Slow rotation of a sphere about its diameter normal to two planes with slip surfaces

The steady creeping flow of an incompressible Newtonian fluid around a slip spherical particle rotating about its diameter perpendicular to one or two slip plane walls is analyzed. To satisfy the Stokes equation for fluid velocity, the general solution consists of the sum of the essential solutions in spherical and cylindrical coordinates. Boundary conditions are implemented first on the plane walls by means of the Hankel transforms and then on the particle surface through a collocation method. The hydrodynamic torque exerted on the particle is obtained with excellent convergence for various values of the pertinent geometrical and stick-slip parameters, and the effect of the slip planes on the rotational motion of the slip particle is interesting. The torque increases with an increase in the stickiness of the walls from the limit of full slip to the limit of no slip. When the stick parameters of the plane walls are larger than some critical values, the hydrodynamic torque is more than that on an identical particle in the unbounded fluid and an increasing function of the stickiness of the particle surface and ratio of the particle radius to distance from the walls. When the stick parameters of the plane walls are smaller than the critical values, on the contrary, the torque is less than that on the particle in the unbounded fluid and a decreasing function of the surface stickiness and relative radius of the particle.