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ISSN 1560-3547, Regular and Chaotic Dynamics, 2011, Vol. 16, No. 6, pp. 671­684. c Pleiades Publishing, Ltd., 2011.

IN MEMORIAM

HASSAN AREF (1950­2011)
Alexey V. Borisov1* , Viatcheslav V. Meleshko2** , Mark Stremler3 *** , and GertJan van Heijst4 ****
Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk 426034, Russia 2 Department of Theoretical and Applied Mechanics, Kiev National University, Volodymyrska 64, 01601 Kiev, Ukraine 3 Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA 4 Fluid Dynamics Laboratory, Department of Physics, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands
Received Novemb er 11, 2011
1

DOI: 10.1134/S1560354711060086

Hassan Aref IUTAM Symposium "150 Years of Vortex Dynamics" Copenhagen, October 2008
* ** *** ****

EEEE-

mail: mail: mail: mail:

borisov@rcd.ru meleshko@univ.kiev.ua stremler@vt.edu G.J.F.v.Heijst@tue.nl

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The world of fluid mechanics and nonlinear dynamics has suffered a great loss -- on 9th Septemb er 2011, Hassan Aref died suddenly at his home in De Land (Illinois, USA) while in his favorite chair, a cup of coffee and his laptop by his side. Thus prematurely passed away one of the most prominent present-day scientists in the field of fluid mechanics, a distinguished scholar in vortex dynamics, the creator of the paradigm of chaotic advection, an excellent teacher and a remarkable human b eing. Hassan Aref was b orn on 28 Septemb er 1950 in Alexandria (Egypt) as the son of Moustapha Aref and Jytte Adolphsen. He graduated from the University of Cop enhagen in 1975 with a Candidatus scientiarum degree (roughly the equivalent of an MS with a thesis) in physics and a minor in mathematics. It was during that time he met Susanne Aref n´ Eriksen, who was to b e his lifelong ee companion and mother of his two sons. Hassan received his PhD degree in physics, with a minor in mechanical and aerospace engineering, from Cornell University in 1980. His research advisor was Eric D. Siggia, a memb er of the US National Academy of Sciences since 2009. A recent up date of the Mathematics Genealogy Pro ject has revealed that Hassan Aref was an academic descendant of Hermann von Helmholtz, who introduced the concept of the p oint vortex, a fitting lineage given Hassan's passion for vortex dynamics. Hassan Aref started his professional academic career in the Division of Engineering at Brown University in 1980, first as an assistant professor from 1980 to 1984, and then as an associate professor from 1984 to 1985. In 1985 Hassan Aref was recruited to the faculty of the University of California, San Diego (UCSD), where in 1988 he was promoted to Professor of Fluid Mechanics (as a successor of John Miles). His app ointment there was split b etween the Department of Applied Mechanics and Engineering Science and the Institute of Geophysics and Planetary Physics. In 1992 Professor Aref b ecame Head of the Department of Theoretical and Applied Mechanics at the University of Illinois at Urbana-Champaign (UIUC), a p osition that he held until 2003. In that year he was app ointed to the p osition of Reynolds Metals Professor in the Department of Engineering Science and Mechanics at Virginia Tech, where he also served as Dean of the College of Engineering from 2003 to 2005. The College of Engineering at Virginia Tech is one of the largest programs of its kind in the USA, with more then 300 faculty, 6,000 undergraduates, 1,800 graduate students, and 200 staff in 11 departments. In addition, from 2006 to 2010 Hassan Aref held the p osition of Niels Bohr Visiting Professor at the Technical University of Denmark, one of six such visiting professorships in different fields of science filled in 2005 through an op en comp etition by the Danish National Research Foundation b etween all institutions of higher education in Denmark. In May 2011 he was awarded the doctor technices honoris causa degree from that University. The citation reads: "Outstanding and highly innovative achievements in fluid dynamics, particularly for his work on chaotic advection and vortex dynamics." Throughout his career Hassan Aref has b een committed to scientific and organizational work, with a sp ecial concern for university education and raising the level of its computerization. Hassan was a theoretician, with a keen interest in and vast knowledge of the literature in mathematics and mechanics, including the pap ers of scientists of the 19th century. Yet, he seamlessly combined his tremendous love of analytical solutions to various nonlinear equations with a vision for practical use of large- and small-scale computing. While at UCSD, he was Chief Scientist at the San Diego Sup ercomputer Center from 1989 to 1992. He was interim Chief Information Officer at UIUC from 1999­2000. In his role as dean at Virginia Tech, he was instrumental in the development of System X, a cluster sup ercomputer constructed in 2003 from 1,100 Apple PowerMac G5 computers. System X was the third sup ercomputer in history to exceed the 10 Tflops/s mark1) and the first academic computer to do so. Professor Aref was a distinguished lecturer at numerous symp osia. His oral presentations and talks were always brilliant in form and profound in content. In 1988 Hassan Aref gave the first of many named lectures, the Stanley Corrsin Lectureship at Johns Hopkins University, and he continued to lecture widely thereafter. These many lectures included the Westinghouse Distinguished Lectureship at University of Michigan in 1991; the Midwest Mechanics Seminar tour of nine institutions in 1992; the Toshiba Keio Lecture at Keio University in Japan in 1994; the Otto Laporte lecture in Washington, D.C. in 2000; and the Zhou-Pei Yuan memorial lecture in Sri Lanka in 2004. In addition, he received a constant stream of invitations to give keynote lectures at congresses, conferences, workshops and summer schools around the world.
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Hassan Aref served as editor for four academic collections in fluid mechanics. He authored over a hundred journal articles, conference proceedings and b ook chapters. A short list of his publications in archival p eer-review journals is given in App endix A; a complete list of all his scientific presentations can b e found at http://nd.ics.org.ru/doc/r/p df/1928/0. Professor Aref received several prestigious awards. In 1985, during his last year at Brown, he was one of the early recipients of the NSF Presidential Young Investigator Award for his prop osed work on "Theoretical and computational fluid mechanics". In 1988 he b ecame a Fellow of the American Physical Society (APS) "for the elucidation of chaotic motion in few-vortex problems and particle advection, and for the development of numerical methods based on many-vortex interactions." He b ecame a Fellow of the American Academy of Mechanics in 2000, and in the same year he received the Otto Lap orte Award of the APS. In 2001 he was named a Fellow of the World Innovation Foundation. Most recently, Hassan Aref was awarded the G. I. Taylor Medal of the Society of Engineering Science (SES) "in recognition of his outstanding research contributions in fluid mechanics, in particular for his seminal applications of dynamical systems theory to fluid mechanics." He was to receive this medal at the annual meeting of the SES in Octob er 2011. Sadly, he passed away a month b efore this event. Hassan Aref served on several prominent scientific committees and advisory b oards. For example, he was chair of the Division of Fluid Dynamics of the APS, he chaired the US National Committee on Theoretical and Applied Mechanics, and he was a memb er of the Executive Committee of the Congress Committee of the International Union of Theoretical and Applied Mechanics (IUTAM). From 2008 he served as Secretary of the Congress Committee, an imp ortant p ost within the Union. He also served on the Board of the Society of Engineering Science. Hassan Aref was an excellent organizer. He organized the IUTAM Symp osium on Stirring and Mixing (La Jolla, August 1990), and the IUTAM Symp osium "150 Years of Vortex Dynamics" (Cop enhagen, Octob er 2008) on the anniversary of Hermann von Helmholtz's seminal publication. He was the president and principal organizer of the 20th ICTAM (International Congress of Theoretical and Applied Mechanics) in Chicago, August 2000. The op ening ceremony of this conference was memorable for the p erformance of a short play written by Hassan Aref that featured Archimedes, Galileo, Newton and "Chicago" as players. Professor Aref was also very actively involved in the preparation of the 23rd ICTAM meeting, to b e held in Beijing in August 2012. Throughout his scientific career Professor Hassan Aref was involved in editorial work. He was associate editor of the Journal of Fluid Mechanics from 1984 to 1994 and of Physics of Fluids from 1998 to 2004. He was founding co-editor (together with the late David Crighton) of the series Cambridge Texts in Applied Mathematics and served as editor from 1986 to 1995. He was editor of Theoretical and Computational Fluid Dynamics from 2006 to 2010, and he was actively co-editing Advances in Applied Mechanics, in which he published two influential review pap ers on vortex dynamics. He also served on the editorial b oards of Applied Mathematics, Informatics and Mechanics; Archives of Mechanics; European Journal of Mechanics B-Fluids; Integrated ComputerAided Engineering; Physical Review E; Regular and Chaotic Dynamics; and Theoretical and Computational Fluid Dynamics. Hassan Aref sup ervised a dozen PhDs who have gone on to successful careers in academia and industry, and he hosted ab out as many p ostdoctoral and visiting scholars, most of whom are also established academics today. He tended to work quite intensely with a small numb er of p eople over a p eriod of many years and to devote a lot of individual time to students, p ostdocs and other collab orators. An overview of Hassan Aref 's contributions to a wide range of fields in fluid mechanics is b est of all outlined by what he indicated himself in his latest CV of 2011: A. Kinematics of an incompressible fluid · Introduction and naming of chaotic advection idea/concept (1982­84). · Demonstration that chaotic advection occurs in Stokes flow b etween eccentric rotating cylinders (with S. Balachandar, 1986) and in flow through a twisted pip e (with S. W. Jones, 1987­88). · Demonstration of how to enhance separation of diffusing particles by using chaotic advection (with S. W. Jones, 1989).
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· Exploration top ological M. A. Strem · Application R. J. Adrian

