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

Michael Dallaston

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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)179-195
Number of pages17
JournalThe Quarterly Journal of Mechanics and Applied Mathematics
Volume72
Issue number2
Early online date4 Feb 2019
DOIs
Publication statusPublished - May 2019

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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. 72, no. 2, pp. 179-195. is available online at: https://dx.doi.org/10.1093/qjmam/hbz001

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

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