Retarded hydrodynamic interaction between two spheres immersed in a viscous incompressible fluid

Abstract

Retarded or frequency-dependent hydrodynamic interactions are relevant for velocity relaxation of colloidal particles immersed in a fluid, sufficiently close that their flow patterns interfere. The interactions are also important for periodic motions, such as occur in swimming. Analytic expressions are derived for the set of scalar mobility functions of a pair of spheres. Mutual hydrodynamic interactions are evaluated in one-propagator approximation, characterized by a single Green function acting between the two spheres. Self-mobility functions are evaluated in a two-propagator approximation, characterized by a single reflection between the two spheres. The approximations should yield accurate results for intermediate and long distances between the spheres. Both translations and rotations are considered. For motions perpendicular to the line of centers there is translationrotation coupling. Extensive use is made of Faxén theorems which yield the hydrodynamic force and torque acting on a sphere in an incident oscillating flow.

Collinear velocity relaxation of two spheres in a viscous incompressible fluid

“…Elsewhere we derived an approximate expression for the mutual mobility function α tt AB (R, ω) based on a one-propagator approximation, which takes account of only a single Green function between the two spheres [1]. The primary flow is a Stokes flow at frequency ω generated by sphere A as if it were moving by itself in infinite fluid.…”

“…The velocity of sphere B as it moves in this flow with zero force is calculated from a Faxén theorem [12]. The resulting mutual mobility function reads [1]…”

“…The approximate self-mobility function α tt AA (R, ω) is more complicated [1]. It is calculated from a single reflection from sphere B, which is freely moving.…”

“…In recent work [1] we investigated the retarded hydrodynamic interaction between two spheres immersed in a viscous incompressible fluid. The corresponding frequency-dependent mobility matrix relates the translational and rotational velocities of the spheres to the hydrodynamic forces and torques exerted by the fluid.…”

“…Our analysis is based on the recently derived approximate expression for the frequencydependent mobility matrix [1]. In the approximation the hydrodynamic interaction between the two spheres is limited to a single Green function, but finite size effects are fully taken into account via the exact expression for the primary frequency-dependent Stokes flow, and a Faxén theorem in the calculation of the secondary velocity.…”