Homogenization and dimension reduction of the Stokes-problem with Navier-Slip condition in thin perforated layers

John Fabricius; Markus Gahn

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arXiv · 2210.12052

Homogenization and dimension reduction of the Stokes-problem with Navier-Slip condition in thin perforated layers

John Fabricius·Markus Gahn

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Abstract

We study a Stokes system posed in a thin perforated layer with a Navier-slip condition on the internal oscillating boundary from two viewpoints: 1) dimensional reduction of the layer and 2) homogenization of the perforated structure. Assuming the perforations are periodic, both aspects can be described through a small parameter $\epsilon>0,$ which is related to the thickness of the layer as well as the size of the periodic structure. By letting $\epsilon$ tend to zero, we prove that the sequence of solutions converges to a limit which satisfies a well-defined macroscopic problem. More precisely, the limit velocity and limit pressure satisfy a two pressure Stokes model, from which a Darcy law for thin layers can be derived. Due to non-standard boundary conditions, some additional terms appear in Darcy's law.

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John Fabricius, Markus Gahn. 2022-10-21. Homogenization and dimension reduction of the Stokes-problem with Navier-Slip condition in thin perforated layers. https://arxiv.org/abs/2210.12052

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