On the use of the incompressibility condition in the Euler and Navier-Stokes equations

TL;DR

This paper critically examines the common practice of replacing the thermal equation with the incompressibility condition in Euler and Navier-Stokes equations, showing it is physically inconsistent and emphasizing that incompressibility must arise from the full system.

Contribution

It demonstrates that the incompressibility condition cannot be externally imposed and must emerge self-consistently from the complete transport equations.

Findings

Replacing thermal equations with div v=0 is physically inconsistent.

Incompressibility cannot be enforced externally; it must arise naturally.

The full transport system is necessary to accurately describe incompressible behavior.

Abstract

The Euler and Navier-Stokes equations both belong to a closed system of three transport equations, describing the particle number density N, the macroscopic velocity v and the temperature T. These sytems are complete, leaving no room for any additional equation. Nonetheless, it is common practice in parts of the literature to replace the thermal equation by the incompressibility condition div v = 0, motivated by the wish to obtain simpler equations. It is shown that this procedure is physically inconsistent in several ways, with the consequence that incompressibility is not a property that can be enforced by an external condition. Incompressible behaviour, if existing, will have to follow self-consistently from the full set of transport equations.