to the viscous stress. In some limits, we may argue from (17) that the divergence of the ﬂux is quite small. (In eﬀect, we are leaving out retardation terms with this approximation.) This is about as simple as we can make this problem, yet much of interest remains as we see when we write the equation for the material ﬂuid as 1 ρ(∂u + u · ∇u) = −∇p − ∇E − ρ∇V. 3

(20)

To this we add the mass conservation equation and the radiative equations which have been reduced to the diagnostic conditions 1 ∇ · F = 0 and ∇E = ρκcF . 3

(21)

Radiatively Induced Instabilities To see in what kind of conditions radiative eﬀects may become important dynamically, consider the simple case of a plane-parallel medium stratiﬁed under gravity. The foregoing equations show that the hydrostatic condition is 1 dp 1 1 = − ∇E − ρ∇V = κcF − ρ∇V ρ dz 3 3

(22)

where z is the upward vertical coordinate. In the typical case where the ﬂow of radiation is outward, hence upward, the radiative force is one of levitation and it balances the gravitational attraction downward when the right side of this equation vanishes. That condition is called the Eddington limit. Near this condition, we ﬁnd instabilities of both sound waves and gravity waves [22]. These are induced by both the thermal and dynamical eﬀects of radiation and there is still room for a better physical understanding of these processes. Those familiar with ﬂuidized beds will recognize a commonality between the Eddington limit and the onset of ﬂuidization, although the medium being traversed by the radiation is a ﬂuid even below the critical condition. As in ﬂuidization, the material layers are rendered unstable by the traversing ﬂuid, though the details of the instability mechanisms do diﬀer. As in those cases where voids form in ﬂuidized beds, the radiative ﬂuid is much less dense than the particles of the medium. This and other arguments suggest that photon bubbles will form in hot stars near to the Eddington limit [17]. Approximate solutions for photon bubbles can be constructed in the way that this is done in the theory of ﬂuidization [20]. Related discussions have been given in the context

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of convection in the chimerical supermassive hot objects [23] and polar caps of pulsars where the magnetic ﬁelds are strong (1012 gauss) [2]. An interesting aspect of this process bears on the question of the lifetimes of objects that exceed the Eddington limit. In the case of ﬂuidized beds, ﬂow through the bed in excess of the value needed for ﬂuidization is observed to escape in voids, or bubbles. We may similarly expect cosmic bodies to survive above the Eddington limit. A modiﬁed limit needs to be calculated but another feature of this problem needs to be addressed ﬁrst.

Photovorticity The hottest objects are relatively rare and there are very few nearby. We see them because they are intrinsically luminous but it is hard to know whether they are like other rotating, turbulent cosmic bodies in forming vortices or concentrated magnetic ﬂux tubes. However, it is clear that the hot objects do rotate rapidly and are turbulent in their outer layers. Moreover, there are observational grounds for supposing that there are spots on hot stars [8] and disks [1]. What can cause spots in such conditions? Hot stratiﬁed media are unstable and, as for ﬂuidized beds, we may expect the formation of (photon) bubbles in objects near the Eddington limit. If we make a vortex in such conditions, we anticipate that, as in many laboratory experiments on rotating turbulence, bubbles are attracted into vortices. Indeed, it is the practice to use small bubbles as markers of vortices in such experiments. On the other hand, bubbles ﬂowing into a vortex will bring with them angular momentum and, when this is of the right sign, the vortex will be intensiﬁed. We have in such a situation the makings of an instability for vortex production analogous to what has been seen in laboratory experiments [24]. Vortex formation in hot media is of interest since it would lead to strong inhomogeneities in the emerging radiation ﬁeld that would have diagnostic implications. It would also be important in the ﬂuid dynamics of hot objects since a vortex of the right kind is a conduit through which radiation may escape from a hot object or disk without disrupting it. A simple calculation reveals the nature of this process. Let us omit the complications of global rotation and consider an isolated vortex in a stratiﬁed polytropic ﬂuid with radiation coming from below [9]. A standard vortex with gravity balancing the pressure gradient in the vertical direction and the centrifugal force in the (horizontal) radial direction is readily constructed. Then, in radiative equilibrium, we have E = aT 4 , where T is the temperature. For radiative prob-

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Figure 2.

Flow through a phortex

lems the parameter value Γ = 4/3 is frequently adopted. This value has the advantage of making ρ3 /T a constant and that greatly simpliﬁes the second of (21). That pair of equations is then readily solved and the streamlines of the radiative energy ﬂux F are as shown in Fig. 2; the contours of T are also indicated. This vortex provides not only a safety valve by which the radiation may escape but such a beam should make itself apparent in observations of hot stars and disks.

References [1] M.A. Abramowicz, A. Lanza, E.A. Spiegel, and E. Szuszkiewicz, Vortices on accretion disks, Nature, Vol. 356, 41, 1991. [2] J. Arons, Photon bubbles – Overstability in a magnetized atmosphere, Astrophys. J., Vol. 399, pp.561–578, 1987. [3] S.A. Balbus and J.S. Hawley, Instability, turbulence and enhanced transport in accretion disks, Rev. Mod. Phys., Vol. 70, pp.1–53, 1998. [4] N.J. Balmforth and E.A. Spiegel, Nonlinear ringing of polytropic disks, Physica D, Vol. 97, pp.1–28, 1996. [5] W.B. Bonnor, Jeans’ formula for gravitational instability, Mon. Not. Roy. Ast. Soc., Vol. 117, 104, 1957. [6] A. Bracco, A. Provenzale, E.A. Spiegel, and P.A. Yecko, Spotted Disks, [in:] Theory of Black Hole Accretion Disks, M. Abramowicz, G. Bjornsen & J. Pringle, [eds.], Cambridge Univ. Press, 1998. [7] A. Bracco, P.-H. Chavanis, A. Provenzale, and E.A. Spiegel, Particle Aggregation in Keplerian Flows, Phys. Fluids Vol. 11, 2280, 1999. [8] J.P. Cassinelli, [in:] The origin of Nonradiative Heating/Momentum in Hot Stars, A.B. Underhill and A.G. Michalitsianos, [eds.], (NASA 2358), pp.2–23, 1985.

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[9] T.E. Dowling and E.A. Spiegel, Stellar and jovian vortices, Ann. N.Y. Acad. Sci., Vol. 617, pp.190–216, 1990. [10] A.S. Eddington, The Mathematical Theory of Relativity, Cambridge Univ. Press., 1924. [11] A. Einstein, Kosmolosiche Betrachtung zur allgemeinen Relativit¨¨atstheorie, Berlin: Sitzungberichte, p.149, 1917. [12] P. Godon and M. Livio, On the nonlinear hydrodynamic stability of thin keplerian disks, Astrophys. J., Vol. 521, 318, 1999. [13] J.H. Jeans, Astronomy and Cosmogony, Cambridge Univ. Press., 1928. [14] P. Ledoux, Sur la Stabilit´ ´e Gravitationelle d’Une Nebuleuse Isotherme, Ann. d’Astrophys., Vol. 14, pp.438–447, 1951. [15] H. Li, S.A. Colgate, B. Wendroﬀ, and R. Liska, Rossby wave instability of thin accretion disks. III. Nonlinear simulations, Astrophys. J., Vol. 551, pp.874–896, 2002. [16] D. Mihalas and B.W. Mihalas, Foundations of Radiation Hydrodynamics, Oxford Univ. Press, 1984. [17] K.H. Prendergast and E.A. Spiegel, Photon Bubbles, Comments on Astrophys. and Space Phys., Vol. 5, 43, 1973. [18] Z.-S. Qian, E.A. Spiegel, and M.R.E. Proctor, The gravitational instability of a gaseous slab, Stability and Applied Analysis of Continuous Media, Vol. 1, 33, 1990. [19] Z.-S. Qian and E.A. Spiegel, Autogravity waves in a polytropic layer, Geophys. & Astrophys. Fluid Dyn., Vol. 74, pp.225–244, 1994. [20] E.A. Spiegel, Photoconvection, [in:] Problems in Stellar Convection, E.A. Spiegel and J.-P. Zahn, [eds.], Springer-Verlag, 1977. [21] E.A. Spiegel, Gravitational screening, On Einstein’s Path: Essays in Honor of Engelbert Schucking, A. Harvey, [ed.] (Springer-Verlag, N.Y.), Chapt. 32, pp.465– 474, 1998. [22] E.A. Spiegel and L. Tao, Photoﬂuid instabilities of hot stellar envelopes, Phys. Rep., Vol. 311, pp.163–176, 1999. [23] V.S. Thorne, Thesis, Univ. of Manchester, 1968. [24] J.S. Turner and D.K. Lilly, The carbonated-water tornado vortex, J. Atmos. Sciences, Vol. 20, pp.468–471, 1963.

MINIATURIZATION OF EXPLOSIVE TECHNOLOGY AND MICRODETONICS D. Scott Stewart University of Illinois Department of Theoretical and Applied Mechanics [email protected]

Abstract

Condensed phase explosives used in conventional explosive systems have a charge size on the order of a meter or a sizable fraction of a meter. We discuss a range of issues, theoretical, computational and experimental, required to scale the size of explosive systems downwards by a factor of one hundred to one thousand, applications and prospects for a ubiquitous new technology.

Keywords: Detonation, shock physics, compressible ﬂow, miniaturization, explosives, new technology

1.

Introduction

A detonation is a chemical reaction driven shock wave in molecularly premixed material called an explosive. The chemical energy released in the reaction zone behind the lead shock is converted into kinetic energy and pressure/volume work done by the reactants. Explosives can be gases, liquids and solids. Detonation pressures in organically based condensed phase explosives (typically made from nitrated hydrocarbons) are in the range of 300–400 Kbar (30–40 GPa), and can potentially induce hundreds of Kbars of pressure in inert materials for fractions of microseconds. Detonation shock speeds are on the order of 3–10 kilometers/sec. The thermodynamic cycle and high pressure, high compression states that can be induced in materials are unlike those that can be obtained with other thermo-mechanical systems, including lasers. Hence detonative processes oﬀer unique methods of altering the state of material surfaces and can serve as a high energy density source for microdevices. Properly engineered, stable explosive detonation fronts work in combination by a principle of synchronicity (i.e. the detonation is a phase-controlled explosion front) and detonations can generate precise 379 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 379–385. © 2005 Springer. Printed in the Netherlands.

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motion-controlled ﬂows that can be used for materials processing and other applications.

2.

Applications

While condensed phase explosives are used in military, mining and demolition applications, other less commonly known applications of explosives include their use for materials processing, precision cutting and pulsed power application. Speciﬁcally, detonation of explosive ﬁlms can be used in surface treatment and hardening of materials. Other material processing applications include cladding and explosive welding, sintering, shock consolidation of powders and shock-induced chemical synthesis. Pulsed power applications include magnetic ﬂux compression, pulsed detonation engines, explosive lasing and the generation of extremely high intensity light pulses. There are biomedical applications for detonation of micro-sized explosive charges that include lithotripsy and localized destruction of pathological tissue [1]. Explosive and pyrotechnic elements pervade satellite and aerospace systems and hence there is interest in the miniaturization of explosive systems for microaerospace and satellite platforms. Suitably controlled detonation fronts represent a basic technology with unique aspects. By establishing the basic parameters of micro-scale explosive systems it should be possible to design micro-scale devices for welding, cladding, pulsed power, surface treatment and so on (as mentioned above) in novel, ubiquitous and unforeseen ways. Micro-explosive systems hold the promise of being a basic enabling technology with wide-spread application. Figure 1 shows a sketch of a experimental conﬁguration being designed at the University of Illinois, in collaboration with Prof. W. Kriven (Material Science and Engineering, UIUC) and Prof. R. Adrian (TAM, UIUC), to study high pressure, temperature synthesis of ceramic materials. The initiator consists of a capacitance discharge unit (CDU) that ﬁres a 10–micron thick wire (typically gold) or metallic ﬁlm embedded in the detonable ﬁlm. The electrical current dump causes the metal to expand from a nominally cylindrical or ﬂat source as plasma and drive a shock wave into the ﬁlm to start the chemical reaction in the ﬁlm. The detonation supported shock wave sweeps across the sample and the detonation shock drives an inert shock into the donor material to do the localized processing of near surface material. Other initiation conﬁgurations include laser driven micro-ﬂyers that induce shocks to start reaction.

Miniaturization of Explosive Technology and Microdetonics

Figure 1.

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Micro-explosive system for materials processing of ceramics

Explosive System Scaling Arguments and Requirements

An explosive system includes the main charge (the secondary explosive), the initiation system (which includes the initiation train and booster made of primary explosives or electrical or optical laser initiators) and the inerts, upon which the explosive products act. Conventional macro-scale explosive system design paradigms exist for explosive systems that have dimensions on the order of a meter or sizable fraction of a meter. The question, what are the scaling principles for smallscale design that are consistent with existing large-scale design, can be addressed by dimensional analysis, based on the Euler equations and consideration of matching shock initiation and propagation experiments of small-scale systems to their to large-scale counterparts. Scaling arguments, [2] show that extreme miniaturization by a scale reduction of current large-scale explosive systems by a factor of 100 to 1000 is possible. To employ existing large-scale design rules, the detonation reaction zone length scale must scale with the device dimension size. Short reaction-zone explosive materials (with small critical diameters) must be used for main charges. This means that one must select the main charge explosives from the list of primary explosives (used in large-scale initiator trains or detonators). Also one might consider using very short

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reaction zone explosives that have never been considered for use in the past, because of safety considerations. The other route to miniaturization uses explosive materials for propagation in sub critical charge dimensions. One expects to experience signiﬁcant transients that do not fall in the existing quasi-steady design paradigms. This route requires a detailed understanding of transient detonation propagation.

4.

A deﬁnition of Microdetonics and the Initiation of Small Systems

This brings us to a deﬁnition of the term “microdetonics” that we attribute to James E. Kennedy of Los Alamos National Laboratory. Kennedy’s deﬁnition of microdetonics is the detonation physics behaviors that are dominated by transient eﬀects such as detonation acceleration, detonation spread and curvature eﬀects that are commonly associated with initiation of explosives by small sources. This deﬁnition includes both small charges and small initiation sources, where transient phenomena is dominant. Reliable and safe initiation systems for miniaturized systems can be built using existing exploding wire and exploding foil initiation systems with existing, well-understood electrical designs. Initiation energy is stored in a standard capacitance discharge unit. It is also possible to build optical initiation systems whereby energy is transmitted through a optical ﬁber to the explosive charge. Initiation system can be placed on chips, [3] and designed with standard photo-lithographic techniques, [4].

5.

New Science Needed to Enable the Technology

In order to deﬁne the properties of the new explosive materials there is a need for a comprehensive linear and nonlinear stability theory for nonideal detonation that can incorporate non-ideal equation of state and realistic reaction rate laws for condensed explosives. Recent eﬀorts are underway to develop novel nano-engineering composite energetic materials and explosives that can be candidates for the miniaturized secondary charge. An entirely new linear stability theory for steady detonation has been developed by us at Illinois, [5] to guide design of miniaturized explosive systems in a rational way that incorporates descriptions of nonideal equation of state and reaction rate kinetics. In order to deﬁne detonation propagation in small dimensions, one must understand the critical conditions required for ignition and propagation of detonation for both ideal and nonideal explosives [6]. This

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includes the development of asymptotic theory for fast and sensitive chemical kinetics. It is important to understand aspects of detonation and shock diﬀraction and how that phenomena aﬀects successful detonation propagation.

High Resolution Multi-Material Simulation Technology is Required Design of integrated systems requires modern high resolution, multidimensional and multi-material, time-dependent simulation. High ﬁdelity simulation is an essential tool that is required to specify the geometry and select materials for miniaturized explosive system. Figure 2 shows a recent simulation of a “corning turning” experiment carried out by E. Ferm of Los Alamos National Laboratory. The corner turning experiment measures the transient eﬀect of the change in conﬁnement of detonation. The initial conﬁguration of Ferm’s experiment has a 6 mm radius, 125 mm long cylindrical stick (donor charge) of explosive PBX-9502 joined to a wider/shorter 25 mm radius, 50 mm long (acceptor charge) cylinder of PBX-9502. The detonation is started and travels as a curved steady detonation in the donor. Once the detonation

–Total length 200 mm - 6mm radius PBX-9502 donor - 150 mm long - 25 mm radius PBX-9502 acceptor - 50 mm long - Density plot near break-out

Detonation shock

"dead zone" with unreacted explosive in diffraction region Figure 2. MULTIMAT Simulation: The density record of a corner turning experiment that shows the eﬀects of detonation diﬀraction and the appearance of dead zones

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in the donor charge enters the acceptor, the detonation must expand into the large acceptor region. Because the lateral boundary of the acceptor charge is perpendicular to the axial propagation direction of the detonation in the donor and that the lateral boundary is unconﬁned, a large depressurization (rarefaction) of the detonation occurs. When the reaction rate of the explosive is pressure sensitive, the depressurization eﬀectively slows or stops the reaction in the region aﬀected by the corner turning diﬀraction event. Hence a large region appears where the explosive does not burn. That region is dubbed a “dead zone”. Instead of reacting, the explosive in the dead zone is simply shocked. For microcharges or for initiation by small sources, the corner turning experiment is a generic conﬁguration that must be studied in detail. Figure 2 is a snapshot of a density record from a simulation carried out by members of our group (D. S. Stewart, B. Wescott and S. Yoo) with our UIUC-code MULTIMAT. Our simulation shows the detonation after the detonation in the donor has entered the acceptor. The explosive charges are initially adjacent to a very low density inert material (shown to the right). As the simulation progresses the material interface between the explosive products and the inert material expands. Our simulation show the appearance of large dead zone preceded by a low pressure shock in the diﬀraction region nearest the corner, connected to a fully emergent detonation in center and conforms closely to Ferm’s experiment. The code MULTIMAT uses high-resolution (4th order in space and 3rd order in time) compressible reactive ﬂow solvers combined with a modern level-set treatment that represent interfaces to enable multi-material simulation required for microdetonic devices.

