Kinetics and nucleation for driven thin film flow

The lubrication theory of thin liquid films, driven by a constant surface stress opposing gravity, is described by a scalar fourth order PDE for the film height h:ht+(h2−h3)x=−γ(h3hxxx)x, in which γ is a positive constant related to surface tension. In this paper, the wave structure of solutions observed in numerical simulations with γ>0 is related to the recent hyperbolic theory of the underlying scalar conservation law, in which γ=0. This theory involves a kinetic relation, describing possible undercompressive shocks, and a nucleation condition, governing the transition from classical to non-classical solution of the Riemann problem. The kinetic relation and nucleation condition are derived from consideration of traveling wave solutions (with γ>0). The kinetic relation is identified with a codimension-one bifurcation of the corresponding vector field, for which there is a traveling wave approximating an undercompressive shock. The nucleation condition is identified as a transition in the vector field at which there is no traveling wave connecting upstream and downstream heights. The thresholds defined by these conditions are incorporated into a Riemann solver map, which is tested for initial value problems for the full PDE. It is found that the parameter γ determines a limit to the applicability of the hyperbolic theory, in which the fourth order diffusion can dominate short-time transients, resulting in long-time convergence to the classical solution when the hyperbolic theory would predict a non-classical solution.