A Historical Overview

Total Page:16

File Type:pdf, Size:1020Kb

A Historical Overview From the Hele-Shaw Experiment to Integrable Systems: A Historical Overview Alexander Vasil’ev to Bj¨orn Gustafsson on the occasion of his 60-th birthday Abstract. This paper is a historical overview of the development of the topic now commonly known as Laplacian Growth, from the original Hele-Shaw experiment to the modern treatment based on integrable systems. Mathematics Subject Classification (2000). Primary 76D7, 01A70; Secondary 30C35, 37K20. Keywords. Hele-Shaw, Polubarinova-Kochina, Kufarev, Richardson, Taylor, Saffman, Conformal Mapping, Integrable Systems. 1. Introduction One of the most influential works in fluid dynamics at the end of the 19-th century was a series of papers written by Henry Selby Hele-Shaw (1854–1941). There Hele- Shaw first described his famous cell that became a subject of deep investigation only more than 50 years later. A Hele-Shaw cell is a device for investigating two- dimensional flow of a viscous fluid in a narrow gap between two parallel plates. This cell is the simplest system in which multi-dimensional convection is present. Probably the most important characteristic of flows in such a cell is that when the Reynolds number based on gap width is sufficiently small, the Navier-Stokes equations averaged over the gap reduce to a linear relation for the velocity similar to Darcy’s law and then to a Laplace equation for the fluid pressure. Different driving mechanisms can be considered, such as surface tension or external forces (e.g., suction, injection). Through the similarity in the governing equations, Hele- Shaw flows are particularly useful for visualization of saturated flows in porous media, assuming they are slow enough to be governed by Darcy’slaw.Nowadays, Supported by the grant of the Norwegian Research Council #177355/V30, and by the European Science Foundation Research Networking Programme HCAA. 2AlexanderVasil’ev the Hele-Shaw cell is used as a powerful tool in several fields of natural sciences and engineering, in particular, soft condensed matter physics, materials science, crystal growth and, of course, fluid mechanics. The modern treatment of the Hele-Shaw evolution is based on the integrable systems and on a general theory of plane contour motion, e.g.,theL¨ownertheory.A mathematical physics perspective, through integrable systems in particular, allows us to look at Hele-Shaw evolution as at a general contour dynamics in the plane included into a dispersionless Toda hierarchy. What happened between these two events? Who contributed to this topic? This paper is my personal account on Hele-Shaw flows and on general con- tour dynamics during the XX-th century from the original Hele-Shaw experiment in 1897 to what is now commonly known as the Laplacian Growth problem, one of the most challenging problems in nonlinear physics. Therewereseveralpersons who influenced me and drew my attention to Hele-Shaw flows. One of them was Yurii Hohlov who organized in 1996 a small seminar in Moscow. Dmitri Prokhorov, Vladimir Gutlyanski˘ı, Konstantin Kornev, and me came to Moscow, and we dis- cussed together with Yurii perspectives of the conformal mapping viewpoint on the Hele-Shaw problem. It is worth mentioning that it was Hohlov who brought Kufarev’s works to the Western audience and revealed the Soviet impact to the development of Hele-Shaw flows. Another person who influencedmewasBj¨orn Gustafsson whose 60-th birthday we celebrated recently, andIverymuchappre- ciate his thorough treatment of weak solutions to the Hele-Shaw problem and his potential-theoretic approach. 10 year’s work on Hele-Shaw flows and 4 years of my collaboration with Bj¨orn resulted in the book [33] published in 2006. Of course, I would also like to mention some earlier surveys covering certain topics or certain time periods [47; 79; 107]. In 2007 Mark Mineev and B¨orn Gustafsson asked me to present a historical overview of Hele-Shaw flows at the BanffInternational Research Station (Canada) meeting. Working on that lecture I discovered many interesting and unknown (for me) facts about persons who contributed to this interesting and challenging topic, first of all about H. S. Hele-Shaw. After discussions with my colleagues at BIRS, I decided to take a risk and to share this lecture with a wider audience adding some information I was given during these discussions. IdonotpretendtocoverallaspectsofHele-Shawflows.SamHowison and Keith Gillow [25] collected more than 560 references on Hele-Shaw flows between 1897-1998. However, much more appeared during the last 10 years. A Google search reveals more than 50 000 references on this topic. My much more modest intent was to draw the reader’s attention to some interestingandbrightpersons who were at the beginning of this boom and who sometimes becameundeservedly forgotten. Among the many documents referenced in this paperIdistinguishan informative obituary of Hele-Shaw, written by H. L. Guy [35],whichIrecommend to an inquisitive reader for independent study. Iwouldliketoexpressmygratitudetomanypersonswhoinfluenced me, with whom we discussed this topic, who gave me some information, mycollaborators Fromthe Hele-ShawExperimentto IntegrableSystems 3 and co-authors. Such a list of persons would occupy the rest ofthispaperso let me keep them in my heart. Special thanks go to Linda Cummings and Bj¨orn Gustafsson for their critical reading of the final version of this manuscript. 