Comparison of Extended Lubrication Theories for Stokes Flow

Lubrication theory leverages the assumption of a long and thin fluid domain to formulate a linearized approximation to the Navier-Stokes equations. Models in extended lubrication theory consider relaxations of the thin film assumption, leading to the inclusion of higher order terms. However, such models are sensitive to large surface gradients which lead the assumptions of the model to break down. In this paper, we present a formulation of extended lubrication theory, and compare our models with several existing models, along with the numerical solution to the Stokes equations. The error in pressure and velocity is characterized for a variety of fluid domain geometries. Our results indicate that the new solution is suitable for a wide range of geometries. Nonetheless, the magnitude of surface variation and the length scale ratio are both important factors influencing the accuracy of the extended lubrication theory models.