Subdiffusive behavior of a dilute non-Brownian suspension under shear

Shear-induced self-diffusion in a dilute suspension of non-Brownian spheres under a simple shear flow is studied in the limit of zero Reynolds number, for different volume fractions φ. Particles are simulated considering the first term in the multipolar expansion to take into account the long-range hydrodynamic interactions, and a repulsive force is added to avoid interpenetration. The final diffusive regime is established after a long time (t(diffusion)~φ(-1.5)), and the diffusion coefficient is proportional to φ(2), as expected in the presence of the short-range repulsive force. Before the diffusive regime is established, there is a rich subdiffusive behavior, particularly in the gradient direction. Each pairwise hydrodynamic interaction is reversible, not leading to streamline migration. Due to the incoherence of the different pair interactions, a plateau in the mean square displacement is first observed, lasting for a period that increases as φ is decreased. Then, a first diffusive regime is established due to three-particle interactions, with a diffusion coefficient of D((1))~φ(2.4). At longer times, a phenomenon similar to caging is observed. Particles diffuse for long times in the vicinity of some positions, and eventually large displacements are produced, moving the particle to a new position and resulting in transient large values of the kurtosis of the displacement distribution. This migration is produced by collisions through the repulsive potential. After several of those large displacements, the final diffusive regime is established.