Unstable van der Waals driven line rupture in Marangoni driven thin viscous films

An intriguing, dramatic and, at present, not fully understood instability often accompanies surfactant driven flows on thin films. This paper investigates a candidate mechanism that could create and drive this instability, van der Waals rupture, via numerical simulations coupled with analytical techniques. The spreading process itself is modelled with a pair of coupled evolution equations for the fluid film thickness and surfactant concentration that are derived in the lubrication approximation. These equations are then linearized about a base state that corresponds to the one-dimensional rupturing solution, and equations for the evolution of the transverse disturbances are derived. These linearized equations are investigated in several ways: numerical simulations where the perturbations are driven by the time evolving base state, or where the base state is frozen at a time t(f) close to the rupture event. The quasistatic initial value problem is also investigated as an eigenvalue problem, where the eigenvalue represents the quasistatic growth rate. We also take advantage of recent similarity scalings and results deduced for rupture, in the absence of surfactant, to motivate some of our numerical investigations. Additionally, we investigate the fully nonlinear equations including the transverse components. Perhaps interestingly, three-dimensional reconstructions of the film profile using the most dangerous mode from linear theory, as well as profiles from direct numerical simulations of the full nonlinear governing equations, that is, including interactions in the transverse direction, assume the form of finger-like patterns. (C) 2002 American Institute of Physics.