Exact Solutions, Asymptotics and Applications for Unequal Spheres in in Viscous Fluid

Hydrodynamic interactions between bodies immersed in viscous fluid have been shown to be important in modelling many complex fluid phenomena. Of particular interest for applications are suspensions of solid particles, e.g., high solid volume fraction cornstarch suspensions and fluidised beds, which exhibit highly non-Newtonian characteristics, such as shear thickening and normal stress differences, posing great challenges in predicting the flow. We study the problem of two spheres approaching each other along their line of centres suspended in viscous fluid, fundamental to understand the dense suspension behaviour. We give, for the first time, the complete formula of the hydrodynamic interaction force between the spheres. We also rigorously derive the behaviour of the forces as the nondimensional separation goes to zero and infinity, reproducing known heuristic results. The formulae are then implemented in the recently developed dynamical density functional theory (DDFT) to obtain novel solutions to problems concerning short range hydrodynamic interactions. Such interactions will be seen to be crucial in capturing phenomena associated with the dynamics of differently-sized and shaped species of particles, the effects of confinement and systems with enforced background flow.