Pré-Publication, Document De Travail Année : 2025

Homogenization of a Stokes problem with non homogeneous Fourier boundary conditions in a thin perforated domain

Résumé

We consider a viscous fluid obeying the 3D Stokes system in a thin layer Ωε ⊂ R3 of thickness 0 < hε ≪ 1, which is perforated by ε-periodically distributed cylinder shaped obstacles of size 0 < ε ≪ 1. On the boundary of the obstacles, we prescribe non homogeneous Fourier boundary conditions with a parameter αε ∈ (0, +∞). Depending on the relation between ε and hε, we prove the convergence of the homogenization process when ε goes to zero for different values of αε. As a result, we derive different 2D Darcy’s laws taking into account the microstructure of the domain by means of local problems. The dimensions of the local problems are 2D or 3D depending on the relation between ε and hε, i.e. if hε ≫ ε or hε ≈ ε, respectively. Moreover, depending on the relation between αε and ε^{−1}, the boundary conditions of the local problems on the reference obstacle are of three types: homogeneous Dirichlet if αε ≫ ε^{−1}, non homogeneous Fourier if αε ≈ ε−1, or homogeneous Fourier boundary conditions if αε ≪ ε^{−1}.

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Dates et versions

hal-04852890 , version 1 (21-12-2024)
hal-04852890 , version 2 (24-09-2025)

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  • HAL Id : hal-04852890 , version 2

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María Anguiano, Francisco J. Suárez-Grau. Homogenization of a Stokes problem with non homogeneous Fourier boundary conditions in a thin perforated domain. 2025. ⟨hal-04852890v2⟩
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