Orientational dynamics and transverse migration of particles settling in vertical shear flow of a viscoelastic fluid

The motion of settling particles in viscoelastic flows plays a role in the dynamics and mixing of bioreactors and naturally occurring microbe-laden fluids. The transverse migration of settling particles in a vertical shear flow provides an opportunity to separate particles of different sizes and shapes, or to separate particles from the fluid. It is also critical in determining the stability of a homogeneous suspension of settling (or fluidized) particles, where lateral disturbances in particle concentration from the homogeneous concentration field induce a vertical shear flow with fluid flowing downward/upward in regions with higher/lower particle concentration. This study employs finite difference computations for general conditions and asymptotic analyses for weak viscoelasticity to determine the rotational motion and lateral migration of prolate spheroidal and spherical particles in an inertia-less viscoelastic fluid. Spherical particles migrate toward downward-flowing fluid (i.e., regions of higher particle concentration) up to moderate Deborah (De) numbers based on the settling velocity, suggesting that a settling suspension of particles would be unstable, as predicted by the small-De theory of Vishnampet and Saintillan (Phys. Fluids 24, 073302, 2012). However, above a critical De, the direction of horizontal migration reverses, and the suspension becomes stable. Prolate spheroids suspended in a weak vertical shear flow achieve an oblique orientation in which the viscoelastic torque favoring vertical alignment competes with the shear-driven torque. The particles stop rotating and migrate toward upward-flowing fluid. Above a critical shear rate, the shear-induced Jeffery rotation overcomes the viscoelastic torque; the particle resumes rotation and migrates toward downward-flowing fluid. Thus, a sedimenting suspension of prolate spheroids is expected to be linearly stable to small concentration perturbations, but unstable to finite-amplitude disturbances.