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 α ε ⪢1 in a ε-periodic lattice of unidirectional cylinders of radius εr ε where r ε ⪡1, and is equal to 1 elsewhere. In the critical regime defined by lim ε→0 ε 2|ln r ε |∈]0,+∞[ and lim ε→0 α ε r ε 2∈]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 lim ε→0 α ε r ε 2=+∞, 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 L 2 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.