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Stokes flow past a pair of stagnant-cap bubbles

Published online by Cambridge University Press:  26 April 2006

L. Lerner  and
J. F. Harper
L. Lerner
Affiliation:
Mathematics Department, Victoria University of Wellington, New Zealand Present address: Research School of Physical Sciences, Australian National University, GPO Box 4, Canberra, ACT 2601, Australia.
J. F. Harper
Affiliation:
Mathematics Department, Victoria University of Wellington, New Zealand
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Abstract

When gas bubbles rise in a surface-contaminated liquid, the upper parts of their surfaces may be almost free of contaminant and shear stress, while on the remainder of their surfaces there is enough contaminant to prevent tangential motion. Sadhal & Johnson solved the problem of Stokes flow of a uniform stream past a single spherical bubble. We extend their method to a single bubble in an arbitrary axially symmetric Stokes flow with the aid of an inversion theorem due to Harper. We also investigate in detail the interaction between two bubbles rising in line, and show how the methods can be made to deal with three or more bubbles, or a line of one or more bubbles rising towards a free surface. We show that a pair of bubbles will remain the same distance apart only if there is a certain relationship between the sizes of the caps on the bubbles. The cap sizes will normally be determined by convective diffusion of a surface-active solute from the bulk liquid in which the bubbles rise. The position of the lower bubble vertically under the upper one is stable to small horizontal displacements, but the upper bubble rises faster and so the distance apart gradually increases.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 232 , November 1991 , pp. 167 - 190
Copyright
© 1991 Cambridge University Press

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References

Adam, N. K. 1968 The Physics and Chemistry of Surfaces. Dover.
Bateman Manuscript Project 1954 Tables of Integral Transforms, vol. 2. McGraw-Hill.
Carslaw, H. S. & Jaeger, J. C. 1959 Conduction of Heat in Solids. Oxford University Press.
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.
Collins, W. D. 1961 On some dual integral series equations and their application to electrostatic problems for spheroidal caps. Proc. Camb. Phil. Soc. 57, 367–384.Google Scholar
Davis, A. M. J., O'Neill, M. E., Dorrepaal, J. M. & Ranges, K. B. 1976 Separation from the surface of two equal spheres in Stokes flow. J. Fluid Mech. 77, 625–644.Google Scholar
Davis, R. E. & Acrivos, A. 1966 The influence of surfactants on the creeping motion of bubbles. Chem. Engng Sci. 21, 681–685.Google Scholar
Gluckman, M. J., Pfeffer, R. & Weinbaum, S. 1971 A new technique for treating multiparticle slow viscous flow: axisymmetric flow past spheres and spheroids. J. Fluid Mech. 50, 705–740.Google Scholar
Harper, J. F. 1970 On bubbles rising in line at large Reynolds numbers. J. Fluid Mech. 41, 751–758.Google Scholar
Harper, J. F. 1972 The motion of bubbles and drops through liquids. Adv. Appl. Mech. 12, 59–129.Google Scholar
Harper, J. F. 1973 On bubbles with small adsorbed films rising in liquids at low Reynolds numbers. J. Fluid Mech. 58, 539–545.Google Scholar
Harper, J. F. 1974 On spherical bubbles rising slowly in dilute surfactant solutions. Q. J. Mech. Appl. Maths 27, 87–100.Google Scholar
Harper, J. F. 1982 Surface activity and bubble motion. Appl. Sci. Res. 38, 343–352.Google Scholar
Harper, J. F. 1983 Axisymmetric Stokes flow images in spherical surfaces with applications to rising bubbles. J. Austral. Math. Soc. B 25, 217–231.Google Scholar
Harper, J. F. & Dixon, J. N. 1974 The leading edge of a surface film on contaminated flowing water. Proc. 5th Australasian Conf. Hydraulics Fluid Mech. Christchurch, NZ, vol. 2, pp. 499–505.
Lerner, L. 1985 The interaction of surface-contaminated drops in Stokes flow. MSc thesis, Victoria University of Wellington, NZ.
LeVan, M. D. & Holbrook, J. A. 1989 Motion of a droplet containing surfactant. J. Colloid Interface Sci. 131, 242–251.Google Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.
Lighthill, M. J. 1950 Contributions to the theory of heat transfer through a laminar boundary layer. Proc. R. Soc. Lond. A 202, 359–377.Google Scholar
Nonweiler, T. R. F. 1984 Computational Mathematics. Chichester: Ellis Horwood and New York: Halsted Press.
Pais, A. 1982 ’Subtle is the Lord⃛’. Oxford University Press.
Payne, L. E. & Pell, W. H. 1960 The Stokes flow problem for a class of axially symmetric bodies. J. Fluid Mech. 7, 529–549.Google Scholar
Pruppacher, H. R. & Klett, J. D. 1978 Microphysics of Clouds and Precipitation. Reidel.
Rushton, E. & Davies, G. A. 1978 The slow motion of two spherical particles along their line of centers. Intl J. Multiphase Flow 4, 357–381.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, 237–250.Google Scholar
Sansone, G. 1959 Orthogonal Functions. Interscience.
Savic, P. 1953 Circulation and distortion of liquid drops falling through a viscous medium. Natl Res. Counc. Can., Div. Mech. Engng Rep. MT-22.
Wacholder, E. & Weihs, D. 1972 Slow motion of a fluid sphere in the vicinity of another sphere or a plane boundary. Chem. Engng Sci. 27, 1817–1828.Google Scholar