Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow

TLDR: Two unequal rigid spheres are immersed in unbounded fluid and are acted on by externally applied forces and couples, which can be described by a set of linear relations between, on the one hand, the forces and spouses exerted by the spheres on the fluid and the translational and rotational velocities of the spheres.

Abstract: Two unequal rigid spheres are immersed in unbounded fluid and are acted on by externally applied forces and couples. The Reynolds number of the flow around them is assumed to be small, with the consequence that the hydrodynamic interactions between the spheres can be described by a set of linear relations between, on the one hand, the forces and couples exerted by the spheres on the fluid and, on the other, the translational and rotational velocities of the spheres. These relations may be represented completely by either a set of 10 resistance functions or a set of 10 mobility functions. When non-dimensionalized, each function depends on two variables, the non-dimensionalized centre-to-centre separation s and the ratio of the spheres’ radii λ. Two expressions are given for each function, one a power series in s−1 and the other an asymptotic expression valid when the spheres are close to touching.