Solutions of the Euler equations and stationary structures in an inviscid fluid

TL;DR

This paper investigates solutions to the two-dimensional Euler equations for inviscid fluids, deriving explicit solutions using elliptic functions and interpreting them as various stationary fluid structures.

Contribution

It introduces a novel method for constructing solutions as rational elliptic functions and explores the quantization of fluid flux in certain nonlinear elliptic equations.

Findings

Solutions include sources, jets, vortex structures

Flux quantization in elliptic Sine-Gordon equation

New method for rational elliptic function solutions

Abstract

The Euler equations describing two-dimensional steady flows of an inviscid fluid are studied. These equations are reduced to one equation for the stream function and then, using the Hirota function, solutions of three nonlinear elliptic equations are found. %: Sine-Gordon, Sinh-Gordon and Tzitz\'{e}ica. The solutions found are interpreted as sources in a rotating fluid, jets, chains of sources and sinks, vortex structures. We propose a new simple method for constructing solutions in the form of rational expressions of elliptic functions. It is shown that the flux of fluid across a closed curve is quantized in the case of the elliptic Sin-Gordon equation.