The Lubrication Approximation for Thin Viscous Films: the Moving Contact Line with a `Porous Media' Cut Off of Van der Waals Interactions

with A. Bertozzi, Nonlinearity, 7(1994)1535-1564.

Abstract

We consider the effect of a second order `porous media' term on the evolution of weak solutions of the fourth order degenerate diffusion equation

$\begin{displaymath} \qquad h_t=-\nabla\cdot(h^n \nabla\Delta h-\nabla h^m)\end{displaymath}$

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 modeling 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 nonnegative weak solutions with increasing support for 0<n<3 but the techniques failed for $n\geq 3$ . This is consistent with the fact that, in this case, nonnegative self-similar source-type solutions do not exist for $n\geq 3$ .

In this work, we discuss a physical justification for the `porous media' term when n=3 and 1<m<2. We propose such behavior as a cut off of the singular `disjoining pressure' modeling long range Van der Waals interactions.

For all n>0 and 1<m<2 we discuss possible behavior at the edge of the support of the solution via leading order asymptotic analysis of traveling wave solutions. This analysis predicts a certain `competition' between the second and fourth order terms. We present rigorous weak existence theory for equation (1) for all n>0 and 1<m<2. In particular, the presence of a second order `porous media' term in equation (1) yields nonnegative weak solutions that converge to their mean as $t \to \infty$ 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' behavior at the edge of the support predicted by asymptotics.

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Some of the computations presented in this paper were performed at the Advanced Computing Research Facility, Mathematics and Computer Science Division, Argonne National Laboratory.

Both authors are supported by NSF postdoctoral fellowships. AB is also partially supported by the Materials Research Laboratory at the Univ. of Chicago and the DOE and MP by NSF grant DMS-9305996.