We study the lubrication process with incompressible fluid taking into account the dependence of the viscosity on the pressure. Assuming that the viscosity-pressure relation is given by the well-known Barus law, we derive an effective model using asymptotic analysis with respect to the film thickness. The key idea is to conveniently transform the governing system and then apply two-scale expansion technique.
"Asymptotic Modeling of the Thin Film Flow with a Pressure-Dependent Viscosity." J. Appl. Math. 2014 1 - 8, 2014. https://doi.org/10.1155/2014/217174