Do viscous flows slip?

Abstract

In this thesis, the Stokes equation is discussed and solved under different boundary conditions. The Stokes equation governs the flow of viscous liquids, for example honey or syrup. The first chapters in the thesis provides an introduction to multivector algebra and analysis, with the aim of presenting the concept of Hodge decompositions. With an application of this theory, the Stokes equation with the Hodge boundary conditions is solved using the finite element method. This is compared to the solution of the Stokes equation under the more standard no-slip condition. It is concluded that the Hodge boundary conditions are natural from a mathematical point of view, although they can not be used to model physical flows. In particular, they are contrary to the known physical fact that viscous flows tend to stick to the boundary. Moreover, it is showed that the Hodge boundary conditions can be interpreted in a way that the friction at the boundary of the domain is solely determined by the curvature.