A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary

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

doi:10.1016/j.crma.2010.07.023

Partial Differential Equations/Mathematical Physics

A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary
[Fluide visqueux dans un domaine de faible épaisseur vérifiant la condition de glissement sur une frontière lègèrement rugueuse]

Présenté par : Evariste Sanchez-Palencia

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

¹ Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 967-971.

Résumé
Abstract

On considère un fluide visqueux de faible épaisseur ε sur un fond rugueux $Γ_{ε}$ , périodique de période $r_{ε}$ et amplitude $δ_{ε}$ , $δ_{ε} ≪ r_{ε} ≪ ε$ , 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 $(δ_{ε} \sqrt{ε}) / (r_{ε} \sqrt{r_{ε}})$ . Si $λ = + \infty$ , le fluide se comporte comme si on aurait imposé la condition d'adhérence sur $Γ_{ε}$ . Ceci justifie la condition usuelle pour un fluide visqueux. Si $λ = 0$ le fluide se comporte comme si $Γ_{ε}$ était plate. Enfin, pour $λ \in (0, + \infty)$ , tout se passe comme si $Γ_{ε}$ était plate, mais avec un coefficient de frottement plus élevé.

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

Reçu le : 2010-06-29
Accepté le : 2010-07-26
Publié le : 2010-08-11

DOI : 10.1016/j.crma.2010.07.023

Affiliations des auteurs :

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

¹ Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

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     author = {Casado-D{\'\i}az, Juan and Luna-Laynez, Manuel and Su\'arez-Grau, Francisco Javier},
     title = {A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {967--971},
     publisher = {Elsevier},
     volume = {348},
     number = {17-18},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.023},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2010.07.023/}
}

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Casado-Díaz, Juan; Luna-Laynez, Manuel; Suárez-Grau, Francisco Javier. A viscous fluid in a thin domain satisfying the slip condition on a slightly rough boundary. Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 967-971. doi : 10.1016/j.crma.2010.07.023. http://www.numdam.org/articles/10.1016/j.crma.2010.07.023/

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