of the nature of stirring with three principles that lead to "maximally ler, 1996­98). of chaotic advection to the design of m , D. J. Beeb e, J. G. Santiago and M. A.

agitators and an explanation of simple chaotic" mixing (with P. Boyland and icrofluidic mixers for MEMS devices (with Stremler, 1998).

B. Vortex dynamics · Rediscovery of Gr¨bli­Synge solution to three-vortex problem on the unb ounded plane o (1979). · Chaos for four identical vortices and for advection of a particle by three vortices (with N. Pomphrey, 1980­82). · Chaos in the collision dynamics of two vortex pairs (with B. Eckhardt, 1988; L. TophÜj, 2008). · Solution of three-vortex problem with zero net circulation (with N. Rott, 1989). · Discovery of a mechanism for linking elliptic vortex rings (with I. Zawadzki, 1991). · Solution of three-vortex problem with zero net circulation in a p eriodic strip and a p eriodic parallelogram (with M. A. Stremler, 1994­99). · Discovery of asymmetric relative equilibria of identical p oint vortices (with D. Vainchtein, 1998). · Theory of "exotic" vortex wakes of bluff b odies (with F. L. Ponta and M. A. Stremler, 2004­ 06). C. Motion of a solid b o dy through an ideal fluid · Chaos of rigid b ody motion in an ideal fluid; control of chaos (with S. W. Jones, 1993­94). · Chaos in b ody-vortex interactions (with J. Roenby 2009­10). D. Vortex-in-cell method for vortex flows and flows with sharp interfaces · Simulations of 2D shear layers, wakes and jets (with E. D. Siggia, 1980). · Simulations of stratified Hele-Shaw flow (with G. Tryggvason, 1983­87). · Simulations of interacting vortex rings (with I. Zawadzki, 1990­91). E. Foam physics · Large-scale simulations of coarsening of 2D dry foam (with T. Herdtle, 1989­92). · Morphological transition in compressible dry foam (with D. L. Vainchtein, 1999­2001). F. Other topics · Scaling solutions to the coagulation equation and applications (with D. O. Pushkin, 2001­ 2004). From the extensive and profound scientific heritage left by Hassan Aref, we focus here only on his contributions to vortex dynamics and chaotic advection. The dynamics of p oint vortices in two-dimensional flow of an ideal, incompressible fluid was a sub ject of constant interest for Hassan. He often said: "I am mostly thinking about point vortices." Appropriately, and p erhaps coincidentally, the first and last of his published pap ers deal with p oint vortex problems. The equations of motion of p oint vortices, or infinitely thin, straight, parallel vortex filaments with constant intensity, b oth in time and along their length, were established by Hermann ¨ von Helmholtz in 1858 in his famous memoir "Ub er Integrale der hydrodynamischen Gleichungen, welche den Wirb elb ewegungen entsprechen" [J. reine angew. Math. 55, 25­55 (1858)]. It is from this work that the discipline of vortex dynamics originates. The solutions of the governing equations for
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Fig. 1. Sketch of tra jectories of two p oint vortices for various typical values of intensities, according to Helmholtz's analytical solution.

