Longitudinal Shear-Induced Diffusion of Spheres in a Dilute Suspension

We present a calculation of the hydrodynamic self-diffusion coefficient of a tagged particle in a dilute mono-disperse suspension of small neutrally buoyant spheres undergoing a steady simple shearing motion. The displacement of the tagged particle parallel to the longitudinal or streamwise direction resulting from a "collision" with one other particle is calculated on the assumption that inertia and Brownian motion effects are negligible. Summing over different pairs leads to a logarithmically divergent integral for the diffusivity which is rendered finite by allowing for the cut-off due to the occasional presence of another pair of particles. The longitudinal shear-induced self-diffusion coefficient is thus found to be 0.267 a^2 gamma {c ln (1/c) + O(c)}, where gamma denotes the applied shear rate, a is the radius of the spheres and c their volume concentration