Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-04-30T14:26:58.612Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady symmetric vortex pairs and rearrangements

Published online by Cambridge University Press:  14 November 2011

G. R. Burton
G. R. Burton
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, U.K.
Get access Rights & Permissions [Opens in a new window]

Synopsis

We prove an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices, and approaching a uniform flow at infinity. The data prescribed are the rearrangement class of the vorticity field, and either the momentum impulse of the vortex pair, or the velocity of the vortex pair relative to the fluid at infinity. The stream function ψ for the flow satisfies the semilinear elliptic equation

in a half-plane bounded by the line of symmetry, where φ is an increasing function that is unknown a priori. The results are proved by maximising the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 3-4 , 1988 , pp. 269 - 290
Copyright
Copyright © Royal Society of Edinburgh 1988

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Agmon, S.. The Lp approach to the Dirichlet problem. Ann. Scuola. Norm. Sup. Pisa. Cl. Sci. (3) 13 (1959), 405–448.Google Scholar
2Benjamin, T. B.. The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics. Applications of methods of functional analysis to problems in mechanics. Lecture Notes in Mathematics 503, pp. 8–29 (Berlin: Springer, 1976).Google Scholar
3Burton, G. R.. Rearrangements of functions, maximization of convex functionals, and vortex rings. Math. Ann. 276 (1987), 225–253.CrossRefGoogle Scholar
4Burton, G. R.. Vortex rings in a cylinder and rearrangements. J. Differential Equations 70 (1987), 333–348.CrossRefGoogle Scholar
5Friedman, A. and Turkington, B.. Vortex rings: existence and asymptotic estimates. Trans. Amer. Math. Soc. 268 (1981), 1–37.CrossRefGoogle Scholar
6Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order (Berlin: Springer, 1977).CrossRefGoogle Scholar
7Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (London: Cambridge University Press, 1934).Google Scholar
8Lamb, H.. Hydrodynamics (London: Cambridge University Press, 6th edn, 1932).Google Scholar
9Norbury, J.. Steady vortex pairs in an ideal fluid. Comm. Pure Appl. Math. 28 (1975), 679–700.CrossRefGoogle Scholar
10Royden, H. L.. Real analysis (New York: Macmillan, 1963).Google Scholar
11Thomson, Sir William (Lord Kelvin). Maximum and minimum energy in vortex motion. Mathematical and physical papers, vol. 4, pp. 172–183 (London: Cambridge University Press, 1910).Google Scholar
12Turkington, B.. On steady vortex flow in two dimensions. Comm. Partial Differential Equations 8 (1983), 1031–1071.CrossRefGoogle Scholar