Second order model in fluid film lubrication

The goal of this Note is to derive the second order model correcting the standard Reynolds equation for fluid film lubrication. Starting from microscopic model described by the Stokes system, we compute an asymptotic expansion for the solution. Instead of computing only the first term, as in the standard Reynolds approximation, we keep first two terms leading to the corrected model. We obtain equations similar to the Brinkman model for porous medium flow.