Similarity solutions for slender dry patches with thermocapillarity

D. Holland, S.K. Wilson, B.R. Duffy

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

We use the lubrication approximation to investigate slender dry patches in an infinitely
wide film of viscous fluid flowing steadily 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.
Similarity solutions describing a thermocapillary-driven flow with a dry patch that
is 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 the
solution contains a free parameter, and for each value of this parameter there is a unique solution representing both a narrowing pendent dry patch and a widening sessile dry patch, whose transverse profile has a monotonically increasing shape. When surface tension effects are strong the solution also contains a free parameter, and for each value
of this parameter there is both a unique solution representing a narrowing dry patch,
whose transverse profile has a monotonically increasing shape, and a one-parameter
family of solutions representing a widening dry patch, whose transverse profile has a
capillary ridge near the contact line and decays in an oscillatory manner far from it.
Similarity solutions are also obtained for both a gravity-driven and a constant surface-
shear-stress-driven flow with a dry patch that is widening or narrowing due to
thermocapillarity on a uniformly heated or cooled substrate. The solutions in both
cases contain a free parameter, and for each value of this parameter there is a unique
solution representing both a narrowing dry patch on a heated substrate and a widening
dry patch on a cooled substrate, whose transverse profile has a monotonically increasing
shape.
LanguageEnglish
Pages369-394
Number of pages26
JournalJournal of Engineering Mathematics
Volume44
Issue number4
DOIs
Publication statusPublished - Dec 2002

Fingerprint

Similarity Solution
Patch
Substrates
Substrate
Transverse
Surface tension
Gravitational effects
Surface Tension
Unique Solution
Thermal gradients
Lubrication
Shear stress
Gravitation
Lubrication Approximation
Contact Line
Fluids
Inclined
Ridge
Shear Stress
Viscous Fluid

Keywords

  • contact line
  • thermocapillarity
  • lubrication approximation
  • gravity
  • dry patch

Cite this

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title = "Similarity solutions for slender dry patches with thermocapillarity",
abstract = "We use the lubrication approximation to investigate slender dry patches in an infinitelywide film of viscous fluid flowing steadily on an inclined plane that is eitherheated or cooled relative to the surrounding atmosphere. Four non-isothermal situationsin which thermocapillary effects play a significant role are considered.Similarity solutions describing a thermocapillary-driven flow with a dry patch thatis widening or narrowing due to either gravitational or surface-tension effects on anon-uniformly heated or cooled substrate are obtained, and we present examples ofthese solutions when the substrate temperature gradient depends on the longitudinalcoordinate according to a general power law. When gravitational effects are strong thesolution contains a free parameter, and for each value of this parameter there is a unique solution representing both a narrowing pendent dry patch and a widening sessile dry patch, whose transverse profile has a monotonically increasing shape. When surface tension effects are strong the solution also contains a free parameter, and for each valueof this parameter there is both a unique solution representing a narrowing dry patch,whose transverse profile has a monotonically increasing shape, and a one-parameterfamily of solutions representing a widening dry patch, whose transverse profile has acapillary ridge near the contact line and decays in an oscillatory manner far from it.Similarity solutions are also obtained for both a gravity-driven and a constant surface-shear-stress-driven flow with a dry patch that is widening or narrowing due tothermocapillarity on a uniformly heated or cooled substrate. The solutions in bothcases contain a free parameter, and for each value of this parameter there is a uniquesolution representing both a narrowing dry patch on a heated substrate and a wideningdry patch on a cooled substrate, whose transverse profile has a monotonically increasingshape.",
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Similarity solutions for slender dry patches with thermocapillarity. / Holland, D.; Wilson, S.K.; Duffy, B.R.

In: Journal of Engineering Mathematics, Vol. 44, No. 4, 12.2002, p. 369-394.

Research output: Contribution to journalArticle

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