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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.
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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

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