the case of two vortices with arbitrary intensities, which were obtained by Helmholtz, are illustrated in Fig. 1. When the numb er of vortices increases to N = 3, the dynamics b ecomes immediately much more complicated. The work that Hassan conducted in vortex dynamics was focused on establishing what happ ens in these more complex situations. Even as early as 1984, in his survey lecture at the 16th ICTAM in Cop enhagen, Aref clearly outlined the general problems of the motion of N p oint vortices in different situations. Working out the details of these motions, however, would occupy the whole of his research career. He examined in great depth the motion of three p oint vortices in an unb ounded fluid. In contrast to the famous three-b ody problem of celestial mechanics, the equations of motion for three p oint vortices are integrable, and an analytical solution of the general problem can b e found, in principle, as was briefly p ointed out in the lectures of H. Poincar´ (1893) on the theory of vortices. Various e constructions of the solution have b een develop ed over the course of more than a century, b eginning with the thesis of W. Gr¨ li in 1877 [see Aref, Rott & Thomann, Ann. Rev. Fluid Mech. 24, 1­20 ob (1992)]. However, it was the work of Hassan that established the most complete solution to this problem. Yet, consistent with his characteristic thoroughness, Hassan was not satisfied with his published treatments of the three-vortex problem. At the time of his death, he was actively exploring the dynamics of the corresp onding vortex triangle geometry, and he left a draft manuscript of this work that is currently b eing prepared for publication. When the total numb er of vortices increases b eyond N = 3, either by taking N 4 on the unb ounded plane, or by taking N 3 in a p eriodic domain, this generally non-integrable problem can lead to chaotic motion of the vortices. In certain cases restrictions can b e made that cause the problem to remain integrable: for example, by taking the sum of the vortex strengths to b e zero for N = 3 in a p eriodic domain, or by taking the angular impulse to b e zero for N = 4 in the unb ounded plane. Hassan was intrigued by these sp ecial cases and explored them quite thoroughly. He was also quite intrigued by the non-integrable cases, and his investigations led to many excellent examples of chaotic dynamics, a scientific area that was clearly formulated only in the last third of the 20th century. As the numb er of vortices is increased b eyond N = 4 in the unb ounded plane, the search for sp ecial solutions reduces to considering configurations in which the relative p ositions of the vortices are constant in time. These relative equilibrium configurations, or vortex crystals, as Hassan liked to call them, consist of vortices that all have the same net translational velocity or that are uniformly rotating ab out the same p oint. This is another classic problem dating back to works published in 1878 by Sir William Thomson (the later Lord Kelvin) [Nature 18, 13­14 (1878)] and A. M. Mayer [Nature 17, 487­488 (1878)] on "vortex atoms", which they visualized via exp eriments with floating magnets. And yet again Hassan produced new findings from a classic problem by recently identifying asymmetric equilibrium configurations and distinct equilibria that are so close as to b e indistinguishable under visual comparison. Hassan Aref sp ent time considering a numb er of other problems in vortex dynamics. He examined the integrable motion of a few p oint vortices in domains with p eriodic b oundary conditions, either in one or two directions, under the constraint that the intensities sum to zero. Recently Hassan was revisiting the nonlinear instability of p oint vortex streets consisting of equal and opp osite vortex intensities with N = 4 in the singly p eriodic domain. He found that third order p erturbation terms lead to nonlinear instability with oscillatory growth, the oscillations growing in amplitude prop ortionally with time. The manuscript describing this work was in preparation at the time of his death. In August 2011, Hassan was preparing an application for a Guggenheim Fellowship in
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order to complete a monograph on p oint vortices. The set-up of the b ook is shown in App endix B; unfortunately, he was not able to fulfill this plan. Hassan Aref 's work in vortex dynamics led him to consider the chaotic motion of a passive particle that is advected by the flow field of two blinking p oint vortices, see Fig. 2. The thought that comes to mind as we look at his pioneering article "Stirring by chaotic advection" [J. Fluid Mech. 143, 1­21 (1984)], is what Lagrange, in his treatise "Analytical Mechanics" (translated and edited by A. Boissannade and V. N. Vagliente, Dordrecht: Kluwer Academic Publishers, 2010, p. 169) wrote ab out Galileo: "The discoveries of the satel lites of Jupiter, the phases of Venus, sunspots, etc., required a telescope and diligence, but it takes an extraordinary genius to resolve and understand the laws of nature in the ever present complexity of the phenomena whose explanation had nevertheless escaped even the research of the philosophers."

Fig. 2. Regular and chaotic advection of (Lagrangian) fluid particles in a two-dimensional laminar (Eulerian) velocity field, formed by "blinking" vortices inside a circle. (H. Aref, Op ening lecture at the workshop "Physics of Mixing", Leiden, The Netherlands, January 2011).

