Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows

A thin liquid film coating a planar horizontal substrate may be unstable to perturbations in the film thickness due to unfavourable intermolecular interactions between the liquid and the substrate, which may lead to finite-time rupture. The self-similar nature of the rupture has been studied before by utilizing the standard lubrication approximation along with the Derjaguin (or disjoining) pressure formalism used to account for the intermolecular interactions, and a particular form of the disjoining pressure with exponent $n=3$ has been used, namely, $\Pi(h)\propto -1/h^{3}$, where $h$ is the film thickness. In the present study, we use a numerical continuation method to compute discrete solutions to self-similar rupture for a general disjoining pressure exponent $n$. We focus on axisymmetric point-rupture solutions and show that pairs of solution branches merge as $n$ decreases, leading to a critical value $n_c \approx 1.485$ below which stable similarity solutions do not appear to exist. We verify that this observation also holds true for plane-symmetric line-rupture solutions for which the critical value turns out to be slightly larger than for the axisymmetric case, $n_c^{\mathrm{plane}}\approx 1.5$. Computation of the full time-dependent problem also demonstrates the loss of stable similarity solutions and the subsequent onset of cascading oscillatory structures.