"A trial free boundary method for computing Batchelor flows"

Steady inviscid fluid flows in which a bounded region of constant vorticity is separated from an external irrotational flow by a vortex sheet are known as Batchelor or Prandtl-Batchelor flows. For two dimensional flows consideration of the stream function gives rise to the following free boundary problem: Determine a curve and a function u that solves the Poisson equation on one side of the curve, is harmonic on the other side and there is a constant jump in the square of |grad u| across the curve. Although similar to free boundary problems considered by Alt, Caffarelli and Friedman, Acker and others, this problem poses additional difficulties, and to date no mathematical proof has been given for the existence of Batchelor flows. In this paper we will present a trial free boundary method for computing Batchelor flows and report on the results of the numerical implementation of the algorithm in the case of flows in bounded domains.