Reiterated homogenization of a cavitation problem in thin films mechanics

Abstract

This paper deals with the coupling of two major problems in lubrication theory: cavitation phenomena and roughness of the surfaces in relative motion: cavitation is defined as the rupture of the continuous film due to the formation of air bubbles, leading to the presence of a liquid-gas mixture. For this, the Elrod-Adams model (which is a pressure-saturation model) is classically used to describe the behavior of a viscous cavitated flow in the lubrication framework. However, in practical situations, the surfaces of the devices are rough, due to manufacturing processes which induce defaults. Thus, we study the behavior of the solution, when highly oscillating roughness effects on the rigid surfaces occur. In particular, we deal with the reiterated homogenization of this elliptic-hyperbolic problem, using periodic unfolding methods. We define a homogenized problem in the most general case, pointing out the fact that it leads to a unusual form (when compared to the initial one). We also state that, under some assumptions on the roughness patterns, the diffculties vanish, leading to a well-posed homogenized problem. A numerical simulation evidences the behavior of the solution: although the pressure tends to a smooth one, the saturation oscillations are not damped. This does not prevent from defining an equivalent homogenized saturation but only points out the anisotropic effects on the saturation function in cavitated area