Three dimensional compressible Navier-Stokes equation - self-similar and traveling wave solutions

TL;DR

This paper explores three-dimensional solutions to the compressible Navier-Stokes equations using self-similar and traveling wave approaches, extending previous non-compressible flow studies to more complex fluid dynamics scenarios.

Contribution

It introduces a three-dimensional generalization of the self-similar Ansatz and applies a traveling wave Ansatz to analyze solutions of the compressible Navier-Stokes equations.

Findings

Development of 3D self-similar solutions for compressible flows

Application of traveling wave solutions to 3D Navier-Stokes equations

Discussion of geometrical interpretations of the solutions

Abstract

We investigate the three dimensional compressible Navier-Stokes and the continuity equations in Cartesian coordinates for Newtonian fluids. The polytropic equation of sate is used as closing condition. The key idea is the three-dimensional generalization of the well-known self-similar Ansatz which was already used for non-compressible viscous flow in our former study (Commun. in Theor. Phys. 56, (2011) 745). In the second method the three dimensional traveling wave Ansatz was applied. The geometrical interpretations of the trial functions are also discussed.