Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Stationary configurations of point vortices

Author: Kevin Anthony O’Neil
Journal: Trans. Amer. Math. Soc. 302 (1987), 383-425
MSC: Primary 76C05; Secondary 58F05, 58F40
MathSciNet review: 891628
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The motion of point vortices in a plane of fluid is an old problem of fluid mechanics, which was given a Hamiltonian formulation by Kirchhoff. Stationary configurations are those which remain self-similar throughout the motion. Results of two types are presented. Configurations which are in equilibrium or which translate uniformly are counted using methods of algebraic geometry, which establish necessary and sufficient conditions for existence. Relative equilibria (rigidly rotating configurations) which lie on a line are studied using a topological construction applicable to other power-law systems. Upper and lower bounds for such configurations are found for vortices with mixed circulations. Arrangements of three vortices which collide in finite time are well known. One-dimensional families of such configurations are shown to exist for more than three vortices. Stationary configurations of four vortices are examined in detail.

References [Enhancements On Off] (What's this?)

  • Hassan Aref, Integrable, chaotic, and turbulent vortex motion in two-dimensional flows, Annual review of fluid mechanics, Vol. 15, Annual Reviews, Palo Alto, Calif., 1983, pp. 345–389. MR 686292
  • ---(1979), Motion of three vortices, Phys. Fluids 22, 393-400.
  • H. Aref and N. Pomphrey, Integrable and chaotic motions of four vortices. I. The case of identical vortices, Proc. Roy. Soc. London Ser. A 380 (1982), no. 1779, 359–387. MR 660416, DOI
  • G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR 1744638
  • O. Friedrichs (1966), Special topics in fluid dynamics, Chapter 19, Gordon and Breach, New York.
  • Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
  • Gràbli (1877), Specielle Probleme uber die Bewegung geradliniger paralleler Wirbelfaden (Zurich: Zurcher und Furrer). H. Havelock (1931), The stability of motion of rectilinear vortices in ring formation, Philos. Mag. 11, 617-633. Helmholtz (1858), On integrals of the hydrodynamical equations which express vortex motion, Philos. Mag., 33, 485-512.
  • Nathan Jacobson, Basic algebra. I, W. H. Freeman and Co., San Francisco, Calif., 1974. MR 0356989
  • Lord Kelvin (1910), Mathematical and physical papers, Vol. IV, Nos. 10, 12, Cambridge Univ. Press, Cambridge, England. R. Kirchhoff (1876), Vorlesungen uber Matematische Physik, Vol. I, Teubner, Leipzig. Lamb (1932), Hydrodynamics, Chapter VII, Dover, New York. A. Novikov (1975), Dynamics and statistics of a system of vortices, Soviet Phys.-JETP 41, 937-943. A. Novikov and Yu. B. Sedov (1979), Vortex collapse, Sov. Phys.-JETP, 50, 297-301.
  • Julian I. Palmore, Relative equilibria of vortices in two dimensions, Proc. Nat. Acad. Sci. U.S.A. 79 (1982), no. 2, 716–718. MR 648066, DOI
  • I. R. Shafarevich, Basic algebraic geometry, Springer Study Edition, Springer-Verlag, Berlin-New York, 1977. Translated from the Russian by K. A. Hirsch; Revised printing of Grundlehren der mathematischen Wissenschaften, Vol. 213, 1974. MR 0447223
  • Sommerfeld (1964), Mechanics of deformable bodies, Chapter IV, Academic Press, New York.
  • J. L. Synge, On the motion of three vortices, Canad. J. Math. 1 (1949), 257–270. MR 30841, DOI
  • J. Thomson (1883), A treatise on the motion of vortex rings, Macmillan, London.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 76C05, 58F05, 58F40

Retrieve articles in all journals with MSC: 76C05, 58F05, 58F40

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society