Thermophoresis of a spherical particle in a microtube

A theoretical study is presented for the axially symmetric thermophoresis of a spherical particle in a microtube filled with a gaseous medium. The uniformly applied temperature gradient is tangential to the tube wall, which is either prescribed with the linear temperature distribution or well insulated. The Knudsen number is small and the fluid motion is characterized by a continuum flow with temperature jump, thermal creep, and frictional slip at the solid surfaces. The general solution to the thermal and aerodynamic governing equations is presented in both spherical and cylindrical coordinates, and the boundary conditions at the particle surface are enforced by a collocation technique. The collocation solutions for the particle's thermophoretic velocity, which are in good agreement with the asymptotic formula resulting from the method of reflections, are obtained for different particle, tube wall, and fluid characteristics. A tube wall prescribed with the far-field temperature distribution and an insulated tube wall influence the thermophoresis of the particle differently. The mobility of a particle confined by a tube wall without thermal creep is a decreasing function of the particle-to-tube radius ratio. When the thermal creep coefficients of the particle and of the tube wall are comparable, the thermoosmotic fluid flow caused by the wall strongly dominates the particle movement and can simply reverse its direction. In general, the influence of the confining tube on thermophoresis is significant.