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 h_t=- Del .(hⁿ Del Delta h- Del h^m) 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<n<3 but the techniques failed for n>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<m<2. We propose such behaviour as a cut off of the singular 'disjoining pressure' modelling long range van der Waals interactions. For all n>0 and 1<m<2, we discuss possible behaviour at the edge of the support of the solution via leading order asymptotic analysis of travelling wave solutions. This analysis predicts a certain 'competition' between the second- and fourth-order terms. We present rigorous weak existence theory for the above equation for all n>0 and 1<m<2. In particular, the presence of a second-order 'porous media' term in the above equation yields non-negative weak solutions that converge to their mean as t to infinity and that have additional regularity. Moreover, we show that there exists a time T* after which the weak solution is a positive strong solution. For n>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.