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Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows
journal contribution
posted on 2017-06-14, 09:03 authored by Michael C. Dallaston, Dmitri TseluikoDmitri Tseluiko, Zhong Zheng, Marco Fontelos, Serafim KalliadasisA 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
NonlinearityVolume
30Pages
2647 - 2666Citation
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 SocietyVersion
- AM (Accepted Manuscript)
Publication date
2017Notes
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
0951-7715eISSN
1361-6544Publisher version
Language
- en