Exploring circulations yielding no vortex equilibria

Abstract

A system of n planar point vortices with prescribed real circulations Γi≠0 may be arranged so that the interaction between vortices causes them all to move with a common velocity. Such solutions are called vortex equilibria, where the required configurations are referred to as stationary or translating depending on whether the velocity is zero or not. The necessary condition, ∑i<jΓiΓj=0, for stationary configurations is not sufficient, for general n. The necessary condition, ∑iΓi=0, for translating configurations is also not sufficient in general. For example, for n≥5, the set of circulations (24−n,⋯,24−n,2(2−n)n(4−n),1) yields no stationary configurations. For n≥4, the set of circulations (−1,⋯,−1,n−3,1) yields no translating configurations. In this paper, we generalize such results and find new classes of sets of circulations satisfying the respective necessary conditions, but yielding no vortex equilibria.Let α and m be two independent parameters of natural numbers. The sets of circulations we consider are of the form (x,…,x,y,1,…,1) with α 1's, no less than three x=−2m, and one y such that ∑i<jΓiΓj=0 or ∑iΓi=0 holds. Using the minimal polynomial systems defined by O'Neil, we obtain new systems involving only part of the variables in the minimal polynomial systems. Sufficient conditions for no vortex equilibria are derived from them. We explore α and m such that the n-vortex system has no vortex equilibria for general n with the help of a computer algebra system. As a byproduct of the derived systems, they also help to find vortex equilibria for some exceptional n's where no information is obtained from our sufficient conditions.