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

Article type: Research Article

Authors: Karek, Chafia^a | Ould-Hammouda, Amar^{b; *}

Affiliations: [a] Mathematics Department, Faculty of Sciences, University 20th August 1955, Skikda, Algeria. E-mail: [email protected] | [b] Laboratory of Physics Mathematics and Applications, ENS, P.O. Box 92, 16050 Kouba, Algiers, Algeria. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We consider the Stokes problem in a domain Ωεδ1δ2 of RN, N⩾3 ε-periodically perforated by holes of size r1(εδ1) and r2(εδ2) (ε, δ1=δ1(ε), δ2=δ2(ε) being small parameters), with r1(εδ1)/ε→0 and r2(εδ2)/ε→0 as ε→0. Our aim is to describe the asymptotic behavior of the velocity and the pressure of the fluid as ε→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).

Keywords: Stokes problem, Robin condition, unfolding method

DOI: 10.3233/ASY-191530

Journal: Asymptotic Analysis, vol. 115, no. 3-4, pp. 147-167, 2019