Homogenization of Reynolds equations and of some parabolic problems via Rothe's method

Sammanfattning: This PhD thesis is focussed on some problems of great interest in applied mathematics. More precisely, we investigate 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 euqipped with scales of Reynolds equations with rapidly oscillating coefficients. In particular, in this PhD thesis we derive the corresponding homogenized (averaged) equations. 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 addition we have successfully developed a reiterated homogenization (with three scales) procedure which makes it possible to efficiently study problems connected to hydrodynamic lubrication including shape, texture and roughness. Furthermore, we solve a linear parabolic initial-boundary value problem with singular coefficients in non-cylindrical domains. We accomplish this feat by developing a variant of Rothe's method to prove the existence and uniqueness of a weak solution to the parabolic problem. By combining the Rothe's method and the technique of two scale convergence we derive a homogenized equation for a linear parabolic problem with time dependent coefficients oscillating rapidly in the space variable. Moreover, we derive a concrete homogenization algorithm for giving a unique and computable approximation of the solution. In Chapter 1 we describe some 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 (varia- tional principle) for the unstationary Reynolds equation and use it to derive a homogenized minimization problem. Moreover, we obtain both the lower and upper bounds for the derived homogenized problem. Chapter 5 is devoted to studying the combined effect that arises due to shape, texture and surface roughness in hydrodynamic lubrication. This is accomplished by first studying a general class of problems that includes the incompressible Reynolds problem in both cartesian and cylindrical coordi- nate forms. In Chapter 6 we prove a homogenization result for the nonlinear equation $\mathrm{div}\,a(x,x/\varpeilson,x/\varepsilon^2,\nabla u_{\varepsilon})=\mathrm{div}\,b(x,x/\varpeilson,x/\varepsilon^2)$, where the coefficients are assumed to be periodic and a is monotone and continuous. This kind of problem has applications in hydrodynamic lubrication of surfaces with roughness on different length scales. In Chapter 7 a variant of Rothe's method is developed, discussed and used to prove existence and uniqueness result for linear parabolic problem with singular coefficients in non-cylindrical domains. In Chapter 8 we combine the Rothe method with a homogenization technique (two-scale convergence) to handle a general time-dependent lin- ear parabolic problem. In particular we prove that both the approximating sequence and the final approximate solution are unique. Finally, we derive a concrete homogenization algorithm on how to compute this homogenized solution.