Homogenization of the Stokes equations with high-contrast viscosity.
type: Item
instance of: scholarly article
title: Homogenization of the Stokes equations with high-contrast viscosity.
zbMATH Open document ID: 1058.35024
DOI: 10.1016/S0021-7824(03)00012-6
author: Marc Briane
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-11-19T00:00:00Z
review text: This paper deals with the homogenization of the Stokes equations in a vertical open cylinder \(\Omega=\Omega^\prime\times (0,1)\) of \(\mathbb R^3\) with varying viscosity and with Dirichlet boundary condition: \[ \begin{cases} -\text{div}(a_\varepsilon \nabla u^\varepsilon)-\nabla p_\varepsilon=f &\text{ in } \Omega,\\ \text{div} (u^\varepsilon)=0 &\text{ in } \Omega,\\ u^\varepsilon=0&\text{ on } \partial\Omega. \end{cases} \] The viscosity \(a_\varepsilon\) is \(\varepsilon\)-periodic, \(a_\varepsilon\) takes a constant value \(\alpha_\varepsilon\gg 1\) in a \(\varepsilon\)-periodic lattice \(\omega_\varepsilon\) of vertical cylinders of radius \(\varepsilon r_\varepsilon\), where \( r_\varepsilon<<1\), and is equal to \(1\) outside \(\omega_\varepsilon\). In the critical regime defined by \(\lim_{\varepsilon\to 0}\varepsilon^2| \ln r_\varepsilon| \in (0,\infty)\) and \(\lim_{\varepsilon\to 0}\alpha_\varepsilon r_\varepsilon^2 \in (0,\infty]\), the limit problem is a coupled Stokes system satisfied by the limit velocity and the limit of the rescaled velocity in the cylinders, which can be read as a nonlocal law of Brinkman type. Moreover, if \(\lim_{\varepsilon\to 0}\alpha_\varepsilon r_\varepsilon^2 =+\infty\), the limit of the rescaled velocity is equal to zero and the Brinkman law is derived as in a paper by \textit{G. Allaire} [in: Appl. Math. Engin. Sci., Proc. 2nd Fr.-Chil. Symp., Santiago de Chile 1989, 7--20 (1991; Zbl 0759.76072)]. In the other regimes the homogenization leads either to classical Stokes problems or to a zero limit velocity. In the critical case the pressure is not bounded in \(L^2\) but only in \(H^{-1}\). Moreover the pressure of the limit problem is not equal to the weak limit of the pressure in \(H^{-1}\).
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76D07
zbMATH DE Number: 2006034
zbMATH Keywords: Brinkman law
zbMATH Keywords: fiber microstructure
zbMATH Keywords: nonlocal effects
reviewed by: Hans-Dieter Alber
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2006034