Particle motions in sheared suspensions: XXX. Rotations of rigid and flexible dumbbells (theoretical)

Abstract

Theoretical models of dumbbells of equal-sized, rigid spheres have been formulated for the case of (i) rigid dumbbells having a fixed interparticle distance with the relative rotation of the spheres hindered by viscous damping, and (ii) flexible dumbbells having variable interparticle distance with the spheres linked by an elastic filament. The general method of calculating forces, torques, and translational and rotational velocities based on the matrix formulation of Brenner and O'Neill was used. In the case of the rigid dumbbell, the additional torque exerted on the spheres is described by a damping coefficient X. It is shown that even a small increase in λ from 0 (the value for freely rotating spheres) results in a marked decrease in the dimensionless period of rotation TG. With the flexible dumbbell it is assumed that the filament only exerts a force on the spheres when the distance between the points of attachment exceeds its resting length. The results show that no matter what the initial conditions of release, the trajectories of the spheres, characterized by an elliptical path of their centers about the mid-point of the dumbbell axis, tend to a limit cycle within a few rotations. It is therefore possible to define a period of rotation which is predicted to decrease with increasing filament stiffness, and to increase with increasing length and diameter, the latter also determining the minimum distance of approach of the sphere surfaces.