High Speed Measurement and Other New Areas of Mechanics Research The events of explosive technology take place at the limits of conventional experimental methods that measure mechanical quantities. Microdetonics is an area whose investigation will stimulate the creation of new measurement technologies. Recently our colleague, R. Adrian and his co-workers are working to develop PIV systems, [7] that can make capture motion events generated by shocks in optically accessible solids. These test solids can be used as a measuring instrument and a “full-ﬁeld” witness plate to capture the energy and momentum transfer from adjacent shock materials that are under investigation. Unlike lower speed PIV systems that take full-ﬁeld velocity measurements in water or air, full ﬁeld measurements in the interior of solids have not been available experimentally. This emerging measurement technology should allow

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for unprecedented improvements in understanding load transfer at the interface between materials. The physics of exploding wire and foils involves the coupling of the mechanics of phase transformations and the magneto hydrodynamics. Target ceramic materials require new thermo-mechanical theories for ceramic formation under rapid high pressure/temperature loading. The theory of critical energy and pulse duration for initiation of detonation is at the heart of fundamental questions in reactive ﬂow science. It is very likely microdetonic devices can be made for use for other basic material property investigations of a fundamental nature. In short, the area of microdetonics and the creation of precisely controlled miniaturized explosive systems will surely be coupled to fundamental advances in thermomechanics and have an impact on areas of technology that can use precision, high energy density output sources applied to materials with precision.

Acknowledgments This work is supported by the US Department of Energy, DOE/LANL and the US Air Force Research Laboratory, AFOSR-Mathematics and Munitions Directorate, Eglin AFB, Florida.

References [1] K. Takayama, T. Saito, Shock Wave/Geophysical and Medical Applications, Annual Review of Fluid Mechanics, Vol.36, pp.347-379, 2004. [2] D.S. Stewart, Toward Miniaturization of Explosive Technology, Shock Waves, Shock Waves, Vol.11, pp.467-473, 2002. [3] T.A. Baginski, S.L. Taliaferro, D.W. Fahey, Novel Electroexplosive Device Incorporating a Reactive Laminated Metallic Bridge, Propulsion and Power, Vol.17, No.1, 2001. [4] A.S. Tappan, A.M. Renlund, G.T. Long, S.H. Kravitz, K.L. Erickson, W.M. Trott, M.R. Baer, Microenergetic processing and testing to determine energetic material properties at the mesoscale, Proceedings of the 12th International Symposium on Detonation, San Diego, USA, 2002. Available online at http://www.sainc.com/onr/detsymp/detsymp2002/technicalProgram.htm. [5] A.R. Kasimov, D.S. Stewart, B.L. Wescott, Sunhee Yoo, Linear Instability Analysis of Non-Ideal, Condensed Phase Detonation, University of Illinois, submitted for publication, 2004. [6] A. Kasimov, D.S. Stewart, Asymptotic theory of the evolution and failure of self-sustained detonations, TAM Report No. 1042 UILU-ENG-2004-6003, ISSN 0073-5264, to appear in the Journal of Fluid Mechanics, 2004. [7] M.J. Murphy, R.J. Adrian, D.S. Stewart, G.S. Elliott, K.A. Thomas, J.E. Kennedy, Visualization of blast waves created by exploding wires, submitted to the Journal of Visualization.

FOAMS IN MICROGRAVITY Denis Weaire and Simon Cox Department of Physics, Trinity College, Dublin 2, Ireland [email protected]

1.

Introduction

Ordinary aqueous foam, which is our main subject in this paper, needs no introduction. Who has not taken a few minutes to study its beautiful structure (Fig. 1) and to watch it change? If you do so, you may study it from at least three diﬀerent perspectives:

Figure 1.

An aqueous foam as seen by the photographer-artist Michael Boran.

387 W. Gutkowski and T.A. Kowalewski (eds.), Mechanics of the 21st Century, 387–394. © 2005 Springer. Printed in the Netherlands.

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as a compacted heap of individual bubbles of widely varying shape and size. as a division of space by conjoined soap ﬁlms, all slightly curved. as a network of lines, the so-called Plateau borders where the ﬁlms meet. The ﬁrst description relates to the manner of formation of the foam, which can even be designed to produce highly monodisperse foams. The second is the key to its stability: how long will the ﬁlms survive without rupture? The third description often comes to the fore in theories of physical properties, such as conductivity, drainage or the mechanics of solidiﬁed foam. Plateau(1873) gave us the ﬁrst coherent account of the basic rules to which the structure must conform, particularly for relatively dry foams, of low liquid content. Underlying these rules is the essential principle of the minimization of surface energy (usually under the constraint of ﬁxed bubble volumes). Indeed, most of the static and quasi-static properties of a foam may be explained by arguments which derive from that principle. It entails the Laplace-Young law (which we should celebrate in 2004/5 as this is its bicentenary year), and Plateau’s rules for the junctions of ﬁlms and borders (Weaire and Hutzler(1999)). As the liquid content is increased, some of Plateau’s strictures are relaxed. Whereas only fourfold conﬂuences of borders are possible for the ideal dry foam, higher numbers may come together in stable junctions of the wet foam. We have not yet grasped the complexities of wet foam structures, except perhaps in two dimensions. Even in the carefully chosen special case of the symmetric eight-fold junction, shown in Fig. 2, progress has been slow since Weaire and Phelan(1996) raised questions

Figure 2. This wet eight-fold junction of Plateau borders is stable until the liquid fraction is reduced to an exceedingly small value, as shown by Brakke. Image courtesy of K. Brakke.

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about it, on the basis of experiment (see in het Panhuis et al.(1998)), neatly following the tradition of Plateau. Very recently, Ken Brakke has completed a masterful analysis which leads to the conclusion that a ﬁnite but exceedingly small liquid fraction Φ is required to stabilize this junction.

2.

The Surface Evolver

Brakke is the originator and chief exponent of the Surface Evolver (Brakke(1992)), a suite of software which has been applied to such static problems. Its impact on this ﬁeld is only part of a wider inﬂuence, whenever surface energy is dominant in physics and engineering (e.g. Collicott and Weislogel(2004)). As the crystallographer Alan Mackay has said The Evolver is a spectacular example of the eﬀects of a gift to science which advances a whole ﬁeld.

Figure 3 gives some further examples of the applications of the Evolver undertaken by our group. The most celebrated of these is the 1994 discovery (Weaire and Phelan(1994)) of a structure of monodisperse dry

Figure 3. Examples of the use of the Surface Evolver. (a) The Weaire-Phelan structure which is conjectured to ﬁll space with the lowest surface area. (b) A ﬁnite cluster of bubbles, used to investigate their local structure (Cox and Graner(2004)). (c) The meniscus surrounding a single bubble trapped between a glass plate and a liquid pool (Vaz et al.(2004)). (d) When two drops of oil are squeezed between parallel plates, there is a symmetry-breaking instability (e) at a certain critical separation (Bradley and Weaire(2001)).

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foam that has a lower surface energy than that conjectured by Lord Kelvin (Thomson(1887); Weaire(1994)). This structure has two diﬀerent bubble shapes that ﬁt together to form a structure of overall (body-centred-) cubic symmetry. It is about to be manifested in a spectacular building for the Beijing Olympics – the Water Cube – illustrated in Fig. 4. Its construction is based on a network of steel beams, corresponding to the Plateau borders of the WeairePhelan structure. In analyzing its stability, the engineers must have repeated the kind of exercise undertaken by materials scientists for opencelled solid foams such as polyurethane (Warren and Kraynik(1991)). It should prove to be an inspiring instance of the harmony of scientiﬁc and aesthetic principles.

Figure 4. The designer’s vision for the Beijing National Swimming Centre for the 2008 Olympics. The interior of its transparent walls consist of the Weaire-Phelan structure of Fig. 3(a). Image courtesy of Arup, PTW and CSCEC.

3.

Debates Over Drainage

As anticipated in the closing chapter of the book of Weaire and Hutzler(1999), the focus of foam physics has moved from static to dynamic properties, related to drainage and rheology. Drainage is the passage of liquid through the foam (mainly through the Plateau borders), driven by gravity or by pressure gradients. Its main properties are captured by a simple continuum theory expressed in a nonlinear partial diﬀerential equation. Suppose we pour liquid steadily into the top of a foam: how fast does it travel downwards under gravity? Assuming Poiseuille ﬂow in the

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borders, a relation may derived for which the velocity is v ∼ Q1/2 ,

(1)

where Q is the ﬂow-rate of added liquid. Although supported by several experiments (Weaire et al.(1997)), this was challenged (Koehler et al.(1999)), in favour of an index of one-third. It transpired that the discrepancy between the old and the new results lay in the use of diﬀerent surfactants to stabilize the foam. Some of these produce relatively rigid surfaces (hence Poiseuille ﬂow) while others leave the surfaces mobile (Durand et al.(1999)). In future, we will be more cautious in asserting generic properties!

4.

Getting Rid of Gravity

Beyond a certain ﬂow-rate, the steady drainage described above becomes unstable, giving rise to a slow convective motion (Hutzler et al. (1998), Vera et al. (2000), Weaire et al. (2003)). It therefore cannot be used as a proxy for the equilibrium structure of a very wet foam, which was part of the original motivation for its study. How then are we to prepare such uniform wet foams? A static foam under gravity has only a very limited height of wet foam (if any) at the bottom. It may be estimated to extend to the height h=

l02 , d

(2)

where d is the bubble diameter and the capillary length is l02 =

σ . g∆ρ

(3)

Here σ is the surface tension, ∆ρ the density diﬀerence between gas and liquid and g the acceleration due to gravity.

Figure 5.

Strategies for preparing and studying wet foams.

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The rest of the foam is essentially dry. Hence the ease of preparing dry foams and our frustration in wishing to study wet samples. Several strategies present themselves, and are summarized in Fig. 5. Of these, density-matched emulsions have been used by Mason et al.(1995) and others to probe static properties which are common to foams and emulsions, in theory. Another strategy is to use small enough bubbles that the wet foam thickness in Eq. (2) above becomes considerable. Thirdly, we may use steady drainage or attempt measurements in a short time-scale, before drainage has developed (Saint-Jalmes and Durian(1999)). All of these suﬀer from limited applicability: hence the appeal of getting rid of gravity altogether, in drop towers, parabolic ﬂights, rockets or space-stations.

5.

Foams under Microgravity

An honoured mention should be made of early space microgravity experiments on foams, particularly by David Noever (1994) and Noever and Cronise (1994), but these isolated eﬀorts did not result in any coherent progress. Following the creation and operation of a Topical Team for this subject by the European Space Agency, there is some hope of a more systematic approach. Currently the subject is being tackled in two ESA-sponsored projects. The ﬁrst aims to utilize microgravity conditions to create homogeneous metallic foams (W¨ u ¨bben et al.(2002)). An example of a metallic foam is shown in Fig. 6. These materials are proving their potential in, for example, the automobile industry. The foamed metal should not suﬀer from drainage while in its liquid state, since this would lead to variations of density within the solid product (Banhart and Weaire(2002); Cox et al.(2001)). Obviously, microgravity sidesteps that limitation. The second project is a study of wet aqueous foams in equilibrium, so that the processes of drainage, rheology and coarsening due to gas diﬀusion can be examined independently (Saint-Jalmes and Langevin(2004)).

6.

Conclusions

For its satisfactory completion, the basic theory of the physics of foams needs to be extended to wet foams, initially in a state of static equilibrium. Describing dynamic wet foams will still be a considerable challenge. But it will reward success: churning, ﬂowing wet foams lie at the heart of the chemical industry. As Weaire and Hutzler(1999) said, throwing down the challenge, a walk by the seaside on a stormy day is

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Figure 6. An example of foamed zinc. The bubbly melt must be solidiﬁed quickly to retain the homogeneous structure, a stricture not required in the microgravity environment. Image courtesy of J. Banhart.

enough to excite curiosity. But it may be that the calmer environment of space is needed for the ﬁrst progress in understanding wet foams.

Acknowledgments Tristram Carfrae (Arup) kindly provided details of the Water Cube (Fig. 4). We wish to thank the European Space Agency for support under ESA Contract 14308/00/NL/SH (AO-99-031) CCN 002 MAP Project AO-99-075.

References [1] J. Banhart and D. Weaire, On the road again: metal foams ﬁnd favor, Physics Today, July:37–42, 2002. [2] G. Bradley and D. Weaire, Instabilities of Two Liquid Drops in Contact, Comp. Sci. Eng., Sept/Oct:16–21, 2001. [3] K. Brakke, The Surface Evolver, Exp. Math., 1:141–165, 1992. [4] S.H. Collicott and M.M. Weislogel, Computing existence and stability of capillary surfaces using surface evolver, AIAA J., 42:289–295, 2004. [5] S.J. Cox, G. Bradley, and D. Weaire, Metallic foam processing from the liquid state: the competition between solidiﬁcation and drainage, Euro. J. Phys: Appl. Physics, 14:87–97, 2001.

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[6] S.J. Cox and F. Graner, Three-dimensional bubble clusters: shape, packing and growth-rate, Phys. Rev. E, 69:031409, 2004. [7] M. Durand, G. Martinoty, and D. Langevin, Liquid ﬂow through aqueous foams: From the Plateau border-dominated regime to the node-dominated regime, Phys. Rev. E, 60:R6307–R6308, 1999. [8] S. Hutzler, D. Weaire, and R. Crawford, Convective instability in foam drainage, Europhysics Lett., 41:461–465, 1998. [10] S.A. Koehler, S. Hilgenfeldt, and H.A. Stone, Liquid ﬂow through aqueous foams: The node-dominated foam drainage equation, Phys. Rev. Lett., 82:4232–4235, 1999. [11] T.G. Mason, J. Bibette, and D.A. Weitz, Elasticity of Compressed Emulsions, Phys. Rev. Lett., 75:2051–2054, 1995. [12] D.A. Noever, Foam Fractionation of Particles in Low Gravity, J. Spacecraft Rockets, 31:319–322, 1994. [13] D.A. Noever and R.J. Cronise, Weightless bubble lattices: A case of froth wicking, Phys. Fluids, 6:2493–2500, 1994. [9] M. in het Panhuis, S. Hutzler, D. Weaire, and R. Phelan, New variations on the soap ﬁlm experiments of Plateau. I. Experiments under forced drainage, Phil. Mag. B, 78(1):1–12, 1998. [14] J.A.F. Plateau, Statique Exp´rimentale ´ et Th´eorique des Liquides Soumis aux Seules Forces Mol´culaires ´ , Gauthier-Villars, Paris, 1873. [15] A. Saint-Jalmes and D.J. Durian, Vanishing elasticity for wet foams: Equivalence with emulsions and role of polydispersity, J. Rheol., 43:1411–1422, 1999. [16] A. Saint-Jalmes and D. Langevin, The foam experimental module for the ISS, Microg. Sci. Tech., in press, 2004. [17] W. Thomson, On the Division of Space with Minimum Partitional Area, Phil. Mag., 24:503–514, 1887. [18] M.F. Vaz, S.J. Cox, and M.D. Alonso, Minimum energy conﬁgurations of small bidisperse bubble clusters, J. Phys.: Condens. Matter, 16:4165–4175, 2004. [19] M.U. Vera, A. Saint-Jalmes, and D.J. Durian, Instabilities in a liquid-ﬂuidized bed of gas bubbles, Phys. Rev. Lett., 84:3001–3004, 2000. [20] W.E. Warren and A.M. Kraynik, The Nonlinear Elastic Behavior of Open-Cell Foams, J. Appl. Mech., 58:376–381, 1991. [21] D. Weaire, editor, The Kelvin Problem, Taylor & Francis, London, 1994. [22] D. Weaire and S. Hutzler, The Physics of Foams, Clarendon Press, Oxford, 1999. [23] D. Weaire, S. Hutzler, S. Cox, N. Kern, M.D. Alonso, and W. Drenckhan, The Fluid Dynamics of Foams, J. Phys.: Condens. Matter, 15:S65–S73, 2003. [24] D. Weaire, S. Hutzler, G. Verbist, and E. Peters, A review of foam drainage, Advances in Chemical Physics, 102:315–374, 1997. [25] D. Weaire and R. Phelan, A counter-example to Kelvin’s conjecture on minimal surfaces, Phil. Mag. Lett., 69:107, 1994. [26] D. Weaire and R. Phelan, Vertex instabilities in foams and emulsions, J. Phys.: Condensed Matter, 8:L 37–L 43, 1996. [27] T. W¨ u ¨ bben, J. Banhart, and S. Odenbach, Production of metallic foam under low gravity conditions during parabolic ﬂights, Microgravity Sci. Tech., 13:36–42, 2002.

Author Index On the following pages authors and co-authors of all presentations are listed. The extended summaries of the presentations can be found on the enclosed CD-ROM. Each name is followed by sequence of entries consisting of session and presentation IDs. For the lectures contained in the present book, the appropriate page number is also given, and the presenting author entry is additionally highlighted by boldface letters. IDs of contributions of presenting authors are printed in upright typeface. Only eligible presentations are listed, i.e. presented during the Congress, excluding multiple presentations and those given by a proxy.

