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 fluid flow in a two-dimensional domain with a rough and free boundary. We assume that the thickness and the roughness are both of order 0 < epsilon << 1. We prove the existence and uniqueness of a solution of this problem for any value of epsilon, and we establish some a priori estimates. Then we use the two-scale convergence technique to derive the limit problem when e tends to zero. Moreover we show that the limit velocity and microrotation fields are uniquely determined via auxiliary well-posed problems and the limit pressure is given as the unique solution of a Reynolds equation.