The fact that there is a direct relationship b etween the Eulerian and Lagrangian velocity field in a fluid, expressed as a (deceptively simple) dynamic system to describ e the motion of individual particles, was known to J. C. Maxwell (1870), E. Riecke (1888), W. Morton (1913), L. M. MilneThomson (1938), Sir G. Darwin (1953), P. Welander (1955), J. Lighthill (1956) and many others later on. In these works the tra jectories of individual particles and the shap e of closed contour lines in a known velocity field have b een constructed, as shown in Fig. 3. Scientifically, Hassan Aref may b e b est known for identifying and naming the phenomenon of chaotic advection, a mechanism for enhanced mixing of viscous fluids that now enjoys applications from planetary physics to microfluidics and the production of nano-materials. This development is based on the observation that in certain situations the simplest non-stationary (often even
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Fig. 3. Typical cases of advection of material lines formed by (Lagrangian) fluid particles in a two-dimensional laminar (Eulerian) velocity field.

time-p eriodic) two-dimensional velocity field of an incompressible fluid, governed by the stream function (x,y ,t), can lead, after solving the (Hamiltonian) system of equations dx = , dt y dy =- dt x

with the initial conditions x = x0 , y = y0 at t = 0, to chaotic motion of a fluid particle. The history of this development is b est conveyed in his article "The development of chaotic advection", Phys. Fluids 14, 1 315­1 325 (2002). For his significant contribution to the development of the paradigm of chaotic advection, Hassan Aref was awarded the Otto Lap orte Award of the American Physical Society (2000) "for his pioneering contributions to the study of chaotic motion in fluids, scientific computation, and vortex dynamics, and most notably for the development of the concept of chaotic advection" and the G. I. Taylor Medal of the Society of Engineering Science (2011) "for seminal applications of dynamical systems theory to fluid mechanics." A January 2011 workshop entitled "Physics of Mixing" held at the Lorentz Center in Leiden, the Netherlands, was something of a 25th anniversary "Festschrift" of chaotic advection, and a great source of Hassan's p ersonal pride and pleasure. He gave invited talks at the op ening and closing of this workshop, which, to our deep regret, turned out to b e the last conference at which Hassan was present. One of the sp eakers, Professor Stefano Cerb elli, has even b egan his talk with the words: "Dostoevsky said that al l of us [Russian literature] came out of "The Overcoat" by Gogol. So I think that al l our results on the stirring and mixing have come out of Hassan Aref 's pioneering JFM 1984 paper." There is a saying attributed to Lamb (or von K´ m´ ) that the first question he would ask ar an God would b e the question of unraveling the problem of turbulence. Hassan was skeptical of this statement, asking "And what exactly is the problem?" b earing in mind that the problem must b e strictly formulated in terms of equations with initial and b oundary conditions. Whatever it was, we think that now he knows all the answers to all questions of vortex and chaotic dynamics. And now we have to get used to living without Hassan Aref, without the b ook on p oint vortices that he didn't have a chance to write, without his kind smile and contagious laughter, without an exchange of views on various scientific (and other) issues and hearing his invariably b enevolent "I see", "right", "correct". Rest in Peace, Hassan.
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On the very day we were to send this obituary to press, we learned of the untimely and tragic death of Professor Viatcheslav V. Meleshko, known to his friends as Slava. Slava Meleshko, a noted sp ecialist in fluid dynamics and the history of science, died on 14 Novemb er 2011 at the age of 60. We extend our sincere condolences to his family and friends who grieve today for their loss. Slava and Hassan, thanks for all these years. You will b e missed very much. Rest in Peace.

Viatcheslav Meleshko and Hassan Aref IUTAM Symposium on Hamiltonian Dynamics, Vortex Structures, Turbulence Moscow, August 2006

