Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-01-07T18:57:06.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Particle migration in suspensions by thermocapillary or electrophoretic motion

Published online by Cambridge University Press:  26 April 2006

A. Acrivos ,
D. J. Jeffrey  and
D. A. Saville
A. Acrivos
Affiliation:
The Levich Institute, Steinman 202, City College of New York. New York, NY 10031, USA
D. J. Jeffrey
Affiliation:
Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada N6A 5B9
D. A. Saville
Affiliation:
Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Two problems of similar mathematical structure are studied: the thermocapillary motion of bubbles and the electrophoresis of colloidal particles. The thermocapillary motion induced in a cloud of bubbles by a uniform temperature gradient is investigated under the assumptions that the bubbles are all the same size that the surface tension is high enough to keep the bubbles spherical, and that the bubbles are non-conducting. In the electrophoresis problem, the particles, identical spheres having a uniform zeta potential, are suspended in an electrolyte under conditions that make the diffuse charge cloud around each particle small when compared with the particle radius. For both problems, it is shown that in a cloud of n particles surrounded by an infinite expanse of fluid, the velocity of each sphere under creeping flow conditions is equal to the velocity of an isolated particle, unchanged by interactions between the particles. However, when the cloud fills a container, conservation of mass shows that this result cannot continue to hold, and the average translational velocity must be calculated subject to a constraint on the mass flux. The computation requires ‘renormalization’, but it is shown that the renormalization procedure is ambiguous in both problems. An extension of Jeffrey's (1974) second group expansion, together with the constraint of conservation of mass, removes the ambiguity. Finally, it is shown that the average thermocapillary or electrophoretic translational velocity of a particle in the cloud is related to the effective conductivity of the cloud over the whole range of particle volume fractions, provided that the particles are identical, non-conducting and, for the thermocapillary problem, inviscid.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 212 , March 1990 , pp. 95 - 110
Copyright
© 1990 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, J. L.: 1985a Droplet interactions in thermocapillary motion. Intl J. Multiphase Flow 11, 813–824.Google Scholar
Anderson, J. L.: 1985b Effect of a non-uniform zeta potential on particle movement in electric fields. J. Colloid Interface Sci. 105, 45–54.Google Scholar
Anderson, J. L.: 1986 Transport mechanisms of biological colloids. Ann. NY Acad. Sci. 469, 166–177.Google Scholar
Balasubramaniam, R. & Chai, An-Ti 1987 Thermocapillary migration of droplets: an exact solution for small Marangoni numbers. J. Colloid Interface Sci. 119, 531–538.Google Scholar
Batchelor, G. K.: 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52, 245–268.Google Scholar
Batchelor, G. K.: 1976 Developments in microhydrodynamics. In Theoretical and Applied Mechanics (ed. W. T. Koiter), pp. 33–55. North-Holland.
Batchelor, G. K. & Green, J. T., 1972 The determination of the bulk stress in a suspension of spherical particles to order c2. J. Fluid Mech. 56, 401–427.Google Scholar
Burgers, J. M.: 1942 On the influence of the concentration of a suspension upon the sedimentation velocity. Proc. Kon. Nederl. Akad. Wet. 44, 1045, 1177; and 45, 9, 126.Google Scholar
Chen, H. S. & Acrivos, A., 1978 The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Intl J. Solids Structures 14, 349–364.Google Scholar
Childress, S.: 1972 Viscous flow past a random array of spheres. J. Chem. Phys. 56, 2527.Google Scholar
Crespo, A. & Manuel, F., 1983 Bubble motion under reduced gravity. In Proc. 4th European Symp. on Materials Science under Microgravity, Madrid, Spain: ESA SP.-191
Einstein, A.: 1906 Eine neue Bestimmung der Molekuldimension. Ann. Phys. 19, 289.Google Scholar
Felderhof, B. U., Ford, G. W. & Cohen, E. D. G. 1982 Cluster expansion for the dielectric constant of a polarizable suspension. J. Statist. Phys. 28, 135–164.Google Scholar
Feuillebois, F.: 1989 Thermocapillary migration of two equal bubbles parallel to their line of centres. J. Colloid Interface Sci. 131, 267–274.Google Scholar
Goddard, J. D.: 1977 Advances in the rheology of particulate dispersions. In Continuum Models of Discrete Systems (ed. J. W. Provan), pp. 605–634. University of Waterloo Press.
