Fixed equilibria of point vortices in symmetric multiply connected domains

Abstract

The paper provides symmetric fixed configurations of point vortices in multiply connected domains in the unit circle with many circular obstacles. When the circular domain is invariant with respect to rotation around the origin by a degree of 2 π / M , a regular M -polygonal ring configuration of identical point vortices becomes a fixed equilibrium. On the other hand, when we assume a special symmetry, called the folding symmetry, on the circular domain, we find a fixed equilibrium in which M point vortices with the positive unit strength and M point vortices with the negative unit strength are arranged alternately at the vertices of a 2 M -polygon. We also investigate the stability of these fixed equilibria and their bifurcation for a special circular domain with the rotational symmetry as well as the folding symmetry. Furthermore, we discuss fixed equilibria in non-circular multiply connected domains with the same symmetries. We give sufficient conditions for the conformal mappings, by which fixed equilibria in the circular domains are mapped to those in the general multiply connected domains. Some examples of such conformal mappings are also provided.