Point vortices and classical orthogonal polynomials

TL;DR

This paper explores the connection between point vortex equilibria in fluid flows and classical orthogonal polynomials, deriving differential equations and showing how Wronskians of these polynomials describe vortex configurations.

Contribution

It introduces a novel link between vortex equilibria and Wronskians of classical orthogonal polynomials, providing a new mathematical framework for analyzing vortex configurations.

Findings

Differential equations for vortex generating polynomials are derived.

Wronskians of classical orthogonal polynomials solve these equations.

Vortex equilibria can be described using these Wronskian polynomials.

Abstract

Stationary equilibria of point vortices with arbitrary choice of circulations in a background flow are studied. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials.