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 Ωε = Ω\Tε . Here Ω ⊂ R3 is a bounded open set and Tε is the union of many disjoint closed sets of size ε3 and density of order 1/ε3, 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 Tε . The result is obtained through an adaptation of the two-scale convergence method.