Sadovskii vortex in a wedge and the associated Riemann-Hilbert problem on a torus

Abstract

Reconstruction of conformal mappings from canonical slit domains onto multiply-connected physical domains with a free boundary is of interest in many different models arising in fluid mechanics. In the present paper, an exact formula for the conformal map from the exterior of two slits onto the doubly connected flow domain is obtained when a fluid flows in a wedge about a Sadovskii vortex. The map is employed to determine the potential flow outside the vortex and the vortex domain boundary provided the circulation $\Gamma$ around the vortex and constant speed $U$ on the vortex boundary are prescribed, and there are no stagnation points on the walls. The map is expressed in terms of a rational function on an elliptic surface topologically equivalent to a torus and the solution to a symmetric Riemann-Hilbert problem on a finite and a semi-infinite segments on the same genus-1 Riemann surface. Owing to its special features, the Riemann-Hilbert problem requires a novel analogue of the Cauchy kernel on an elliptic surface. Such a kernel is proposed and employed to derive a closed-form solution to the Riemann-Hilbert problem and the associated Jacobi inversion problem. The final formula for the conformal map possesses a free geometric parameter and two model parameters, the wedge angle $\alpha$ and $\Gamma/U$. It is shown that when $\alpha<\pi$ the solution exists and the vortex has two cusps, while the solution does not exist when the wedge angle exceeds $\pi$.