Pressure-dipole solutions of the thin-film equation

M. Bowen, T. P. Witelski

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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.

Original languageEnglish
Pages (from-to)358-399
Number of pages42
JournalEuropean Journal of Applied Mathematics
Issue number2
Publication statusPublished - 2019 Apr 1


  • Second-kind solutions
  • Sign-changing solutions
  • Similarity solutions
  • Thin-film equation

ASJC Scopus subject areas

  • Applied Mathematics


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