Similarity solutions for slender rivulets with thermocapillarity

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

Research output: Contribution to journalArticle

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117 Downloads (Pure)

Abstract

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).
Original languageEnglish
Pages (from-to)411-439
Number of pages29
JournalQuarterly Journal of Mechanics and Applied Mathematics
Volume56
Issue number3
DOIs
Publication statusPublished - Aug 2003

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Similarity Solution
Substrate
Surface Tension
Substrates
Surface tension
Gravity
Gravitation
interfacial tension
Transverse
gravitation
Shear Stress
shear stress
Shear stress
profiles
Gravitational effects
Lubrication Approximation
gravitational effects
steady flow
viscous fluids
Steady flow

Keywords

  • thermocapillarity
  • applied mathematics
  • rivulets
  • transverse profiles

Cite this

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abstract = "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).",
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Similarity solutions for slender rivulets with thermocapillarity. / Holland, D.; Wilson, S.K.; Duffy, B.R.

In: Quarterly Journal of Mechanics and Applied Mathematics, Vol. 56, No. 3, 08.2003, p. 411-439.

Research output: Contribution to journalArticle

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