Homogenisation of the Stokes equations for evolving microstructure

  • We consider the homogenisation of the quasi-stationary Stokes equations in a porous medium that evolves over time. The evolution is a priori given. At the interface of the pore space and the solid part, we prescribe an inhomogeneous Dirichlet boundary condition, which enables a no-slip boundary condition at the evolving boundary. We pass rigorously to the homogenisation limit employing the two-scale transformation method. In order to derive uniform a priori estimates, we show a Korn-type inequality for the two-scale transformation method. The homogenisation result is a new version of Darcy's law. It features a time- and space-dependent permeability tensor, which accounts for the local pore structure, and a macroscopic inhomogeneous divergence condition, which induces a new source term for the pressure. In the case of a no-slip boundary condition at the interface, this source term relates to the change of the local pore volume.

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Author:David WiedemannGND, Malte A. PeterORCiDGND
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/112231
Parent Title (English):Journal of Differential Equations
Place of publication:Amsterdam
Year of first Publication:2024
Publishing Institution:Universität Augsburg
Release Date:2024/03/21
Tag:Analysis; Applied Mathematics
First Page:172
Last Page:209
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehr- und Forschungseinheit Angewandte Analysis
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
Licence (German):CC-BY 4.0: Creative Commons: Namensnennung (mit Print on Demand)