The kinetics of flowing dispersions: VIII. Doublets of rigid spheres (theoretical)

Abstract A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutral, equal-sized spheres in a viscous fluid undergoing uniform shear flows is presented. The method is based on the matrix formulation of the hydrodynamic resistances by Brenner and O'Neill, and it is illustrated for simple shear and planar and axisymmetric extensional flows, for which the trajectories of pairs of sphere centers are calculated. It is shown that, in simple shear, trajectories are either open or closed; in extensional flows, all trajectories must be open. This has interesting implications in the dispersion of aggregates by shear. Although the translational and rotational behavior of interacting spheres is characterized by the type of flow, the behavior of the spheres in each flow is calculable from flow-independent quantities. Special emphasis is placed on the collision behavior of doublets. Initially separate spheres can never touch, but they can come into very close proximity, especially during equatorial encounters. In simple shear, translating spheres rotate about their own axes, which, in a doublet, causes one sphere to roll relative to the other except when they touch or are rigidly connected. The rotations of free and connected doublets and dumbbells are related to those of prolate spheroids.

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