Steady vortex-patch flows with circulation past an obstacle in an infinity long strip

TL;DR

This paper constructs steady vortex-patch flows around an obstacle in an infinitely long strip, analyzing how obstacle shape, circulation, and boundary conditions influence flow stability and existence.

Contribution

It introduces a new Green's function and establishes conditions for the existence of vortex patch flows influenced by obstacle and circulation parameters.

Findings

Existence of steady vortex-patch flows depends on obstacle and circulation parameters.

New Green's function aids in analyzing flow stability.

Stable minimum points of Kirchhoff-Routh functions are crucial for flow existence.

Abstract

We consider the steady Euler flows past an obstacle in an infinity long strip with horizontal constant velocity at infinity, prescribed circulation around the obstacle and sharply concentrated patch-type vorticity. The construction of these flows are based on the structure of a new Green's function, the existence of stable minimum points of the Kirchhoff-Routh functions and the existence of maximizers of the kinetic energy for the vorticity. We mainly focus on the effect from the obstacle, the velocity at infinity and the circulation around the obstacle on the existence of minimum points of the Kirchhoff-Routh function, and hence on the existence of vortex patch flows.