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Regular and complex singularities of the generalized thin film equation in two dimensions

Published online by Cambridge University Press:  23 April 2021

M.C. Dallaston ,
M.A. Fontelos ,
M.A. Herrada Open the ORCID record for M.A. Herrada [Opens in a new window] ,
J.M. Lopez-Herrera Open the ORCID record for J.M. Lopez-Herrera [Opens in a new window]  and
J. Eggers Open the ORCID record for J. Eggers [Opens in a new window]
M.C. Dallaston
Affiliation:
School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD4001, Australia
M.A. Fontelos
Affiliation:
Instituto de Ciencias Matemáticas, (ICMAT, CSIC-UAM-UCM-UC3M), Campus de Cantoblanco, 28049Madrid, Spain
M.A. Herrada
Affiliation:
E.S.I., Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Spain
J.M. Lopez-Herrera
Affiliation:
E.S.I., Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Spain
J. Eggers*
Affiliation:
School of Mathematics, University of Bristol, Fry Building, Woodland Road, BristolBS8 1UG, UK
*
†Email address for correspondence: [email protected]
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Abstract

We use a generalized version of the equation of motion for a thin film of liquid on a solid, horizontal substrate as a model system to study the formation of singularities in space dimensions greater than one. Varying both the exponent controlling long-ranged forces, as well as the exponent of the nonlinear mobility, we predict the structure of the singularity as the film thickness goes to zero. The spatial structure of rupture may be either ‘pointlike’ (approaching axisymmetry) or ‘quasi-one-dimensional’, in which case a one-dimensional singularity is unfolded into two or higher space dimensions. The scaling of the profile with time may be either strictly self-similar (the ‘regular’ case) or discretely self-similar and perhaps chaotic (the ‘irregular’ case). We calculate the phase boundaries between these regimes, and confirm our results by detailed comparisons with time-dependent simulations of the nonlinear thin film equation in two space dimensions.

JFM classification

Interfacial Flows (free surface): Thin films Nonlinear Dynamical Systems: Pattern formation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 917 , 25 June 2021 , A20
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Dallaston et al. supplementary movie

Time evolution of the inverse thickness as in Fig.12

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