Axisymmetric thermophoretic motion of two spheres

An exact analytical study is presented for the thermophoretic motion of two spheres in a uniform prescribed temperature gradient along their line of centers. The particles may differ in radius, in thermal conductivity and in surface properties, but the Knudsen numbers are assumed to be small so that the fluid flow is described by a continuum model with a thermal creep and a hydrodynamic slip at the particle surfaces. The appropriate energy and momentum equations are solved in the quasisteady situation using spherical bipolar coordinates and the thermophoretic velocities of the particles are calculated for various cases. The interaction between particles can be significant when the surface-to-surface spacing approaches zero. The influence of the interaction, in general, is stronger on the smaller particle than on the larger one. For the thermophoresis of two identical spheres, both migrate faster than the velocity they would possess if isolated. The thermophoretic motion of a sphere in the direction normal to a plane wall is also studied for the case that the gas-solid surfaces may have different properties. In general, the particle-particle and particle-wall interaction effects in thermophoresis are much weaker than for sedimentation.