The lubrication approximation for thin viscous films: The moving contact line wih a “porous media” cut-off of van der Waals interactions. (English) Zbl 0811.35045

Summary: We consider the effect of a second-order ‘porous media’ term on the evolution of weak solutions of the fourth-order degenerate diffusion equation \[ h_ t = - \nabla \cdot (h^ n \nabla \Delta h - \nabla h^ m) \] 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.
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 \(t \to \infty\) 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’ behaviour at the edge of the support predicted by asymptotics.


35K55 Nonlinear parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K65 Degenerate parabolic equations
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