Self-diffusion in sheared suspensions

Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either macroscopically quiescent or in linear flow at arbitrary Peclet number Pe = gamma over dot a(2)/2D, where gamma over dot is the shear rate, a is the particle radius, and D = k(B)T/6 pi eta a is the diffusion coefficient of an isolated particle. Here, k(B) is Boltzmann's constant, T is the absolute temperature, and eta is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by k(B)T times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction phi = 4/3 pi a(3)n and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension, it perturbs the average microstructure, and the long-time self-diffusion tensor, D-infinity(s), is given by the sum of D-0(s) and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D-0(s), and D-infinity(s) are anisotropic, in general, with the anisotropy of D-0(s) due solely to that of the steady microstructure. The influence of flow upon D-infinity(s) is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations. The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to O(phi Pe(3/2)) for Pe much less than 1 and phi much less than 1, both with and without hydrodynamic interactions between the particles. The leading dependence upon flow of D-0(3) is 0.22D phi Pe (E) over cap, where (E) over cap is the rate-of-strain tensor made dimensionless with gamma over dot. Regardless of whether or not the particles interact hydrodynamically, flow influences D-infinity(s) at O(phi Pe) and O(phi Pe(3/2)). In the absence of hydrodynamics, the leading correction is proportional to phi PeD (E) over cap. The correction of O(phi Pe(3/2)), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to phi Pe(3/2)DI; when hydrodynamics are included, the correction is given by two terms, one proportional to (E) over cap, and the second a non-isotropic tensor. At high phi a scaling theory based on the approach of Brady (1994) is used to approximate D-infinity(s). For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of (P) over bar e = gamma over dot a(2)/2D(0)(s)(phi). At small (P) over bar e the dependence on (P) over bar e is the same as at low phi.