Solutions of a free boundary problem in a doubly connected domain via a circular-arc polygon

This paper addresses a free boundary problem for a steady, uniform patch of vorticity surrounding a single flat plate of zero thickness and finite length. Exact solutions to this problem have previously been found in terms of conformal maps represented by Cauchy-type integrals. Here, however, it is demonstrated how, by considering an associated circular-arc polygon and using ideas from automorphic function theory, these maps can be expressed in a simple non-integral form.