THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: THE MOVING CONTACT LINE WITH A 'POROUS MEDIA' CUT OFF OF VAN DER WAALS INTERACTIONS

We consider the effect of a second-order 'porous media' term on the evolution of weak solutions of the fourth-order degenerate diffusion equation ht=- Del .(hn Del Delta h- Del hm) in one space dimension. The equation without the second-order term is derived from a 'lubrication approximation' and models surface tension dominated motion of thin viscous films and spreading droplets. Here h (x,t) is the thickness of the film, and the physical problem corresponds to n=3. For simplicity, we consider periodic boundary conditions which has the physical interpretation of modelling a periodic array of droplets. In a previous work we studied the above equation without the second-order 'porous media' term. In particular we showed the existence of non-negative weak solutions with increasing support for 0 or=3. This is consistent with the fact that, in this case, non-negative self-similar source-type solutions do not exist for n>or=3. In this work, we discuss a physical justification for the 'porous media' term when n=3 and 1 0 and 1 0 and 1 3/2, we show that the regularity of the weak solutions is in exact agreement with that predicted by the asymptotics. Finally, we present several numerical computations of solutions. The simulations use a weighted implicit-explicit scheme on a dynamically adaptive mesh. The numerics suggest that the weak solution described by our existence theory has compact support with a finite speed of propagation. The data confirms the local 'power law' behaviour at the edge of the support predicted by asymptotics.