### Abstract

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

Pages | 233-252 |

Number of pages | 20 |

Journal | European Journal of Applied Mathematics |

Volume | 12 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2001 |

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### Keywords

- applied mathematics
- gravity
- inclined plane
- geometry

### Cite this

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*European Journal of Applied Mathematics*, vol. 12, no. 3, pp. 233-252. https://doi.org/10.1017/S095679250100417X

**On a slender dry patch in a liquid film draining under gravity down an inclined plane.** / Wilson, S.K.; Duffy, B.R.; Davis, S.H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On a slender dry patch in a liquid film draining under gravity down an inclined plane

AU - Wilson, S.K.

AU - Duffy, B.R.

AU - Davis, S.H.

PY - 2001

Y1 - 2001

N2 - In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an infinitely wide liquid film draining under gravity down an inclined plane are obtained. The first solution, which predicts that the dry patch has a parabolic shape and that the transverse profile of the free surface always has a monotonically increasing shape, is appropriate for weak surface-tension effects and far from the apex of the dry patch. The second solution, which predicts that the dry patch has a quartic shape and that the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly vertical) and near (but not too close) to the apex of the dry patch. With the average volume flux per unit width (or equivalently with the uniform height of the layer far from the dry patch) prescribed, both solutions contain a free parameter. For each value of this parameter there is a unique solution in the first case and either no solution or a one-parameter family of solutions in the second case. The solutions capture some of the qualitative features observed in experiments.

AB - In this paper two similarity solutions describing a steady, slender, symmetric dry patch in an infinitely wide liquid film draining under gravity down an inclined plane are obtained. The first solution, which predicts that the dry patch has a parabolic shape and that the transverse profile of the free surface always has a monotonically increasing shape, is appropriate for weak surface-tension effects and far from the apex of the dry patch. The second solution, which predicts that the dry patch has a quartic shape and that the transverse profile of the free surface has a capillary ridge near the contact line and decays in an oscillatory manner far from it, is appropriate for strong surface-tension effects (in particular, when the plane is nearly vertical) and near (but not too close) to the apex of the dry patch. With the average volume flux per unit width (or equivalently with the uniform height of the layer far from the dry patch) prescribed, both solutions contain a free parameter. For each value of this parameter there is a unique solution in the first case and either no solution or a one-parameter family of solutions in the second case. The solutions capture some of the qualitative features observed in experiments.

KW - applied mathematics

KW - gravity

KW - inclined plane

KW - geometry

U2 - 10.1017/S095679250100417X

DO - 10.1017/S095679250100417X

M3 - Article

VL - 12

SP - 233

EP - 252

JO - European Journal of Applied Mathematics

T2 - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

IS - 3

ER -