Self-diffusion in sheared suspensions

Abstract

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 raacroscopically quiescent or in linear flow at arbitrary Peclet number $Pe = \dot{\gamma}a^2/2D$, where $\dot{\gamma}$ is the shear rate, a is the particle radius, and D = k_BT/6πa is the diffusion coefficient of an isolated particle. Here, k_B is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by k_BT times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction $\phi = \frac{4}{3}\pi a^3n$ 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_∞^s, is given by the sum of D₀^s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D₀^s and D_∞ are anisotropic, in general, with the anisotropy of D₀^s due solely to that of the steady microstructure. The influence of flow upon D_∞^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(øPe^3/2) for Pe ≤ 1 and ø ≤ 1, both with and without hydro-dynamic interactions between the particles. The leading dependence upon flow of D₀^s is 0.22DøPeÊ, where Ê is the rate-of-strain tensor made dimensionless with $\dot{\gamma}$. Regardless of whether or not the particles interact hydrodynamically, flow influences D_∞^s at O(øPe) and O(øPe^3/2). In the absence of hydrodynamics, the leading correction is proportional to øPeDÊ. The correction of O(øPe^3/2), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to øPe^3/2DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê, and the second a non-isotropic tensor.

At high ø a scaling theory based on the approach of Brady (1994) is used to approximate D_∞^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 $\bar{P}e = \dot{\gamma}a^2/2D^s_0(\phi)$. At small $\bar{P}e$ the dependence on $\bar{P}e$ is the same as at low ø.