# 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.
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https://hal.archives-ouvertes.fr/hal-03098222
Contributor : Florian Feppon Connect in order to contact the contributor
Submitted on : Tuesday, January 5, 2021 - 4:38:56 PM
Last modification on : Tuesday, June 1, 2021 - 2:34:10 PM
Long-term archiving on: : Wednesday, April 7, 2021 - 9:29:22 AM

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• HAL Id : hal-03098222, version 1

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Florian Feppon, Wenjia Jing. High order homogenized Stokes models capture all three regimes. 2021. ⟨hal-03098222⟩

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