2. Hele-Shaw and his experiment Hele-Shaw (1854–1941) was one of the most prominent engineering researchers at the edge of the XIX and XX centuries, a pioneer of tech- nical education, a great organizer, President of several engineering societies, including the Royal Institution of Mechanical Engineers, Fellow of the Royal Society, and sadly, an example of one of the many undeservedly forgotten great names in Science and Engineering. Hele–Shaw was born on 29 July 1854 at Billericay (Essex). The son of a successful solic- itor Mr Shaw, he was a very religious person, influenced by his mother from whom he adopted her family name ‘Hele’ in his early twenties. At Figure 1. H. S. Hele-Shaw the age of 17 he finished a private education and was apprenticed at the Mardyke Engineer- ing Works, Messr Roach & Leaker in Bristol. His brother PhilipE.Shaw(Lec- turer and then Professor in Physics, University College Nottingham) testifies: ‘... Hele’s life from 17 to 24 was asustainedepic:10hrspracticalwork by day followed by night classes’. Hele- Shaw applied for a 3 year Whitworth Scholarship in Bristol and he was a lead- ing candidate in the list before an exam, when the congestion of lungs happened and the effort and exposure would be dangerous. Nevertheless, he went by cab to the examination and again headed Figure 2. Hele-Shaw’s birthplace the list and got the highest award of £740. It is interesting that later in 1923 he founded the Whitworth Society. 2.1. 1876–1885 In 1876 he entered the University College Bristol (founded in1872)andin1878he was offered a position of Lecturer in Mathematics and Engineering under Professor J. F. Main. In 1880 he got a Miller Scholarship from the Institution of Civil Engineers for a paper on Small motive power. 4AlexanderVasil’ev In 1882 Main left the College and Hele-Shaw was appointed as Professor of Engineering while the Chair in Mathematics was dropped. Atthattimehe organized his first Department of Engineering at the age of 27 and became its first professor. 2.2. 1885–1904 In 1885 Hele-Shaw was invited to organize the Department of Engineering at the University College Liverpool (founded in 1881), his second department, where he served as a Professor of Engineering until 1904 when he moved to South Africa. During this period Hele-Shaw carried out his seminal experiments at University College Liverpool, designing the cell that bears his name. 2.3. 1904–1906 In 1904 Hele-Shaw became the first Pro- fessor of Civil, Mechanical and Electri- cal Engineering of the Transvaal Tech- nical Institute (founded in 1903) which then gave rise to the University of Jo- hannesburg and the University of Pre- toria. It became his third department. In 1905 he was appointed as a Principal of the Institute and an organizer of Tech- Figure 3. Transvaal Technical Insti- nical Education in the Transvaal. Hele- tute Shaw thus became one of the pioneers of technical education not only in the metropolitan area but also in the colonies. Moreover, he was an exceptional teacher and his freehand drawing always attracted special attention. He always tried to present difficult experiments in an easier way, creating new devices in order to visualize certain phenomena. 2.4. 1906–1941 Upon returning from South Africa, Hele-Shaw abandoned academic life, setting up as a consulting engineer in Westminster, concerning with development and exploitation of his own inventions. In 1920 Hele-Shaw becametheChairmanof the Educational Committee of the Institution of Mechanical Engineers, the British engineering society, founded in 1847 by the Railway ‘father’GeorgeStephenson.In 1922 Hele-Shaw became the President of the Institution of Mechanical Engineers. Hele-Shaw took a very active part in the professional and technical life of the Great Britain. He was • President of the Liverpool Engineering Society (1894); • President of the Institution of Automobile Engineers (1909); • President of the Association of Engineers in Charge (1912); Fromthe Hele-ShawExperimentto IntegrableSystems 5 Figure 4. Tay Bridge disaster • President of Section G of the British Association for the Advancement of Science (1915); • President of the Institution of Mechanical Engineers (1922); • Fellow of the Royal Society (1899). One of his greatest contributions to Technical Education wasthefoundationof ‘National Certificates’ in Mechanical Engineering. He was joint Chairman of the corresponding Committee (1920–1937). Hele-Shaw was a man of great mental and physical alertness, ofgreatenergy and of great courage. He was a self-made person and was successful and recognized during his professional life.
