A Brinkman law in the homogenization of the stationary Navier–Stokes system in a non-periodic porous medium

Abstract

We study the asymptotic behavior of the Navier–Stokes system with Dirichlet boundary conditions posed in a domain Ω ε = Ω ∖ T ε . Here Ω ⊂ R 3 is a bounded open set and T ε is the union of many disjoint closed sets of size ε 3 and density of order 1 ∕ ε 3 , with ε a small positive parameter. Similarly to the periodic case we get a limit system corresponding to a Brinkman flow. The difference is that now the Brinkman matrix is not homogeneous, it depends on the density of the closed sets composing T ε . The result is obtained through an adaptation of the two-scale convergence method.