Homogenization of Reynolds Equations

Abstract

This Licentiate thesis is focused on some new questions in homogenization. theory, which have been motivated by some concrete problems in tribology. From the mathematical point of view these questions are equipped with scales of Reynolds equations with rapidly oscillating coeffcients. In particular, in this Licentiate thesis we derive the corresponding homogenized (averaged) equation. We consider the Reynolds equations in both the stationary and unstationary forms to analyze the effect of surface roughness on the hydrodynamic performance of bearings when a lubricant is flowing through it. In Chapter 1 we describe the possible types of surfaces a bearing can take. Out of these we select two types and derive the appropriate Reynolds equations needed for their analysis. Chapter 2 is devoted to the derivation of the homogenized equations associated with the stationary forms of the compressible and incompressible Reynolds equations. We derive these homogenized equations by using the multiple scales expansion technique. In Chapter 3 the homogenized equations for the unstationary forms of the Reynolds equations are considered and some numerical results based on the homogenized equations are presented. In Chapter 4 we consider the equivalent minimization problem for the unstationary Reynolds equation and use it to derive a homogenized minimization problem. Finally, we obtain both the lower and upper bounds for the derived homogenized problem