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Title: Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows
Authors: Dallaston, Michael C.
Tseluiko, Dmitri
Zheng, Zhong
Fontelos, Marco
Kalliadasis, Serafim
Keywords: Self-similar behaviour
Finite-time singularity formation
Thin-film flows
Issue Date: 2017
Publisher: © IOP Publishing & London Mathematical Society
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.
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 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.
Description: This paper is closed access until 19th May 2018.
Sponsor: 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.
Version: Accepted for publication
DOI: 10.1088/1361-6544/aa6eb3
URI: https://dspace.lboro.ac.uk/2134/25372
Publisher Link: http://dx.doi.org/10.1088/1361-6544/aa6eb3
ISSN: 0951-7715
Appears in Collections:Closed Access (Maths)

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