Thermophoretic Motion of a Slightly Deformed Sphere Through a Viscous Fluid

Abstract

This paper provides a general approach to the solution of the problem of nonisothermal Stokes flow relative to a heat-conducting particle having the shape of a slightly deformed sphere, taking account of Maxwell's [J. C. Maxwell, Philos. Trans. R. Soc. Lond., 170 (1879), pp. 231-256] thermal creep condition at the surface of the particle. The results, which are of interest in connection with the phenomenon of thermophoresis, have potential applications in aerosol technology, and in the nonisothermal transport and processing of particulate matter. For the specific case of thermally nonconducting particles, the results obtained herein accord with Morrison's [F. A. Morrison, J. Colloid Interface Sci., 34 (1970), pp. 210-214] proof in the comparable electrophoretic case that the phoretic velocity of a nonconducting, force-and torque-free, nonspherical particle undergoing electrophoresis in a fluid that is otherwise at rest is independent of the size, shape and orientation of the particle, and is identical to that of a sphere.

Diffusiophoresis of colloidal particles in neutral solute gradients at finite Péclet number

“…Specifically, the colloidal translates at a velocity U = M e E, where E is the imposed electric field and M e = εζ /η is a scalar 'electrophoretic mobility', where ε and η are, respectively, the permittivity and viscosity of the fluid, and ζ is the zeta potential. Morrison's result also holds for diffusiophoresis under a neutral solute gradient Anderson 1989) and thermophoresis due to an imposed temperature gradient (Mohan & Brenner 2006): in each case, the rectilinear particle velocity equals the imposed gradient multiplied by an appropriate scalar mobility that is independent of particle size and shape. Indeed, experiments on thermophoresis in liquids suggest that the thermophoretic mobility is independent of particle size (McNab & Meisen 1973;Braibanti et al 2008).…”

Fluid mechanics in fluids at rest

“…[16]) is frequently cited. In regard to particle geometry, the velocity given by (4.1) is found to be independent of the particle's size and shape, and (when nonspherical [25]) also of the particle's orientation relative to either of the confining walls or, equivalently, relative to the direction of the externally imposed vector temperature gradient. For example, Eq.…”

“…In contrast, the particle's noninert velocity, (4.2), valid for the case of spherical particles, is thus now dependent upon several of the particle's transport properties, namely k S , as well as the particle's shape. [The shape effect is evidenced in the work of Mohan and Brenner [25], which derives the generalization of (4.2) for a heat-conducting, slightly deformed sphere, showing that the body's consequent anisotropy renders the particle's velocity -U T dependent upon the particle's orientation relative to the temperature gradient.] In contrast, in the thermally inert limit k S = 0 the particle's thermophoretic velocity U T as given by (4.1) was formally confirmed by these same authors to be independent of the nonspherical particle's orientation relative to the temperature gradient, despite the particle's geometric anisotropy.…”

“…valid for the case where the particle is spherical in shape, and only for that shape [25]. When k S k this formula reduces to (4.1).…”

Thermophoretic motion of slightly deformed aerosol spheres

“…Earlier, the axisymmetric thermophoresis of a spheroidal particle along its axis of revolution without temperature jump and frictional slip at its surface was also analytically studied (Leong, 1984;Williams, 1986). This analysis has been generalized to a spheroid (Keh & Ou, 2004) and a particle departing slightly in shape from a sphere (Mohan & Brenner, 2006) with an arbitrary orientation relative to the imposed temperature gradient. Although the thermophoretic motion of a general axisymmetric particle with the effects of temperature jump, thermal slip and frictional slip along its axis of revolution was numerically examined to some extent by using a boundary collocation method , the problem of thermophoresis of particles of a general shape with the jump/slip conditions at the particle surface in an arbitrary direction has not been analytically solved yet, mainly due to the fact that, if the temperature jump and/or frictional slip is included, a separable solution of the temperature and/or fluid stream function is not feasible for most orthogonal curvilinear coordinate systems, such as the prolate and oblate spheroidal ones.…”