An interpretation of Temam's extra force in the quasi-incompressible Navier-Stokes system

TL;DR

This paper explores the physical meaning of Temam's extra force in a quasi-incompressible Navier-Stokes system, revealing it as a manifestation of inertia, and discusses its role in approximating incompressible flow solutions.

Contribution

The paper provides a physical interpretation of Temam's extra force, connecting it to inertia, and analyzes its role in the quasi-incompressible Navier-Stokes system.

Findings

The extra force  is interpreted as inertia.

The extra force helps approximate incompressible flow solutions.

The analysis clarifies the physical meaning of Temam's force.

Abstract

We discuss the role of the extra force $$ \mathbf f_{\rm e}=-\frac 1 2(\nabla\cdot\mathbf v)\mathbf v $$ in the system of partial differential equations \begin{equation*} \left\{ \begin{aligned} &\frac{\partial\mathbf v}{\partial \rm t}+(\mathbf v\cdot\nabla)\mathbf v+\nabla \mathrm p-\frac 1 {\rm Re}\Delta\mathbf v=\mathbf f+\mathbf f_{\rm e},\\ &\frac 1 {\mathrm K}\frac{\partial \mathrm p}{\partial \rm t}+\nabla\cdot\mathbf v=0,\qquad \mathrm K>>1, \end{aligned}\right. \end{equation*} whose weak solution has been proved in [Arch. Rat. Mech. Analysis, 32:135-153] to approximate, in the limit $\mathrm K\to\infty$, the weak solution of the incompressible Navier-Stokes system. Taking the cue from [Ann. Mat. Pura Appl. 172:103-124], we provide a physical interpretation of the extra force $\mathbf f_{\rm e}$, showing that it is a manifestation of inertia.