Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation

Abstract

We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth-order and destabilising second-order term. It has previously been shown that blow up is only possible when the exponent in the second-order term is above a certain critical value (dependent on the exponent in the fourth-order term). We show that in the limit that the critical value is approached from above, the primary branch of similarity profiles exhibits a well-defined structure consisting of a peak near the origin, and a thin, algebraically decaying tail, connected by an inner region equivalent (to leading order) to a generalised version of the Landau–Levich ‘drag-out’ problem in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between a parameter representing the height of the peak and the distance from the criticality threshold. The asymptotic results are supported by numerical computations found using continuation.