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

TL;DR

This study investigates finite-time rupture in thin liquid films by numerically analyzing self-similar solutions across different disjoining pressure exponents, revealing a critical exponent below which stable rupture solutions vanish.

Contribution

It extends previous work by numerically computing self-similar rupture solutions for a range of disjoining pressure exponents, identifying a critical value where stability is lost.

Findings

Stable similarity solutions exist only for exponents above approximately 1.485.

Pairs of solution branches merge as the exponent decreases, leading to loss of stability.

Time-dependent simulations show the disappearance of stable solutions and emergence of oscillatory structures.

Abstract

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.