Dynamics of three-dimensional thin film rupture

We consider the problem of thin film rupture driven by van der Waals forces. A fourth-order nonlinear PDE governs the low Reynolds number lubrication model for a viscous liquid on a solid substrate. Finite-time singularities in this equation model rupture leading to formation of dry spots in the film. Our study addresses the problem of rupture in the full three-dimensional geometry. We focus on stability and selection of the dynamics determined by the initial conditions on small finite domains with planar and axisymmetric geometries. We also address the final stages of the dynamics — self-similar dynamics for point, line, and ring rupture. We will demonstrate that line and ring rupture are unstable and will generically destabilize to produce axisymmetric rupture at isolated points.