Draining of a 2D thin viscous film on a solid particle

Abstract

In this thesis we derive and solve equations governing the flow of a thin film on a bounded particle with any given geometry, provided it is not cornered or cusped. Our method is to solve the Navier-Stokes, continuity and free surface equations by using asymptotic expansions in powers of the ratio of film thickness to the appropriate length scale of the system using lubrication theory. In the first chapter (1) we describe some of the many industrial processes in which such flows are important, and summarize related work that has been carried out by other authors. In the second chapter (2) the governing lubrication equations and boundary conditions are derived for a simple flat film on an unbounded substrate. The calculations are then extended to a film on a particle using a local normal-tangential coordinate system with independent variables measured by arc length along the solid surface and by the distance normal to the surface. The governing (continuity and Navier-Stokes) equations, the free surface, and no-slip boundary conditions are then rescaled and expanded using lubrication theory in order to derive the final evolution equation for the film thickness (chapter 3). In the fourth chapter (4) particular example of the dynamics of a thin film on an elliptically shaped particle is considered. A MATLAB code is developed to simulate flow dynamics on the elliptical particles. A circle and five different ellipses have been chosen as the solid substrates and the evolution equation has been integrated over each one of them to evaluate the film thickness in time. Finally, we draw our conclusions and suggest further work in the final chapter (5).