High Order Homogenized Stokes Models Capture all Three Regimes

Abstract

This article is a sequel to our previous work [13] concerned with the derivation of high-order homogenized models for the Stokes equation in a periodic porous medium. We provide an improved asymptotic analysis of the coefficients of the higher order models in the low-volume fraction regime whereby the periodic obstacles are rescaled by a factor $\eta$ which converges to zero. By introducing a new family of order $k$ corrector tensors with a controlled growth as $\eta\rightarrow 0$ uniform in $k\in\N$, we are able to show that both the infinite order and the finite order models converge in a coefficient-wise sense to the three classical asymptotic regimes. Namely, we retrieve the Darcy model, the Brinkman equation or the Stokes equation in the homogeneous cubic domain depending on whether $\eta$ is respectively larger, proportional to, or smaller than the critical size $\eta_{\rm crit}\sim \epsilon^{2/(d-2)}$. For completeness, the paper first establishes the analogous results for the perforated Poisson equation, considered as a simplified scalar version of the Stokes system.