Motion of Particles in a Parabolic Flow Near a Slip Wall

Consider a suspension of dilute spherical particles transported in the flow of a viscous fluid along a wall. A Navier slip condition applies on the wall, in view of applications to hydrophobic walls in microchannels. The Stokes flow problem for a sphere in a quadratic shear flow is solved here by the method of bispherical coordinates. It is shown how the infinite linear system for four coupled series of coefficients may be solved with a cost that is simply proportional to the number of coefficients. Results are the force and torque on a fixed sphere and the velocities of translation and rotation of a freely moving sphere in a quadratic flow near a slip wall. The stresslet, that is the symmetric moment of surface stresses on the sphere, is also derived in view of applications to suspension rheology. Finally, expansions of the various quantities for a large distance from the wall and for a low slip length are performed on the basis of the analytical solutions, using computer algebra. Pade approximants calculated therefrom provide a good approximation.