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Stokes flow past bubbles and drops partially coated with thin films. Part 2. Thin films with internal circulation – a perturbation solution

Published online by Cambridge University Press:  20 April 2006

Robert E. Johnson
Affiliation:
Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2983
S. S. Sadhal
Affiliation:
Department of Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453

Abstract

In the present study we examine the steady axisymmetric creeping flow due to the motion of a liquid drop or a bubble which is partially covered by a thin immiscible fluid layer or film. The analysis is based on the assumption that surface-tension forces are large compared with viscous forces which deform the drop, and that the circulation in the film is weak. The latter assumption is satisfied provided that the film-fluid viscosity is not too small. A perturbation scheme based on the thinness of the fluid layer is used to construct the solution.

One of the principal results is an expression for the drag force on the complex drop. We also find that the extent to which the drop or bubble is covered the film has a maximum value depending on the magnitude of the driving force on the film. In addition, we find the rather interesting result that when the ratio of the primary drop viscosity and bulk fluid viscosity is greater than ½, the circulation within the film may have a double-cell structure.

Type
Research Article
Copyright
© 1983 Cambridge University Press

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References

Collins, W. D. 1961 On some dual series equations and their application to electrostatic problems for spheroidal caps Proc. Camb. Phil. Soc. 57, 367384.Google Scholar
Dussan, V., E, B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Ann. Rev. Fluid Mech. 11, 371400.Google Scholar
Johnson, R. E. 1981 Stokes flow past a sphere coated with a thin fluid film J. Fluid Mech. 110, 217238.Google Scholar
Mori, Y. H. 1978 Configurations of gas-liquid two-phase bubbles in immiscible liquid media Int. J. Multiphase Flow 4, 383396.Google Scholar
Sadhal, S. S. & Johnson, R. E. 1983 Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film – exact solution J. Fluid Mech. 126, 237250.Google Scholar
Sneddon, I. N. 1966 Mixed Boundary Value Problems in Potential Theory. Wiley.
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number J. Fluid Mech. 18, 466476.Google Scholar
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