High order homogenized Stokes models capture all three regimes
Résumé
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.
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