Pressure-dipole solutions of the thin-film equation

Journal Article (Journal Article)

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.

Full Text

Duke Authors

Cited Authors

  • Bowen, M; Witelski, TP

Published Date

  • April 1, 2019

Published In

Volume / Issue

  • 30 / 2

Start / End Page

  • 358 - 399

Electronic International Standard Serial Number (EISSN)

  • 1469-4425

International Standard Serial Number (ISSN)

  • 0956-7925

Digital Object Identifier (DOI)

  • 10.1017/S095679251800013X

Citation Source

  • Scopus