396

Abbott J.R. — FM16 11725

Abdalla H.M. — FSM6 11806 Abdou L. — SM2 12082 Abdul-Latif A. — SM4 10456 Abdunabi T.A. — SM25 11082 Abed F.H. — SM18 10393 Abedi R. — SM1 12441 Abkarian M. — FM1 12423 Aboudi J. — MS1 12011 Acartuerk A. — SM15 11194 Acharya A. — SM18 11820 Acrivos A. — FM16 10361 FM4 10753 Adachi S. — FM25 12041 Adachi T. — FM5 11864 Adams N.A. — FM24 10455 FM24 11256 FM24 11802 FM24 12564 FSM6 10399 Adda-Bedia M. — SM9 10338 Addas K. — MS2 10709 (p. 355) D’Addetta G.A. — SM4 11414 Aderogba K. — SM13 10745 Adrian R.J. — FM25 12161 FM25 12196 MS4 11607 Agapov D. — SM17 11688 Agarwal R.K. — FM11 10872 Aghalovyan L.A. — SM6 11526 Ahcene B. — FSM4 13050 Ahlawat A.S. — SM3 10098 Ahmetolan S. — SM11 10891 Aider A.A. — FM13 10185 Aidun C.K. — FM1 12124 Aifantis K. — SM13 11782 Aitta A. — FM21 11328 Aksel N. — FM14 10642 FM14 10928 Akylas T.R. — FM26 10762 Al-Abduljabbar A. — SM9 12798 Albers B. — SM11 10072 Alderliesten R. — SM8 12217 Alexander J.I.D. — MS5 12447 Alexeev A. — FM14 12293 FSM1 12516 Ali R. — FM2 11887 Alibiglu A. — SM13 13009 Alleborn N. — FM14 11413 Allix O. — SM1 12237 Almeida S.F.M. de — SM13 12148 Altenbach H. — SM27 11368 Altobelli S.A. — FM18 10983 Alvarez J.O. — FSM1 12477 Amalia P. — SM17 11352 Amberg G. — FM21 12312 Ambr´ ´ osio J. — SL1 11040 (p. 61) Ammann M. — SM15 10792 Amyot O. — FM12 11343 Anderson D.M. — FM21 12768

ICTAM04 Andersson L.-E. — SM2 10686 Andreev O. — FM19 11120 Andreeva T.A. — FSM1 12149 Andrejczuk M. — FM9 12592 Andrews M.G. — SM13 11162 Andrianov I.V. — SM13 12588 Antkowiak A. — FM13 12483 Antretter T. — SM14 10088 Anweiler S. — FM20 11745 Arakelian V. — SM16 10553 Arbez P. — FM6 12510 Aref H. — FM25 11676 FSM7 10003 Argyriadi K. — FM14 10557 Arias I. — SM9 12171 Ario I. — SM22 12688 Ariza M.P. — FSM6 11471 Arora M. — FM8 11999 Arroyo M. — SM1 12455 Asai M. — FM13 10489 Aschenbrenner C. — FM13 10188 Ashawesh G.M. — SM25 11082 Ashida F. — MS1 12018 Ashmore J. — FM21 12276 Askes H. — SM12 11277 SM1 12924 Asmolov E.S. — FM16 11480 Astley J. — FSM1 11249 Atakhorrami M. — MS2 10709 (p. 355) Aubouy M. — FSM5 11095 Auregan Y. — FM24 12740 Aurnou J.M. — FM7 11440 Auslender F. — SM13 11507 Aussillous P. — FM21 11320 Avalos-Zuniga R.A. — FM19 12120 Avramov K.V. — SM25 12989 Awrejcewicz J. — SM25 12989 SM2 11481

Baaijens F.P.T. — SM18 11344

Baars A. — FM10 13048 Babinsky H. — FM5 10984 Babushkin I.A. — FM7 12267 Bachorz P. — SM25 12770 Badel P.-B. — SM4 12913 Bai X. — SM9 12174 Bai Y. — FSM7 11765 SM7 11764 Baillis C. — SM22 11174 Bajer K. — FM25 12275 SL2 10042 (p. 89) Balamurugan V. — MS1 12147 Balandin D.V. — SM3 11219 Baldessari F. — FM8 11289 Balint A.M. — FM21 11362 Balint D.S. — MS3 12386

397

Author Index Balint S. — FM21 11362 Baltov A.I. — SM18 11953 Balueva A.V. — SM9 11856 Bambrey R.R. — MS6 11336 Banach Z. — FSM6 11237 Banaszak J. — SM15 13006 Banaszek J. — FM21 11364 Banerjee A. — SM17 11375 Banta R.M. — MS6 10513 Bar-Lev O. — FM17 12308 Bar-Yoseph P.Z. — FM13 11461 Baracho Neto O.G.P. — SM25 11494 Baradat C. — SM16 10553 Barba L.A. — FM6 11937 Barbe F. — SM14 12421 SM14 12747 Barber T.J. — FM14 11477 Bargmann S. — SM1 12924 Barkley D. — FM13 12431 Barmin A.A. — FM9 11703 Barnea D. — FM20 11655 Baroud C.N. — FM1 11013 MS4 12047 SL18 10495 Barral S. — FM19 12363 Barrau J.-J. — SM13 11180 Bartel T. — SM14 12077 Barth´ ´es-Biesel D. — MS2 11961 Batra R.C. — SM10 12594 Baty H. — FM19 11429 Bauer D. — SM15 12614 Bauruelle J.C. — SM22 11174 Baweja M. — SM3 12756 Bayer I.S. — FM4 11603 Bazant M.Z. — MS4 11777 Bazin B. — FM12 10234 Baldyga J. — FM22 12415 Becache E. — SM2 12382 Becker M. — FSM6 12118 Becker W. — SM15 11463 SM9 11484 Beda P.B. — SM12 10314 Bednarz T. — FM7 13022 Behringer R.P. — FM17 12399 Behringer R. — FM12 12617 Belcher S.E. — FM26 10746 Belzons M. — FM17 11751 Ben Dhia H. — SM11 12763 Ben Hamida A. — SM1 12727 Benallal A. — SM12 10265 Benilov E.S. — FM26 11468 Benilov E. — FM25 11467 Bennacer R. — FM15 11386 Bennett D.J. — FM4 10753 Benzerga A.A. — SM1 12737 Berbenni S. — SM18 11498 Beresneva E.N. — FM7 10538

Berezovski A. — SM10 11592 SM14 11393 Berﬁeld T. — MS1 11685 Berg A. van den — FM1 12400 Bergdorﬀ M. — FM6 12583 Bergeon A. — FM22 12731 Bergerot A. — SM13 11180 Bergman L.A. — SM25 10408 Bergmann R. — FM10 11909 FM17 10253 Bergougnoux L. — FM16 11588 FM16 12345 Bernardo M. di — FSM2 11363 Beron-Vera F.J. — MS6 12776 Berton G. — FM17 12883 Bertram A. — SM18 12549 Beskos D.E. — SM1 12845 Bessonnet G. — SM17 12539 Betelu S.I. — FM8 12548 Beysens D.A. — SL3 10880 (p. 117) Bezpalcova K. — FM9 10467 Bhattacharya K. — MS1 11731 SM14 10721 Bi W. — FM17 12883 Bialecki R.A. — SM24 11137 Bidulya A. — SM17 11688 Bielski J. — SM4 11796 Bielski P. — FM8 12311 Bielski W.R. — FM12 11248 Bigoni D. — SM12 10265 Billant P. — FM25 11391 FM25 12108 MS6 12294 Billardon R. — SM4 10974 Bin G. — SM1 10559 Binding D.M. — FM4 12254 Bing-Zheng G. — SM10 11572 Bioul F. — FM15 12591 Biros G. — FSM1 12691 Bisgaard A. — FM23 10247 Blab R. — SM27 11981 Blachut J. — SM22 10378 Blackmore D. — FM25 11683 Blajer W. — SM17 10026 Blanc X. — FSM6 12336 Blasinska A. — FM4 11184 Blekhman I.I. — SM25 11055 SM25 11719 Blekhman I. — SM25 11055 Blachowski B.D. — SM3 10457 Blonski ´ S. — FM17 11169 FM4 11184 FM8 12873 Blume H. — SM2 12086 Blyth M.G. — FM18 10766 Bo W. — SM24 13013 Bobaru F. — SM24 12789 Bobylov A.A. — SM2 11933 Bocciarelli M. — SM18 10797

398 Bochenek B. — SM24 12075 Bodnar A. — SM4 11665 Boeck T. — FM19 12090 FM8 11723 Boehlke T. — SM18 12549 Boehm H.J. — SM13 12321 Boehm R. — SM13 12576 Boer P. de — FM11 12258 Boer R. de — SM15 10198 Boerner E.F.I. — SM1 12584 Bogacz R. — SM22 12931 Bogaert N. van den — FM15 12591 Boguslawski A. — FM3 11914 FM6 12076 Bohatier C. — SM17 11361 Bokhove O. — MS6 10716 (p. 103) Bolotnik N.N. — SM3 11219 Bolotnova R.Kh. — FM8 12190 Bolzon G. — SM18 10797 Bonamy D. — SM9 12378 Bonn D. — FM8 13025 Bonnecaze R.T. — FM16 11991 FM16 12574 Bonnevie Harbitz C. — FM9 12270 Bonsel J.H. — SM25 11783 Bontoux P. — FM6 13051 FM15 11893 FM25 12080 Bontozoglou V. — FM14 10557 Borg U. — SM13 12214 Borges L.S.A. — SM10 13018 Borisov A.V. — SM1 12351 Bornert M. — SM13 11507 SM13 11805 Borodich F.M. — SM2 11135 SM2 11215 Borodulin V.I. — FM24 12675 FM2 10275 Boronska K. — FM7 10914 Boro´ n ´ski P. — FM6 10921 Borrell M. — FM8 11289 Borve S. — FM19 11809 Bos F. van der — FM6 13030 Bostrom A. — FSM7 12603 Bottaro A. — FM11 12602 Bottausci F. — FM22 12761 Bouchet G. — FM24 12740 Bouch V. — FM7 11972 Boudaoud A. — FM8 13025 Boudifa M. — SM4 11885 Bouizi O. — FM15 12636 Boukpeti N. — SM20 11667 Boulanger P.S. — FSM3 11818 SM11 11812 Bouruet-Aubertot P. — MS6 12601 Boussa H. — SM1 12727 Bouvet C. — SM13 11180 SM14 12225

ICTAM04 Bouville M. — FSM6 11476 Boyland P. — FM22 12442 Bozhko A.A. — FM7 10538 Brady J.F. — FM16 12452 SL4 12160 Braescu L. — FM21 11362 Bragov A.M. — SM20 10374 Brancher J.P. — FM13 10185 Brancher P. — FM13 12483 Branicki M. — FM18 12164 FSM6 11862 Braud P. — FM5 13016 Brauer H. — FM19 12605 Braun S. — FM2 11084 Braunbrueck A. — SM11 11620 Bravo-Castillero J. — SM13 11176 Brechet Y. — SM1 12737 Brekelmans W.A.M. — FSM6 11830 SM18 11344 SM18 11963 SM1 12723 Bremond N. — FM8 11317 Brenner H. — FM16 11725 Brenner M.P. — FM8 11330 Breuer K.S. — MS4 11604 Breysse D. — SM18 12634 Brinckmann S. — SM8 11927 Brinson C.L. — MS1 12027 Brizuela E. — FM3 11888 Broeckhoven T. — FM3 11888 Brons M. — FM23 10247 Brovchenko I.A. — FM26 12268 Brown E.N. — MS1 13010 MS1 13011 Browne D.J. — FM21 11364 Bruls O. — SM17 12208 Brun C. — FM24 12740 Brunet E. — MS4 12130 Brunig M. — SM4 11406 Bruyn J.R. de — FM17 12338 Buehler O. — MS6 10977 Buhl T. — SM24 12079 Bulgakova G.T. — FM12 10548 Bulgarelli U.P. — FM8 12547 Burachik R. — SM24 10594 Burczy´ n ´ ski T.S. — SM24 11137 Burde G.I. — FM26 12813 Burde G. — FM2 11984 Burnett D.S. — FSM1 11373 Burton D. — MS1 12027 Buryachenko V. — SM6 12685 Buschmann M.M. — FM2 10948 Busilas A. — SM16 12546 Busse F. — FM19 10931 Byrne H.M. — FM23 11060 Byrtus M. — SM25 11759 Byskov E. — SM22 12297 Byun Y.-H. — FM6 11569

399

Author Index Blawzdziewicz J. — FM8 11411 Buhler ¨ O. — MS6 12157

Caﬀrey J.P. — MS1 13007

Cailletaud G. — SM14 10088 Calhoun R. — FM9 12738 Callegari G. — FM4 10140 Calloch S. — SM14 12225 SM18 12469 Camotim D. — SM22 12558 Campana E.F. — FM8 12547 Campregher R. — FM6 11456 Cannata G. — FM24 11151 Cantat I. — FSM5 12052 Carabineanu A.S. — FM1 11943 Carbonneau X. — FM6 12510 Cardin P. — FM19 12330 Cardon A. — SM13 12818 Cardonne C. — FM22 12761 Carey M. — FM12 12617 Carlotti P. — MS6 10513 Carnasciali M.-I. — MS5 12651 Carneiro C.A.V. — SM10 13018 Carpen I.C. — FM16 12452 Carpenter P.W. — FM2 11887 Carpinteri A. — FSM6 11224 Carriere P. — FM13 11186 Cartraud P. — SM1 12359 Casalis G. — FM13 12508 Casandjian C. — SM4 10011 Case S. — SM23 10242 Casey J. — FSM3 11096 Cassar C. — FM16 11749 Cassardo C. — MS6 13017 Castellanos A. — FM4 12437 Castro F. — FM8 11670 Cataldi-Spinola E. — FSM1 12586 Caughey T.K. — MS1 13007 Caulliez G. — FM13 12233 FM26 11230 Cazacu O. — SM18 11930 SM20 12779 Cazalbou J.B. — FM6 12510 Cebers A. — MS2 11102 Cenedese A. — FM7 11673 Cercignani C. — MS4 10723 Chabert E. — SM13 11805 Chaboche J.L. — SM4 11885 Chadli M. — SM4 10456 Chakraborty P. — FM25 12161 Challamel N. — SM4 10011 Chang C.-C. — FM25 11741 Chang F.-R. — SM16 12712 Chang H.-H. — SM10 10364 Chang J.-T. — SM1 11193 Chang K.T. — FM11 11043 Chang S.-Y. — SM24 11826

Chao G. — SM11 11271 Chaplin J.R. — FSM4 12557 Charrier Mojtabi M.C. — MS5 10980 Chassaing P. — FM6 12510 Chassiakos A.G. — MS1 13007 Chatterjee A. — SM25 12814 Chatzidai N. — FM18 12859 Chatzigeorgiou G. — SM4 12749 Chauve M.-P. — FM24 10506 Chazallon C. — SM18 12634 Chen B. — SM24 11532 Chen C.F. — FM21 11058 Chen F. — FM9 11294 Chen H.-C. — FM25 11741 Chen J. — SM25 10166 Chen L.-Q. — FSM2 12674 Chen P. — FM13 11995 Chen S. — SM14 11395 Chen T. — SM6 11960 Chenchiah I. — SM14 10721 Cheng G.D. — SM24 13013 Cheong K.B. — FM8 12596 Chern M.-J. — FSM4 11668 Cherniy D.I. — FM23 12868 Chernoray V.G. — FM10 11455 FM2 11339 Chernousko F.L. — SM3 11243 Chew H.K. — FM8 12596 Chiba N. — SM18 10652 Chickichev I. — SM20 12364 Chien W.Z. — FM21 11208 Chini G.P. — FM14 12155 Chiskis A. — MS3 12432 Chizhik S.A. — SM2 10999 Chmielniak T.J. — FM6 11578 FM10 11916 Cho J.-H. — SM27 11827 Cho Y.-S. — SM1 11817 Choi C.K. — FM21 13008 Choi H. — FM11 11636 FM19 11637 Choi J. — FM11 11636 Choi S.T. — MS3 12778 Cholet C. — SM17 11361 Chomaz J.-M. — FM13 12425 FM13 12711 FM25 11391 FM25 12108 MS6 12294 Chow C.L. — SM4 10961 Christophorou C. — FM24 12965 Chro´ ´scielewski J. — SM19 11890 Chrzanowski M. — SM4 11665 Chu C.-C. — FM25 11741 Chucholowski C. — SM26 11672 Chung C.A. — FM21 11208 Chung S.-H. — SM1 12441 Churochkin D.V. — FSM1 11508 Chwa M. — MS3 12251

