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.