Kevin O'Neil

Relations satisfied by the roots of the Loutsenko sequence of polynomials are derived. These roots are known to correspond to families of stationary and uniformly translating point vortices with two vortex strengths in ratio $-2$. The relations are analogous to those satisfied by the roots of the Adler–Moser polynomials, corresponding to equilibria with ratio $-1$. The proof uses an analysis of the differential equation that these polynomial pairs satisfy.

An exact method is presented for obtaining uniformly translating distributions of vorticity in a two-dimensional ideal fluid, or equivalently, stationary distributions in the presence of a uniform background flow. These distributions are generalizations of the well-known vortex dipole and consist of a collection of point vortices and an equal number of bounded vortex sheets. Both the vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a simple rational function in which the point vortex positions and strengths appear as parameters. The vortex sheets lie on heteroclinic streamlines of the flow. Dipoles and multipoles that move parallel to a straight fluid boundary are also obtained. By setting the translation velocity to zero, equilibrium configurations of point vortices and vortex sheets are found.

The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way.