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Pressure-dipole solutions of the thin-film equation

Published online by Cambridge University Press:  02 April 2018

M. BOWEN  and
T. P. WITELSKI
M. BOWEN
Affiliation:
International Center for Science and Engineering Programs, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan email: bowen@waseda.jp
T. P. WITELSKI
Affiliation:
Department of Mathematics, Duke University, 295 Physics Building, Box 90320, Durham, NC 27708-0320, USA email: witelski@math.duke.edu
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Abstract

We investigate self-similar sign-changing solutions to the thin-film equation, ht = −(|h|nhxxx)x, on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = hxx = 0 imposed at the origin. In particular, we identify classes of first- and second-kind compactly supported self-similar solutions (with a free-boundary x = s(t) = Ltβ) and consider how these solutions depend on the mobility exponent n; multiple solutions can exist with the same number of sign changes. For n = 0, we also construct first-kind self-similar solutions on the entire half-line x ≥ 0 and show that they act as limiting cases for sequences of compactly supported solutions in the limit of infinitely many sign changes. In addition, at n = 1, we highlight accumulation point-like behaviour of sign-changes local to the moving interface x = s(t). We conclude with a numerical investigation of solutions to the full time-dependent partial differential equation (based on a non-local regularisation of the mobility coefficient) and discuss the computational results in relation to the self-similar solutions.

Keywords

Thin-film equationSimilarity solutionsSecond-kind solutionsSign-changing solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Issue 2 , April 2019 , pp. 358 - 399
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†MB acknowledges the support of a Waseda University Grant for Special Research Projects (Project number: 2017B-161).

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