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Pressure-dipole solutions of the thin-film equation

Publication ,  Journal Article
Bowen, M; Witelski, TP
Published in: European Journal of Applied Mathematics
April 1, 2019

We investigate self-similar sign-changing solutions to the thin-film equation, h t = -(|h| n h xxx ) x , on the semi-infinite domain x ≥ 0 with zero-pressure-type boundary conditions h = h xx = 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) = Lt β ) 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.

Duke Scholars

Published In

European Journal of Applied Mathematics

DOI

EISSN

1469-4425

ISSN

0956-7925

Publication Date

April 1, 2019

Volume

30

Issue

2

Start / End Page

358 / 399

Related Subject Headings

  • Applied Mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics
 

Citation

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Bowen, M., & Witelski, T. P. (2019). Pressure-dipole solutions of the thin-film equation. European Journal of Applied Mathematics, 30(2), 358–399. https://doi.org/10.1017/S095679251800013X
Bowen, M., and T. P. Witelski. “Pressure-dipole solutions of the thin-film equation.” European Journal of Applied Mathematics 30, no. 2 (April 1, 2019): 358–99. https://doi.org/10.1017/S095679251800013X.
Bowen M, Witelski TP. Pressure-dipole solutions of the thin-film equation. European Journal of Applied Mathematics. 2019 Apr 1;30(2):358–99.
Bowen, M., and T. P. Witelski. “Pressure-dipole solutions of the thin-film equation.” European Journal of Applied Mathematics, vol. 30, no. 2, Apr. 2019, pp. 358–99. Scopus, doi:10.1017/S095679251800013X.
Bowen M, Witelski TP. Pressure-dipole solutions of the thin-film equation. European Journal of Applied Mathematics. 2019 Apr 1;30(2):358–399.
Journal cover image

Published In

European Journal of Applied Mathematics

DOI

EISSN

1469-4425

ISSN

0956-7925

Publication Date

April 1, 2019

Volume

30

Issue

2

Start / End Page

358 / 399

Related Subject Headings

  • Applied Mathematics
  • 4901 Applied mathematics
  • 0102 Applied Mathematics