Thin Viscous Films on Curved Geometries

Abstract

The topic of this thesis is the evolution of thin viscous films on curved substrates. Using techniques from differential geometry, namely the exterior calculus of differential forms, and from optimization theory, in particular the theory of saddle point problems and the shape calculus, we reduce a variational form of the Stoke equations, which govern the flow, to a two dimensional optimization problem with a PDE constraint on the substrate. This reduction is analogous to the lubrication approximation of the classic thin film equation. We study the well-posedness of a, suitably regularised, version of this reduced model of the flow, using variational techniques. Furthermore, we study the well-posedness and convergence of time- and space-discrete versions of the model. The time discretization is based on the idea of the natural time discretization of a gradient flow, whereas the spatial discretization is done via suitably chosen finite element spaces. Finally, we present a particular implementation of the discrete scheme on subdivision surfaces, together with relevant numerical results.