Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows
File(s)
Accepted version
Author(s)
Dallaston, MC
Tseluiko, D
Zheng, Z
Fontelos, MA
Kalliadasis, S
Type
Journal Article
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 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, $\Pi(h)\propto -1/h^{3}$ , 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 (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 $n_c \approx 1.485$ . 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, $n_c^{{\rm plane}}\approx 1.499$ . Computation of the full time-dependent problem also demonstrates the loss of stable similarity solutions and the subsequent onset of cascading, increasingly small structures.
Date Issued
2017-05-19
Date Acceptance
2017-04-24
Citation
Nonlinearity, 2017, 30 (7), pp.2647-2666
ISSN
0951-7715
Publisher
IOP Publishing
Start Page
2647
End Page
2666
Journal / Book Title
Nonlinearity
Volume
30
Issue
7
Copyright Statement
© 2017 IOP Publishing Ltd & London Mathematical Society
Sponsor
Engineering & Physical Science Research Council (EPSRC)
Engineering & Physical Science Research Council (EPSRC)
Identifier
http://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000402413000004&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=1ba7043ffcc86c417c072aa74d649202
Grant Number
EP/K008595/1
EP/L020564/1
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Physics, Mathematical
Mathematics
Physics
self-similar behaviour
finite-time singularity formation
thin-film flows
LONG-WAVE INSTABILITIES
LIQUID-FILMS
VISCOUS FILMS
RUPTURE
DYNAMICS
STABILITY
MOTION
SIMULATION
EVOLUTION
DROPLET
Publication Status
Published