400 Cichocki B. — FM16 10993 FM18 11300 Ciechanowski J. — FM9 12266 Cieslik J. — SM25 12498 Cieszko M. — FM12 12611 Cilliers J.J. — FSM5 12851 Claire D. — SM4 12021 Clamond D. — FM26 11417 Clanet C. — FM8 11317 Clavel M. — SM8 12228 Clayton J.D. — SM18 10955 Clercx H.J.H. — FM25 11964 Clercx H. — FM22 12633 Cleri F. — FSM6 12334 Cliﬀord M.J. — SM13 11387 Cloirec M. — SM1 12359 Cloitre M. — FM16 12574 Cocks A.C.F. — SM15 12903 Cocou M. — SM2 12382 Coelho I. — SM12 12682 Coelho P.J. — FM3 12348 Cohen J. — FM13 11461 Cohen-Addad S. — FSM5 12578 Coiﬀet F. — FM5 13016 Coker D. — SM2 11987 Cola B.A. — FM22 12158 Colette A. — FM25 11391 Collicott S.H. — MS5 12447 Colombo L. — FSM6 12334 Colonius T. — FM11 12387 Coman C.D. — SM22 12967 Combescure A. — SM1 11797 SM22 11174 Comte P. — FM19 11429 FM24 12740 Constantin P. — FM23 10938 Constantinescu D.M. — SM9 10864 Cooker M.J. — FM26 11345 Coons J. — FSM5 12435 Cordeiro Fernandes P. — FM20 12139 Cornwell P. — FSM7 12642 Costa M. — FM3 12348 Costanzo F. — FSM7 12642 Cottet G.-H. — FM6 12583 Cottron M. — SM9 10429 Courrech du Pont S. — FM17 13029 Cox B.N. — SM13 11162 Cox S.M. — FM22 10497 FM23 11060 Cox S. — MS5 10598 (p. 387) Craig W. — FM26 11548 Cramer A. — FM15 12349 Crandall S.H. — SM25 10863 Cristescu N.D. — SM18 11930 SM20 12779 Crochet M.W. — SM15 12771 Croll J.G.A. — FM9 12072

ICTAM04 Cudzilo S. — FM3 12745 Cugy P. — SM18 12469 Cummins S.J. — FM8 12163 Cunningham J.C. — SM15 12917 Currie P.K. — FSM5 12986 Cuypers Y. — FM24 11303 Czarnecki S. — SM24 11760 Czechowski L. — FM21 12743 Czerwi´ n ´ ska J. — FM24 11802 FSM6 10399

D

ahan M. — SM6 11901 Dahl S.R. — FM17 12073 Dail B. — FM15 10363 Dal Pont S. — SM15 12114 Dalziel S.B. — FM25 12051 Dam D.B. van — FM14 11509 Dam R. — SM22 12297 Daneshmand F. — FM9 10733 Daneshmehr A. — SM13 13009 Danishevs’kyy V.V. — SM13 12588 Dankowicz H. — SM17 11020 Danner T. — FM8 11669 Daru V. — FM8 11979 Darve E. — FM16 12005 Daube O. — FM13 10640 Davaille A. — FM7 12126 Davide B. — SM12 11254 Davidson P.A. — FM19 11124 Davis R.H. — FM16 11473 FM8 12329 FM8 12741 Deblaise D. — SM16 10553 Debatin K. — FM5 11864 Debin S. — SM1 10559 Decamp S. — FM9 12410 Degre G. — MS4 12130 Dekajlo K. — FM9 10576 Delannay R. — FM17 12883 FSM5 12052 Delgado A. — MS2 12031 Dell’Aversana P. — MS5 12651 Delsaute B. — FM15 12591 Delville J. — FM5 13016 Dems K. — SM24 11349 Denda M. — MS1 12786 Denier J.P. — FM2 10103 Denis A. — SM18 12634 Denkov N.D. — FM8 11669 Deprince X. — SM18 11110 Derby J.J. — FM15 10363 Derksen J. — FM20 12397 Deshpande V.S. — SM5 11022 Destrade M. — FSM3 11950 Detournay E. — SM15 12713 SM25 12535 Dettmar J. — FSM6 12112 Di G. Sigalotti L. — FM6 10857 Dias F. — FM26 12089

401

Author Index Dick E. — FM6 11602 Didelle H. — MS6 12494 Dijksman J.F. — FM4 12894 Dimakopoulos Y. — FM14 12858 Ding E.-J. — FM1 12124 Dinkler D. — SM25 12785 Dmitrochenko O. — SM17 10893 Dobler W. — FM7 10306 Dobovsek I. — SM12 12419 Dobrolyubov A.I. — FM26 11144 Doche O. — FM11 11394 Dodge A. — MS4 12536 Doerﬀer P.P. — FM5 12997 Doghri I. — SM4 11886 SM18 13038 Dohnal F. — SM25 12026 Dolezel I. — SM2 10211 Dollet B. — FSM5 11095 Domaas U. — FM9 12270 Doma´ n ´ ski W. — FSM3 11840 Domaradzki A.J. — FM24 10149 Domenico C. — SM12 11254 Dominguez J. — SM8 10698 Dominguez J. — SM17 12071 SM9 11321 Dommelen J.A.W. van — SM18 11344 Dongen M.E.H. van — SM11 11271 Dorfman K.D. — FM4 11209 Dorfmann L.A. — FSM3 11227 Dormieux L. — SM15 11941 Doudard C. — SM18 12469 Doval-Montes L. — SM13 11176 Dovgal A.V. — FM2 11339 Dovgiy S.A. — FM23 12868 Dowling A.P. — FSM1 12527 Dragon A. — SM18 11110 Drazer G. — FM16 10361 Dreiden G.V. — SM11 10114 Drescher A. — SM20 11667 Dritschel D.G. — MS6 11336 Drobi´ n ´ ski P. — MS6 10513 Drobniak S. — FM3 11914 Drugan W.J. — SM12 12025 Du J. — SM24 10433 Dual J. — FSM1 12586 MS3 12384 Dubois F. — SM17 11361 Duchemin L. — FM8 12388 Duck P.W. — FM13 10916 FM2 10103 Dudeck M. — FM19 12363 Dufour G. — FM6 12510 Duineveld P.C. — FM4 12894 Duluc M.C. — FM8 11979 Dumais J. — MS2 11581 Dumontet H. — SM1 12727 Dupere I.D.J. — FSM1 12527

Duplat J. — FM22 12326 Dupont R. — FM13 12233 Dupret F. — FM15 12591 Durgin W.W. — FSM1 12149 Duschlbauer D. — SM13 12321 Dusek J. — FM13 12766 Duszy´ n ´ski P. — FM6 12565 Duysinx P. — SM17 12208 Dvorak V. — FM5 12219 Dlugosz A. — SM24 11137

Eberhard P. — SM17 12133

Eberhardsteiner J. — SM27 11981 Eck C. — FSM6 11340 Eckert K. — FM21 12365 Eckert S. — FM21 12365 Eckhardt B. — FM13 10313 Edwards N.R. — FM26 12424 Eggers J. — FM14 11089 Ehlers W. — SM15 10792 SM15 11194 Ehrenstein U. — FM13 12372 Ehrlacher A. — SM15 12114 SM15 12614 Eidelman A. — FM10 12070 Ekiel-Je˙z˙ ewska M.L. — FM16 11409 FM18 11300 El Abd A. — SM18 12634 El G. — FM26 11134 El Ganaoui M. — FM15 11386 FM15 11893 El Maihy A. — FM22 10130 Elaguine D. — SM2 11365 Eldredge J.D. — FSM1 12180 Elias F. — FSM5 11095 Elmabrok A.M. — SM25 11082 Elperin T. — FM10 12070 FM7 11053 Emelin V.K. — SM7 11556 Emmerich H. — FSM6 11340 Engel R.S. — SM27 10906 Engelbrecht J. — SM10 11592 SM11 11813 Engelen R.A.B. — SM4 11415 England A.H. — SM10 12273 Epstein M. — SM6 12122 Eremeyev V.A. — SM19 10287 Erenburg V. — FM3 12199 Erickson J. — SM1 12441 Ermakov M.K. — FSM7 11869 MS5 11843 Ern P. — FM20 12139 Erofeyev V.I. — SM11 12203 Escalona J.L. — SM17 12071 Esveld C. — SM24 11015 Etling D. — MS6 12035 Etnyre J.B. — FM23 11677

402 Evans D.L. — FSM7 12642 Evans G.M. — FM15 11903 Evers L.P. — SM18 11963

F

aciu C. — SM14 11852 Faisst H. — FM13 10313 Faivre M. — FM1 12423 Falcovitz J. — FM5 11049 Falk M. — FSM6 11476 Falkovich G. — FM22 12048 Fan J. — SM27 11506 Fan Y. — SM1 12441 Fang D. — MS1 12016 Fayzrakhmanova I.S. — FM21 12278 Fedenkova A.A. — FM2 10275 Feissel P. — SM1 12237 Felderhof B.U. — FM16 11236 Feng L. — SM24 12287 Feng X.-Q. — MS3 12314 Feng X. — MS1 12016 Fenghuan S. — MS3 12181 Fenili A. — SM3 13004 Feraille T. — FM13 12508 Fernandes A. — SM10 12093 Fernandez-Feria R. — FM9 11346 Fernando H.J.S. — FM9 10576 FM9 12738 Ferrarese S. — MS6 13017 Ferreira M.J.F. — FM20 10690 Ferri F. — FM10 10445 Feuillebois F. — FM16 10248 Fey R.H.B. — SM25 11783 Fidlin A. — SM25 11679 SM25 12694 Finn M.D. — FM22 10497 FM23 11060 Firsova A.D. — SM25 11712 Fischer F.D. — SM14 10088 Flanagan J.D. — FM26 11468 Fleck N.A. — SM13 12680 SM5 11022 Fleck N. — MS2 10988 Floryan J.M. — FM13 10489 Flukiger F. — FM12 11343 Fl´ ´ or J.-B. — MS6 11829 Fochesato C. — FM26 12089 Fomin N.A. — FM10 12213 Fontelos M.A. — FM8 12548 Fornalik E. — FM7 12173 Forterre Y. — FM17 11775 MS2 11581 Foure T.M. — FM13 10188 Fourment C. — FM5 13016 Fowler P.W. — SM1 11482 Foysi H. — FM24 11116 Fraigneau Y. — FM13 10188 Frana K. — FM19 12587 Franciosi P. — SM18 11498

ICTAM04 Francius M. — FM26 11417 Francois M.M. — FM8 12163 Frankel I. — FM7 12850 Franklin J. — FM16 11884 Franz S. — FM24 11256 Fredriksson P. — SM18 11410 Freidin A. — SM14 11837 Freund J.B. — FSM1 10907 Friedrich R. — FM24 11116 Frischmuth K. — SM11 11853 Frohlich ¨ J. — FM6 11348 Frolova N.Yu. — SM7 12272 Frost T. — SM22 12297 Fructus D. — FM26 11417 Fruend E.-T. — FM1 11851 Fu S. — FM24 12652 Fu Y. — SM14 11837 Fuchs P. — SM22 12084 Fuenmayor J. — SM8 10698 Fuetterer C. — FM4 11209 Fujinuma S. — SM18 11295 Fujita K. — SM17 12582 Fukumoto Y. — FM25 11615 SM25 12593 Fukunishi Y. — FM11 12491 Funakoshi M. — FM23 12484 Furutani Y. — SM12 11432

G

abrielli P.G. — FSM6 12334 Gad-el-Hak M. — FM11 10128 FM2 10948 Gadaj S.P. — SM14 10046 SM14 11308 Galaktionov O.S. — FM1 12063 Galambos P.C. — FM4 10753 Galanov B.A. — SM2 11135 Gallaire F. — FM13 12372 FM13 12425 Gallerano F. — FM24 11151 Galperin B. — MS6 11152 MS6 11942 Galtier A. — SM18 12469 Galvanetto U. — FSM2 11241 Galvin J.E. — FM17 12073 Gamallo P. — FSM1 11249 Gambarotta L. — MS2 13049 SM4 12298 Gambaryan-Roisman T. — FM14 12293 Gambin B. — SM13 11895 SM13 12667 Gambin W.L. — SM18 11975 Ganczarski A. — SM4 12335 Gandzha I.S. — FM26 11591 MS2 12059 Ganguly P. — SM1 12458 Ganqing F. — SM24 12287

403

Author Index Gao C.-F. — MS1 11857 Gao H. — SL5 10772 (p. 131) Gao X. — MS1 12027 Garcia-Sanchez F. — SM9 11321 Garcia-Villalba M. — FM6 11348 Gardeniers H. — FM1 12400 Garikipati K. — FSM6 11476 Garstecki A. — SM24 12628 Garstecki P. — MS4 12966 Gaspar Z. — SM22 10915 Gautier P.-E. — SM17 11361 Gavrilov S.N. — SM6 12324 Gavrilova E. — FSM4 10520 Galka A.A. — SM13 12732 Galka A. — SM13 11895 SM13 12667 Geers M.G.D. — FSM6 11830 SM14 11395 SM18 11963 SM1 12131 SM1 12723 SM4 11415 SM4 12341 Geﬀroy P. — FM11 12244 Gelder B. van — FSM5 12373 Gendelman O.V. — SM25 11079 Geoﬀroy S. — FM12 11343 Georgelin M. — FM21 12543 Gerbeth G. — FM11 12068 FM13 12542 FM15 12349 FM19 12613 Gerkema T. — FM26 12087 Gerland P. — SM16 12417 Germay C. — SM25 12535 Gerstmayr J. — SM17 12029 Getling A.V. — FM7 10306 Geubelle P.H. — SM10 11177 Geurts B.J. — FM6 13030 Gharib M. — FM1 12784 Ghicini S. — FM16 11588 FM16 12345 Ghrist R.W. — FM23 11677 Gibson A.N. — FM10 11455 Giessen E. van der — MS2 10627 SM1 12737 SM8 11927 Gilat R. — MS1 12011 Gilewicz J. — FSM3 11642 Gill S.P.A. — MS3 12627 Gilles L. — SM4 12277 Ginalski M.K. — FM1 12409 Girgis I.G. — FM13 10487 Girish J. — SM13 12531 Giusti A. — FM20 13012 Gladden L.F. — FM21 11320 Glema A. — SM5 12949 Glicksman M.E. — FSM5 11240 Gloaguen J.M. — SM18 12339 Glocker C. — FSM2 11183 Gloria A. — FSM6 12078 Godard V. — SM4 12913

Goddard J.D. — FM17 10371 Godoy-Diana R. — FM25 12108 Goektepe S. — SM4 12116 Goetsch M. — FSM1 12586 Goldhirsch I. — FM17 12308 Goldobin D.S. — FM13 12280 Golﬁer F. — FM12 10234 Golinval J.-C. — SM17 12208 Gollub J.P. — FM17 12445 Goncalves P.B. — SM22 12420 Goncharova O.N. — MS5 11757 Gondret P. — FM17 13029 Gong L. — FSM5 11605 Gonik M.A. — FM21 11331 Gontarowskiy P. — SM2 10211 Gonthier K.A. — SM15 12771 Gonthier Y. — SM2 12773 Gonzalez A. — FM4 12437 Goo B. — SM8 11737 Gorbatikh L. — SM2 12412 Goryacheva I. — SM2 11196 Goto T. — FM1 11441 Goujon-Durand S. — FM11 12736 Goulitski K. — FM26 10245 Gourjii A.A. — FM22 12110 FM25 12196 Govindarajan R. — FM2 12631 Govorukhin V.N. — FM12 10199 Graf T. — SM15 10792 Graham A.L. — FM16 11725 Gramstad O. — FSM4 12956 Graner F. — FSM5 11095 Grants I. — FM13 12542 Gravouil A. — SM1 11797 Gray G.L. — FSM7 12642 Grebe R. — SM15 12614 Green M.L. — SM20 12779 Green N.G. — FM4 12437 Grekova E.F. — SM6 12434 Griﬃths R. — MS6 11499 Griggs A.J. — FM8 12741 Grimshaw R.H.J. — SM11 11242 Gristchak V.Z. — SM25 10656 Grondin F. — SM1 12727 Grossman-Clarke S. — FM9 12738 Grue J. — FM26 11417 FSM4 12956 Grunefeld G. — FM10 12070 Gryschka M. — MS6 12035 Gu Y. — SM24 11532 Guazzelli E. — FM16 11588 FM16 11850 FM16 12345 Guba P. — FM21 11270 Guddati M.N. — FSM1 11973 Gudmundson P. — SM18 11410 Guedes de Carvalho J.R.F. — FM20 10690

404 Guegan A. — FM11 12305 Guerses E. — SM1 12115 Guest S.D. — SM1 11482 SM22 12669 Gueydan F. — SM12 12751 Guinovart-Diaz R. — SM13 11176 Guitong Y. — MS3 12181 Gundrum T. — FM19 12613 Guo J.-G. — SM16 10510 Guo L. — SM10 11551 Guo Z. — FSM4 12521 Gupta N.K. — SM5 12982 Guse N. — SM17 11223 Gutﬁnger C. — FSM1 12516 Gutkowski W. — SM3 10457 SM3 10458 Guz I.A. — SM9 10824 Guzina B. — SM20 12364

Haber R.B. — SM1 12441

Haberkorn M. — FM24 12740 Hachemi A. — SM18 12058 Hackl K. — SM14 12077 SM24 12074 Haddad J. — FM4 11648 Hadjiconstantinou N.G. — MS4 11437 Hagedorn P. — SM25 11261 Hahn S. — FM11 11636 Hajzman M. — SM25 11759 Halbedel B. — FM19 11906 Halkjaer S. — SM24 11744 Halley P. — FSM5 12435 Hallworth M.A. — FM17 10524 Hammerton P.W. — FM2 12241 FSM1 12497 Hanada S. — SM11 12418 Hanagud S. — SM10 12968 Hansen J.S. — SM13 12148 Hansen L.V. — FSM1 12533 Hao S. — SM9 10676 Haque A. — MS3 12705 Hara T. — FM26 10746 Harambat F. — FM22 11190 Harrison P. — SM13 11387 Hartmann C. — MS2 12031 FM10 13048 Hasegawa S. — FM5 11560 Hata T. — SM11 10344 Hattori Y. — FM19 11458 FM25 11615 Haughton D.M. — SM6 11220 Hayashi S. — MS1 10821 Hayes M.A. — FSM3 11818 SM11 11812 Haynes P. — MS6 10987 (p. 139) Hazel A.L. — FM1 12295 He J. — FM24 12652

