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

摘要

We consider the Stokes problem in a domain Omega(epsilon)delta(1)delta(2) of R-N, N = 3 epsilon-periodically perforated by holes of size r(1)(epsilon delta(1)) and r(2)(epsilon delta(2)) (epsilon, delta(1) = delta(1) (epsilon), delta(2) = delta(2)(epsilon) being small parameters), with r(1)(epsilon delta(1))/epsilon - 0 and r(2)(epsilon delta(2))/epsilon - 0 as epsilon - 0. Our aim is to describe the asymptotic behavior of the velocity and the pressure of the fluid as 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).