APPENDIX A Hassan Aref 's Journal Articles 1. Aref, H., Motion of Three Vortices. Phys. Fluids, 1979, vol. 22, pp. 393­400. 2. Aref, H. and Siggia, E. D., Vortex Dynamics of the Two-Dimensional Turbulent Shear Layer. J. Fluid Mech., 1980, vol. 100, pp. 705­737. 3. Aref, H. and Pomphrey, N., Integrable and Chaotic Motions of Four Vortices. Phys. Lett. A, 1980, vol. 78, 297­300. 4. Siggia, E. D. and Aref, H., Scaling and Structures in Fully Turbulent Flows. Ann. New York Acad. Sci., 1980, vol. 357, pp. 368­376. 5. Siggia, E. D. and Aref, H., Point Vortex Simulation of the Inverse Energy Cascade in Twodimensional Turbulence. Phys. Fluids, 1981, vol. 24, pp. 171­173. 6. Aref, H. and Siggia, E. D., Evolution and Breakdown of a Vortex Street in Two Dimensions. J. Fluid Mech., 1981, vol. 109, pp. 435­463. 7. Aref, H. and Pomphrey, N., Integrable and Chaotic Motions of Four Vortices. I: The Case of Identical Vortices. Proc. R. Soc. A, 1982, vol. 380, pp. 359­387. 8. Aref, H., Point Vortex Motions with a Center of Symmetry. Phys. Fluids, 1982, vol. 25, pp. 2 183­2 187. 9. Aref, H., Integrable, Chaotic and Turbulent Vortex Motion in Two-Dimensional Flows. Annu. Rev. Fluid Mech., 1983, vol. 15, pp. 345­389.
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10. Tryggvason, G. and Aref, H., Numerical Exp eriments on Hele-Shaw Flow with a Sharp Interface. J. Fluid Mech., 1983, vol. 136, pp. 1­30. 11. Aref, H., Stirring by Chaotic Advection. J. Fluid Mech., 1984, vol. 143, pp. 1­21. 12. Aref, H. and Daripa, P. K., Note on Finite Difference Approximations to Burgers' Equation. SIAM J. Sci. Stat. Comput., 1984, vol. 5, pp. 856­864. 13. Aref, H. and Tryggvason, G., Vortex Dynamics of Passive and Active Interfaces. Physica D, 1984, vol. 12, pp. 59­70. 14. Aref, H. and Flinchem, E. P., Dynamics of a Vortex Filament in a Shear Flow. J. Fluid Mech., 1984, vol. 148, pp. 477­497. 15. Tryggvason, G. and Aref, H., Finger Interaction Mechanisms in Stratified Hele-Shaw Flow. J. Fluid Mech., 1985, vol. 154, pp. 287­301. 16. Aref, H. and Balachandar, S., Chaotic Advection in a Stokes Flow. Phys. Fluids, 1986, vol. 29, pp. 3 515­3 521. 17. Aref, H., The Numerical Exp eriment in Fluid Mechanics. J. Fluid Mech., 1986, vol. 173, pp. 15­41. 18. Aref, H., Jones, S. W., and Tryggvason, G., On Lagrangian Asp ects of Flow Simulation. Complex Systems, 1987, vol. 1, pp. 544­558 (1987). 19. Jones, S. W. and Aref, H., Chaotic Advection in Pulsed Source-Sink Systems. Phys. Fluids, 1988, vol. 31, pp. 469­485. 20. Pumir, A. and Aref, H., A Model of Bubble Dynamics in a Hele-Shaw Cell. Phys. Fluids, 1988, vol. 31, pp. 752­763. 21. Aref, H. and Kamb e, T., Rep ort on the IUTAM Symp osium: Fundamental Asp ects of Vortex Motion. J. Fluid Mech., 1988, vol. 190, pp. 571­595. 22. Aref, H., Kadtke, J. B., Zawadzki, I., Campb ell, L. J., and Eckhardt, B., Point Vortex Dynamics: Recent Results and Op en Problems. Fluid Dyn. Research, 1988, vol. 3, 63­74. 23. Aref, H., Jones, S. W., and Thomas, O. M., Computing Particle Motions in Fluid Flows. Computers in Physics, 1988, vol. 2, no. 6, pp. 22­27. 24. Eckhardt, B. and Aref, H., Integrable and Chaotic Motions of Four Vortices. I I: Collision Dynamics of Vortex Pairs. Phil. Trans. R. Soc. A, 1988, vol. 326, pp. 655­696. 25. Aref, H. and Tryggvason, G., Model of Rayleigh­Taylor Instability. Phys. Rev. Lett., 1989, vol. 62, pp. 749­752. 26. Aref, H. and Jones, S. W., Enhanced Separation of Diffusing Particles by Chaotic Advection. Phys. Fluids A, 1989, vol. 1, pp. 470­474. 27. Aref, H., Three-vortex Motion with Zero Total Circulation: Addendum. Z. Angew. Math. Phys., 1989, vol. 40, pp. 495­500. 28. Aref, H., Jones, S. W., Mofina, S., and Zawadzki, I., Vortices, Kinematics and Chaos. Physica D, 1989, vol. 37, pp. 423­440. 29. Jones, S. W., Thomas, O. M., and Aref, H., Chaotic Advection by Laminar Flow in a Twisted Pip e. J. Fluid Mech., 1989, vol. 209, 335­357. 30. Kimura, Y., Zawadzki, I., and Aref, H., Vortex Motion, Sound Radiation, and Complex Time Singularities. Phys. Fluids A, 1990, vol. 2, pp. 214­219. 31. Aref, H., Chaotic Advection of Fluid Particles. Phil. Trans. R. Soc. A, 1990, vol. 333, pp. 273­ 289. 32. Herdtle, T., and Aref, H., On the Geometry of Comp osite Bubbles. Proc. R. Soc. A, 1991, vol. 434, pp. 441­447. 33. Aref, H., Stochastic Particle Motion in Laminar Flows. Phys. Fluids A, 1991, vol. 3, pp. 1 009­ 1 016. 34. Zawadzki, I. and Aref, H., Mixing During Vortex Ring Collision. Phys. Fluids A, 1991, vol. 3, pp. 1 405­1 410. 35. Herdtle, T. and Aref, H., Relaxation of Fractal Foam. Phil. Mag. Lett., 1991, vol. 64, pp. 335­ 340.
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36. Aref, H. and Zawadzki, I., Linking of Vortex Rings. Nature, 1991, vol. 354, pp. 50­53. 37. Aref, H., Rott, N., and Thomann, H., Grobli's Solution of the Three-Vortex Problem. Annu. ¨ Rev. Fluid Mech., 1992, vol. 24, pp. 1­20. 38. Herdtle, T. and Aref, H., Numerical Exp eriments on Two-Dimensional Foam. J. Fluid Mech., 1992, vol. 241, 233­260. 39. Weaire, D., Bolton, F., Herdtle, T., and Aref, H., The Effect of Strain up on the Top ology of a Soap Froth. Phil. Mag. Lett., 1992, vol. 66, pp. 293­299. 40. Aref, H., and Jones, S. W., Chaotic Motion of a Solid through Ideal Fluid. Phys. Fluids A, 1993, vol. 5, pp. 3 026­3 028. 41. Aref, H., Chaotic Advection in Persp ective. Chaos, Solitons and Fractals, 1994, vol. 4, pp. 745­748. 42. Mahadevan, L., Aref, H., and Jones, S. W., Comment on "Behavior of a Falling Pap er". Phys. Rev. Lett., 1995, vol. 75, 1 420. 43. Aref, H., On the Equilibrium and Stability of a Row of Point Vortices. J. Fluid Mech., 1995, vol. 290, pp. 167­181. 