Homogenization theory with applications in tribology

Abstract: Homogenization is a mathematical theory for studying differential equations with rapidly oscillating coefficents. Many important problems in physics with one or several microscopic length scales give rise to this kind of equations. Hence there is a need for methods that enable an efficient treatment of such problems. To this end several homogenization techniques exist, ranging from the fairly abstract ones to those that are more oriented towards applications. This thesis is concerned with two such methods, namely the "asymptotic expansion method", also known as the "method of multiple scales", and multiscale convergence. The former method, sometimes referred to as the "engineering approach to homogenization" has, due to its versatility and intutive appeal, gained wide acceptance and popularity in the applied fields. However, it is not rigorous by mathematical standards. Multiscale convergence, introduced by Nguetseng in 1989, is a notion of weak convergence in Lp spaces that is designed to take oscillations into account. Although not the most general method around, multiscale convergence has become widely used by homogenizers because of its simplicity. In spite of its success, the multiscale theory is not yet sufficiently developed to be used in connection with certain nonlinear problems with several microscopic scales. In Paper A we extend some previously obtained results in multiscale convergence that enable us to homogenize a nonlinear problem with three scales. In Appendix to Paper A we present in more detail some results that were used in the proof of some of the main theorems in Paper A. Tribology is the science of bodies in relative motion interacting through a mechanical contact. An important aspect of tribology is to explain the principles of friction, lubrication and wear. Tribological phenomena are encountered everywhere in nature and technology and have a huge economical impact on society. An important example is that of two sliding solid surfaces interacting through a thin film of viscous fluid (lubricant). Hydrodynamic lubrication occurs when the pressure generated within the lubricant, through the viscosity of the fluid, is able to sustain an externally applied load. Many common bearings, e.g. journal bearings or slider bearings, operate according to this principle. As a branch of fluid dynamics, the mathematical foundations of lubrication theory are given by the Navier-Stokes equations, describing the motion of a viscous fluid. Because of the thin film assumption several simplifications are possible, leading to various reduced equations named after Osborne Reynolds, the founding father of lubrication theory. The Reynolds equation is used by engineers to compute the pressure distribution in various situations of thin film lubrication. For extremely thin films, it has been observed that the surface micro topography is an important factor in hydrodynamic performance. Hence it is important to understand the influence of surface roughness with small characteristic wavelength upon the pressure solution. Since the 1980s such problems have been increasingly studied by homogenization theory. The idea is to replace the original equation with a homogenized equation where the roughness effects are "averaged out". One problem consists of finding an algorithm that gives the homogenized equation. Another problem, consists of showing, by introducing the appropriate mathematical defintions, that the homogenized equation really is the correct one. Papers B and C investigate the effects of surface roughness by means of multiscale expansion of the pressure in various situations of hydrodynamic lubrication. Paper B, for which Paper A constitutes a rigorous basis, considers homogenization of the stationary Reynolds equation and roughness with two characteristic wavelengths. This leads to a multiscale problem and adds to the complexity of the homogenization process. To compare the homogenized solution to the solution of the unaveraged Reynolds equation, some numerical examples are also included. Paper C is devoted to homogenization of a variational principle which is a generalization of the unstationary Reynolds equation (both surfaces are rough). The advantage of adopting the calculus of variations viewpoint is that the recently introduced "variational bounds" can be computed. Bounds can be seen as a "cheap" alternative to computing the realtively costly homogenized solution. Several numerical examples are included to illustrate the utility of bounds.