Tribology using homogenization

Abstract: Most of physical and engineering problems are described by mathematical
models that involves partial differential equations PDE's. Because of the
structure of materials involved in this problems, some PDE's presents
rapidly oscillating coeffcients that makes them not easy to solve even
numerically.
Homogenization a technique used to transform this type of problems in ano-
ther problems easily solvable and that present a solution which is very
close to solution from original problem.

Since the fifties a very rapid progress in the development of numerical me-
thods has taken place within the engineering sciences in cooperation with
numerical analysts and mathematicians. In the light of this, homogeniza-
tion theory provides a good example of modern mathematics development
by mathematicians aware of recent numerical methods and with an interest
in questions in fields where classical mathematical methods have not been
successful in formulating problems in a form that suits existing numerical
techniques. In this thesis we consider homogenization theory, and the way it
can be applied to solve problems from Tribology.

This Master thesis is composed by five chapters: in the first chapter we
make an brief introduction of homogenization theory. The second chapter
we motivate the Reynolds equations by using notions of fluid motion. In the
chapter three we present the homogenization procedure for the case where
the fluid is incomprehensible considering the presence and absence of surface
roughness effect. Numerical methods to compare and show the behavior of
our solutions and compare the solutions from real problem with the solu-
tions from the homogenized problem are developed in chapter four. In the
chapter five we motivate the expression which describe the friction force
and we evaluate the effect of hydrodynamic pressure built from lubricant
flowing between two thin surfaces into the friction force.