### Abstract

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 language | English |
---|---|

Pages (from-to) | 1-42 |

Number of pages | 42 |

Journal | European Journal of Applied Mathematics |

DOIs | |

Publication status | Accepted/In press - 2018 Apr 2 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*European Journal of Applied Mathematics*, 1-42. https://doi.org/10.1017/S095679251800013X

**Pressure-dipole solutions of the thin-film equation.** / Bowen, Mark; WITELSKI, T. P.

Research output: Contribution to journal › Article

*European Journal of Applied Mathematics*, pp. 1-42. https://doi.org/10.1017/S095679251800013X

}

TY - JOUR

T1 - Pressure-dipole solutions of the thin-film equation

AU - Bowen, Mark

AU - WITELSKI, T. P.

PY - 2018/4/2

Y1 - 2018/4/2

N2 - 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.

AB - 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.

KW - Second-kind solutions

KW - Sign-changing solutions

KW - Similarity solutions

KW - Thin-film equation

UR - http://www.scopus.com/inward/record.url?scp=85044651543&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044651543&partnerID=8YFLogxK

U2 - 10.1017/S095679251800013X

DO - 10.1017/S095679251800013X

M3 - Article

SP - 1

EP - 42

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

ER -