Singularities Motion Equations in 2-Dimensional Ideal Hydrodynamics of Incompressible Fluid

TL;DR

This paper derives motion equations for singularities like vortices and dipoles in 2D ideal hydrodynamics, proving integrability for systems involving these singularities and clarifying the limitations on higher-order multipoles.

Contribution

It introduces motion equations for point vortices and dipoles, demonstrating their Hamiltonian structure and integrability, and shows higher multipoles are not exact solutions.

Findings

Motion equations for vortices and dipoles are Hamiltonian.

Two-particle systems with vortex and dipole are completely integrable.

Higher-order multipoles are not solutions of 2D ideal hydrodynamics.

Abstract

In this paper, we have obtained motion equations for a wide class of one-dimensional singularities in 2-D ideal hydrodynamics. The simplest of them, are well known as point vortices. More complicated singularities correspond to vorticity point dipoles. It has been proved that point multipoles of a higher order (quadrupoles and more) are not the exact solutions of two-dimensional ideal hydrodynamics. The motion equations for a system of interacting point vortices and point dipoles have been obtained. It is shown that these equations are Hamiltonian ones and have three motion integrals in involution. It means the complete integrability of two-particle system, which has a point vortex and a point dipole.