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

We consider the Stokes problem in a domain with holes periodically distributed with a period epsilon. The size of the holes is of the order of epsilon, a small parameter going to zero. On the boundary of the holes we prescribe a Robin-type condition depending on a parameter gamma. The aim is to give the asymptotic behavior of the velocity and of the pressure of the fluid as epsilon 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, Serie 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 gamma (Darcy, Brinkmann or Stokes type problems).