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.
| Author: | David WiedemannGND, Malte A. PeterORCiDGND |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-1122311 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/112231 |
| ISSN: | 0022-0396OPAC |
| Parent Title (English): | Journal of Differential Equations |
| Publisher: | Elsevier |
| Place of publication: | Amsterdam |
| Type: | Article |
| Language: | English |
| Year of first Publication: | 2024 |
| Publishing Institution: | Universität Augsburg |
| Release Date: | 2024/03/21 |
| Tag: | Analysis; Applied Mathematics |
| Volume: | 396 |
| First Page: | 172 |
| Last Page: | 209 |
| DOI: | https://doi.org/10.1016/j.jde.2024.02.056 |
| 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 |
