Homogenization of the Stokes equations with high-contrast viscosity

Abstract This paper deals with the homogenization of the Stokes equations in a cylinder with varying viscosity and with Dirichlet boundary condition. The viscosity is equal to αe⪢1 in a e-periodic lattice of unidirectional cylinders of radius ere where re⪡1, and is equal to 1 elsewhere. In the critical regime defined by lime→0e2|lnre|∈]0,+∞[ and lime→0αere2∈]0,+∞], the limit problem is a coupled Stokes system satisfied by the limit velocity and the limit of the rescaled velocity in the cylinders, which can be read as a nonlocal law of Brinkman type. Moreover, if lime→0αere2=+∞, the limit of the rescaled velocity is equal to 0 and the Brinkman law is derived as in [G. Allaire, Arch. Rational Mech. Anal. 13 (1991) 209–259]. In the other regimes the homogenization leads either to classical Stokes problems or to a zero limit velocity. In the critical case the pressure is not bounded in L2 but only in H−1. Moreover, the pressure of the limit problem is not equal to the weak limit of the pressure in H−1.