Vortex structures with complex points singularities in the two-dimensional Euler equation. New exact solutions

TL;DR

This paper introduces a new class of exact stationary solutions to the 2D Euler equations featuring complex point singularities with higher indices, expanding understanding of vortex structures.

Contribution

It presents explicit elementary-function solutions with complex singularities of index three, a novel addition to known vortex solutions.

Findings

Solutions include complex singularities with index three.

Solutions are expressed in elementary functions.

Conditions for stationary complex vortex configurations are discussed.

Abstract

In this work we found the new class of exact stationary solutions for 2D-Euler equations. Unlike of already known solutions, the new one contain complex singularities. We consider as complex, point singularities which have the vector field index greater than one. For example, the dipole singularity is complex because its index is equal to two. We present in explicit form a large class of exact localized stationary solutions for 2D-Euler equations with the singularity which index is equal to three. The obtained solutions are expressed in terms of elementary functions. These solutions represent complex singularity point surrounded by vortex satellites structure. We discuss also motion equation of singularities and conditions for singularity point stationarity which provides the stationarity of complex vortex configuration.