Asymptotic behavior of the Stokes problem with Robin condition in a domain with small holes

Abstract

We consider the Stokes problem in a domain ${\mathrm{\Omega }}_{\mathit{\epsilon }{\mathit{\delta }}_1{\mathit{\delta }}_2}$ of ${\mathbb{R}}^{\mathit{N}}$ , $\mathit{N}⩾3$ ε-periodically perforated by holes of size ${\mathit{r}}_1\mathrm{(}\mathit{\epsilon }{\mathit{\delta }}_1\mathrm{)}$ and ${\mathit{r}}_2\mathrm{(}\mathit{\epsilon }{\mathit{\delta }}_2\mathrm{)}$ (ε,  ${\mathit{\delta }}_1={\mathit{\delta }}_1\mathrm{(}\mathit{\epsilon }\mathrm{)}$ , ${\mathit{\delta }}_2={\mathit{\delta }}_2\mathrm{(}\mathit{\epsilon }\mathrm{)}$ being small parameters), with ${\mathit{r}}_1\mathrm{(}\mathit{\epsilon }{\mathit{\delta }}_1\mathrm{)}\mathrm{/}\mathit{\epsilon }\to 0$ and ${\mathit{r}}_2\mathrm{(}\mathit{\epsilon }{\mathit{\delta }}_2\mathrm{)}\mathrm{/}\mathit{\epsilon }\to 0$ as $\mathit{\epsilon }\to 0$ . Our aim is to describe the asymptotic behavior of the velocity and the pressure of the fluid as $\mathit{\epsilon }\to 0$ and give, if possible, a limit (“homogenized”) problem. To do so, we use the periodic unfolding method introduced by Cioranescu, Damlamian and Griso (C. R. Acad. Sci. Paris, Ser. I 335 (2002) 99–104; SIAM J. of Math. Anal. 40(4) (2008) 1585–1620). It allow us to consider a general geometric framework and so, to extend the results from Cioranescu, Donato and Ene (Math. Models and Methods in Appl. Sciences 19 (1996) 857–881) and Capatina and Ene (European J. of Math. 22 (2011) 333–345).