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The energetics of the breakup of a sheet and of a rivulet on a vertical substrate in the presence of a uniform surface shear stress

Published online by Cambridge University Press:  18 March 2011

S. K. WILSON ,
J. M. SULLIVAN  and
B. R. DUFFY
S. K. WILSON*
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK
J. M. SULLIVAN
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK
B. R. DUFFY
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, UK
*
†Email address for correspondence: [email protected]
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Abstract

We use the lubrication approximation to obtain a complete description of the energetics of the breakup both of a fluid sheet of uniform thickness into a periodic array of infinitely many identical thin rivulets and of a single thin rivulet into one or more identical sub-rivulets on a vertical substrate in the presence of a prescribed uniform longitudinal shear stress at the free surface of the fluid by comparing the total energies of the different states. For both problems the situation when the volume flux is positive is relatively straightforward (and, in particular, qualitatively the same as that in the case of no prescribed shear stress), but when the volume flux is negative it is more complicated. However, whatever the value of the prescribed shear stress, there is always a critical thickness below which it is energetically favourable for a sheet to break up into rivulets and a critical semi-width below which it is energetically favourable for a rivulet to remain as a single rivulet, and a critical thickness above which it is energetically favourable for a sheet to remain as a sheet and a critical semi-width above which it is energetically favourable for a rivulet to break up into sub-rivulets.

JFM classification

Interfacial Flows (free surface): Capillary flows Low-Reynolds-number flows: Lubrication theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 281 - 306
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Bankoff, S. G. 1971 Minimum thickness of a draining liquid film. Intl J. Heat Mass Transfer 14, 2143–2146.CrossRefGoogle Scholar
Chung, J. C. & Bankoff, S. G. 1980 Initial breakdown of a heated liquid film in cocurrent two-component annular flow: II. rivulet and drypatch models. Chem. Engng Commun. 4, 455–470.CrossRefGoogle Scholar
Doniec, A. 1988 Flow of a laminar liquid film down a vertical surface. Chem. Engng Sci. 43, 847–854.CrossRefGoogle Scholar
Doniec, A. 1991 Laminar flow of a liquid rivulet down a vertical solid surface. Can. J. Chem. Engng 69, 198–202.CrossRefGoogle Scholar
El-Genk, M. S. & Saber, H. H. 2001 Minimum thickness of a flowing down liquid film on a vertical surface. Intl J. Heat Mass Transfer 44, 2809–2825.CrossRefGoogle Scholar
El-Genk, M. S. & Saber, H. H. 2002 An investigation of the breakup of an evaporating liquid film, falling down a vertical, uniformly heated wall. J. Heat Transfer 124, 39–50.CrossRefGoogle Scholar
Hartley, D. E. & Murgatroyd, W. 1964 Criteria for the break-up of thin liquid layers flowing isothermally over solid surfaces. Intl J. Heat Mass Transfer 7, 1003–1015.CrossRefGoogle Scholar
Hobler, T. 1964 Minimum surface wetting. Chemia Stosowana 2B, 145–159 (in Polish).Google Scholar
Hughes, D. T. & Bott, T. R. 1998 Minimum thickness of a liquid film flowing down a vertical tube. Intl J. Heat Mass Transfer 41, 253–260.CrossRefGoogle Scholar
Mikielewicz, J. & Moszynski, J. R. 1976 Minimum thickness of a liquid film flowing vertically down a solid surface. Intl J. Heat Mass Transfer 19, 771–776.CrossRefGoogle Scholar
Mikielewicz, J. & Moszynski, J. R. 1978 An improved analysis of breakdown of thin liquid films. Arch. Mech. 30, 489–500.Google Scholar
Myers, T. G., Liang, H. X. & Wetton, B. 2004 The stability and flow of a rivulet driven by interfacial shear and gravity. Intl J. Non-Linear Mech. 39, 1239–1249.CrossRefGoogle Scholar
Robertson, A. C., Taylor, I. J., Wilson, S. K., Duffy, B. R. & Sullivan, J. M. 2010 Numerical simulation of rivulet evolution on a horizontal cable subject to an external aerodynamic field. J. Fluids Struct. 26, 50–73.CrossRefGoogle Scholar
Saber, H. H. & El-Genk, M. S. 2004 On the breakup of a thin liquid film subject to interfacial shear. J. Fluid Mech. 500, 113–133.CrossRefGoogle Scholar
Schmuki, P. & Laso, M. 1990 On the stability of rivulet flow. J. Fluid Mech. 215, 125–143.CrossRefGoogle Scholar
Sullivan, J. M., Wilson, S. K. & Duffy, B. R. 2008 A thin rivulet of perfectly wetting fluid subject to a longitudinal surface shear stress. Q. J. Mech. Appl. Math. 61, 25–61.CrossRefGoogle Scholar
Wilson, S. K. & Duffy, B. R. 2005 a When is it energetically favorable for a rivulet of perfectly wetting fluid to split? Phys. Fluids 17, 078104.CrossRefGoogle Scholar
Wilson, S. K. & Duffy, B. R. 2005 b Unidirectional flow of a thin rivulet on a vertical substrate subject to a prescribed uniform shear stress at its free surface. Phys. Fluids 17, 108105.CrossRefGoogle Scholar