Hashin, Z. & Shtrikman, S., 1963 A variational approach to the theory of the effective magnetic permeability of multiphase materials. J. Appl. Phys. 33, 1514–1517.Google Scholar
Hinch, E. J.: 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695–720.Google Scholar
Howells, I. D.: 1974 Drag due to motion of a Newtonian fluid through a sparse random array of small fixed rigid objects. J. Fluid Mech. 64, 449–475.Google Scholar
Hunter, R. J.: 1981 Zeta Potential in Colloid Science. Academic.
Jeffrey, D. J.: 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. A 335, 355–367.Google Scholar
Jeffrey, D. J.: 1974 Group expansions for the bulk properties of a statistically homogeneous, random suspension. Proc. R. Soc. Lond. A 338, 503–516.Google Scholar
Jeffrey, D. J.: 1977 The physical significance of non-convergent integrals in expressions for effective transport properties. In Continuum Models of Discrete Systems (ed. J. W. Provan), pp. 653–674. University of Waterloo Press.
McCoy, J. J. & Beran, M., 1976 Effective thermal conductivity of a random suspension of spheres. Intl J. Engng Sci. 14, 7–18.Google Scholar
Meyyappan, M. & Subramanian, R. S., 1984 The thermocapillary motion of two bubbles oriented arbitrarily relative to a thermal gradient. J. Colloid Interface Sci. 97, 291–294.Google Scholar
Meyyappan, M., Wilcox, W. R. & Subramanian, R. S., 1983 The slow axisymmetric motion of two bubbles in a thermal gradient. J. Colloid Interface Sci. 94, 243–257.Google Scholar
O'Brien, R. W.: 1979 A method for the calculation of the effective transport properties of suspensions of interacting particles. J. Fluid Mech. 91, 17–39.Google Scholar
Pyun, C. W. & Fixman, M., 1964 Frictional coefficient of polymer molecules in solution. J. Chem. Phys. 41, 937.Google Scholar
Rayleigh, Lord: 1892 On the influence of obstacles arranged in rectangular order on the properties of the medium. Phil. Mag. 34, 481.Google Scholar
Reed, L. D. & Morrison, F. A., 1976 Hydrodynamic interaction in electrophoresis. J. Colloid Interface Sci. 54, 117–133.Google Scholar
Russel, W. B., Saville, D. A. & Schowalter, W. R., 1989 Colloidal Dispersions. Cambridge University Press.
Saffman, P. G.: 1973 Settling speed of free and fixed suspensions. Stud. Appl. Maths 52, 115–127.Google Scholar
Subramanian, R. S.: 1985 The Stokes force on a droplet in an unbounded fluid medium due to capillary effects. J. Fluid Mech. 153, 389–400.Google Scholar
Torquato, S.: 1987 Thermal conductivity of disordered heterogeneous media from the microstructure. Rev. Chem. Engng 4, 151–204.Google Scholar
Willis, J. R. & Acton, J. R., 1976 Overall elastic moduli of a dilute suspension of spheres. Q. J. Mech. Appl. Maths 29, 163–177.Google Scholar
Young, N. O., Goldstein, J. S. & Block, M. J., 1959 The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6, 350–356.Google Scholar
Zukoski, C. F.: 1984 Studies of electrokinetic phenomena in suspensions. Ph.D. thesis, Princeton University.
Zukoski, C. F. & Saville, D. A., 1987 Electrokinetic properties of particles in concentrated suspensions. J. Colloid Interface Sci. 115, 422–436.Google Scholar
Zukoski, C. F. & Saville, D. A., 1989 Electrokinetic properties of particles in concentrated suspensions: heterogeneous systems. J. Colloid Interface Sci. 132, 220–229.Google Scholar