Asymptotic analysis of a micropolar fluid flow in a thin domain with a free and rough boundary

Motivated by lubrication problems, we consider a micropolar uid ow in a 2D domain with a rough and free boundary. We assume that the thickness and the roughness are both of order 0 < " << 1. We prove the existence and uniqueness of a solution of this problem for any value of " and we establish some a priori estimates. Then we use the two-scale convergence technique to derive the limit problem when " tends to zero. Moreover we show that the limit velocity and micro-rotation elds are uniquely determined via auxiliary well-posed problems and the limit pressure is given as the unique solution of a Reynolds equation.