Abstract
This thesis presents a comparison of models for an incompressible fluid with negligible inertia: classical lubrication theory, extended and perturbed lubrication theories, and Stokes flow. In contrast with Stokes flow, lubrication theories are sensitive to large surface gradients and surface discontinuities, which cause the lubrication assumptions of a long and thin domain and a small scaled Reynolds number to break down. Here we present a linear time solution to the Reynolds equation of lubrication theory, which is an exact solution in the case of piecewise linear heights and a second order accurate approximation in general. We also present numerical finite difference methods for the various lubrication theories and Stokes flow. For a comparison of lubrication theories with Stokes flow, we consider a variety of two dimensional geometries, including a backward facing step (BFS), piecewise linear and smooth approximations to the BFS, and several triangular textures. We analyze the pressure and velocity solutions in these examples, and examine the extent to which lubrication theories reasonably approximate the solution of Stokes flow. We determine that the overall length scale ratio of the domain, and the magnitude of surface variation relative the height, are both important factors affecting the accuracy of the lubrication models.