THERMOPHORESIS DUE TO STRONG TEMPERATURE GRADIENTS

We consider a standard thermophoretic configuration, wherein an insulating spherical particle is suspended within a gaseous domain which is bounded between two parallel walls. The walls are maintained at two different temperatures, thereby generating a nonuniform temperature field within the gas. Due to thermal slip, the particle drifts toward the cold wall. Conventional analyses of this problem, starting with the classical work of Epstein [Z. Physik., 54 (1929), pp. 537-563], employ the small-temperature-difference limit. Then, if the particle is small enough, the problem becomes quasi-steady, and the animating effect of the two walls can be represented by a uniform far-field temperature gradient. The corresponding unbounded problem is identical to other slip-generated problems, such as electrophoresis. We focus here upon the general case where the temperature difference is not small. The dependence of the pertinent flow variables upon the absolute temperature prohibits a transformation to a quasi-steady description, whence the transport problem is governed by an unsteady nonlinear process. The small-particle limit is a singular one, wherein the walls cannot be represented by effective far-field conditions. Moreover, the unique structure of the thermal-slip condition implies that inertia and heat-convection effects are of comparable scaling to wall effects. The singular limit is analyzed using inner-outer expansions. In the outer domain, the temperature field is steady to leading-order, but is not described by a uniform gradient. In the inner particle-scale domain, the flow problem is governed by the steady Stokes equations only in the leading order. The transformation between the inner and outer coordinates involves the particle velocity, itself a dependent variable. Using symmetry arguments, we avoid the detailed calculation of the leading-order flow correction, and focus instead upon its effect on the particle thermophoretic velocity. Due to a fortuitous cancellation of terms, Epstein's result remains valid to leading-order analysis. It has been proposed by Kogan, Galkin, and Fridlender [Sov. Phys. Usp., 19 (1976), pp. 420-428] that thermal stresses must be incorporated into all continuum descriptions which apply to flows driven by O(1) temperature differences. Using symmetry arguments, we also analyze the effects of these stresses in the present configuration.