Partial Differential Equations/Mathematical Physics
A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 967-971.

We consider a viscous fluid of small height ε on a periodic rough bottom $Γε$ of period $rε$ and amplitude $δε$, $δε≪rε≪ε$, where we impose the slip boundary condition. When ε tends to zero we obtain a Reynolds system depending on the limit λ of $(δεε)/(rεrε)$. If $λ=+∞$, the fluid behaves as if we would impose the adherence condition on $Γε$. This justifies why this is the usual boundary condition for viscous fluids. If $λ=0$ the fluid behaves as if $Γε$ was plane. Finally, for $λ∈(0,+∞)$ it behaves as if $Γε$ was flat but with a higher friction coefficient.

On considère un fluide visqueux de faible épaisseur ε sur un fond rugueux $Γε$, périodique de période $rε$ et amplitude $δε$, $δε≪rε≪ε$, où on impose la condition de glissement. Quand ε converge vers zéro on obtient un système de type Reynolds qui dépend de la limite λ de $(δεε)/(rεrε)$. Si $λ=+∞$, le fluide se comporte comme si on aurait imposé la condition d'adhérence sur $Γε$. Ceci justifie la condition usuelle pour un fluide visqueux. Si $λ=0$ le fluide se comporte comme si $Γε$ était plate. Enfin, pour $λ∈(0,+∞)$, tout se passe comme si $Γε$ était plate, mais avec un coefficient de frottement plus élevé.

Accepted:
Published online:
DOI: 10.1016/j.crma.2010.07.023

Juan Casado-Díaz 1; Manuel Luna-Laynez 1; Francisco Javier Suárez-Grau 1

1 Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain
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Juan Casado-Díaz; Manuel Luna-Laynez; Francisco Javier Suárez-Grau. A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 967-971. doi : 10.1016/j.crma.2010.07.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.023/

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