Asymptotic analysis of the lubrication problem with nonstandard boundary conditions for microrotation

Abstract

The purpose of this paper is to propose a new effective model describing lubrication process with incompressible micropolar fluid. Instead of usual zero Dirichlet boundary condition for the microrotation, we consider more general (and physically justified) type of boundary condition at the fluid-solid interface, linking the velocity and microrotation through a so-called boundary viscosity. Starting from the linearized micropolar equations, we derive the second-order effective model by means of the asymptotic analysis with respect to the film thickness. The resulting equations, in the form of the Brinkman-type system, clearly show the influence of new boundary conditions on the effective flow. We also discuss the rigorous justification of the obtained asymptotic model.