Analytical solutions for rotating vortex arrays involving multiple vortex patches

Abstract

A continous two-parameter family of analytical solutions to the Euler equations are presented representing a class of steadily rotating vortex arrays involving N + 1 interacting vortex patches where N ≥ 3 is an integer. The solutions consist of a central vortex patch surrounded by an N-fold symmetric alternating array of satellite point vortices and vortex patches. One of the parameters governs the size of the central patch, the other governs the size of the N satellite patches. In the limit where the areas of the satellite vortex patches tend to zero, the solutions degenerate to the exact solutions of Crowdy