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

27 Citations (Scopus)
148 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)233-252
Number of pages20
JournalEuropean Journal of Applied Mathematics
Volume12
Issue number3
DOIs
Publication statusPublished - 2001

Keywords

  • applied mathematics
  • gravity
  • inclined plane
  • geometry

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