ICTAM04 He Y. — SM27 11506 Healey J.J. — FM13 12150 Heijst G.J.F. van — FM22 12110 FM22 12633 FM25 11964 FM25 12051 Heil M. — FM1 11013 FM1 12295 Heintz A. — FM6 11763 Hellmich C. — SM11 12313 Herakovich C.T. — FSM7 10226 Herczy´ n ´ski A. — FM2 11984 FSM4 11986 Herrada M.A. — FM25 10683 Herrmann K.P. — SM9 10824 SM9 12249 Herskovits J. — SM24 10594 Herve I. — FM11 12244 Hickel S. — FM24 11256 FM24 12564 Hierck B.P. — FM1 11900 Higashi A. — SM25 11066 Hikihara T. — FSM2 12014 Hild F. — SM18 12469 SM4 12021 Hilgenfeldt S. — FM1 12400 FSM5 10597 FSM5 12373 Hinch J. — FM16 11588 FM16 12345 Hino R. — SM24 12870 Ho K.H. — FM21 11208 Ho Y.G. — SM24 12185 Hochlenert D. — SM25 11261 Hodson H.P. — FSM5 12395 Hogan J.S. — FSM2 10499 Hohe J. — SM15 11463 SM9 11484 Hohler R. — FSM5 12578 Hojjati H.M. — SM1 11291 Holeman J.E. — FM9 12738 Holm D.D. — FM6 13030 Holnicki-Szulc J. — MS1 13015 Holobut P. — SM3 10458 Holopainen S. — SM24 12200 Homayoun Heidari A. — FSM1 11973 Homsy G.M. — MS5 11133 Hong Y. — SM8 12391 Hopkins M.M. — FM18 10983 Horanyi S. — FM19 10931 Horimoto H. — FSM1 10294 Hornowski T. — SM15 12567 Hornych P. — SM18 12634 Horst E. — SM25 12026 Hosseini-Godarzi A. — SM9 10646 Hosseini-Tehrani P. — SM9 10646 Howle L. — FM12 12617 Howlin C.P. — FM26 11468 Hoyer K.W. — FM25 12821

405

Author Index Hrenya C.M. — FM17 12073 Hribersek M. — FSM4 10826 Hu X.X. — SM19 12224 Hu X. — FM24 11802 Huang H. — FM26 11126 FM4 12894 Huang I.-D. — SM1 11193 Huang P. — MS4 11604 Huang R.F. — FM11 11043 Huang R. — MS3 10699 Huang X. — FM4 11165 Huang Y. — MS3 12314 Huerre P. — FM11 12305 FM13 12425 Hufenbach W. — SM13 12576 Hughes G. — MS6 11499 Hughes T.J.R. — SL6 13003 (p. 153) Hulbert G.M. — SM11 12772 Hulin J.-P. — FM20 12092 Hulsbaek L. — SM22 12297 Hulsenberg D. — FM19 11906 Hunt G.W. — SM22 12688 Hunter J.K. — FM5 11730 Huppert H.E. — FM17 10524 FM21 11320 FM21 11328 FM9 10366 Hussein M.I. — SM11 12772 Hutchinson J.W. — MS3 12386 Hutter K. — FSM6 12040 Huu Nam T. — SM1 12844 Hwang I.G. — FM21 13008 Hwang K.-C. — MS1 12016 MS3 12314 Hwang P.A. — FM26 11230 Hwang W.-S. — SM24 11858

I

afrati A. — FM8 12547 Ignaczak J. — SM13 10347 Ilison O. — SM11 12568 Im S.H. — MS3 10699 Im S. — SM1 11817 Imai K. — SM15 10790 SM18 11295 Imanishi E. — SM17 12537 Imielowski S. — SM22 12931 Indinger T. — FM24 12564 Inoue T. — SM14 10434 Insperger T. — FSM2 10252 Ionita A. — SM13 11014 Iooss G. — FM26 11290 Ippolito M. — FSM6 12334 Irannejad R.H. — SM2 10718 Irgens F. — FM9 12270 Irschik H. — MS1 10784 SM17 12029 Isermann R. — SM16 12758 Ishihara M. — MS1 11857

Ishii K. — FM25 12041 Isoda H. — FM1 12522 Itihara M. — SM17 12582 Ivanov A.V. — FM2 10275 Ivanov I.B. — FM8 11669 Ivanov M. — FM5 10531 Ivanov T.P. — SM6 10972 Iwai K. — FM15 10860 Iwankiewicz R. — FSM2 12972 Iwnicki S.D. — SM26 10909 Izawa S. — FM11 12491

Jacqmin D. — FM4 10753

Jacques N. — SM18 11309 Jacquin L. — FM11 12244 Jaeger A. — SM27 11981 Jakobsen B. — FM23 10247 Jakubiak B. — FM9 12592 James C.D. — FM4 10753 James R.D. — MS1 10722 Jang G.-W. — SM24 10048 Jankowski R. — SM25 12659 Janour Z. — FM9 10467 Jansen E.L. — SM19 12346 Jarza A. — FM9 12266 Jarz¸ebowski A. — SM20 12848 Jasi´ n ´ ska M. — FM22 12415 Jasi´ n ´ ski M. — SM1 11389 Jasiuk I.M. — SM6 12189 Jaszczur M. — FM22 12610 Javaitis I. — MS2 11102 Jemiolo S. — MS2 12100 Jenkins A.D. — MS6 12463 Jensen H.M. — MS3 11070 (p. 165) Jensen J.S. — SM24 12222 Jensen O.E. — FM14 12155 Jeon W.-P. — FM11 11636 Jiang H. — SM24 12789 Jiang J.-B. — FM24 11488 Jiang L.J. — FM22 12209 Jie M. — SM4 10961 Jinnouchi Y. — SM7 10691 Joensson P.-A. — SM26 11795 John A. — SM1 12562 Johnson M.E. — FSM1 12497 Jolivet L. — SM12 12751 Joly P. — SM2 12382 Jones A.S. — MS1 13011 MS1 13010 Jones N. — SM5 11107 Jones R.B. — FM18 10565 Jones W.P. — FM3 11914 Jop P. — FM17 11775 Jordan P. — FM5 13016 Joseph D.D. — FM20 10843 Joseph P. — MS4 12673 Josserand C. — FM8 11614

406 Joubert P. — FM7 12220 Jullien C. — MS4 12536 Jun S. — SM1 11817 Jung J.-Y. — MS1 10985 Jung S. — SL18 10495 Juric D. — FM8 11979

Kabouya N. — FM13 12766

Kachanov Y.S. — FM24 12675 FM2 10275 Kaczmarek M. — SM15 12566 SM15 12567 SM15 12577 Kaczy´ n ´ ski A. — SM9 11980 Kagemoto H. — FSM4 11758 Kaiser I. — SM25 12322 Kaliszky S. — SM24 12573 Kalliadasis S. — FM14 10220 Kallivokas L.F. — FSM1 12691 FSM1 12717 Kaltayev A. — FM3 11031 Kam Liu W. — SM1 12167 Kambe T. — FM23 11166 Kameyama T. — FM15 10860 Kan K. — FM14 11739 Kanarska Y. — MS6 12408 Kanaun S.K. — SM11 11989 Kaneda M. — FM7 13022 Kaneko Y. — FM17 11876 Kang L.-C. — SM1 10270 Kanno Y. — SM2 12646 Kapania R.K. — FSM7 12752 Kapla´ n ´ ski F.B. — FM25 12032 Kappl K. — SM27 11981 Karagiozova D. — SM5 11107 Karaiwa M. — SM13 11426 SM9 11425 Karajan N. — SM15 11194 Karakasidis T.E. — FSM2 12482 Karapetsas G. — FM18 12859 Karas M.S. — SM1 12340 Karcher C. — FM19 12290 Karihaloo B.L. — FSM6 11806 Karpenko E.E. — SM25 10470 Karpov E.G. — SM1 12167 Kashtalyan M. — SM9 10824 Kasperski G. — FM15 12636 Kataoka T. — FM26 11746 Katsuyama T. — FSM2 11535 Kawamoto A. — SM24 12302 Kawazoe H. — FM23 12484 Kazakevitch M.I. — SM25 12632 Kazemzadeh-Parsi M.J. — FM9 10733 Ka´ ´zmierczak B. — MS2 10838 Ke F. — SM7 11764 Keane R.J. — FM22 12545 Keer L.M. — SM2 11215 Kent E.F. — FM18 11915

ICTAM04 Keppens R. — FM19 11429 Kere P. — SM13 10667 Kerr R.M. — FM24 12430 Kerschen E.J. — FSM1 12477 Keunings R. — SL7 10512 Khabakhpasheva T.I. — FSM4 10911 Kharif C. — FM26 11417 Khasanov M.M. — FM12 10548 Khelidj A. — SM4 12749 Kheradvar A. — FM1 12784 Khoei R.A. — SM2 10718 Khom’yak T.V. — SM17 10779 Khoo B.C. — FM24 11802 Khoshravan M.R. — SM13 10189 Khotyanovsky D.V. — FM6 11485 Khotyanovsky D. — FM5 10531 Khusid B. — FM16 10361 FM4 10753 Khusnutdinova K.R. — SM11 11242 Kiger K.T. — FM1 11900 Kim B.-K. — SM27 11827 Kim J.-H. — MS1 10985 Kim J. — FM11 11636 FM11 12013 FM19 11637 Kim K.-S. — MS3 12778 SM8 12034 Kim Y.-M. — FM1 12655 Kim Y.Y. — FSM7 12752 SM24 10048 Kimura Y. — FM23 11660 Kinell M. — FM5 10789 King G.P. — FM22 12545 King J.R. — FM14 12155 King M.P. — FM7 10438 King R. — FM11 12624 Kireenkov A.A. — SM2 11778 Kishino Y. — SM20 12462 Kit E. — FM26 10245 Kitagawa H. — MS3 11122 Kityk A. — FM8 11462 Kiyono K. — FSM2 11535 Klapp J. — FM6 10857 Klarbring A. — MS2 10670 SM2 10686 Kleeorin N. — FM10 12070 FM7 11053 Klepaczko J.R. — SM11 11360 SM14 10046 SM18 10734 Klingbeil D. — SM4 10865 Kloker M.J. — FM13 12381 Kluwick A. — FM2 11083 FM2 11084 FM5 10789 Knap J. — SM1 11154 SM9 12171 Knight D. — FM5 11560 Kobayashi S. — FM5 11864 Kobiera A. — FM3 12843

407

Author Index Koca´ n ´ da D. — SM8 12914 Kocourek V. — FM19 12290 Koenderinck G.H. — MS2 10709 (p. 355) Koikari S. — FM23 11660 Kolandavel M.K. — FM1 11851 Kolcavova Sirkova B. — FSM3 11247 Kolesnikov Y. — FM19 11120 FM19 12090 Kolk K. — SM9 12044 Kolmychkov V.V. — FM15 11893 Komarova V.Y. — FM2 10275 Kondic L. — FM17 12399 Kondo D. — SM15 11941 Kondrachuk A.V. — MS2 12059 Konieczny P. — FM11 12602 Kononov Y.N. — SM17 10779 Kopiev V.F. — FM11 12156 Koplik J. — FM16 10361 Korczyk P.M. — FM10 12855 Korepanov V.V. — SM6 12289 Kornev K.G. — FM4 10140 Korvink J.G. — SM24 11316 Koseki T. — SM17 12582 Kosi´ n ´ ski W. — SM11 11853 Kositsyn A. — FSM7 11724 Kothe D.B. — FM8 12163 Kotucha G. — SM24 12074 Koudella C. — MS6 12601 Koumoutsakos P. — FM24 11256 FM6 12583 Kouznetsova V. — FSM6 11830 SM14 11395 Kovacs F. — SM1 11482 Kovaleva A. — SM3 11265 Kovaleva L. — FM12 12528 Kowalczyk P. — SM24 12679 Kowalczyk P. — FSM2 11363 Kowalczyk P.G. — FM6 11763 Kowalczyk-Gajewska K. — MS2 11808 SM6 10442 Kowalewski T.A. — FM10 12855 FM14 11477 FM21 11331 FM4 11184 FM7 11160 FM7 12173 FM8 12873 FM9 10576 Kowalewski Z.L. — SM27 11957 Kowalski S.J. — SM15 13006 Kozien M.S. — SM25 12511 Kozlov A.A. — FM7 12267 Kozlov V.V. — FM2 11339 Krasnikovs A. — SM23 10242 Krasnopolskaya T.S. — FM22 12257 Krasovskyy V.L. — SM19 11815 Krauklis A.V. — FM10 12213 Krause E. — FM25 12690 Kravtsova M.A. — FM2 11780 Krawczyk J. — FM9 12592

Kraynik A.M. — FSM5 10597 Krein A. — FM10 12070 Kreuzer E.J. — SL8 10544 (p. 173) FSM4 11912 Krieger U. — FM19 11906 Kristiansen O. — FM26 11417 Krivtsov A.M. — SM11 12481 Kroeger M. — SM2 12086 Kruszka L. — SM20 10374 Kruyt N.P. — SM20 10866 Kruzelecki J. — SM24 11634 Kubacki S. — FM6 12076 Kubair D.V. — SM10 11177 Kubik J. — SM15 12566 SM15 12577 Kucaba-Pi¸etal A. — MS4 10971 Kuczma M.S. — MS1 10136 Kudela H. — FM25 12836 Kudryavtsev A.N. — FM6 11485 Kudryavtsev A. — FM5 10531 Kuhl A. — FM3 12745 Kuhl E. — SM12 11277 SM1 11211 SM1 12924 Kuhn G. — SM9 12044 Kuilekov M. — FM19 12605 Kuistiala A. — SM4 10960 Kujawski D. — SM9 10334 Kulesh M.A. — SM6 12289 Kullander F. — FSM1 12533 Kulvietis G. — SM16 10900 Kumagai T. — FM8 11299 Kumar A. — FM4 10753 Kuna M. — SM9 11045 Kuna-Ciska H. — SM4 11796 Kurashige M. — SM15 10790 SM18 11295 Kurkin A.A. — FM9 11433 Kurnik W. — SM22 12931 SM22 12934 Kuroda M. — FM11 12491 Kurowski M. — FM25 12275 Kurowski P. — SM7 12799 Kurpa L.V. — SM19 12350 Kursa M. — SM24 11760 Kurzydlowski K.J. — SM7 12666 Kurzyna J. — FSM2 12927 Kus W. — SM24 11137 Kwon Y.-I. — FM15 10363 Kyriakides S. — FSM5 11605 SM22 11600

Llorca J. — SM13 11080

Labiausse V. — FSM5 12578 Labuz J.F. — SM20 11845 Lac E. — MS2 11961 Lacaze L. — FM13 11396 Lackner R. — SM27 11981

408 Lacor C. — FM3 11888 Ladd A.J.C. — FM16 11418 Ladd T. — FM12 11994 Ladev´ ´ eze P.J. — SL9 10508 (p. 187) SM1 11769 SM25 11246 Lagha M. — FM13 11647 Lamanna G. — FM10 10445 Lambros J. — SM10 11177 Lampis M. — MS4 10723 Landa M. — MS1 12637 Lane-Serﬀ G.F. — FM9 11182 Lange C. — SM2 12773 Lange U. — FM8 12388 Langevin D. — FSM5 12212 Langkamp A. — SM13 12576 Langre E. de — FSM4 11801 Lanos C. — SM4 10011 Larecki W. — FSM6 11237 Larsson R. — SM15 12518 Larue de Tournemine A. — FM20 12383 Lathrop D.P. — FM19 11681 Lauga E. — FM8 11330 Lavinskaya E.I. — FM10 12213 Lavrenteva O.M. — FM8 11250 Le Bars M. — FM7 12126 Le Bris C. — FSM6 12078 FSM6 12336 Le Dizes S. — FM13 11396 FM13 12509 Le Gal P. — FM13 11396 FM24 10506 Le Grognec P. — SM22 11962 Le Quere P. — FM7 12220 FM8 11979 Le van A. — SM22 11962 Leal G.L. — FM8 11289 Lebensohn R.A. — SM18 11005 Leblanc S. — FM13 10525 Leblond J.-B. — SM4 12913 SM9 12393 Lecomte-Beckers J. — SM1 11298 Lee C. — FM6 11569 Lee D.-H. — SM24 11858 Lee D. — FM11 11636 Lee H.T. — FSM5 12851 Lee J.-W. — FM6 11569 Lee J. — FM16 11418 Lee S.-J. — FM1 12655 Lee S.M. — FM9 12738 Lee W.-S. — SM27 11827 Leﬁk M. — SM13 12970 Legarth B.N. — SM13 11794 Legendre D. — FM8 11911 Legoll F. — FSM6 12336 Leine R.I. — FSM2 11183 Leiner W. — FM7 12173

