THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS - THE MOVING CONTACT LINE WITH A POROUS-MEDIA CUTOFF OF VAN-DER-WAALS INTERACTIONS

We consider the effect of a second-order 'porous media' term on the ev olution of weak solutions of the fourth-order degenerate diffusion equ ation h(t) = -DEL . (h(n)DEL DELTAh - DELh(m)) in one space dimension. The equation without the second-order term is derived from a 'lubrica tion approximation' and models surface tension dominated motion of thi n viscous films and spreading droplets. Here h(x,t) is the thickness o f the film, and the physical problem corresponds to n = 3. For simplic ity, we consider periodic boundary conditions which has the physical i nterpretation of modelling a periodic array of droplets. In a previous work we studied the above equation without the second-order 'porous m edia' term. In particular we showed the existence of non-negative weak solutions with increasing support for 0 < n < 3 but the techniques fa iled for n greater-than-or-equal-to 3. This is consistent with the fac t that, in this case, non-negative self-similar source-type solutions do not exist for n greater-than-or-equal-to 3. In this work, we discus s 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 'dis joining 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 'competiti on' between the second- and fourth-order terms. We present rigorous we ak 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 t o their mean as t --> infinity and that have additional regularity. Mo reover, we show that there exists a time T after which the weak solut ion is a positive strong solution. For n > 3/2, we show that the regul arity 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 so lution described by our existence theory has compact support with a fi nite speed of propagation. The data confirms the local 'power law' beh aviour at the edge of the support predicted by asymptotics.