44. Aref, H. and Stremler, M. A., On the Motion of Three Point Vortices in a Periodic Strip. J. Fluid Mech., 1996, vol. 314, pp. 1­25. 45. Meleshko, V. V. and Aref, H., A Blinking Rotlet Model for Chaotic Advection. Phys. Fluids, 1996, vol. 8, pp. 3 215­3 217; Erratum, 1998, vol. 10, p. 1 543. 46. Aref, H. and BrÜns, M., On Stagnation Points and Streamline Top ology in Vortex Flows. J. Fluid Mech., 1998, vol. 370, pp. 1­27. 47. Aref, H. and Vainchtein, D. L., Point vortices exhibit asymmetric equilibria. Nature, 1998, vol. 392, pp. 769­770. 48. Stremler, M. A. and Aref, H., Motion of Three Vortices in a Periodic Parallelogram. J. Fluid Mech., 1999, vol. 392, 101­128. 49. Aref, H. and Stremler, M. A., Four-Vortex Motion with Zero Total Circulation and Impulse. Phys. Fluids, 1999, vol. 11, 3 704­3 715. 50. Aref, H., Order in chaos. Nature, 1999, vol. 401, pp. 756­758. 51. Aref, H. and Vainchtein, D. L., The Equation of State of a Foam. Phys. Fluids, 2000, vol. 12, pp. 23­28. 52. Boyland, P. L., Aref, H., and Stremler, M. A., Top ological Fluid Mechanics of Stirring. J. Fluid Mech., 2000, vol. 403, pp. 277­304. 53. Liu, R. H., Sharp, K. V., Olsen, M. G., Stremler, M. A., Santiago, J. G., Adrian, R. J., Aref, H., and Beeb e, D. J., A Passive Three-dimensional `C-shap e' Helical Micromixer. J. Microelectromech. Systems, 2000, vol. 9, pp. 190­197. 54. Vainchtein, D. L. and Aref, H., Morphological Transition in Compressible Foam. Phys. Fluids, 2001, vol. 13, pp. 2 152­2 160. 55. Beeb e, D. J., Adrian, R. J., Olsen, M. G., Stremler, M. A., Aref, H., and Jo, B. H., Passive Mixing in Microchannels: Fabrication and Flow Exp eriments. M´ anique & Industries, 2001, ec vol. 2, pp. 343­348. 56. Aref, H., Dunaeva, T. A., and Meleshko, V. V., Chaotic Advection by Two-Dimensional Stokes Flow in a Circle. Int. J. Fluid Mech. Research, 2002, vol. 29, 525­533. 57. Pushkin, D. O. and Aref, H., Self-Similarity Theory of Stationary Coagulation. Phys. Fluids, 2002, vol. 14, pp. 694­703. 58. Aref, H., The Development of Chaotic Advection. Phys. Fluids, 2002, vol. 14, pp. 1 315­1 325. 59. Aref, H., A Transformation of the Point Vortex Equations. Phys. Fluids, 2002, vol. 14, pp. 2 395­2 401. 60. Aref, H., Newton, P. K., Stremler, M. A., Tokieda, T., and Vainchtein, D. L., Vortex Crystals. Adv. Appl. Mech., 2003, vol. 39, 1­79. 61. Boyland, P. L., Stremler, M. A., and Aref, H., Top ological Fluid Mechanics of Point Vortex Motions. Physica D, 2003, vol. 175, pp. 69­95.
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62. Pushkin, D. O. and Aref, H., Bank Mergers as Scale-Free Coagulation. Physica A, 2004, vol. 336, 571­584. 63. Stremler, M. A., Haselton, F. R., and Aref, H., Designing for Chaos: Applications of Chaotic Advection at the Microscale. Phil. Trans. R. Soc. A, 2004, vol. 362, pp. 1 019­1 036. 64. Ponta, F. L. and Aref, H., The Strouhal­Reynolds Numb er Relationship for Vortex Streets. Phys. Rev. Lett., 2004, vol. 93, 084501, 4pp. 65. Ponta, F. L. and Aref, H., Vortex Synchronization Regions in Shedding from an Oscillating Cylinder. Phys. Fluids, 2005, vol. 17, 011703, 4pp. 66. Aref, H. and van Buren, M., Vortex Triple Rings. Phys. Fluids, 2005, vol. 17, 057104, 21pp. 67. Ponta, F. L. and Aref, H., Numerical Exp eriments on Vortex Shedding From an Oscillating Cylinder. J. Fluids & Structures, 2006, vol. 22, pp. 327­344. 68. Aref, H., Stremler, M. A., and Ponta, F. L., Exotic Vortex Wakes -- Point Vortex Solutions. J. Fluids & Structures, 2006, vol. 22, pp. 929­940. 69. Aref, H., Vortices and Polynomials. Fluid Dyn. Research, 2007, vol. 39, 5­23. 70. Meleshko, V. V. and Aref, H., A Bibliography of Vortex Dynamics, 1858­1956. Adv. Appl. Mech., 2007, vol. 41, pp. 197­292. 71. Aref, H., Point Vortex Dynamics -- a Classical Mathematics Playground. J. Math. Phys., 2007, vol. 48, 065401, 23pp. 72. Aref, H., Hutzler, S., and Weaire, D., Toying with Physics. EuroPhysics News, 2007, vol. 38, no. 3, pp. 23­26. 73. Aref, H., A Note on the Energy of Relative Equilibria of Point Vortices. Phys. Fluids, 2007, vol. 19, 103603, 7pp. 74. Aref, H., Meleshko, V. V., Guba, A. A., and Gourjii, A. A., Uniformly Rotating Configurations of Point Vortices. Prykladna Gidromekhanika, 2007, vol. 9, no. 2­3, pp. 5­24 [in Russian.] 75. Aref, H., Something Old, Something New. Phil. Trans. R. Soc. A, 2008, vol. 366, pp. 2 649­ 2 670. 76. TophÜj, L. and Aref, H., Chaotic Scattering of Two Identical Point Vortex Pairs Revisited. Phys. Fluids, 2008, vol. 20, 093605, 10pp. 77. Aref, H., Stability of Relative Equilibria of Three Vortices. Phys. Fluids, 2009, vol. 21, 094101, 22pp. 78. Aref, H., 150 Years of Vortex Dynamics. Theor. Comp. Fluid Dyn., 2010, vol. 24, pp. 1­7. 79. Roenby, J. and Aref, H., Chaos in Body-Vortex Interactions. Proc. R. Soc. A, 2010, vol. 466, pp. 1 871­1 891. 80. Roenby, J. and Aref, H., On the Atmosphere of a Moving Body. Phys. Fluids, 2010, vol. 22, 057103, 6pp. 81. Aref, H., Roenby, J., TophÜj, L., and Stremler, M. A., Nonlinear Excursions of Particles in Ideal 2D Flows. Physica D, 2011, vol. 240, pp. 199­207. 82. Aref, H., Self-similar Motion of Three Point Vortices. Phys. Fluids, 2010, vol. 22, 057104, 12pp. 83. Roenby, J. and Aref, H., Chaotic Dynamics of a Body-Vortex Pair. J. Fluids & Structures, 2011, vol. 27, pp. 768­773. 84. Aref, H., Relative Equilibria of Point Vortices and the Fundamental Theorem of Algebra. Proc. R. Soc. A, 2011, vol. 467, pp. 2 168­2 184. 85. Dirksen, T. and Aref, H., Close Pairs of Relative Equilibria for Identical Point Vortices. Phys. Fluids, 2011, vol. 23, 051706, 4pp.