ICTAM04 Lekszycki T. — MS2 12315 Lele S.K. — FSM1 11561 Lemak M. — SM17 11375 Lenci S. — FSM2 12129 Lengyel A. — SM1 12101 Lenormand R. — FM12 10234 Lentzen S. — MS1 11752 Leonard A. — FM6 11937 Leonardi E. — FM14 11477 FM15 11386 Leong S.S. — FM14 11477 Leorat J. — FM19 11621 Lerbet J. — SM17 11674 Leroy Y.M. — SM12 12751 Lesieur M. — SL10 10731 (p. 203) Letser Y.A. — SM2 11933 Leung R.C.K. — FM6 10163 Leungvichcharoen S. — SM11 12514 Levin V.M. — SM11 11989 Levitas V.I. — SM14 11325 Levitsky S.P. — FM4 11648 Levy Y. — FM3 12199 FM3 12348 Lewandowski J. — FSM1 12555 Lewi´ n ´ski T. — SM24 10052 SM24 11760 Lewis D.M. — FM1 11908 Lewis S.R. — MS6 12494 Lexcellent C. — SM14 12225 Li H. — SM9 10676 Li J. — FM2 10103 FSM7 12752 SM25 10166 SM4 12036 Li K. — SM7 10492 Li Q. — SM9 11934 Li X.M. — FM6 10163 Li Y. — MS6 12194 Liakopoulos A.B. — FSM2 12482 Liang B. — SM26 10909 Liao X. — FM7 12168 MS6 12169 Libersky L.D. — SM1 10829 Liechti K.M. — MS3 12132 (p. 217) Lijuan L. — SM24 12287 Lik Chan C. — FM21 11058 Lim C.W. — SM19 12224 Lim J. — FM11 12013 Lim K.-M. — SM1 12901 Lim K.-W. — FSM1 11973 Lima R. — FM8 11026 Lin C.Y. — SM25 11867 Lin J. — SM11 11099 Lin Y. — SM1 10559 Linden P.F. — MS6 10261 Lindner M. — SM2 12086 Ling Z. — SM13 12242 Lipniacki T. — FM24 11161

409

Author Index Lippermann F. — FSM5 13033 Lipson S. — FM16 11884 Lisitsin Y.V. — SM2 10342 Lisowski W. — SM7 12799 Lister J. — FM8 12388 Litewka A. — SM4 11228 Liu J.T.C. — FM13 10487 Liu T. — SM24 11532 Liu X. — SM6 11998 Liu Y.-C. — FM3 10367 Liu Y.-L. — FM24 11488 Liu Y. — SM18 11005 Lo Jacono D. — FM22 12731 FM3 10918 Loboda V.V. — SM9 12249 Lobov N.I. — FM7 12344 Loefdahl L. — FM2 11339 Loehnert S. — SM1 12534 SM1 12584 Lofdahl L. — FM10 11455 Loginov M.S. — FM24 10455 Loglisci N. — MS6 13017 Logo J. — SM24 11623 SM24 12573 Logvinova K. — FM12 12376 Lohse D. — FM10 11909 FM17 10253 FM1 12400 Loimer T. — FM12 11487 Lomunov A.K. — SM20 10374 Long A.C. — SM13 11387 Long S.G. — SM13 12496 Longere P. — SM18 11110 Longhetto A. — MS6 13017 Longmao Z. — MS3 12181 Lopes S.R.X. — SM22 12420 Lopez-Lopez E. — SM13 11176 Louge M.Y. — MS5 10714 (p. 229) Love B.M. — SM10 12594 Lozia Z. — SM26 11832 Lu J.-Z. — FSM4 11668 Lu K. — FM4 11165 Lu T.J. — FSM1 12527 FSM5 12395 Lu W. — SM12 11432 Lu X. — SM10 12968 Lube G. — FM17 10524 Lubowiecka I. — SM19 11890 Lucey A.D. — FSM4 11736 Luczak M. — SM25 11273 Luding S. — SM17 12133 Lukes V. — SM10 11671 Lukomsky V.P. — FM26 11591 MS2 12059 Lund E. — SM24 12098 SM24 12916 Lund F. — FSM1 12643 Lundberg B. — SM7 12551

Lusseyran F. — FM13 10188 Lvov G.I. — SM19 10071 Lyakh V.V. — SM6 12575 Lyly M. — SM13 10667 Lysenko V.V. — SM25 10656 Lyubimov D.V. — FM13 12280 FM7 12344 MS5 12402 Lyubimova T.P. — FM21 12278 FM7 12344 MS5 12402

L

obocki L. — FM9 12592 Lodygowski T. — SM5 12949 Lukasiewicz S.A. — SM1 11291

M

a J. — SM8 11686 Ma L. — SM10 11551 Mac Sithigh G. — SM6 12354 MacKintosh F.C. — MS2 10709 (p. 355) MacMartin D.G. — FM11 12387 Maciejewski J. — SM20 12848 Mader M. — FM1 10741 Maderich V.S. — FM26 12268 Maderich V. — MS6 12408 Maeda T. — SM11 10422 SM25 12593 Magara Y. — SM9 11425 Magariyama Y. — FM1 11441 Magatti D. — FM10 10445 Magnaudet J. — FM20 12139 FM8 11911 Mahadevan L. — MS2 11581 Maier G. — SM18 10797 Mailybaev A.A. — SM25 11536 Maj M. — SM18 11185 Majewski J. — FM6 12618 Majewski T. — SM25 12984 Majkut M.M. — FM10 11916 Majorkowska-Knap K. — SM8 11371 Makhovskaya Y. — SM2 11196 Makin V.K. — FM26 11230 Makinde O.D. — FM3 10747 Makipelto J. — SM1 12310 Makowski K. — FM19 12363 Malinowski S.P. — FM10 12855 Malinowski S. — FM9 12592 Maliwan K. — MS5 10980 Malki-Epshtein L. — FM9 10366 Maluleke G.H.-S. — SM6 12039 Mamaev I.S. — SM1 12351 Mandal P. — SM19 10233 Mandelis A. — FM10 12719 Manela A. — FM7 12850 Mang H.A. — SM11 12313 SM19 11021 Manneville P. — FM13 11647 Mansukh M. — MS1 12786 Mansur S.S. — FM6 11456

410 Manucharyan G.V. — SM25 12989 Marah D. — MS6 10261 Marchioli C. — FM20 13012 Margulies S.S. — MS2 10933 Marian J. — SM1 11154 Marie Habraken A. — SM1 11298 Marinho W. — FM6 11456 Marinova D. — SM3 11036 Markert B. — SM15 11194 Markine V.L. — SM24 11015 Marmottant P. — FM1 12400 Marquillie M. — FM13 12372 Marsavina L. — SM9 10864 Marsik F. — FM11 12258 FM2 12062 MS1 12637 Martinand D. — FM13 11186 Martins J.A.C. — SM12 12682 SM2 12646 Maruschak P.O. — SM8 11920 Marvalova B. — SM1 12844 Marze S. — FSM5 12212 Maslov B.P. — SM27 10018 Mason D.P. — SM6 12039 Masri R. — SM5 12043 Masri S.F. — MS1 13007 Massabo R. — SM13 11162 Massart T.J. — SM4 12341 Masuda M. — FSM1 10294 Matalon M. — FM3 10918 Matas J.P. — FM16 11850 Matioukevitch S.I. — SM9 10897 Matsuda H. — SM19 12224 Matsui R. — MS1 10821 Matsumoto Y. — MS2 12192 Mattioni A. — FSM6 12334 Matveyenko V.P. — SM6 12289 Matvienko A. — FM10 12719 Matysiak S.J. — SM9 11980 Matyukhin Y. — SM2 10211 Maugin G.A. — FSM3 11347 SM10 11592 SM14 11393 SM6 12434 Maurel A. — FM24 11303 FSM1 12643 Maurine P. — SM16 10553 Mavletov M. — FM12 12528 Maxworthy T. — FM13 11995 Ma´ ´ zdziarz M. — MS2 11808 SM20 11584 Mazeika D. — SM16 10900 Mazhorova O.S. — FM15 11893 Mazza E. — MS2 12064 Mazzilli C.E.N. — SM25 11494 McElwaine J.N. — FM9 12396 McFarland M.D. — SM25 10408 McGlashan S. — FSM5 12435

ICTAM04 McIntyre M.E. — MS6 10977 MS6 12157 McPhee J. — SM2 12773 McPhie M.G. — FM16 11431 Mebarki A. — SM2 12082 Medeiros M.A.F. — FM13 12381 Medyanik S.N. — SM1 12167 Meer D. van der — FM10 11909 FM17 10253 Meftah F. — SM2 12082 Meftah S. — SM14 12747 Megaridis C.M. — FM4 11603 Mehandia V. — FM16 10603 Mehshikov Y.L. — SM25 11716 Meiburg E. — FM13 11995 FM13 12425 Meijaard J.P. — SM25 11338 Meinhart C. — FM22 12761 Meironke H. — FM7 11905 Mejak G. — SM1 11334 Meleshko V.V. — FM22 12110 FM22 12257 FM22 12633 FM23 12868 FM25 12196 Melnik O.E. — FM9 11703 Melo J. — FM3 12348 Melville W.K. — MS6 12722 Men S. — FM19 12605 Mendez C. — FM8 11670 Mendonca M.T. — FM13 12381 Mendoza G. — FM6 10857 Mendrok K. — SM7 12799 Menetrier L. — MS4 12047 Menon N. — FM5 11072 Menshykov O.V. — SM9 11195 Menzel A. — FSM3 12392 Mercier J.F. — FSM1 12643 Mercier S. — SM18 11309 Mergheim J. — SM1 11211 Merodio J. — SM12 11705 Mestel J.A. — FM19 12979 Meunier P. — FM22 10532 Meysman F.J.R. — FM1 12063 Mezic I. — FM22 12439 FM22 12761 FSM2 12468 Mezyk A. — SM25 12770 Micciche P. — FM10 10445 Michaltsos G.T. — SM22 11172 Michalek T. — FM7 11160 Michalowski R.L. — FM17 12790 Michel U. — FM21 12365 Micunovic M. — SM18 10383 Middelburg J.J. — FM1 12063 Miehe C. — FSM6 12112 FSM6 12113 FSM6 12118 SM1 12115 SM4 12116 Miettinen A. — SM2 10459 Mikhalchenko G. — SM26 11589

411

Author Index Mikhayalov D.N. — FM12 11115 Mikkelsen R. — FM17 10253 Millet C. — FM13 12509 Miloh T. — FM20 12342 Min B.-Y. — FM6 11569 Minc N. — FM4 11209 Minier J.-P. — FM24 10564 Miozzi M. — FM7 11673 Miskiewicz M.A. — SM7 12666 Mitsumoji T. — FM13 12291 Miura K. — SM17 12464 Miyamoto H. — FM9 11164 Miyazaki T. — FM24 10937 MS6 12194 Mizerski K.A. — FM19 12107 Mizuguchi F. — SM25 11066 Mizuno M. — SM18 11350 Moctezuma M. — FM8 11026 Modarres-Sadeghi Y. — FSM4 11078 Moe A. — FM9 12270 Moes N. — SM1 12359 Moﬀatt H.K. — FM18 12164 FM19 11458 FSM7 10016 Moﬀatt K. — FSM6 11862 Mohan J.A. — FM5 11072 Moisy F. — FM24 11814 Mojtabi A. — MS5 10980 Mokos V.G. — SM1 10665 Molenaar D. — FM25 11964 Molinari A. — SM14 11852 SM18 11309 Moller H.T. — SM24 12916 Mondy L.A. — FM18 10983 Monetto I. — SM4 12298 Moniuk W. — FM8 12311 Monkewitz P.A. — FM13 11186 FM24 11454 FM3 10918 Monkewitz P. — FM24 12965 Moo Koh H. — SM8 12034 Morch K.A. — FM8 11999 Moreau J.-J. — SM17 11361 Moreau R. — FM15 10860 Morel A. — MS2 11961 Moreno M.J. — FM20 11189 Morgan H. — FM4 12437 Morita C. — SM19 12224 Morize C. — FM24 11814 Morland L.W. — FM18 10297 Moroni M. — FM7 11673 Morozov N.F. — SM9 10897 Morris J.F. — FM16 11850 Morro A. — SM11 11596 Morzy´ n ´ski M. — FM11 12624 FM11 12736 FM24 11454 SM24 12413 Mouaze D. — FSM4 12557 Moulia B. — FSM4 11801

Moulin F. — MS6 11829 Mounajed G. — SM1 12727 Mousavi S. — SM7 12551 Movchan A.B. — FSM1 11896 Movchan N.V. — FSM1 11896 Mr´ o ´z Z. — SM18 10064 SM24 11349 SM2 11074 Muc A. — MS3 12251 SM13 11457 SM13 12560 Mueller A. — SM9 11484 Mueller U. — FM19 10931 Mullarney J. — MS6 11499 Muller-Slany H.H. — SM1 10423 Murai M. — FSM4 11758 Musalimov V.M. — SM2 10342 Muth B. — SM17 12133 Mutschke G. — FM11 12068 Myagkov N.N. — SM11 10426

Na S.-W. — FSM1 12691

FSM1 12717 ¨ G. — FM16 11431 Nagele FM18 11300 Nagata M. — FM13 12291 Nagib H. — FM24 12965 Nagy P. — MS5 12651 Nait Abdelaziz M. — SM18 12339 Nakamura M. — FM1 12522 Nakano H. — MS6 11152 Nakas A.A. — SM25 11082 Nakata K. — FM1 11441 Nakatani A. — MS3 11122 Nalepka K. — MS3 11444 Nalepka P. — MS3 11444 Narayanan S. — MS1 12147 Narita F. — SM13 11426 SM13 11610 SM9 11425 Naumavicius R. — SM16 12546 Naumenko K. — SM27 11368 Nava A. — MS2 12064 Navarro C. — SM8 10698 Nayak H.V. — FM16 11991 Nazarenko L. — SM13 10555 Needleman A. — SM1 12737 SM2 11987 SM9 10910 Neel M.-C. — FM12 12376 Neethling S.J. — FSM5 12851 Neimark A.V. — FM4 10140 Neishtadt A. — FM22 12439 SL11 10551 (p. 241) Neitzel G.P. — MS5 12651 Nepomnyashchy A.A. — FM14 10543 Nerinckx K. — FM6 11602 Nesteruk I. — FM2 11017 Netto T.A. — SM22 11600 Newsom R.K. — MS6 10513 Ngan A.H.W. — FSM5 11663

412 Nguyen A.V. — FM15 11903 Nguyen Q.P. — FSM5 12986 Nicolas A. — MS2 12811 (p. 329) Nicolas M. — FM16 11749 FM17 11751 Nicolleau F. — FM22 10130 Nigmatulin R.I. — FM8 12190 Nihei T. — FM25 12041 Nijmeijer H. — SM25 11783 Nikiﬁrov S. — FM5 10531 Nikiforovich E.I. — FM25 12196 Nikitin N.V. — FM15 11929 Nikitin S.A. — FM15 11929 FSM7 11869 Nikolaevskiy V.N. — FM12 11109 Nikrityuk P.A. — FM21 12365 Niordson C.F. — SM18 11790 Nir A. — FM8 11250 Nishimura J. — SM3 10879 Nishimura M. — FM1 11441 Nishimura T. — SM10 12461 SM17 12464 Niss K. — FM23 10247 Niziol J. — SM25 12511 Noack B.R. — FM11 12624 FM11 12736 FM24 11454 Noda N. — MS1 11857 Nogarede B. — FM11 12602 Noguchi H. — SM1 12177 SM22 10692 Nohguchi Y. — FSM7 12488 Nore C. — FM13 10640 Norman J.T. — FM16 11991 Norris A.N. — SM25 12404 Noskowicz S.H. — FM17 12308 Nott P.R. — FM16 10603 Nouar C. — FM13 12766 Nouy A. — SM1 11769 Novak V. — MS1 12637 Novosiadliy V.A. — FM7 13005 Nowacki W.K. — SM14 10046 SM14 11308 Nowak A.J. — FM1 12409 Nowak M. — SM24 12413 Nowak Z. — SM19 12328 SM4 12333 Nycander J. — MS6 12210

Oberste-Brandenburg C. —

SM14 12121 Obrecht H. — SM22 12084 Oertel H. Jr. — FM5 11864 Ogasawara N. — SM18 10652 Ogden R.W. — FSM3 11227 SM12 11705 SL12 10143 (p. 263) Ohkitani K. — FM23 10938 Ohl C.-D. — FM8 11999

ICTAM04 Ohno N. — SM1 12177 SM22 10692 Okada M. — SM17 11700 Okandan M. — FM4 10753 Okkels F. — MS4 12047 MS4 12130 MS4 12536 Okulov V.L. — FM13 13019 Okumura D. — SM1 12177 SM22 10692 Olascoaga M.J. — MS6 12776 Olhoﬀ N. — SM24 10433 Oliferuk W. — SM18 11185 Olshevsky A.A. — SM2 12559 SM2 12559 Olsson T. — MS2 10670 Omang M. — FM19 11809 Onck P. — MS2 10627 Ong R. — MS1 11685 Ootao Y. — SM10 12465 Orantek P. — SM1 12562 Orlov S.V. — SM2 10342 Oron A. — FM14 10543 Ortiz M. — FSM6 11471 SM1 11154 SM9 12171 Osipov M.N. — SM7 11556 Osipov V. — FSM1 11508 Osiptsov A.A. — FM9 11703 Ospennikov N.A. — FM7 12344 Ostachowicz W.M. — MS1 10680 (p. 275) Ostoja-Starzewski M. — SM11 11682 Ostrowska-Maciejewska J. — SM6 10442 Ostrowski Z. — SM24 11137 Otheguy P. — MS6 12294 Ottino J.M. — FM22 11293 Ovsyannikov S.V. — SM7 12272 Ozawa H. — MS6 11732 Ozoe H. — FM7 12173 FM7 13022

