Local polynomial solutions of steady Euler equations for planar ideal fluids

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.