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

S.K. Wilson, B.R. Duffy, S.H. Davis

Research output: Contribution to journalArticle

25 Citations (Scopus)

Abstract

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.
LanguageEnglish
Pages233-252
Number of pages20
JournalEuropean Journal of Applied Mathematics
Volume12
Issue number3
DOIs
Publication statusPublished - 2001

Fingerprint

Liquid films
Inclined
Patch
Gravity
Gravitation
Liquid
Apex
Surface tension
Surface Tension
Free Surface
Transverse
Predict
Contact Line
Similarity Solution
Ridge
Quartic
Unique Solution
Fluxes
Vertical
Decay

Keywords

  • applied mathematics
  • gravity
  • inclined plane
  • geometry

Cite this

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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.

In: European Journal of Applied Mathematics, Vol. 12, No. 3, 2001, p. 233-252.

Research output: Contribution to journalArticle

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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.

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