### Abstract

Language | English |
---|---|

Pages | 365-402 |

Number of pages | 38 |

Journal | Quarterly Journal of Mechanics and Applied Mathematics |

Volume | 62 |

Issue number | 4 |

Early online date | 26 Jun 2009 |

DOIs | |

Publication status | Published - Nov 2009 |

### Fingerprint

### Keywords

- volatile liquid droplets
- similarity solutions
- solid surface
- contact line
- films
- evaporation
- drops
- thermocapillarity
- instability

### Cite this

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*Quarterly Journal of Mechanics and Applied Mathematics*, vol. 62, no. 4, pp. 365-402. https://doi.org/10.1093/qjmam/hbp014

**Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate.** / Dunn, G.J.; Duffy, B.R.; Wilson, S.K.; Holland, D.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate

AU - Dunn, G.J.

AU - Duffy, B.R.

AU - Wilson, S.K.

AU - Holland, D.

PY - 2009/11

Y1 - 2009/11

N2 - We investigate theoretically the problem of the quasi-steady spreading or contraction of a thin two-dimensional sessile or pendent ridge of viscous fluid with temperature-dependent surface tension on a planar horizontal substrate that is uniformly heated or cooled relative to the atmosphere. We derive an implicit solution of the leading-order thin-film equation for the free-surface profile of the ridge and use this to examine the quasi-steady evolution of the ridge, the dynamics of the moving contact lines being modelled by a 'Tanner law' relating the velocity of the contact line to the contact angle; in particular, we obtain a complete description of the possible forms that the evolution may take. In both the case of a (sessile or pendent) ridge on a heated substrate and the case of a pendent ridge on a cooled substrate when gravitational effects are relatively weak, there is one stable final state to which the ridge may evolve. In the case of a pendent ridge on a cooled substrate when gravitational effects are stronger, there may be one or two stable final states; moreover, the contact angles may vary non-monotonically with time during the evolution to one of these states. In the case of a pendent ridge on a cooled substrate when gravitational effects are even stronger, there may be up to three stable final states with qualitatively different solutions; moreover, the ridge may evolve via an intermediate state from which quasi-steady motion cannot persist, and so there will be a transient non-quasi-steady adjustment (in which the contact angles change rapidly, with the positions of the contact lines unaffected), after which quasi-steady motion is resumed. Lastly, we consider the behaviour of the ridge in the asymptotic limits of strong heating or cooling of the substrate and of strong or weak gravitational effects.

AB - We investigate theoretically the problem of the quasi-steady spreading or contraction of a thin two-dimensional sessile or pendent ridge of viscous fluid with temperature-dependent surface tension on a planar horizontal substrate that is uniformly heated or cooled relative to the atmosphere. We derive an implicit solution of the leading-order thin-film equation for the free-surface profile of the ridge and use this to examine the quasi-steady evolution of the ridge, the dynamics of the moving contact lines being modelled by a 'Tanner law' relating the velocity of the contact line to the contact angle; in particular, we obtain a complete description of the possible forms that the evolution may take. In both the case of a (sessile or pendent) ridge on a heated substrate and the case of a pendent ridge on a cooled substrate when gravitational effects are relatively weak, there is one stable final state to which the ridge may evolve. In the case of a pendent ridge on a cooled substrate when gravitational effects are stronger, there may be one or two stable final states; moreover, the contact angles may vary non-monotonically with time during the evolution to one of these states. In the case of a pendent ridge on a cooled substrate when gravitational effects are even stronger, there may be up to three stable final states with qualitatively different solutions; moreover, the ridge may evolve via an intermediate state from which quasi-steady motion cannot persist, and so there will be a transient non-quasi-steady adjustment (in which the contact angles change rapidly, with the positions of the contact lines unaffected), after which quasi-steady motion is resumed. Lastly, we consider the behaviour of the ridge in the asymptotic limits of strong heating or cooling of the substrate and of strong or weak gravitational effects.

KW - volatile liquid droplets

KW - similarity solutions

KW - solid surface

KW - contact line

KW - films

KW - evaporation

KW - drops

KW - thermocapillarity

KW - instability

UR - http://qjmam.oxfordjournals.org/content/by/year

U2 - 10.1093/qjmam/hbp014

DO - 10.1093/qjmam/hbp014

M3 - Article

VL - 62

SP - 365

EP - 402

JO - Quarterly Journal of Mechanics and Applied Mathematics

T2 - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

IS - 4

ER -