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APPENDIX B Hassan Aref 's Plan For Guggenheim Fellowship The plan for the Guggenheim Fellowship, should I receive it, is to complete the manuscript of a monograph on p oint vortex dynamics that is currently ab out 75% done. "Completion" here means somewhat more than just writing down known results. There are some thorny issues that I want to resolve, or try to resolve, to make the b ook as self-contained as p ossible. Assuming I am successful, these results will also b e rep orted in research pap ers. My first published journal article (see "Publications" #1) was on the three-vortex problem, and this problem has b een a continuing interest of mine b ecause of its richness and, I argue, its imp ortance. The monograph takes the classic but consistently overlooked 1877 dissertation on the three-vortex problem by a largely unknown Swiss mathematician Walter Gr¨ li (1852­ ob 1903) as its p oint of departure, but it is really a text on what we know ab out p oint vortex dynamics written with a healthy dose of historical p ersp ective. The history of the three-vortex problem, in particular, displays an intriguing pattern of discovery and re-discovery. Key results were obtained and then forgotten only to b e re-obtained decades later. The interested researcher must, therefore, piece together an account from pap ers in diverse journals in several languages written over more than a century. The cast of contributors includes several of the great fluid mechanicians and applied mathematicians such as Helmholtz, Kirchhoff, Kelvin, Poincar´ Kochin e, and Zhukovskii, but the key contributions are really Grobli's thesis, and a largely ignored 1949 pap er ¨ by the distinguished applied mathematician J. L. Synge, published in the first volume of Canadian Journal of Mathematics. Lamb mentions Grobli's work in his classic textb ook Hydrodynamics but ¨ with the understated comment that the pap er "contain[s] other interesting examples of rectilinear vortex-systems." Batchelor discusses three-vortex motion in his influential 1967 text, but his brief (one paragraph) synopsis is imprecise. In a 1979 letter Batchelor acknowledged that "[t]hat section of my text b ook will soon b e rather b ehind the times." Sometime in 1985 I received a telephone call from Nicholas Rott, a well known Swiss aerodynamicist retired from ETH-Zurich and then working at Stanford University. Rott had read ¨ my 1983 review article in which there is a brief mention of the history of the three-vortex problem as I knew it then. He told me that when he was a student at the ETH, his professor Jakob Ackeret, one of the grand old men of aerodynamics, had regularly included in his lectures a description of the three-vortex problem and had shown illustrations from the work by Grobli. This contact ¨ and mutual interest led to discussions on how one might make Gr¨ li's work more accessible and ob rescue it from obscurity. It was ultimately agreed that publication of a translation of a century old dissertation was probably not of sufficient interest to modern day readers -- some indication of more recent developments was required. A detailed historical study was probably not appropriate either (although Gr¨ li's accomplishments in mountain climbing, and his death in an avalanche, make for ob a fascinating tale), nor were we the right individuals to undertake it. In the end, with the help of some detective work by Hans Thomann, Rott's successor at the ETH, a historical appreciation of Grobli's work was prepared for Annual Review of Fluid Mechanics (#37). ¨ The monograph attempts to give Grobli's contribution the recognition and accessibility it ¨ deserves and to provide the necessary up-to-date account of what has happ ened since, allowing the modern reader to see the three-vortex problem, and some asp ects of what we know ab out the N -vortex problem for N 4, in a richer context. In doing so one encounters a sp ectrum of phenomena, including stationary patterns of vortices in sup erfluids and electron plasmas, chaotic motions of a few vortices, asp ects of the dynamics of wakes and shear layers, and the ubiquitous problem of two-dimensional turbulence. Since it was desirable to quote from Gr¨ li's dissertation (after translation) without reproducing ob the entire work, and since there were then several other passages by other authors that one could and should cite, a format of quotations connected by original text has b een used, at least for the earlier chapters. The freshness of some of the century-old exp ositions is remarkable. These excerpts are bracketed on one side by Helmholtz's seminal 1858 pap er on vortex motion and on the other by present day research, i.e., a span of ab out a century and a half. Some of this material will primarily b e of interest to research students working on problems in vortex dynamics, but much of it has b een incorp orated into broader seminars and graduate courses that I have given and seems to enjoy general interest from engineers, physicists and applied mathematicians alike.
