Sedimentation of axisymmetric particles in shear flows

Axisymmetric particles possessing fore-and-aft symmetry undergo tumbling motions, known as Jeffery orbits, when suspended in shear flows. The sedimentation velocity of such a particle depends on its instantaneous orientation and, hence, varies during an orbit. Closed-form analytical expressions are derived for the time-averaged sedimentation velocity of a particle as it tumbles through one full orbit, with the gravity vector being oriented arbitrarily with respect to the flow direction. This orbit velocity is then integrated numerically over the distribution of orbits in order to obtain the average sedimentation velocity in dilute suspensions, for several different orbit distributions corresponding to the short-time limit with random initial orientations, the long-time limit for dilute suspensions with weak Brownian motion, measured steady distributions for dilute suspensions of non-Brownian particles with particle–particle interactions, and singular distributions of particles aligning such that viscous dissipation is minimized. Specific calculations are made for prolate and oblate ellipsoids, cylindrical rods and disks, and a pair of touching spheres. The average sedimentation velocity depends strongly on the aspect ratio of the particles, and on the orientation of the gravity vector relative to the flow field, but only weakly on the assumed form of the distribution of orbits.