Existence of steady symmetric vortex pairs on a planar domain with an obstacle
T. V. Badiani · Mathematical Proceedings of the Cambridge Philosophical Society · 1998
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∘ψin 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.