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Existence of steady symmetric vortex pairs on a planar domain with an obstacle

Published online by Cambridge University Press:  01 March 1998

T. V. BADIANI
T. V. BADIANI
Affiliation:
School of Mathematical Sciences, University of Bath, Claverton Down, Avon, BA2 7AY
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Abstract

We prove an existence theorem for a steady planar flow of an ideal fluid past an obstacle, containing a bounded symmetric vortex pair and approaching a uniform flow at infinity. The vorticity is a rearrangement of a prescribed function.

The stream function ψ for the flow satisfies the equation

−Δψ=ϕ∘ψ

in a region bounded by the line of symmetry, where ϕ is an increasing function that is unknown a priori.

The result is obtained from a variational principle in which a functional related to the kinetic energy is maximised over all flows whose vorticity fields are rearrangements of a prescribed function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 123 , Issue 2 , March 1998 , pp. 365 - 384
Copyright
Cambridge Philosophical Society 1998

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Footnotes

This paper contains material included in a thesis submitted to the University of Bath for which the author was supported by a studentship from the Engineering and Physical Sciences Research Council. This material was revised into the form of a paper while the author was supported by a ‘Human Capital and Mobility’ Institutional Fellowship (no. ERBCHRXCT940494) at the Scuola Normale Superiore in Pisa.