Pagneux V. — FM24 12740

Paidoussis M.P. — FSM4 11078 Pakiela Z. — SM7 12666 Pakleza J. — FM8 12873 Pakula M. — SM15 12566 SM15 12577 Palaniappan D. — FM18 12981 Pamplona D.C. — SM22 12420 Pandolﬁ A. — SM9 12171 Pantelyat M. — SM2 10211 Papageorgiou D. — FM26 12715 Papas P. — FM24 11454 FM3 10918 Papoulia K. — SM1 12458 Parau E. — FM26 11345 Park J. — FM19 11637

413

Author Index Park N. — FM17 12790 Park S.-J. — SM17 10893 Parland H. — SM2 10459 Parnes R. — MS3 12432 Parra M.T. — FM8 11670 Parsons A.T. — FM2 12320 Pasero E. — FM24 11151 Pasol L. — FM16 10248 Pasqualino I.P. — SM1 12607 Passarel W. — SM24 10594 Pastoor M. — FM11 12624 Patterson M.D. — FM25 12051 FM9 12396 Paulino G.H. — SM10 12665 Paulino G. — SM10 10135 Pavlidis M. — FM18 12859 Pavlovskaia E.E. — FSM2 11302 SM11 12481 Pavlovskaia E. — SM25 10470 Pawlowski P. — MS1 13015 Payne D.A. — MS1 11685 P´ a ´czelt I. — SM2 11074 Peacock T. — FM26 11199 Pecquet E. — SM1 11298 Pedersen E.M. — FM1 11851 Pedersen N.L. — SM24 10669 SM24 11936 Pedersen P. — SM24 10669 Pedersen S.L. — SM17 12569 Peerlings R.H.J. — SM4 11415 SM4 12341 Pegushin A.G. — SM11 12203 Pellegrino S. — SM22 11097 Peradzy´ n ´ski Z. — FM19 12363 FSM2 12927 MS4 10971 Perelmuter M.N. — MS3 11599 Perov V. — MS6 11942 Perrin B. — FM17 13029 FM20 12092 Perzyna P. — SM5 12949 Petermann M. — FM22 12610 Petersen R. — FM23 10247 Petit S. — SM13 11180 Petrova V.E. — SM9 10579 Petryk H. — SM12 10996 Pettermann H.E. — SM13 12321 P¸e¸cherski R.B. — MS3 11444 SM18 10064 Pfeiﬀer F.G. — SM16 10809 Phedina M.E. — SM4 11779 Philippidis T. — SM13 12818 Phillips O.M. — FM9 10366 Phillips W.R.C. — FM26 11179 Photiadis D.M. — SM25 12404 Picandet V. — SM4 12749 Picciotto M. — FM20 13012 Pichler B. — SM11 12313

Pichler U. — MS1 10784 Piech´ ´ or K. — MS2 10838 Pieczyska E.A. — SM14 11308 Piedboeuf J.-C. — SM2 12773 Piekarski J. — MS2 11808 Pienkowska I.T. — FM18 11922 Pierard O. — SM4 11886 SM18 13038 Pierre L. — SM4 12277 Pietraszkiewicz W. — SM19 10287 Piiroinen P. — FSM2 11363 Pijaudier-Cabot G. — SM4 12749 Pilgun G.V. — SM19 12350 Pinto da Costa A. — SM12 12682 SM2 12646 Piotrowski Z. — FM9 12592 Pivovarov M. — SM25 11699 Placidi L. — FSM6 12040 Plamenevskii B.A. — SM9 10897 Plotnikov P. — FM26 11290 Plourabou´ ´e F. — FM12 11343 Plourabou F. — FM22 12731 Plunian F. — FM19 12120 Pocheau A. — FM21 12543 FM22 11190 Pocwierz M. — FM6 12565 Podgorski T. — FM1 10741 Pogorelov D. — SM17 10893 SM17 11640 SM17 11688 SM26 11589 Pohorecki R. — FM8 12311 Polach P. — SM17 10647 Polezhaev V.I. — FM15 11929 FSM7 11869 Poloukhina O.E. — FM9 11433 Polyakov N.V. — SM25 11716 Polyzos D. — SM1 12845 Pommier S. — SM8 12228 Poncet S. — FM24 10506 Ponta F. — FM25 11676 Ponte Casta˜ n ˜ eda P. — SM18 11005 Ponter A.R.S. — SM27 11201 Pop O. — SM9 10429 Popov Y.P. — FM15 11893 Popp K. — SM25 12322 SM2 12086 Pouget J. — SM10 12093 Pouliquen O. — FM16 11749 FM17 11751 FM17 11775 Powers T.R. — MS2 11866 Pozorski J. — FM24 10564 Pozorski Z. — SM24 12628 Pozrikidis C. — FM18 10766 Prat M. — FM12 11343 Preumont A. — SL13 10930 (p. 287) Price W.G. — FM2 12320 FSM4 11833 SM25 11834

414 Princevac M. — FM9 12738 Prioris S. — FM5 10984 Proczek M. — FM21 11364 Prokunin A.N. — FM18 11460 Prosperetti A. — FM10 11909 Prunet-Foch B. — FM4 11954 Pukhnachov V.V. — MS5 11757 Pumir A. — FM22 12048 Purini R. — MS6 13017 Putelat T. — SM2 12563 Putin G.F. — FM7 10538 FM7 12267 Puzzi S. — FSM6 11224 Py C. — FSM4 11801 Pyryev Y. — SM2 11481 Pyrz R. — SM13 10244 SM13 12489

Q

iu X. — FM24 11488 Qiu Z. — FM4 10753 Qu J. — SM8 11686 Querin O.M. — SM24 11623 Quey R. — SM14 12421 Quilliet C. — FSM5 11095 Quintard M. — FM12 10234

Rabaud M. — FM17 13029

FM24 11814 Radkowski S. — SM25 12977 Radler K.-H. — FM19 12120 Rafai S. — FM8 13025 Raftoyiannis I.G. — SM22 11172 Raghu Prasad B.K. — SM9 10419 Rajasekhar G.P. — FM17 11876 Rajchenbach J. — FM17 10959 Ramachandra L.S. — SM13 12531 Ramaswamy A. — SM3 10098 Rambod E. — FM1 12784 Ramm E. — SM4 11414 Ramos A. — FM4 12437 Randles P.W. — SM1 10829 Rao R.R. — FM18 10983 Raszillier H. — FM14 11413 Ravasoo A. — SM11 11620 Ravi-Chandar K. — SM9 12378 Razi Y.P. — MS5 10980 Read P.L. — FM22 12545 MS6 12494 Reardon P.T. — FM16 11725 Rebow M. — FM21 11364 Redelsperger J.-L. — MS6 10513 Rega G. — FSM2 12129 Regnier V. — FM15 12591 Regucki P. — FM25 12836 Reinaud J.N. — MS6 11336 Reinelt D.A. — FSM5 10597 Reinl A. — FM4 12254

ICTAM04 Rejniak K.A. — MS2 10689 Rekik A. — SM13 11507 Ren M. — FM7 11251 Ren W. — SM7 10492 Ren Z. — FSM4 10826 Renotte A. — FM11 12602 Resagk C. — FM19 12605 Resseguier T. de — SM18 11110 Rethore J. — SM1 11797 Reusch F. — SM4 10865 Rhines P.B. — SL14 12781 Ribeiro P. — SM25 12759 Ricci S. — SM4 11406 Richard P. — FM17 12883 Richard T. — SM25 12535 Richiardone R. — MS6 13017 Richterova J. — FSM3 11247 Ricken T. — SM15 10198 Ricoeur A. — SM9 11045 Riedel J.J. — SM20 11845 Rill G. — SM26 11672 Rindt C.C.M. — FM7 11251 Ringgaard S. — FM1 11851 Rinoshika A. — FM24 12004 Riou H. — SM25 11246 Ris V.V. — FM7 12730 Risbet M. — SM8 12228 Risso F. — FM20 12139 Rochinha F.A. — SM10 13018 Rodi W. — FM6 11348 Rodriguez M.A. — FM8 11670 Rodriguez-Ramos R. — SM13 11176 Rodzewicz M. — SM8 11371 Rogachevskii I. — FM10 12070 FM7 11053 Rohan E. — SM10 11671 Roig V. — FM20 12383 Rojiani K.B. — FSM7 12752 Rosakis A.J. — SM2 11987 Rosenthal B. — SM22 12084 Rossky P.J. — MS3 12132 (p. 217) Rothenburg L. — SM20 10866 Rother M.A. — FM8 12329 Roumi F. — SM13 10483 Roux S. — SM4 12021 Roy A. — SM18 11820 Roy D. — SM25 10750 Rozhkov A. — FM4 11954 Rozvany G.I.N. — SM24 11623 Ruan H.H. — SM5 11051 Ruban A.I. — FM2 11397 Rubin M.B. — SM1 12584 Rubinstein A.A. — SM9 11626 Rudi Y.A. — FM25 12032 Rudnyi E.B. — SM24 11316 Rueberg T. — SM15 11119 Ruimy C. — SM6 11901

415

Author Index Ruith M.R. — FM13 11995 Ruith M. — FM13 12425 Ruo A.C. — FM9 11294 Ruoﬀ R.S. — MS3 11594 (p. 303) Rusinek A. — SM14 10046 SM18 10734 Rymuza Z. — SM2 10999 Ryzhak E.I. — SM12 10867

Saada R.A. — SM2 12082

Saanouni K. — SM4 11885 Sabina F.J. — SM13 11176 Sackmann E. — SL15 11327 (p. 313) Sadlej K. — FM16 10993 Sadowski T. — SM7 11617 Saez A. — SM9 11321 Safarik P. — FM5 12219 Safran S.A. — MS2 12811 (p. 329) Sahn D. — FM1 12784 Saida S. — FM8 11299 Saif T. — MS3 12705 Saint-Jalmes A. — FSM5 12212 Saintillan D. — FM16 12005 Saito T. — FM5 11049 Sakalo V.I. — SM2 12559 Sakiyama T. — SM19 12224 Saksala T. — SM24 10621 Salalha W. — FM4 12319 Salem A. — FM13 12766 Salin D. — FM20 12092 Salmon J.-B. — MS4 12047 Salupere A. — SM11 11813 SM11 12568 Sam C.-H. — SM1 12458 Sam Han J. — SM24 11316 Samborski S. — SM7 11617 Samsonov A.M. — SM11 10114 Samsonowicz J. — SM25 12977 Sanches C.T. — SM25 11494 Sanchez U. — FM20 11189 Sanchis A. — FSM4 12956 Sano O. — FM17 11876 FM17 11883 Sanomura Y. — SM18 11350 Santaoja K. — SM4 10960 Santiago J.G. — MS4 12961 (p. 343) Sapountzakis E.J. — SM1 10665 Sarkar S. — FM24 11116 Sarler B. — FM7 11160 Sarout J. — SM15 12713 Sartorius D. — FM24 12675 Sato K. — SM15 10790 SM19 10770 Sato M. — SM7 10691 Saussine G. — SM17 11361 Sauzay M. — SM18 12697 Savic L. — FM2 12030

Saville D.A. — SL16 10529 Savova R. — SM6 10972 Sayir M.B. — SM11 11085 Sburlati R. — SM6 12987 Scarella G. — SM2 12382 Schaeﬀer N. — FM19 12330 Schanz M. — SM15 11119 Scheichl B. — FM2 11083 Scheimberg S. — SM24 10594 Schenkel T. — FM5 11864 Schiehlen W. — SM17 11223 SM17 12134 Schjoedt-Thomsen J. — SM13 12489 Schlogl S.M. — SM14 11625 Schmid M. — FM9 12626 Schmid P. — FM11 12305 Schmidt C.F. — MS2 10709 (p. 355) Schmidt M.J. — SM20 12779 Schmidt R. — MS1 11752 Schneider L.C.R. — SM15 12903 Schnerr G.H. — FM5 12066 Schranz C. — SM19 11021 Schreﬂer B.A. — SM15 12114 Schreurs P.J.G. — SM1 12131 Schroeder W. — FM25 12690 Schuette H. — SM4 12088 Schulte H. — SM16 12417 Schulze D. — FM19 12290 Schulze T.P. — FM21 12768 Schumacher J. — FM22 10896 Schuster M. — FM6 12337 Schwarz U. — MS2 12811 Scott R.A. — SM11 12772 Sederman A.J. — FM21 11320 Sedlak P. — MS1 12637 Segev R. — SM6 11523 Seguin P. — SM17 12539 Segurado J. — SM13 11080 Seidel C. — SM25 12785 Seiden G. — FM16 11884 Seifried R. — SM17 12134 Sejnoha M. — SM23 12380 Selezov I. — SM11 11401 Self B.P. — FSM7 12642 Sellier A. — FM19 10025 Semenov Y.A. — FM8 10580 Semenova I.V. — SM11 10114 Semler C. — FSM4 11078 Semma E. — FM15 11386 Seon T. — FM20 12092 Serebryakov V. — FM5 12066 Sergeichev I.V. — SM20 10374 Sergent A. — FM7 12220 Serre E. — FM25 12080 Seto T. — SM10 12461 Seyranian A.P. — SM22 11939 Shakeri M. — SM13 13009

416 Shaqfeh E.S.G. — FM16 12005 Shardakov I.N. — SM6 12289 Sharma P. — MS3 11474 Sharp R.S. — SM26 10701 Shasholko D.I. — SM2 10999 Shchennikov V.V. — SM7 12272 Shemer L. — FM20 11655 FM26 10245 Shen C. — SM1 11580 Sheng P. — FM4 11165 Sherbaum V. — FM3 12199 Shergold O. — MS2 10988 Sheu T.W. — FM1 11187 Shevchenko I.V. — FM12 10199 Shevchenko K.V. — SM2 12559 Shevchuk V. — SM2 11697 Shevtsov I.Y. — SM24 11015 Shi D.-L. — MS3 12314 Shibutani Y. — SM23 12571 Shillor M. — SM2 11452 Shimizu M. — FM8 11299 Shimokawa S. — MS6 11732 Shimomura Y. — FSM6 11862 Shimoyama K. — FM9 11164 Shindo Y. — SM13 11426 SM13 11610 SM9 11425 Shiono Y. — SM17 12582 Shishkina E. — SM25 11055 Shklyaev S.V. — FM7 12344 Shleykel A.L. — FM7 13005 Shodja H.M. — SM13 10483 Shrira V.I. — FM13 12233 FM26 12087 Shrivastava S.C. — SM22 11653 Shtern V. — FM19 10235 FM25 10683 Shulzhenko M. — SM2 10211 Shy S.S. — FM22 12209 Shyshkanova G. — SM2 11611 Sichermann W.M. — FSM4 11912 Sideman S. — MS2 11148 Sidoroﬀ F. — SM14 12747 Sielamowicz I. — FM17 11169 Siemaszko A. — SM1 12579 Sigmund O. — SM24 11744 Silva E.C.N. — SM10 12665 Silveira Neto A. — FM6 11456 Silvestre N. — SM22 12558 Simons G. — MS3 12384 Simpson J.E. — FM25 12051 Simes F.M.F. — SM12 12682 Sinclair G.B. — FSM3 11704 Sinka C.I. — SM15 12903 SM15 12917 Sira E. — FM6 10857 Siso-Nadal F. — FM19 11124 Sittner P. — MS1 12637

ICTAM04 Skali S. — FM13 10185 Skews B.W. — FM5 11072 Skotheim J. — MS2 11581 Skrzypek J.J. — SM4 11796 Sladek J. — SM10 10836 SM19 11238 Sladek V. — SM10 10836 SM19 11238 Slobozhanin L.A. — MS5 12447 Sawianowski J.J. — FSM3 12406 Slowicka A. — FM4 12910 Smas P. — SM24 11634 Smeulders D.M.J. — SM11 11271 Smirnov E.M. — FM7 12730 Smirnov S. — FM3 11888 Smirnovskii A.A. — FM7 12730 Smith F.T. — FM2 11210 Smits A. — SM13 12818 Smyth A.W. — MS1 13007 So R.M.C. — FM6 10163 Soares M.E.S. — SM25 11494 Sobczyk K. — CL 10697 (p. 19) Sobieczky H. — FM5 10945 Sofonea M. — SM2 11452 Sokoowska R. — SM25 12984 Sokoowski J. — SM24 10052 SM24 11486 Soldati A. — FM20 13012 Sommeria J. — FM9 12410 MS6 11547 MS6 12494 MS6 12601 Soomere T. — FM9 12099 Sorensen J.N. — FM13 13019 Sorokin S. — FSM1 11337 Sotera M.R. — FSM2 12972 Sottos N.R. — MS1 11685 MS1 13011 MS1 13010 Sousa J.M.M. — FM3 12348 Souza L.C.G. — SM3 13004 Souza L.F. — FM13 12381 Sparks R.S. — FM17 10524 Spelt P.D.M. — FM8 12106 Spencer A.J.M. — SM10 12273 Spiegel E. — SL17 10158 (p. 365) Spiegl M. — SM27 11981 Squires T.M. — FM16 12452 MS4 11777 Sreenivasan K.R. — FM22 10896 Srigiriraju S.V. — MS2 11866 Staalhand J. — MS2 10670 Stachurski A. — SM4 12333 Staicu A.D. — FSM5 12373 Staquet C. — FM26 12424 Staroszczyk R. — FM18 10297 Stavroulakis G. — SM3 11036 Steblyanko P.A. — SM1 11202 Steen M. — FM12 12617

