Darcy's law as low Mach and homogenization limit of a compressible fluid in perforated domains

TL;DR

This paper investigates the homogenization limit of compressible Navier-Stokes equations in perforated domains, showing convergence to Darcy's law under specific scaling regimes and providing new pressure estimates.

Contribution

It extends homogenization results to compressible fluids, estimating pressure deficits and proving convergence for all barotropic exponents above 1.5.

Findings

Convergence to Darcy's law in compressible fluids under specific scaling.

New pressure estimates using Bogovski
2f operator.

Validates Darcy's law for a broader range of barotropic exponents.

Abstract

We consider the homogenization limit of the compressible barotropic Navier-Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is increasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy's law. The range of sizes of particles we consider are exactly the same which lead to Darcy's law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovski\u{{\i}} operator This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents $\gamma> \frac32$.