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The manuscript has b een in fitful preparation for several years. During that time the b ook by P. K. Newton, The N -vortex Problem: Analytical Techniques (Springer-Verlag, 2001), has app eared. It overlaps my treatment in several places. Nevertheless, where it overlaps the emphasis in Newton's b ook tends to b e more formal, and there are treatments and topics in b oth his b ook and mine that are entirely complementary. I see the existence of two recent b ooks on p oint vortices as a renaissance for this delightful sub ject which has languished over the years. As a road map to how I would sp end my time if granted a Fellowship, I give a list of contents of the monograph with some indication of where the writing currently stands. In those places where the writing is incomplete, there is often an unresolved issue, the resolution of which may lead to a research pap er in addition to the monograph treatment. Several of my pap ers in recent years, e.g., journal articles # 59, 66, 73, 77, 82, 84 and 85 in the list of publications, have arisen from attempts to resolve issues highlighted by the writing of the monograph. I. SETTING THE STAGE (Kinematics of vorticity. The vorticity equation. The two-dimensional case. The p oint vortex model. Applications of the p oint vortex model.) All this is written. I I. THE HAMILTONIAN FORMULATION (Complex coordinates. Physical interpretation of the integrals of motion. The Kirchhoff­Routh function. The circle theorem. A transformation of the p oint vortex equations with application to the three-vortex problem. Moment relations.) All this is written, although an elab oration of one piece is active as a student research pro ject. I I I. INTEGRABILITY (Two-vortex problem. Integrability of the three-vortex problem. Introduction of Poisson brackets and Poisson bracket algebra. Pole decomp osition. Discrete symmetries.) All is largely written, although the section on Poisson brackets seems to keep growing in interesting ways! IV. THREE IDENTICAL VORTICES (Geometrical interpretation. Evolution of the triangle area. Reduction by canonical transformations. Determining the absolute motion. Some illustrations.) This is largely written, with some more thought to b e given to the illustrations. V. VORTEX TRIPLES WITH CIRCULATIONS OF OPPOSITE SIGN (The three-vortex problem (, , -). Collisions b etween a pair and a single vortex. Trapping in an equilateral triangle configuration and in a collinear state. The scattering cross-section. Exp eriments on pairvortex scattering. Solution for vanishing total circulation.) Largely written, although there is a refinement of the expression for the scattering cross-section to b e done that should result in a research pap er. VI. CLASSIFICATION OF THREE-VORTEX MOTION (Equations of relative motion. The phase plane. Physically accessible regions and phase plane tra jectories. Discussion of various cases. Diagrams of Synge.) Largely written based on journal article #1, although after #77 some changes may b e necessary. VI I. RELATIVE EQUILIBRIA AND SELF-SIMILAR MOTIONS (Classification problem. Equilateral triangle relative equilibria. Collinear relative equilibria of three vortices. Graphical representation. Linear stability of equilateral triangle and collinear relative equilibria. Self-similar motion and vortex collapse. Central configurations. Linear stability of self-similar motions.) This chapter largely follows (and in one place corrects) journal articles #77 and #82. VI I I. VORTEX STATICS (Identical vortices on a line. Vortex p olygons. Beyond vortex p olygons. Formulae for nested regular p olygons. Double rings. Triple rings. Numerical explorations. Linear stability of relative equilibria of identical vortices. Stability calculations for simple configurations. Energy calculations. Stationary vortex patterns and Tkachenko's equation. Translating vortex patterns.) The stability formalism is new and still requires work. Ab out 80% written. Some of this material is based on rather recent journal articles, e.g., #73 and #84. IX. VORTICES IN PERIODIC DOMAINS (Vortices in a p eriodic strip. A row of identical vortices. The double-alternate row or vortex street. Stability of the vortex row. Stability of the vortex street. Integrability of three-vortex motion in a p eriodic strip and a p eriodic parallelogram. The ratios of vortex strengths are rational. Irrational strength ratios and vortex quasi-crystals.) Later sections are mostly written based on Journal articles #44 and #48. The nonlinear stability
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of the vortex street, trying to cast Kochin's remarkable mid-century work in a modern context, is still in progress. X. THE FIELD OF FLOW AROUND THE VORTICES (Elementary results on stagnation p oints. Morton's equation. Instantaneous streamline top ology for three vortices. Stagnation p oints of twoand three-vortex flow via pro jective geometry. Regular and chaotic advection. Advection by p oint vortices. The restricted four-vortex problem. The notion of top ological chaos.) This chapter is still to b e written for the most part, although the basic material is largely develop ed in the research literature, including several of my own pap ers. The restricted four-vortex problem (three vortices and a passive particle) was, of course, the origin of the field of chaotic advection, which has since taken on a life of its own. XI. FOUR-VORTEX symmetry. Paired vortices full discussion of two-pair some 17 years later. Gr¨ ob PROBLEMS (Vanishing total circulation and linear impulse. Central with a common axis.) In progress. Remarkably, Gr¨ li's thesis includes a ob leapfrogging, an analysis usually attributed to a pap er by Love published li's analysis is probably more comprehensive than Love's.

I have sp ent some 30 years thinking ab out the three-vortex problem and its ramifications, off and on, and intersp ersed with p eriods of thinking ab out other things. I am rather confident that I know things ab out this problem that very few others know just by b eing immersed in it for so long. I feel strongly that now is the time to get this accumulated p ersp ective, and the many details that go with it, written down in an accessible form for p osterity. The Guggenheim Fellowship would provide a most welcome opp ortunity to push all else aside and focus on this task.

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