### Abstract

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

Pages (from-to) | 411-439 |

Number of pages | 29 |

Journal | Quarterly Journal of Mechanics and Applied Mathematics |

Volume | 56 |

Issue number | 3 |

DOIs | |

Publication status | Published - Aug 2003 |

### Fingerprint

### Keywords

- thermocapillarity
- applied mathematics
- rivulets
- transverse profiles

### Cite this

}

*Quarterly Journal of Mechanics and Applied Mathematics*, vol. 56, no. 3, pp. 411-439. https://doi.org/10.1093/qjmam/56.3.411

**Similarity solutions for slender rivulets with thermocapillarity.** / Holland, D.; Wilson, S.K.; Duffy, B.R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Similarity solutions for slender rivulets with thermocapillarity

AU - Holland, D.

AU - Wilson, S.K.

AU - Duffy, B.R.

N1 - added document

PY - 2003/8

Y1 - 2003/8

N2 - We use the lubrication approximation to investigate the steady flow of slender non-uniform rivulets of a viscous fluid on an inclined plane that is either heated or cooled relative to the surrounding atmosphere. Four non-isothermal situations in which thermocapillary effects play a significant role are considered. We derive the general equations for a slender rivulet subject to gravity, surface tension, thermocapillarity and a constant surface shear stress. Similarity solutions describing a thermocapillary-driven rivulet widening or narrowing due to either gravitational or surface-tension effects on a non-uniformly heated or cooled substrate are obtained, and we present examples of these solutions when the substrate temperature gradient depends on the longitudinal coordinate according to a general power law. When gravitational effects are strong there is a unique solution representing both a narrowing pendent rivulet and a widening sessile rivulet whose transverse profile always has a single global maximum. When surface-tension effects are strong there is a one-parameter family of solutions representing both a narrowing and a widening rivulet whose transverse profile has either a single global maximum or two equal global maxima and a local minimum. Unique similarity solutions whose transverse profiles always have a single global maximum are also obtained for both a gravity-driven and a constant-surface-shear-stress-driven rivulet widening or narrowing due to thermocapillarity on a uniformly heated or cooled substrate. The solutions in both cases represent both a narrowing rivulet on a heated substrate and a widening rivulet on a cooled substrate (albeit with infinite width in the gravity-driven case).

AB - We use the lubrication approximation to investigate the steady flow of slender non-uniform rivulets of a viscous fluid on an inclined plane that is either heated or cooled relative to the surrounding atmosphere. Four non-isothermal situations in which thermocapillary effects play a significant role are considered. We derive the general equations for a slender rivulet subject to gravity, surface tension, thermocapillarity and a constant surface shear stress. Similarity solutions describing a thermocapillary-driven rivulet widening or narrowing due to either gravitational or surface-tension effects on a non-uniformly heated or cooled substrate are obtained, and we present examples of these solutions when the substrate temperature gradient depends on the longitudinal coordinate according to a general power law. When gravitational effects are strong there is a unique solution representing both a narrowing pendent rivulet and a widening sessile rivulet whose transverse profile always has a single global maximum. When surface-tension effects are strong there is a one-parameter family of solutions representing both a narrowing and a widening rivulet whose transverse profile has either a single global maximum or two equal global maxima and a local minimum. Unique similarity solutions whose transverse profiles always have a single global maximum are also obtained for both a gravity-driven and a constant-surface-shear-stress-driven rivulet widening or narrowing due to thermocapillarity on a uniformly heated or cooled substrate. The solutions in both cases represent both a narrowing rivulet on a heated substrate and a widening rivulet on a cooled substrate (albeit with infinite width in the gravity-driven case).

KW - thermocapillarity

KW - applied mathematics

KW - rivulets

KW - transverse profiles

U2 - 10.1093/qjmam/56.3.411

DO - 10.1093/qjmam/56.3.411

M3 - Article

VL - 56

SP - 411

EP - 439

JO - Quarterly Journal of Mechanics and Applied Mathematics

JF - Quarterly Journal of Mechanics and Applied Mathematics

SN - 0033-5614

IS - 3

ER -