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

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posted on 14.06.2017, 09:03 by Michael C. Dallaston, Dmitri TseluikoDmitri Tseluiko, Zhong Zheng, Marco Fontelos, Serafim Kalliadasis
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 finitetime rupture. The self-similar nature of the rupture has been studied before by utilising 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, Π(h) ∝ −1/h3, 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 (not necessarily equal to 3), which has not been previously performed. We focus on axisymmetric point-rupture solutions and show for the first time that pairs of solution branches merge as n decreases, starting at nc ≈ 1.485. 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, nplane c ≈ 1.499. Computation of the full time-dependent problem also demonstrates the loss of stable similarity solutions and the subsequent onset of cascading, increasingly small structures.

Funding

We acknowledge financial support from the Engineering and Physical Sciences Research Council (EPSRC) of the UK through Grants No. EP/K008595/1 and EP/L020564/1. The work of DT was partly supported by the EPSRC through Grant No. EP/K041134/1.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Nonlinearity

Volume

30

Pages

2647 - 2666

Citation

DALLASTON, M.C. ... et al, 2017. Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows. Nonlinearity, 30 (7), pp. 2647-2666.

Publisher

© IOP Publishing & London Mathematical Society

Version

AM (Accepted Manuscript)

Publication date

2017

Notes

This is an author-created, un-copyedited version of an article published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1361-6544/aa6eb3.

ISSN

0951-7715

eISSN

1361-6544

Language

en