Stationary Configurations of Point Vortices on a Cylinder
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In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ1 and Γ2 (Γ2 = −μΓ1) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.
Keywordspoint vortices stagnation points stationary configuration vortices on a cylinder polynomial solution of differential equation
MSC2010 numbers33D45 76M23
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