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High order homogenized Stokes models capture all three regimes

by F. Feppon and W. Jing

(Report number 2021-02)

Abstract
This article is a sequel to our previous work 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 $k\in\mathbb{N}$, 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.

Keywords: Homogenization, higher order models, perforated Poisson problem, Stokes system, low volume fraction asymptotics, strange term.

@Techreport{FJ21_944,
  author = {F. Feppon and W. Jing},
  title = {High order homogenized Stokes models capture
all three regimes},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-02},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-02.pdf },
  year = {2021}
}

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