Local polynomial solutions of steady Euler equations for planar ideal fluids

TL;DR

This paper derives local polynomial solutions up to degree four for steady two-dimensional Euler equations of ideal fluids, providing a foundation for understanding analytical solutions in fluid dynamics.

Contribution

It introduces a method using tensorial and complex variable notation to explicitly find local polynomial solutions of degree up to four for steady Euler equations.

Findings

Explicit polynomial solutions up to degree four are obtained.

The general form of higher-degree polynomial solutions is discussed.

Effects of viscous terms from Navier-Stokes equations are considered.

Abstract

Exploring the general analytical solutions to the Euler equations for ideal fluids holds significant theoretical and practical importance. The steady flows in two-dimensional spaces are considered whether there is an analytical solution in the form of finite polynomials defined in the local region. By employing the tensorial representation and the complex variable notation, we successfully work out the local analytical solutions in terms of polynomials with highest degree up to four, and the general form of solutions for higher-degree polynomials can also be anticipated. Several examples are illustrated and some discussion comments on the effect of the viscous term from the Navier-Stokes (N-S) equations are presented.