We investigate self-similar solutions of the dipole problem for the one-dimensional thin film equation on the half-line {x >= 0}. We study compactly supported solutions of the linear moving boundary problem and show how they relate to solutions of the nonlinear problem. The similarity solutions are generally of the second kind, given by the solution of a nonlinear eigenvalue problem, although there are some notable cases where first-kind solutions also arise. We examine the conserved quantities connected to these first-kind solutions. Difficulties associated with the lack of a maximum principle and the non-self-adjointness of the fundamental linear problem are also considered. Seeking similarity solutions that include sign changes yields a surprisingly rich set of ( coexisting) stable solutions for the intermediate asymptotics of this problem. Our results include analysis of limiting cases and comparisons with numerical computations.

We consider the evolution of a thin Viscous fluid sheet subject to thermocapillary effects. Using a lubrication approximation we find, for symmetric interfacial deflections, coupled evolution equations for the interfacial profile, the streamwise component of the fluid velocity and the temperature variation along the surface. Initial temperature profiles change the initial flow field through Marangoni-induced shear stresses. These changes then lead to preferred conditions For rupture prescribed by the initial temperature distribution. We show that the time to rupture may be minimized by varying the phase difference between the initial velocity profile and the initial temperature profile. For sufficiently large temperature differences, the phase difference between the initial velocity and temperature profiles determines the rupture location.

We consider the problem of a thin film driven by a thermal gradient with an opposing gravitational force. Under appropriate conditions, an advancing film front develops a leading undercompressive shock followed by a trailing compressive shock. Here, we investigate the nonlinear dynamics of these shock structures that describe a surprisingly stable advancing front. We compare two-dimensional simulations with linear stability theory, shock theory, and experimental results. The theory/experiment considers the propagation of information through the undercompressive shock towards the trailing compressive shock. We show that a local perturbation interacting with the undercompressive shock leads to nonlocal effects at the compressive shock. (c) 2005 Elsevier B.V. All rights reserved.

We investigate self-similar solutions of the thin film equation in the case of zero contact angle boundary conditions on a finite domain. We prove existence and uniqueness of such a solution and determine the asymptotic behaviour as the exponent in the equation approaches the critical value at which zero contact angle boundary conditions become untenable. Numerical and power-series solutions are also presented.

Alternating Direction Implicit (ADI) schemes are constructed for the solution of two-dimensional higher-order linear and nonlinear diffusion equations, particularly including the fourth-order thin film equation for surface tension driven fluid flows. First and second-order accurate schemes are derived via approximate factorization of the evolution equations. This approach is combined with iterative methods to solve nonlinear problems. Problems in the fluid dynamics of thin films are solved to demonstrate the effectiveness of the ADI schemes. (C) 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

This paper is based on a series of four lectures, by the first author, on thin films and moving contact lines. Section 1 presents an overview of the moving contact line problem and introduces the lubrication approximation. Section 2 summarizes results for positivity preserving schemes. Section 3 discusses the problem of films driven by thermal gradients with an opposing gravitational force. Such systems yield complex dynamics featuring undercompressive shocks. We conclude in Section 4 with a discussion of dewetting films.

We investigate similarity solutions of the thin lm equation. In particular we look at solutions on the half-line x greater than or equal to 0 with compact support and zero contact angle boundary conditions in x = 0. uch dipole solutions feature an anomalous exponent and are therefore called similarity solutions of the second kind. Using a combination of phase space analysis and numerical simulations, we numerically construct trajectories representing these solutions, at the same time obtaining broader insight into the nature of the four-dimensional phase space. Additional asymptotic analysis provides further information concerning the evolution to self-similarity.

A variety of mass preserving moving boundary problems for the thin film equation, u(t) = -(u(n)u(xxx))(x), are derived (by formal asymptotics) from a number of regularisations, the case in which the substrate is covered by a very thin pre-wetting film being discussed in most detail. Some of the properties of the solutions selected in this fashion are described, and the full range of possible mass preserving non-negative solutions is outlined.

We investigate the extinction behaviour of a fourth order degenerate diffusion equation in a bounded domain, the model representing the flow of a viscous fluid over edges at which zero contact angle conditions hold. The extinction time may be finite or infinite and we distinguish between the two cases by identification of appropriate similarity solutions. In certain cases, an unphysical mass increase may occur for early time and the solution may become negative; an appropriate remedy for this is noted. Numerical simulations supporting the analysis are included.