Semi-analytic solution of the motion of two spheres in arbitrary shear flow

The semi-analytic solution for the motion of two spheres suspended in arbitrary, unbounded shear flow is developed. The solution is a generalization of classical bispherical-coordinate solutions for two spheres moving along or perpendicular to the line of centers, rotating about the centerline in a quiescent liquid, and suspended in linear flow. The semi-analytic solution is highly efficient and can be used to study dilute sphere interactions in low Reynolds number flows. Several application problems are considered including the net particle migration of rough spheres in Couette and Poiseuille flow and the self-diffusivity of spheres in linear and nonlinear flow. Although it has been established previously that rough sphere pairs in nonlinear shear flow migrate towards the low shear rate region of the flow field, the physics behind this migration is elucidated. It is also shown that, under certain circumstances, a rough sphere pair undergoing short period oscillatory Poiseuille flow can actually migrate towards the high-shear rate region of the flow field. Finally, new results are generated for the effect of particle radius ratio on the migration phenomenon.