Qualitative derivation of a density dependent incompressible Darcy law
Danica Basarić, Florian Oschmann, Jiaojiao Pan
TL;DR
The paper addresses the formation of a density-dependent Darcy law as the homogenized limit of the 3D non-homogeneous incompressible Navier–Stokes system in perforated domains with supercritical hole sizes $a_\varepsilon=\varepsilon^{\alpha}$, $1<\alpha<3$. It introduces a relative energy framework and a local problem yielding a resistance matrix $M_0$, and it proves convergence of the density and velocity to a Darcy state with quantitative rates in both toroidal and bounded domains. The work also establishes the convergence of the extended pressure and demonstrates the existence of strong solutions for the limiting Darcy system, highlighting an in-depth quantitative understanding of the homogenization process in the presence of non-constant density. These results extend classical Darcy-limit theory to non-homogeneous incompressible flows in highly perforated media and provide explicit error bounds that depend on the perforation scale and domain geometry, contributing both theory and potential applications in porous-media modeling.
Abstract
This paper provides the first study of the homogenization of the 3D non-homogeneous incompressible Navier--Stokes system in perforated domains with holes of supercritical size. The diameter of the holes is of order $\varepsilon^α \ (1<α<3)$, where $\varepsilon > 0$ is a small parameter measuring the mutual distance between the holes. We show that as $\varepsilon\to 0$, the asymptotic limit behavior of velocity and density is governed by Darcy's law under the assumption of a strong solution of the limiting system. Moreover, convergence rates are obtained. Finally, we show the existence of strong solutions to the inhomogeneous incompressible Darcy law, which might be of independent interest.