The Euler and Navier-Stokes equations revisited

TL;DR

This paper reexamines the derivation of Euler and Navier-Stokes equations from kinetic theory, clarifying their valid conditions and challenging common assumptions like incompressibility, emphasizing their coupling with temperature equations.

Contribution

It provides a fundamental derivation showing Euler and Navier-Stokes equations are valid only for ideal gases with small deviations from equilibrium, and clarifies their coupling with temperature equations.

Findings

Euler and Navier-Stokes derived from kinetic theory only valid for ideal gases.

Incompressibility condition cannot replace the full set of transport equations.

The equations are inherently coupled with temperature, not separable.

Abstract

The present paper is motivated by recent mathematical work on the incompressible Euler and Navier-Stokes equations, partly having physically problematic results and unrealistic expectations. The Euler and Navier-Stokes equations are rederived here from the roots, starting at the kinetic equation for the distribution function in phase space. The derivation shows that the Euler and Navier-Stokes equations are valid only if the fluid under consideration is an ideal gas, and if deviations from equilibrium are small in a defined sense, thereby excluding fully nonlinear solutions. Furthermore, the derivation shows that the Euler and Navier-Stokes equations are unseparably coupled with an appertaining equation for the temperature, whereby, in conjunction with the continuity equation, a closed system of transport equations is set up which leaves no room for any additional equation, with the consequence that the frequently used incompressibility condition $\nabla\cdot{\bf v}=0$ can, at best, be applied to simplify these transport equations, but not to supersede any of them.