Thermophoretic deposition of a sphere normal to a plane surface

An analytical study is presented for the thermophoresis of a sphere in a constant applied temperature gradient perpendicular to a plane surface. The Knudsen number is 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 surface. The asymptotic formulas for the temperature and velocity fields in the quasisteady situation are obtained by using a method of reflections. The plane surface may be a solid wall or a free surface. The boundary effect on the thermophoretic motion is found to be weaker than that on the motion driven by a gravitational force, Even so, the interaction between the plane and the particle can be very strong when the gap thickness approaches zero. For the motion of a particle normal to a solid plane, the effect of the plane surface is to reduce the thermophoretic velocity of the particle, For the case of particle migration toward a free surface owing to thermophoresis, the particle velocity can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity and the surface properties of the particle and its relative distance from the plane, Also, the deposition time for a particle translating perpendicularly through the thermophoretic boundary layer is integrated by considering its thermophoretic mobility, Generally speaking, a free surface exerts less influence on particle movement than a solid surface does.