Slow axisymmetric rotation of a sphere in a circular tube with slip surfaces

The steady rotation of a slip spherical particle about a diameter lying along the longitudinal axis of a slip circular tube filled with an incompressible Newtonian fluid at low Reynolds numbers is analyzed. To solve the Stokes equations for the fluid flow, the solution is constituted by the summation of general solutions in both cylindrical and spherical coordinates. The boundary conditions are implemented first along the tube wall via the Fourier cosine transform and then over the particle surface through a collocation method. Results of the resisting torque acting on the particle are obtained for various values of the relevant dimensionless parameters. The effect of the confining tube on the axisymmetric rotation of the particle with slip surfaces is interesting. The torque increases monotonically with an increase in the stickiness of the tube wall, keeping the other parameters unchanged. When the stickiness of the tube wall is greater than a critical value, the torque is greater than that on the particle in an unbounded identical fluid and increases with increases in the stickiness of the particle surface and particle-to-tube radius ratio. When the stickiness of the tube wall is less than the critical value, conversely, the torque is smaller than that on the unconfined particle and decreases with increases in the particle stickiness and radius ratio.