Stationary configurations of point vortices
Author:
Kevin Anthony O’Neil
Journal:
Trans. Amer. Math. Soc. 302 (1987), 383425
MSC:
Primary 76C05; Secondary 58F05, 58F40
MathSciNet review:
891628
Fulltext PDF Free Access
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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 selfsimilar 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 powerlaw 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. Onedimensional families of such configurations are shown to exist for more than three vortices. Stationary configurations of four vortices are examined in detail.
 [H]
Hassan
Aref, Integrable, chaotic, and turbulent vortex motion in
twodimensional flows, Annual review of fluid mechanics, Vol. 15,
Annual Reviews, Palo Alto, Calif., 1983, pp. 345–389. MR 686292
(84h:76024)
 1.
(1979), Motion of three vortices, Phys. Fluids 22, 393400.
 [H]
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
(83g:76041), http://dx.doi.org/10.1098/rspa.1982.0047
 [G]
G.
K. Batchelor, An introduction to fluid dynamics, Second
paperback edition, Cambridge Mathematical Library, Cambridge University
Press, Cambridge, 1999. MR 1744638
(2000j:76001)
 [K]
O. Friedrichs (1966), Special topics in fluid dynamics, Chapter 19, Gordon and Breach, New York.
 [P]
Phillip
Griffiths and Joseph
Harris, Principles of algebraic geometry, WileyInterscience
[John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
(80b:14001)
 [W]
Gràbli (1877), Specielle Probleme uber die Bewegung geradliniger paralleler Wirbelfaden (Zurich: Zurcher und Furrer).
 [T]
H. Havelock (1931), The stability of motion of rectilinear vortices in ring formation, Philos. Mag. 11, 617633.
 [H]
Helmholtz (1858), On integrals of the hydrodynamical equations which express vortex motion, Philos. Mag., 33, 485512.
 [N]
Nathan
Jacobson, Basic algebra. I, W. H. Freeman and Co., San
Francisco, Calif., 1974. MR 0356989
(50 #9457)
 2.
Lord Kelvin (1910), Mathematical and physical papers, Vol. IV, Nos. 10, 12, Cambridge Univ. Press, Cambridge, England.
 [G]
R. Kirchhoff (1876), Vorlesungen uber Matematische Physik, Vol. I, Teubner, Leipzig.
 [H]
Lamb (1932), Hydrodynamics, Chapter VII, Dover, New York.
 [E]
A. Novikov (1975), Dynamics and statistics of a system of vortices, Soviet Phys.JETP 41, 937943.
 [E]
A. Novikov and Yu. B. Sedov (1979), Vortex collapse, Sov. Phys.JETP, 50, 297301.
 [J]
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 (83b:58034)
 [I]
I.
R. Shafarevich, Basic algebraic geometry, Springer Study
Edition, SpringerVerlag, Berlin, 1977. Translated from the Russian by K.
A. Hirsch; Revised printing of Grundlehren der mathematischen
Wissenschaften, Vol. 213, 1974. MR 0447223
(56 #5538)
 [A]
Sommerfeld (1964), Mechanics of deformable bodies, Chapter IV, Academic Press, New York.
 [J]
J.
L. Synge, On the motion of three vortices, Canadian J. Math.
1 (1949), 257–270. MR 0030841
(11,61c)
 [J]
J. Thomson (1883), A treatise on the motion of vortex rings, Macmillan, London.
 [H]
 Aref (1983), Integrable, chaotic and turbulent vortex motion in twodimensional flows, Ann. Rev. Fluid Mech. 15. MR 686292 (84h:76024)
 1.
 (1979), Motion of three vortices, Phys. Fluids 22, 393400.
 [H]
 Aref and N. Pomphrey (1982), Integrable and chaotic motions of four vortices, Proc. Roy. Soc. London Ser. A 380, 359387. MR 660416 (83g:76041)
 [G]
 K. Batchelor (1967), An introduction to fluid dynamics, Cambridge Univ. Press, Cambridge, England, Chapter 7. MR 1744638 (2000j:76001)
 [K]
 O. Friedrichs (1966), Special topics in fluid dynamics, Chapter 19, Gordon and Breach, New York.
 [P]
 Griffiths and J. Harris (1978), Principles of algebraic geometry, Wiley, New York. MR 507725 (80b:14001)
 [W]
 Gràbli (1877), Specielle Probleme uber die Bewegung geradliniger paralleler Wirbelfaden (Zurich: Zurcher und Furrer).
 [T]
 H. Havelock (1931), The stability of motion of rectilinear vortices in ring formation, Philos. Mag. 11, 617633.
 [H]
 Helmholtz (1858), On integrals of the hydrodynamical equations which express vortex motion, Philos. Mag., 33, 485512.
 [N]
 Jacobson (1974), Basic algebra. I, Freeman, San Francisco, Calif. MR 0356989 (50:9457)
 2.
 Lord Kelvin (1910), Mathematical and physical papers, Vol. IV, Nos. 10, 12, Cambridge Univ. Press, Cambridge, England.
 [G]
 R. Kirchhoff (1876), Vorlesungen uber Matematische Physik, Vol. I, Teubner, Leipzig.
 [H]
 Lamb (1932), Hydrodynamics, Chapter VII, Dover, New York.
 [E]
 A. Novikov (1975), Dynamics and statistics of a system of vortices, Soviet Phys.JETP 41, 937943.
 [E]
 A. Novikov and Yu. B. Sedov (1979), Vortex collapse, Sov. Phys.JETP, 50, 297301.
 [J]
 I. Palmore (1982), Relative equilibria of vortices in two dimensions, Proc. Nat. Acad. Sci. U.S.A. 79, 716718. MR 648066 (83b:58034)
 [I]
 Shafarevich (1977), Basic algebraic geometry, SpringerVerlag, New York. MR 0447223 (56:5538)
 [A]
 Sommerfeld (1964), Mechanics of deformable bodies, Chapter IV, Academic Press, New York.
 [J]
 L. Synge (1949), On the motion of three vortices, Canad. J. Math. 1, 257270. MR 0030841 (11:61c)
 [J]
 J. Thomson (1883), A treatise on the motion of vortex rings, Macmillan, London.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708916281
PII:
S 00029947(1987)08916281
Article copyright:
© Copyright 1987 American Mathematical Society
