Homogenization of a Stokes problem in a porous medium by the periodic unfolding method

Abstract

We consider the Stokes problem in a domain with holes periodically distributed with a period ε. The size of the holes is of the order of ε, a small parameter going to zero. On the boundary of the holes we prescribe a Robin-type condition depending on a parameter γ. The aim is to give the asymptotic behavior of the velocity and of the pressure of the fluid as ε goes to zero. The study for a problem of this type was done in Math. Meth. Appl. Sci. 19 (1996), 857–881, via classical homogenization methods. In this work we use the periodic unfolding method in perforated domains (see C. R. Acad. Sci. Paris, Série 1 342 (2006), 469–474; Portugaliae Mathematica 63(4) (2006), 467–496; Asymptotic Analysis 53(4) (2007), 209–235; in: Multiple Scales in Problems in Biomath., Mech., Physics and Numeric, Gakuto Int. Series, Math. Sci. App., Vol. 31, Gakkokotosho, 2009, pp. 37–68, and SIAM J. Math. Anal. 44(2) (2012), 718–760), which allows us to consider a general geometric framework. We give the limit problems corresponding to different values of γ (Darcy, Brinkmann or Stokes type problems).