417

Author Index Steenhoven A.A. van — FM7 11251 Stefanelli R. — FSM1 12586 Stefani F. — FM19 12613 Stegmann J. — SM24 12098 Steigenberger J. — SM25 11699 Steinberg L. — FSM3 11140 Steindl A. — SM22 11816 Steinmann P. — FSM3 12392 SM12 11277 SM1 11211 SM1 12924 Steinrueck H. — FM2 12030 Stepan G. — FSM2 10252 Stepanova L.V. — SM4 11779 Stephan P. — FM14 12293 Stevanovic-Hedrih K. — SM17 10624 Stewart D.S. — MS4 12976 (p. 379) Stichel S. — SM26 11795 Stieglitz R. — FM19 10931 Stijnman M. — FM10 11909 Stiller J. — FM19 12587 Stokes Y.M. — FM8 11855 Stolpe M. — SM24 12302 Stone H.A. — FM16 12056 FM1 12423 MS4 12966 Storakers B. — SM2 11365 Stoychev G.B. — SM18 11953 Stremler M.A. — FM22 12158 Striz B. — FSM3 11247 Strozik M.D. — FM10 11916 Struzewska J. — FM9 12592 Stulov A. — SM1 10728 Stupkiewicz S. — SM2 12553 Styczek A. — FM6 12565 Su A. — FM3 10367 Sudak L. — SM6 12122 Sudhakar V. — SM25 12984 Sugano N. — SM17 12537 Sugii T. — MS2 12192 Sugimoto N. — FSM1 10294 Sugimoto T. — FM1 10132 Sugiura T. — SM11 12418 Sugiura Y. — SM25 12593 Suiker A.S.J. — SM13 12680 SM14 10648 Sukoriansky S. — MS6 11152 MS6 11942 Sullivan J.M. — FSM5 10597 Sun H. — FM10 10445 Sun L. — SM10 10135 Sun M. — FM5 11049 Sun Q. — SM14 12881 Sun Y. — SM8 11686 Sundaresan S. — FM20 12397 Sung Lee H. — SM8 12034 Suponitsky V. — FM13 11461 Suzuki T. — FM11 12387 FSM1 11561

Svendsen B. — SM4 10865 Swaters G.E. — MS6 10634 Swevers J. — SM16 11282 Swift F.J. — SM26 10909 Swinney H.L. — SL18 10495 Symens W. — SM16 11282 Sypeck D.J. — FSM5 12395 Szabo R. — FM21 11362 Szalai R. — FSM2 10252 Sze K.Y. — SM9 11568 Szekrenyes A. — SM13 11747 Szmyd J.S. — FM7 12173 FM7 13022 Szmyd J. — FM22 12610 Szojda L. — SM4 11228 Szumbarski J. — FM6 12565 Szumowski A. — FM11 11918 Szwaba R. — FM5 12997 Szymczak P. — FM12 11994 Szymczyk J.A. — FM21 11331 FM7 11905 Szyszkowski W. — SM3 12756 ´ Swito´ n ´ ski E. — SM25 12770

Tabaei A. — FM26 10762

FM26 11199 Tabeling P. — MS4 12047 MS4 12130 MS4 12536 MS4 12673 MS4 12960 Taberlet N. — FM17 12883 Tadmor G. — FM11 12624 FM24 11454 Tagawa T. — FM7 13022 Taillard K. — SM14 12225 Takada T. — MS1 10821 Takagi M. — SM14 11308 Takagi S. — MS2 12192 Takahashi N. — FM24 10937 Takayama K. — FM5 11049 Takeda T. — SM13 11610 Talagrand O. — MS6 10803 Taleb L. — SM14 12421 SM14 12747 Tamai K. — FM5 11049 Tamuzs V. — FSM7 12259 Tan L.H. — FM14 11477 Tan S.H.N. — SM13 10412 Tan V.B.C. — SM13 10412 Tang J.X. — MS2 10709 Tang M. — FM24 12652 Tang Y. — SM9 11626 Tanigawa Y. — SM10 12465 Tanizawa K. — SM17 11700 SM3 10879 Tanno H. — FM5 11049 Tardu S. — FM11 11394

418 Tarn J.-Q. — SM10 10364 Tarnai T. — SM1 11482 SM25 11384 Tartar M. — FM13 10640 Tatsumi T. — FM24 12151 Tauchert T.R. — MS1 12018 Tay A.A.O. — SM1 12901 Tay T.-E. — SM13 10412 Taya M. — SM10 12978 Tcholakova S.S. — FM8 11669 Tejchman J. — SM15 12358 Telega J.J. — FM12 11248 FSM3 11642 MS2 12100 SM13 11895 SM13 12667 SM13 12732 SM1 12562 SM2 11452 Ten Hagen T.L.M. — FM1 11900 Teodorczyk A. — FM3 12911 Terletska K.V. — SM11 11377 Tesdall A.M. — FM5 11730 Teymur M. — SM11 10891 Thermann K. — SM12 10996 Thess A. — FM19 11120 FM19 11906 FM19 12090 Thiria B. — FM11 12736 Thite S. — SM1 12441 Thivolle-Cazat E. — MS6 11547 Thomas P.J. — MS6 10261 Thomer O. — FM25 12690 Thompson A.F. — FM21 11328 Thomsen J.J. — SM25 12694 SM25 11679 Tian J. — FSM5 12395 Tian Z. — SM9 12174 Tigoiu V.M. — FSM3 11638 Tihon J. — FM14 12656 Tilgner A. — FM19 12125 Timchenko V. — FM15 11386 Ting E.C. — SM1 10270 Ting L. — FM25 11683 Tkachev P.V. — SM12 12223 Tobushi H. — MS1 10821 SM14 11308 Tokarzewski S. — FSM3 11642 SM13 11895 SM13 12667 Toland J.F. — FM26 11290 Toll S. — SM15 12518 Tomita Y. — SM12 11432 Topolnikov A.S. — FM8 12190 Toropov V. — SM24 12870 Touhei T. — SM1 10274 Tran-Cong T. — FSM5 12435 Travnicek Z. — FM11 12258 Tr¸¸ebicki J. — SM11 11682 Trevelyan P. — FM14 10220 Troger H. — SM22 11816 Trostinetsky E. — FM20 11655

ICTAM04 Trotsyuk A. — FM5 10531 Trulsen J. — FM19 11809 Trumel H. — SM18 11110 Trzci´ n ´ ski R. — FM8 12873 Trzebicki M. — SM13 12560 Tsai J.C. — FM17 12445 Tsai P.-S. — SM16 12712 Tsamopoulos J. — FM14 12858 FM18 12859 Tsekhmister Y.V. — FM26 11591 Tsemakh D. — FM8 11250 Tsepoura K.G. — SM1 12845 Tsubota K.-I. — FM1 12522 Tsubota K. — MS2 12453 Tsuru T. — SM23 12571 Tsutahara M. — FM26 11746 Tsypkin G.G. — FM21 11926 Tuck E.O. — FM8 11855 Tucker P.G. — FM24 12430 Tuckerman L.S. — FM13 10640 FM13 12431 Tuckerman L. — FM6 10921 FM7 10914 Tuliszka-Sznitko E. — FM25 12080 Tumin A. — FSM1 12477 Tur M. — SM8 10698 Turner M.R. — FM2 12241 Turska E. — SM24 12679 SM9 11910 Turteltaub S. — SM14 10648 Tutty O.R. — FM2 12320 Tvergaard V. — SM9 10910 Tylikowski A. — SM10 10045 Tyliszczak A. — FM3 11914 Tyrkiel E. — FSM2 11416

U

aliev Z. — FM3 11031 Ualiyev G. — SM17 10533 Ualiyev Z.G. — SM17 10533 Ubachs R.L.J.M. — SM1 12131 Udwdia F.E. — SM3 11163 Ugawa A. — FM17 11883 Uhl T. — SM7 12799 Uj J. — SM13 11747 Ulbrich R. — FM20 11745 FM20 12886 Ulrych B. — SM2 10211 Ungarish M. — FM16 11884 MS6 11585 Urban D. — SM5 12043 Ursem N.T.C. — FM1 11900 Ustohalova V. — SM15 10198

Vadillo J.L. — FM11 10872

Vainchtein D. — FM22 12439 Vakakis A.F. — SM25 10408 Vakhitova N.K. — FM8 12190

419

Author Index Vakhrouchev A.V. — SM1 12054 Valance A. — FM17 12883 Valdek U. — SM7 12551 Valle V. — SM9 10429 Valtorta D. — MS2 12064 Valverde J. — SM17 12071 Van Brussel H. — SM16 11282 Van Hemelrijck D. — SM13 12818 Vanden-Broeck J.-M. — FM26 11345 FM26 12715 Vanneste J. — MS6 12104 Varghese S. — FM9 12980 Varhsney K. — FM15 12349 Varyanychko M.A. — SM19 11815 Vasiljev P. — SM16 10900 Vasudeva Murthy A.S. — FM9 12980 Velasco Fuentes O.U. — FM25 12748 Vennemann P. — FM1 11900 Vereecke B. — SM4 10974 Veres I.A. — SM11 11085 Vereshchaka S.M. — SM19 10071 Veron F. — MS6 12722 Versluis M. — FM17 10253 FM1 12400 Vesenjak M. — FSM4 10826 Viallat A. — FM1 12423 Viatkina E.M. — SM1 12723 Viba J. — FSM7 12259 Vidya Sagar R. — SM9 10419 Vierendeels J. — FM6 11602 Vignes-Adler M. — FM4 11954 Villafruela J.M. — FM8 11670 Villermaux E. — FM22 10532 FM22 12326 FM8 11317 Vimmr J. — FM6 11702 Vinod N. — FM2 12631 Viovy J.-L. — FM4 11209 Vit T. — FM11 12258 FM2 12062 Vitkova V. — FM1 10741 Vlachogiannis M. — FM14 10557 Voigt L.K. — FM23 10247 Volkov K.N. — FM20 10416 Volkova V.E. — SM25 12632 Volles R. — SM1 11298 Vollmann J. — MS3 12384 Vorobev A.M. — MS5 12402 Voyiadjis G.Z. — SM18 10393 Vu-Delcarte C.D. — FM15 12636

Waarsing J.H. — MS2 11808

Waclawczyk M. — FM24 10564 Wada S. — FM1 12522 MS2 12453 Wadley H.N.G. — FSM5 12395 Wagner C. — FM8 11462 Wagner S. — FM24 12675

Wahi P. — SM25 12814 Wajnryb E. — FM16 11409 FM18 11300 FM8 11411 Walenta Z.A. — FM3 12911 FM4 12910 MS4 10971 Walker P.G. — FM1 11851 Wall D.P. — FM13 12291 Walsh A.M. — FM17 12338 Wan F.S. — FM8 12596 Wan Y. — MS1 12016 Wang C.-Y. — SM1 10270 Wang C. — FM8 12596 SM1 11580 Wang D.W. — FM26 11230 Wang H.-T. — SM9 11568 Wang H. — SM1 11720 Wang J. — SM11 11099 Wang L.-S. — SM16 12712 Wang M. — MS3 12132 (p. 217) Wang P. — SM1 11720 SM9 12174 Wang R.-Z. — SM1 10270 Wang Z. — SM9 12174 Weaire D. — MS5 10598 (p. 387) Weber J.E. — FM26 11803 Weber M. — SM16 12758 Wei M. — FSM1 10907 Wei Q. — MS6 12194 Wei Y. — MS3 11279 Weichen S. — SM10 10501 Weichert D. — SM13 12588 SM18 12058 Weidman P. — FM26 11199 FM2 11984 FSM4 11986 Weier T. — FM11 12068 Weigand B. — FM10 10445 Weir G.J. — SM2 11008 Weiss D.A. — FM4 12254 Weitsman Y.J. — SM13 11014 Weiwen L. — SM24 12287 Welch K. — SM7 12551 Wen W. — FM4 11165 Weronko J. — SM22 12934 Wesfreid J.E. — FM11 12736 Westerweel J. — FM10 11088 FM1 11900 Wheeler L. — MS3 11474 White J.M. — MS3 12132 (p. 217) White S.R. — MS1 13010 MS1 13011 Whitesides G.M. — MS4 12966 Widmann R. — SM15 10198 Wiercigroch M. — FSM2 11302 SM11 12481 SM25 10470 Wierschem A. — FM14 10642 FM14 10928 Wieteska R. — FM6 12618

420 Wi¸eckowski Z. — SM20 10231 Wiggins S. — FM22 11293 Wijeyewickrema A.C. — SM11 12514 Wijngaarden L. van — OL 10498 (p. 1) Willers B. — FM21 12365 Williams P.D. — MS6 12494 Willis J.R. — SM13 11782 SM2 12563 Wilma´ n ´ ski K. — SM11 10612 Wilson H.J. — FM16 11473 Wilson M. — FM7 10438 Wingerde A.M. van — SM13 12818 Winter R.M. — MS3 12132 (p. 217) Wi´ ´sniewski K. — SM19 12328 SM24 12679 SM9 11910 Witkowski A.S. — FM10 11916 Witkowski W. — FM3 12911 SM19 11890 Wnuk M.P. — SM9 10250 Wojciechowski J. — FM11 11918 Wojnar R. — SM13 12732 Wola´ n ´ ski P. — FM3 12745 FM3 12843 Wong Y.W. — SM22 11097 Woods A.W. — FM21 11926 Worster M.G. — FM21 11270 FM21 11320 FM21 11328 FM21 12276 Woude D. van der — FM22 12633 Woznica K. — SM18 12339 Wriggers P. — SM1 12534 SM1 12584 Wu D. — SM9 12492 Wu J. — SM4 12036 Wu L. — FM15 12591 SM10 11551 Wuest A. — FM9 12626 Wurz W. — FM24 12675 Wysocki M. — SM15 12518

Xia M. — SM7 11764

Xiao Q.Z. — FSM6 11806 Xing T.J. — FSM4 11833 SM25 11834 Xiong A.-K. — FM11 12491 Xiong P.Y. — SM25 11834 Xu H. — MS5 10714 (p. 229) Xu X. — SM7 11764

Yadav A. — FM3 12348

Yamaguchi E. — MS4 11607 Yamaguchi T. — FM1 12522 MS2 12453 Yamamoto K. — SM17 11700 SM3 10879

ICTAM04 Yamashita K. — FSM1 10294 Yamasue K. — FSM2 12014 Yamazaki Y.H. — MS6 12494 Yang J. — SM1 11580 Yang S. — FM4 11165 SM8 11737 Yang X. — FM24 10149 Yang Y. — FSM4 12521 Yano J.-I. — MS6 12140 Yao Z. — SM1 11720 Yaremchuk V.P. — FSM7 11869 Yarin A.L. — FM4 10932 Yasinskyi A. — SM2 11697 Yasniy P.V. — SM8 11920 Yau J.-D. — SM3 11287 Yazdi S.S.H. — SM25 11904 Yazykov V. — SM26 11589 Yevdokymov D.V. — MS5 10958 Yin H. — SM10 10135 Yokosawa S. — FM1 12522 Yokota K. — FM8 11299 Yoo W.-S. — SM17 10893 Yoon Y. — FM8 11289 Yoshida F. — SM24 12870 SM27 12450 Yoshimura T. — FM24 12151 Yoshinaga T. — FM14 11739 Youn S.-K. — SM24 11826 SM27 11827 Young K.Y. — SM24 12185 Young T.H. — SM25 11867 Yourdkhani A. — SM13 10189 Yu S. — SM9 11934 Yu T.X. — SM5 11051 Yuan T.H. — FM22 12209

Z

abielski L. — FM19 12979 Zachara A. — FM8 12873 Zaera R. — SM18 10734 Zahn M. — FM4 11982 Zairi F. — SM18 12339 Zaj¸ac D. — FM20 12886 Zakharov V.E. — FM26 10861 Zaleski S. — FM8 11614 FM8 11723 Zametaev V.B. — FM2 11780 Zammali C. — SM11 12763 Zampolli M. — FSM1 11373 Zangeneh M.S. — SM25 11904 Zaoui A. — SM13 11507 SM13 11805 Zardecki D. — SM26 11832 Zavala-Garay J. — MS6 12776 Zavaliangos A. — SM15 12917 ´ G. — SM1 11389 Zboinski Zeidis I. — SM25 11699 Zeilstra C. — FM17 10253

Author Index Zelinger Z. — FM9 10467 Zeman J. — SM23 12380 Zeman V. — SM25 11759 Zenit R. — FM8 11026 Zenkovskaya S.M. — FM7 13005 Zhang C. — SM10 10836 Zhang K. — FM7 12168 MS6 12169 Zhang N.-H. — FSM2 12674 Zhang R. — SM8 11686 Zhang R. — SM27 10906 Zhang W.W. — FM8 12429 Zhang X. — FM26 11680 Zhao X. — SM17 11020 Zhao Y.-P. — SM16 10510 Zheltovodov A.A. — FM24 10455 Zheng L. — SM9 12174 Zheng X.J. — MS3 12486 Zheng X. — SM6 11998 Zhou C. — SM8 12391 Zhou M. — MS3 10105 Zhou W. — SM1 12901 Zhou Y.C. — MS3 12486 SM13 12496 SM9 12492 Zhou Y. — FM24 12004 FSM4 12521 Zhu H.X. — FSM5 11292 Zilitinkevich S. — FM7 11053 Zimmermann K. — SM25 11699 Zinchenko A.Z. — FM8 12329 FM8 12741 Ziopaja K. — SM24 12628 Zi´ olkowski M. — FM19 12605 Zitha P.L.J. — FSM5 12986 Zmitrowicz A. — SM2 11613 Zouhar G. — FM21 12365 Zussman E. — FM4 10932 FM4 12319 ˙ Zochowski A. — SM24 11486 ˙ Zukowski J.M. — FM10 11916 ˙ Zyczkowski M. — SM24 12075

421

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