Recommended publications
  • Harmonic and Complex Analysis and Its Applications
    Trends in Mathematics Alexander Vasil’ev Editor Harmonic and Complex Analysis and its Applications Trends in Mathematics Trends in Mathematics is a series devoted to the publication of volumes arising from conferences and lecture series focusing on a particular topic from any area of mathematics. Its aim is to make current developments available to the community as rapidly as possible without compromise to quality and to archive these for reference. Proposals for volumes can be submitted using the Online Book Project Submission Form at our website www.birkhauser-science.com. Material submitted for publication must be screened and prepared as follows: All contributions should undergo a reviewing process similar to that carried out by journals and be checked for correct use of language which, as a rule, is English. Articles without proofs, or which do not contain any significantly new results, should be rejected. High quality survey papers, however, are welcome. We expect the organizers to deliver manuscripts in a form that is essentially ready for direct reproduction. Any version of TEX is acceptable, but the entire collection of files must be in one particular dialect of TEX and unified according to simple instructions available from Birkhäuser. Furthermore, in order to guarantee the timely appearance of the proceedings it is essential that the final version of the entire material be submitted no later than one year after the conference. For further volumes: http://www.springer.com/series/4961 Harmonic and Complex Analysis and its Applications Alexander Vasil’ev Editor Editor Alexander Vasil’ev Department of Mathematics University of Bergen Bergen Norway ISBN 978-3-319-01805-8 ISBN 978-3-319-01806-5 (eBook) DOI 10.1007/978-3-319-01806-5 Springer Cham Heidelberg New York Dordrecht London Mathematics Subject Classification (2010): 13P15, 17B68, 17B80, 30C35, 30E05, 31A05, 31B05, 42C40, 46E15, 70H06, 76D27, 81R10 c Springer International Publishing Switzerland 2014 This work is subject to copyright.
    [Show full text]
  • Mathematicians Fleeing from Nazi Germany
    Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D.
    [Show full text]
  • David Hilbert
    Complex Numbers and Colors As is our tradition, we bring you “Complex Beauties,” providing you with a look into the wonderful world of complex functions and the life and work of mathematicians who contributed to our under- standing of this field. We would like to reach a diverse audience: We present both relatively simple concepts and a few results that require some mathematical background on the part of the reader. Ne- vertheless, we hope that non-mathematicians find our “phase portraits” exciting and that the pictures convey a glimpse of the richness and beauty of complex functions. Besides the calendar team, this year we have two contributions from guest authors: Bengt Forn- berg presents radial basis functions and Jorg¨ Liesen’s contribution is about rational harmonic functi- ons and gravitational lensing. To understand the representations of these functions, it is helpful to know something about com- plex numbers. The construction of the phase portraits is based on the interpretation of complex numbers z as points in the Gaussian plane. The horizontal coordinate x of the point representing z is called the real part of z (Re z) and the vertical coordinate y of the point representing z is called the imaginary part of z (Im z); we write z = x + iy. Alternatively, the point representing z can also be given by its distance from the origin (jzj, the modulus of z) and an angle (arg z, the argument of z). The phase portrait of a complex function f (appearing in the picture on the left) arises when all points z of the domain of f are colored according to the argument (or “phase”) of the value w = f (z).
