Wall Effect on Lateral Migration of Spherical Particles in a Linear Shear Flow

Inertia induced lateral migration of particles suspended in a fluid flow is important with regard to the theological properties of suspensions in both physical and industrial processes. In the present work, this phenomenon is studied theoretically. Specifically, we consider the problem of a spherical particle immersed in a linear shear flow bounded by two parallel plane walls. The particle is assumed to be spinning and translating axially with respect to the undisturbed flow. Our main goal is then to evaluate the lateral force on the sphere. Assuming a low Reynolds number flow, the problem is solved via perturbation methods. In general, the problem, being singular, requires two regions of expansion. In the present work, we use the method of matched asymptotic expansions with the assumption that the walls are located in the outer region of expansion. The lateral force (to first order) on the sphere is evaluated for arbitrary gaps between the walls. It is found that for a given gap, this force is maximum at the center and minimum near the walls. Saffman's result, concerning the lateral force on a small sphere immersed in an infinite linear shear flow, becomes a limiting case to our results. A second contribution to the lateral force, which takes into account the effects of local shear and finite size of the sphere, is obtained. The sphere is first replaced in the outer region by a distribution of Stokeslets proportional to the axial component of the slip velocity generated by the motion of the particle around its center. Then, a second term is generated by summing up the contributions of the individual point forces of the distribution. This additional lift is larger near the walls than at the gap center. Actually, for large distances between the sphere and the walls, this second contribution becomes negligibly small which implies that it is essentially a wall effect. Finally, the above theoretical results are applied to Halow's experimental data for the lateral migration of spheres suspended in Couette flow between two concentric cylinders. Good agreement between theory and experiment is obtained.