Asymptotic Analysis of a Micropolar Fluid Flow in a Thin Domain with a Free and Rough Boundary

Abstract

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 $0<\varepsilon \ll 1$. We prove the existence and uniqueness of a solution of this problem for any value of $\varepsilon$, and we establish some a priori estimates. Then we use the two-scale convergence technique to derive the limit problem when $\varepsilon$ 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.