Particle migration in suspensions by thermocapillary or electrophoretic motion

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.

[1]  A. Einstein Eine neue Bestimmung der Moleküldimensionen , 1905 .

[2]  M. Meyyappan,et al.  The slow axisymmetric motion of two bubbles in a thermal gradient , 1983 .

[3]  F. Feuillebois Thermocapillary migration of two equal bubbles parallel to their line of centers , 1989 .

[4]  R. Subramanian The Stokes force on a droplet in an unbounded fluid medium due to capillary effects , 1985, Journal of Fluid Mechanics.

[5]  J. L. Anderson,et al.  Transport Mechanisms of Biological Colloids a , 1986, Annals of the New York Academy of Sciences.

[6]  M. Fixman,et al.  Frictional Coefficient of Polymer Molecules in Solution , 1964 .

[7]  J. Willis,et al.  THE OVERALL ELASTIC MODULI OF A DILUTE SUSPENSION OF SPHERES , 1976 .

[8]  D. Jeffrey,et al.  Group expansions for the bulk properties of a statistically homogeneous, random suspension , 1974, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  M. Beran,et al.  On the effective thermal conductivity of a random suspension of spheres , 1976 .

[10]  G. Batchelor,et al.  The determination of the bulk stress in a suspension of spherical particles to order c2 , 1972, Journal of Fluid Mechanics.

[11]  A. Acrivos,et al.  The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations , 1978 .

[12]  E. J. Hinch,et al.  An averaged-equation approach to particle interactions in a fluid suspension , 1977, Journal of Fluid Mechanics.

[13]  J. S. Goldstein,et al.  The motion of bubbles in a vertical temperature gradient , 1959, Journal of Fluid Mechanics.

[14]  L. Rayleigh,et al.  LVI. On the influence of obstacles arranged in rectangular order upon the properties of a medium , 1892 .

[15]  S. Shtrikman,et al.  A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials , 1962 .

[16]  R. Balasubramaniam,et al.  Thermocapillary migration of droplets: an exact solution for small Marangoni numbers , 1987 .

[17]  D. Jeffrey,et al.  Conduction through a random suspension of spheres , 1973, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[18]  C. Zukoski,et al.  Electrokinetic properties of particles in concentrated suspensions , 1987 .

[19]  M. Meyyappan,et al.  The thermocapillary motion of two bubbles oriented arbitrarily relative to a thermal gradient , 1984 .

[20]  S. Childress Viscous Flow Past a Random Array of Spheres , 1972 .

[21]  C. Zukoski,et al.  Electrokinetic properties of particles in concentrated suspensions: Heterogeneous systems , 1989 .

[22]  J. L. Anderson Droplet interactions in thermocapillary motion , 1985 .

[23]  F. A. Morrison,et al.  Hydrodynamic interactions in electrophoresis , 1976 .

[24]  R. W. O'Brien,et al.  A method for the calculation of the effective transport properties of suspensions of interacting particles , 1979, Journal of Fluid Mechanics.

[25]  B. U. Felderhof,et al.  Cluster expansion for the dielectric constant of a polarizable suspension , 1982 .

[26]  G. Batchelor Sedimentation in a dilute dispersion of spheres , 1972, Journal of Fluid Mechanics.

[27]  John L. Anderson,et al.  Effect of nonuniform zeta potential on particle movement in electric fields , 1985 .

[28]  I. Howells Drag due to the motion of a Newtonian fluid through a sparse random array of small fixed rigid objects , 1974, Journal of Fluid Mechanics.

[29]  S. Torquato Thermal Conductivity of Disordered Heterogeneous Media from the Microstructure , 1987 .

[30]  P. Saffman,et al.  On the Settling Speed of Free and Fixed Suspensions , 1973 .