A homotopy continuation approach for analysing finite-time singularities in thin liquid films

摘要

We consider self-similar solutions related to rupture of thin liquid films on a solid substrate that evolve solely under the stabilizing influence of surface tension and the destabilizing influence of effective van der Waals interactions between the free surface of the film and the substrate. Such solutions have been previously analysed in the literature and various numerical approaches to obtain such solutions have been proposed. Such approaches are based either on shooting or finite-difference schemes and require well-chosen initial guesses for solutions. We propose an alternative numerical method, which is based on a homotopy approach and continuation techniques and allows one to reach self-similar solutions from analytically known small-amplitude steady solutions of the related thin-film equation. We argue that this method is more robust than previously proposed methods and does not require initial guesses to obtain solutions. Although the present study focuses on the particular case of self-similar solutions related to planar rupture that have square-root far-field behaviour, our approach can also be used to obtain planar solutions having a different far-field behaviour and radially symmetric self-similar solutions for the considered thin-film equation. We expect the approach to be also valid for other equations of similar type that show a subcritical primary bifurcation and finite-time singularities.