Vortex generated fluid flows in multiply connected domains

TL;DR

This paper develops a method to construct the stream function for fluid flows generated by point vortices in multiply connected domains, providing explicit convergence rates and applications to ocean eddy flows and Laplace's equation problems.

Contribution

It introduces a new approach using the method of images to compute the stream function in complex domains with explicit convergence analysis.

Findings

Explicit formulas for the stream function in multiply connected domains.

Convergence speed of the functional sequence is determined.

Applicable to solving Dirichlet problems for Laplace's equation.

Abstract

A fluid flow in a multiply connected domain generated by an arbitrary number of point vortices is considered. A stream function for this flow is constructed as a limit of a certain functional sequence using the method of images. The convergence of this sequence is discussed, and the speed of convergence is determined explicitly. The presented formulas allow for the easy computation of the values of the stream function with arbitrary precision in the case of well-separated cylinders. The considered problem is important for applications such as eddy flows in the oceans. Moreover, since finding the stream function of the flow is essentially identical to finding the modified Green's function for Laplace's equation, the presented method can be applied to a more general class of applied problems which involve solving the Dirichlet problem for Laplace's equation.