Concentration instability of sedimenting spheres in a second-order fluid

The slow sedimentation of a dilute suspension of spherical particles in a second-order fluid is investigated using theory and numerical simulations. We first analyze the motion of a single isolated spherical particle sedimenting under gravity when placed in a linear flow field. In the limit of weak viscoelasticity (low Deborah number), the velocity of the particle is calculated, and the nonlinear coupling of the settling motion with the local flow field is shown to result in a lateral drift in a direction perpendicular to gravity. By the same effect, the mean flow driven by weak horizontal density fluctuations in a large-scale suspension of hydrodynamically interacting particles will also result in a horizontal drift, which has the effect of reinforcing the fluctuations as we demonstrate using a linear stability analysis. Based on this mechanism, an initially homogeneous suspension is expected to develop concentration fluctuations, a prediction supported by previous experiments on sedimentation in polymeric liquids. We further confirm this prediction using large-scale weakly nonlinear numerical simulations based on a point-particle model. Concentration fluctuations are indeed found to grow in the simulations, and are shown to result in an enhancement of the mean settling speed and velocity fluctuations compared to the Newtonian case.