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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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