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

## TL;DR

This paper derives analytic expressions for frequency-dependent hydrodynamic interactions between two spheres in a viscous fluid, considering both translation and rotation, with approximations suitable for intermediate and long distances.

## Contribution

It introduces a set of scalar mobility functions for two spheres, evaluated using one- and two-propagator approximations, advancing understanding of retarded hydrodynamic interactions.

## Key findings

- Derived analytic scalar mobility functions for sphere pairs.
- Evaluated mutual and self-mobility functions using propagator approximations.
- Considered translation, rotation, and translation-rotation coupling effects.

## 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 translation-rotation coupling. Extensive use is made of Fax\'en theorems which yield the hydrodynamic force and torque acting on a sphere in an incident oscillating flow.