Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation

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Abstract

We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth-order and destabilising second-order term. It has previously been shown that blow up is only possible when the exponent in the second-order term is above a certain critical value (dependent on the exponent in the fourth-order term). We show that in the limit that the critical value is approached from above, the primary branch of similarity profiles exhibits a well-defined structure consisting of a peak near the origin, and a thin, algebraically decaying tail, connected by an inner region equivalent (to leading order) to a generalised version of the Landau–Levich ‘drag-out’ problem in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between a parameter representing the height of the peak and the distance from the criticality threshold. The asymptotic results are supported by numerical computations found using continuation.
Original languageEnglish
Pages (from-to)(In-press)
Number of pages17
JournalThe Quarterly Journal of Mechanics and Applied Mathematics
Volume(In-press)
Early online date4 Feb 2019
DOIs
Publication statusE-pub ahead of print - 4 Feb 2019

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Matched Asymptotics
Thin Film Equation
Asymptotic analysis
Asymptotic Analysis
Blow-up
Lubrication
Drag
exponents
Thin films
Fourth Order
Critical value
Term
lubrication
thin films
profiles
Exponent
drag
Criticality
Numerical Computation
Continuation

Bibliographical note

This is a pre-copyedited, author-produced version of an article accepted for publication in The Quarterly Journal of Mechanics and Applied Mathematics following peer review. The version of record [Dallaston, M 2019, 'Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation' The Quarterly Journal of Mechanics and Applied Mathematics, vol. (In-press), pp. (In-press)] is available online at https://dx.doi.org/10.1093/qjmam/hbz001

Cite this

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title = "Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation",
abstract = "We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth-order and destabilising second-order term. It has previously been shown that blow up is only possible when the exponent in the second-order term is above a certain critical value (dependent on the exponent in the fourth-order term). We show that in the limit that the critical value is approached from above, the primary branch of similarity profiles exhibits a well-defined structure consisting of a peak near the origin, and a thin, algebraically decaying tail, connected by an inner region equivalent (to leading order) to a generalised version of the Landau–Levich ‘drag-out’ problem in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between a parameter representing the height of the peak and the distance from the criticality threshold. The asymptotic results are supported by numerical computations found using continuation.",
author = "Michael Dallaston",
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