Point vortices with a rational necklace: New exact stationary solutions of the two-dimensional Euler equation

Abstract

In this paper we find a new class of explicit exact stationary solutions of the two-dimensional (2D) Euler equation which describe vortex patterns of necklace type with N+1-fold symmetry in rotational shear flow. The point vortex with a strength equal to ±4πN (where N-integer number) is situated in the center of the vortex structure. The vorticity distribution outside of the center is smooth and is described by a two-parametric family of rational functions which are known in explicit form for any N. In the centers of vortex satellites the vorticity does not have any singularity and remains of finite value. When N is increasing, the solutions describe the transition layer in 2D–rotational shear flow.