Particle migration in suspensions by thermocapillary or electrophoretic motion

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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.