Homogenized elliptic equations and variational inequalities with oscillating parameters. Application to the study of thin flow behavior with rough surfaces

Double scale homogenization is used to average stationary equation or inequalities in which both highly oscillating variables and parameters appear. We demonstrate how the limit is obtained using a two-stage procedure, firstly by carrying out a classical homogenization process by freezing the oscillating parameter, then by averaging the result with respect to this parameter. These results allow us to average the pressure for a newtonian fluid in a narrow gap between two rough unstationary surfaces. Numerical results are given.