Slow Motion of a Spherical Particle Perpendicular to Two Planar Walls with Slip Surfaces

The quasi-steady creeping flow of a viscous fluid around a slip sphere translating perpendicular to one or two large slip planar walls at arbitrary relative positions is analyzed. To solve the axisymmetric Stokes equation for the fluid flow, we construct a general solution using fundamental solutions in spherical and cylindrical coordinate systems. Boundary conditions are first applied to the planar walls using the Hankel transform and then to the particle surface using a collocation method. Numerical results of the drag force exerted by the fluid on the particle are obtained for different values of the relevant stickiness/slip and configuration parameters. Our force results agree well with existing solutions for the motion of a slip sphere perpendicular to one or two nonslip planar walls. The hydrodynamic drag force acting on the particle is a monotonic increasing function of the stickiness of the planar walls and the ratio of its radius to distance from each planar wall. With other parameters remaining constant, this drag force generally increases with increasing stickiness of the particle surface. The influence of the slip planar walls on the axisymmetric translation of a slip sphere is significantly stronger than its axisymmetric rotation.