We consider mathematical modeling of thin film flow between two rough surfaces which are in relative motion. For example such flows take place in different kinds of bearings and gears when a lubricant is used to reduce friction and wear between the surfaces. The mathematical foundations of lubrication theory is given by the Navier--Stokes equation, which describes the motion of viscous fluids. In thin domains several approximations are possible which lead to the so called Reynolds equation. This equation is crucial to describe the pressure in the lubricant film. When the pressure is found it is possible to predict vorous important physical quantities such as friction (stresses on the bounding surfaces), load carrying capacity and velocity field. In hydrodynamic lubrication the effect of surface roughness is not negligible, because in practical situations the amplitude of the surface roughness are of the same order as the film thickness. Moreover, a perfectly smooth surface does not exist in reality due to imperfections in the manufacturing process. Therefore, any realistic lubrication model should account for the effects of surface roughness. This implies that the mathematical modeling leads to partial differential equations with coefficients that will oscillate rapidly in space and time. A direct numerical computation is therefore very difficult, since an extremely dense mesh is needed to resolve the oscillations due to the surface roughness. A natural approach is to do some type of averaging. In this PhD thesis we use and develop modern homogenization theory to be able to handle the questions above. Especially, we use, develop and apply the method based on the multiple scale expansions and two-scale convergence. The thesis is based on five papers (A-E), with an appendix to paper A, and an extensive introduction, which puts these publications in a larger context. In Paper A the connection between the Stokes equation and the Reynolds equation is investigated. More precisely, the asymptotic behavior as both the film thickness <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cepsilon" /> and wavelength <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmu" /> of the roughness tend to zero is analyzed and described. Three different limit equations are derived. Time-dependent equations of Reynolds type are obtained in all three cases (Stokes roughness, Reynolds roughness and high frequency roughness regime). In paper C we extend the work done in Paper A where we compare the roughness regimes by numeric computations for the stationary case. In paper B we present a mathematical model that takes into account cavitation, surfaces roughness and compressibility of the fluid. We compute the homogenized coefficients in the case of unidirectional roughness.In the paper D we derive a mathematical model of thin film flow between two close rough surfaces, which takes into account cavitation, surface roughness and pressure dependent density. Moreover, we use two-scale convergence to homogenize the model. Finally, in paper E we prove the existence of solutions to a frequently used mathematical model of thin film flow, which takes cavitation into account.

Surfaces are rough, unless special care is taken to make them atomically smooth. Roughness exists at all scales, and any surface-producing operation affects the roughness in certain degrees, specific to the production process. When two surfaces are brought close to each other, contact is established at many isolated locations. The number and size of these contact islands depend on the applied load, material properties of the surfaces and the nature of roughness. These contact islands affect the tribological properties of the contacting surfaces. The real contact area, which is the sum total of the area of contacting islands, is much smaller than the apparent contact area dictated by the macroscopic geometry of the contacting surfaces. Since the total load is supported by these contact islands, the local contact pressure will be very high, and dependent on the local microscopic geometry of the roughness. Thus understanding the deformation behaviour of the rough surfaces will lead to better understanding of friction and wear properties of the surfaces. In this work, the interaction of these contact islands with each other is studied when two surfaces are in contact and sliding past each other. Asperities can be thought of as basic units of roughness. The geometry and the distribution of heights of asperities can be used to define the roughness. For example, one of the earliest models of roughness is that of hemispherical asperities carrying smaller hemispherical asperities on their back, which in turn carry smaller asperities, and soon. In the present study the asperities are assumed to be of uniform size, shape and distribution. Normal and tangential loading response of these asperities with a rigid indenter is studied through elastic-plastic plane strain finite element studies. As a rigid indenter is loaded onto a surface with a regular array of identical asperities, initial contact is established at a single asperity. The plastic zone is initially confined within the asperity. When the load is increased ,the elastic-plastic boundary moves towards the free surface of the asperity, and the contact pressure decreases. The geometry and spacing are determined when the neighbouring asperities come into contact. The plastic zone in these asperities is constrained, and hence contact pressure sustained by these asperities is larger. As the indentation progresses, more asperities come into contact in a similar way. If a tangential displacement is now applied to the indenter, the von Mises stress contours shift in the direction of indenter displacement. As the tangential displacement increases, the number of asperities in contact with the indenter decreases gradually before reaching a steady sliding state. The tangential sliding force experienced by the indenter arises from two components. One is the frictional resistance between the contacting surfaces and the other is due to the plastic deformation of the substrate. If the surface is completely elastic, it has been seen that the sliding force is purely due to the specified friction coefficient. For the smooth surface, as the subsurface makes the transition from purely elastic to confined plastic zone, plasticity breaks out on the free surface, hence the sliding force increases. For surfaces with asperities, even at very small load, the asperities deform plastically and hence the sliding force is considerably higher. The frictional force is experimentally measured by sliding a spherical indenter on smooth and rough surfaces. These experimental results are qualitatively compared with two dimensional finite element results. It has been observed that for rough surface, sliding force is considerablyhigherthanthesmoothsurface,asisobservedinsimu-lations at lower loads. In contrast to the simulations, the sliding force decreases at higher loads for both the smooth and rough surfaces.