Relative equilibria of point vortices and linear vortex sheets

Abstract

A new exact method is presented for obtaining uniformly rotating distributions of vorticity in a two dimensional ideal fluid. The vorticity is confined to the union of a straight line and a finite collection of points; i.e., the distribution is a collection of point vortices together with a number of vortex sheets lying on the common line. The vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a rational function in which the point vortex positions and strengths appear as parameters. For many values of these parameters, the vortex sheet portion of the distribution is not unique, and there is a continuous family of vortex sheets which combine with the point vortices to form relative equilibria. Several examples are worked out in detail.