Interaction of point sources and vortices for incompressible planar fluids

TL;DR

This paper introduces a novel differential equation system modeling interactions of point sources and vortices in incompressible planar fluids, highlighting its Hamiltonian structure and collision behavior.

Contribution

It presents a new gradient and locally Hamiltonian system for point sources, extending classical vortex interaction models with new analytical insights.

Findings

Binary collisions are easily blown up due to first-order equations.

The model can be generalized to include source-vortex interactions.

The system retains some integrals from the classical vortex problem.

Abstract

We consider a new system of differential equations which is at the same time gradient and locally Hamiltonian. It is obtained by just replacing a factor in the equations of interaction for N point vortices, and it is interpreted as an interaction of N point sources. Because of the local Hamiltonian structure and the symmetries it obeys, it does possess some of the first integrals that appear in the N vortex problem. We will show that binary collisions are easily blown up in this case since the equations of motion are of first order. This method may be easily generalized to the blow up of higher order collisions. We then generalize the model further to interactions of sources and vortices.