Exploration of Partial Differential Equations Governing the Motion of Ruptured Thin Films

Abstract

In this thesis we study the motion of a thin film after rupture. The thin film is modeled using the non-linear Brenner-Gueyffier equations. The derivation of Brenner-Gueyffier equations and relevant boundary conditions was reviewed. The thin films were simulated using central finite difference and the Crank-Nicolson method. The numerical simulations independently verified claims regarding the retraction speed of a thin film with finite length. The validity of Brenner-Gueyffier equations in approximating Navier-Stokes equations for the finite thin film is established in this study. Thin films with infinite lengths were also simulated, and some asymptotic predictions were verified. The equations were analyzed, and a hidden conserved quantity was discovered in the highly viscous limit. The discovery subsequently reduced the non-linear coupled equations into linear heat equations for the height profile of the thin film. A simple way of calculating the velocity profile from the height profile was also found. Furthermore, the velocity profile is shown to follow Burgers’ equation. Several new asymptotic predictions are made, and previous claims are reproduced from the analysis.