    [Show full text]
  • Emmy Noether and Bryn Mawr College Qinna Shen
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Scholarship, Research, and Creative Work at Bryn Mawr College | Bryn Mawr College... Bryn Mawr College Scholarship, Research, and Creative Work at Bryn Mawr College German Faculty Research and Scholarship German 2019 A Refugee Scholar from Nazi Germany: Emmy Noether and Bryn Mawr College Qinna Shen Let us know how access to this document benefits ouy . Follow this and additional works at: https://repository.brynmawr.edu/german_pubs Part of the German Language and Literature Commons This paper is posted at Scholarship, Research, and Creative Work at Bryn Mawr College. https://repository.brynmawr.edu/german_pubs/19 For more information, please contact [email protected]. A Refugee Scholar from Nazi Germany: Mathematician Emmy Noether at Bryn Mawr College Qinna Shen It is everywhere incumbent upon university faculties . to maintain their historic duty of welcoming scholars, irrespective of race, religion and political opinion, into academic society, of protecting them in the interest of learning and human understanding, and of conserving for the world the ability and scholarship that might otherwise disappear. —Emergency Committee in Aid of Displaced German Scholars On April 7, 1933, two months after Hitler came to power, the new Civil Service Law barred non- Aryan Germans, including university professors and researchers of Jewish descent, from working in the public sector.1 The Institute of International Education, which was founded in New York City in 1919, quickly responded by establishing the Emergency Committee in Aid of Displaced German Scholars in June 1933. The committee’s mission was to place the suddenly unemployed and imperiled academics in institutions outside of Germany.
    [Show full text]
  • Yvette Kosmann-Schwarzbach the Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E
    Sources and Studies in the History of Mathematics and Physical Sciences Yvette Kosmann-Schwarzbach The Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E. Schwarzbach ! "# $ # % "& & &%' & & & #&& %())% ) )*+*, Emmy Noether (1882–1935) (photograph courtesy of the Emmy Noether Foundation, Bar Ilan University) Yvette Kosmann-Schwarzbach The Noether Theorems Invariance and Conservation Laws in the Twentieth Century Translated by Bertram E. Schwarzbach 123 Yvette Kosmann-Schwarzbach Bertram E. Schwarzbach Centre de Mathématiques Laurent Schwartz (Translator) École Polytechnique 91128 Palaiseau France [email protected] "#$ %&'((*'&('&'+&(+ ("#$ %&'((*'&('&'+'(* ," &-%&'((*'&('&'+'(* $ . / 0 1 0 2 3 0 Mathematics Subject Classification (2010): 01-02, 22-03, 22E70, 49-03, 49S05, 70-03, 70H03, 70H33, 83-03 < =# 0 33 > 0 ? / 4 2 0 0 5 2 9 =# 0 33 ** $ . / $. * : ; 5 2 5 ; > 4 4 ( 4 5 5 5 0 > 0 5 24 0 0 4 / 5 0 > 0 5 200 ? 2 5 0 0 / > / 0 > 5 4 0 8 0 2 / ; 5 4 27 4 @ 0 0(5 5 =# 0 9 : Ö In memory of Yseult who liked science as well as history Preface What follows thus depends upon a combination of the methods of the formal calculus of variations and of Lie's theory of groups. Emmy Noether, 1918 This book is about a fundamental text containing two theorems and their converses which established the relation between symmetries and conservation laws for varia- tional problems. These theorems, whose importance remained obscure for decades, eventually acquired a considerable influence on the development of modern theo- retical physics, and their history is related to numerous questions in physics, in me- chanics and in mathematics.
    [Show full text]
  • Between Two Evils Mikhail Shifman
    INFERENCE / Vol. 5, No. 3 Between Two Evils Mikhail Shifman his we know: any number of great scientists were exiled to Siberia, where they were left to die during the forced to flee Nazi Germany: Albert Einstein, John Siberian winter. Collectivization resulted in the massive von Neumann, Kurt Gödel, Hans Bethe, Felix famine that struck the Ukraine and other areas of the TBloch, Max Born, James Franck, Otto Frisch, Fritz London, USSR in 1932 –1933. Indirect data provide evidence that Lise Meitner, Erwin Schrödinger, Otto Stern, Leo Szilard, the Holodomor, as it is now known, killed two to three mil- Edward Teller, Victor Weisskopf, and Eugene Wigner, a lion victims. few at first and then a flood. Most chose the UK or the USA The Holodomor, although terrible, was limited; and as their refuge. A few preferred the Soviet Union. Among Soviet citizens living beyond the Ukraine could regard the the latter were Fritz Noether, a famous mathematician and unfolding catastrophe with a sense that, if they had little to the brother of Emmy Noether, and Hans Hellmann, one of eat, they had, at least, little to fear. the founding fathers of quantum chemistry. The situation changed dramatically in the summer of Their destinations became their destinies. 1937. The Great Terror had begun. Arrests, executions, By the fall of 1932, the Nazi Party had entered fully into and the Gulag began to affect virtually every family in the German political mainstream, receiving, in the Novem- the Soviet Union. In 1937 and 1938, the slaughter grew to ber elections, 196 seats in the Reichstag.
