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Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film – exact solution

Published online by Cambridge University Press:  20 April 2006

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

In this investigation the creeping flow due to the motion of a liquid drop or a bubble in another immiscible fluid is examined when the interface is partially covered by a stagnant layer of surfactant. The associated boundary-value problem involves mixed boundary conditions at the interface, which lead to a set of dual series equations. An inversion of these equations yields the exact solution to the stagnant cap problem.

Several useful results are obtained in closed form. Among these are the expressions for the drag force, the difference between the maximum and the minimum interfacial tensions, and the amount of adsorbed surfactant. A shifting of the centre of the internal vortex is observed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 126 , January 1983 , pp. 237 - 250
Copyright
© 1983 Cambridge University Press

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References

Beitel, A. & Heideger, W. J. 1971 Surfactant effects on mass transfer from drops subject to interfacial instability Chem. Engng Sci. 26, 711.Google Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic.
Collins, W. D. 1961 On some dual series equations and their application to electrostatic problems for spheroidal caps Proc. Camb. Phil. Soc. 57, 367.Google Scholar
Davies, J. T. 1963 Mass transfer and interfacial phenomena Adv. Chem. Engng 4, 1.Google Scholar
Davis, R. E. & Acrivos, A. 1966 The influence of surfactants on creeping motion of bubbles Chem. Engng Sci. 21, 681.Google Scholar
Elzinga, E. R. & Banchero, J. T. 1961 Some observations on the mechanics of drops in liquid–liquid systems A.I.Ch.E. J. 7, 394.Google Scholar
Garner, F. H. & Skelland, A. H. P. 1955 Some factors affecting droplet behavior in liquid–liquid systems Chem. Engng Sci. 4, 149.Google Scholar
Gibbs, J. W. 1878 On the equilibrium of hetrogeneous substances. Trans. Conn. Acad. 3, 343 (in Scientific Papers, vol. 1, p. 272, Dover 1961).Google Scholar
Griffith, R. M. 1962 The effect of surfactants on the terminal velocity of drops and bubbles Chem. Engng Sci. 17, 1057.Google Scholar
Hadamard, J. 1911 Mouvement permanent lent d'une sphère liquide et visqueuse dans un liquide visqueux C.R. Acad. Sci. Paris 152, 1735.Google Scholar
Harper, J. F. 1972 The motion of drops and bubbles through liquids Adv. Appl. Mech. 12, 59.Google Scholar
Harper, J. F. 1973 On bubbles with small immobile adsorbed films rising in liquids at low Reynolds numbers J. Fluid Mech. 58, 539.Google Scholar
Harper, J. F. 1982 Surface activity and bubble motion. (Review paper presented at IUTAM Symp. on Mechanics and Physics of Bubbles in Fluids, Pasadena, 15–19 June 1981.) Appl. Sci. Res. 38, 343.Google Scholar
Horton, T. J., Fritsch, T. R. & Kintner, R. C. 1965 Experimental determination of circulation velocities inside drops Can. J. Chem. Engng 43, 143.Google Scholar
Huang, W. S. & Kintner, R. C. 1969 Effect of surfactants on mass transfer inside drops A.I.Ch.E. J. 15, 735.Google Scholar
Johnson, R. E. 1981 Stokes flow past a sphere coated with a thin fluid film J. Fluid Mech. 110, 217.Google Scholar
Levan, M. D. 1981 Motion of a droplet with a Newtonian interface J. Colloid Interface Sci. 83, 11.Google Scholar
Levan, M. D. & Newman, J. 1976 The effect of surfactant on terminal and interfacial velocities of a bubble or drop A.I.Ch.E. J. 22, 695.Google Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics, p. 395. Prentice-Hall.
Li, N. N. & Asher, W. J. 1973 Blood oxygenation by liquid membrane permeation Chem. Engng in Medicine, Adv. Chem. Ser. 118, 1.Google Scholar
Lucassen, J. & Giles, D. 1975 Dynamic surface properties of nonionic surfactant solutions J. Chem. Soc. Faraday Trans. 171, 217.Google Scholar
Mori, Y. H. 1978 Configurations of gas–liquid two-phase bubbles in immiscible liquid media Int. J. Multiphase Flow 4, 383.Google Scholar
Newman, J. 1967 Retardation of falling drops Chem. Engng Sci. 22, 83.Google Scholar
Payne, L. E. & Pell, W. H. 1960 The Stokes flow problem for a class of axially symmetric bodies J. Fluid Mech. 7, 529.Google Scholar
Rybczynski, W. 1911 On the translatory motion of a fluid sphere in a viscous medium (in German). Bull. Int. Acad. Pol. Sci. Lett. Cl. Sci. Math. Nat. Ser. A, 40–46.Google Scholar
Savic, P. 1953 Circulation and distortion of liquid drops falling through a viscous medium. Nat. Res. Counc. Can., Div. Mech. Engng Rep. MT-22.Google Scholar
Saville, D. A. 1973 The effects of interfacial tension gradients on the motion of bubbles or drops Chem. Engng J. 5, 251.Google Scholar
Schechter, R. S. & Farley, R. W. 1963 Interfacial tension gradients and droplet behavior Can. J. Chem. Engng 41, 103.Google Scholar
Sideman, S. & Taitel, Y. 1964 Direct-contact heat transfer with change of phase: evaporation of drops in an immiscible liquid medium. Int. J. Heat Mass Transfer 7, 1273.Google Scholar
Wasserman, M. L. & Slattery, J. C. 1969 Creeping flow past a fluid globule when a trace of surfactant is present A.I.Ch.E. J. 15, 533.Google Scholar