Equilibrium singularity distributions in the plane

TL;DR

This paper characterizes all fixed equilibrium point singularity distributions in the plane, linking their existence to the nullspace of a skew-symmetric matrix derived from the system's geometry, and classifies equilibria using eigenvalues and entropy.

Contribution

It provides a linear algebra framework for identifying and classifying equilibrium singularity distributions in planar fluid flow systems.

Findings

Equilibria exist for odd number of singularities due to the kernel of the configuration matrix.

The existence of equilibria for even number of singularities depends on geometric configuration.

Eigenvalue analysis enables classification of equilibrium states and their entropy.

Abstract

We characterize all fixed equilibrium point singularity distributions in the plane of logarithmic type, allowing for real, imaginary, or complex singularity strengths \Gamma . The dynamical system follows from the assumption that each of the N singularities moves according to the flowfield generated by all the others at that point. For strength vector {\Gamma} from R^N, the dynamical system is the classical point vortex system obtained from a singular discrete representation of the vorticity field from incompressible fluid flow. When {\Gamma} is purely imaginary, it corresponds to a system of sources and sinks, whereas when {\Gamma} from C^N the system consists of spiral sources and sinks discussed in Kochin et. al. (1964). We formulate the equilibrium problem as one in linear algebra, A\Gamma = 0, where A is a NxN complex skew-symmetric configuration matrix which encodes the geometry of the system of interacting singularities. For an equilibrium to exist, A must have a kernel. {\Gamma} must then be an element of the nullspace of A. We prove that when N is odd, A always has a kernel, hence there is a choice of {\Gamma} for which the system is a fixed equilibrium. When N is even, there may or may not be a non-trivial nullspace of A, depending on the relative position of the points in the plane. We describe a method for classifying the equilibria in terms of the distribution of the non-zero eigenvalues (singular values) of A, or equivalently, the non-zero eigenvalues of the associated covariance matrix A'A, from which one can calculate the Shannon entropy of the configuration.