Slow motion of a slip spherical particle perpendicular to two plane walls

The problem of the quasisteady motion of a spherical fluid or solid particle with a slip-flow surface in a viscous fluid perpendicular to two parallel plane walls at an arbitrary position between them is investigated theoretically in the limit of small Reynolds number. To solve the axisymmetric Stokes equation for the fluid velocity field, a general solution is constructed from the superposition of the fundamental solutions in both circular cylindrical and spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Hankel transform and then on the particle surface by a collocation technique. Numerical results for the hydrodynamic drag force exerted on the particle are obtained with good convergence for various values of the relative viscosity or slip coefficient of the particle and of the relative separation distances between the particle and the confining walls. For the motions of a spherical particle normal to a single plane wall and of a no-slip sphere perpendicular to two plane walls, our drag results are in good agreement with the available solutions in the literature for all relative particle-to-wall spacings. The boundary-corrected drag force acting on the particle in general increases with an increase in its relative viscosity or with a decrease in its slip coefficient for a given geometry, but there are exceptions. For a specified wall-to-wall spacing, the drag force is minimal when the particle is situated midway between the two plane walls and increases monotonically when it approaches either of the walls. The boundary effect on the particle motion normal to two plane walls is found to be significant and much stronger than that parallel to them.