    [Show full text]
  • Mathematicians Going East
    MATHEMATICIANS GOING EAST PASHA ZUSMANOVICH ABSTRACT. We survey emigration of mathematicians from Europe, before and during WWII, to Russia. The emigration started at the end of 1920s, the time of “Great Turn”, and accelerated in 1930s, after the introduction in Germany of the “non-Aryan laws”. Not everyone who wanted to emigrate managed to do so, and most of those who did, spent a relatively short time in Russia, being murdered, deported, or fleeing the Russian regime. After 1937, the year of “Great Purge”, only handful of emigrant mathematicians remained, and even less managed to leave a trace in the scientific milieu of their new country of residence. The last batch of emigrants came with the beginning of WWII, when people were fleeing eastwards the advancing German army. INTRODUCTION A lot is written about emigration of scientists, and mathematicians in particular, from Germany and other European countries between the two world wars. First and foremost, one should mention a de- tailed study [Sie2], concentrating mainly on emigration to US, but also briefly covering emigration to other countries; then there are a lot of papers concentrating on emigration to specific countries: [Bers], [Rein] (US), [Fl], [NK] (UK), [Ri] (both US and UK), [Sø] (Denmark), and [EI] (Turkey). In absolute figures, the largest number of mathematicians emigrated to US (over 100 by many accounts), while in relative numbers (say, proportional to the size of the population, or to the number of actively working mathematicians in the host country) the first place belongs to UK. Concerning emigration to Russia, besides separate accounts of the fate of individual mathematicians scattered over the literature (a recent interesting contribution being [O]), there is only a very brief survey in [Sie2, pp.
    [Show full text]
  • Siegmund-Schultze R. Mathematicians Fleeing from Nazi Germany
    Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D.
    [Show full text]
  • Classical and Stochastic Löwner–Kufarev Equations
    Classical and Stochastic Löwner–Kufarev Equations Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal, and Alexander Vasil’ev Abstract In this paper we present a historical and scientific account of the development of the theory of the Löwner–Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K. Löwner in 1923 to recent generalizations and stochastic versions and their relations to conformal field theory. Keywords Brownian motion • Conformal mapping • Evolution family • Inte- grable system • Kufarev • Löwner • Pommerenke • Schramm • Subordination chain Mathematics Subject Classification (2000). Primary 01A70 30C35; Secondary 17B68 70H06 81R10 Filippo Bracci () Dipartimento Di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133 Roma, Italy e-mail: [email protected] M.D. Contreras S. Díaz-Madrigal Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n 41092 Sevilla, Spain e-mail: [email protected]; [email protected] A. Vasil’ev Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway e-mail: [email protected] A. Vasil’ev (ed.), Harmonic and Complex Analysis and its Applications, 39 Trends in Mathematics, DOI 10.1007/978-3-319-01806-5__2, © Springer International Publishing Switzerland 2014 40 F. Bracci et al. 1 Introduction The Löwner theory went through several periods of its development. It was born in 1923 in the seminal paper by Löwner [165], its formation was completed in 1965 in the paper by Pommerenke [189] and was finally formulated thoroughly in his monograph [190] unifying Löwner’s ideas of semigroups and evolution equations and Kufarev’s contribution [144, 146]ofthet-parameter differentiability (in the Carathéodory kernel) of a family of conformal maps of simply connected domains ˝.t/ onto the canonical domain D, the unit disk in particular, and of PDE for subordination Löwner chains of general type.
    [Show full text]