A New Model of Diphasic Fluids in Thin Films

In this work, we are interested in the modelling of diphasic fluids flows in thin films. The diphasic aspect is described by a diffuse interface model, the Cahn-Hilliard equation. The specific geometry (thin domain) allows to replace heuristically the usual Navier-Stokes equations by an asymptotic approximation, a modified Reynolds equation (in which the pressure and the velocity are uncoupled), where the viscosity depends on the composition of the mixture. An existence result on the limit system is stated. since the boundary conditions are chosen in order to model the injection phenomenon, previous results on the Cahn-Hilliard equation cannot be applied, and new estimates have to be obtained. Moreover, we present numerical simulations for lubrication applications to improve the understanding of the cavitation phenomenon.