This observation immediately suggests a simplified description of colloidal hydrodynamic coupling. Consider a sphere labelled , located at within a suspension. A force applied to this sphere excites a flow at displacement . This linear relationship is guaranteed by the Stokes equation's linearity. is known as the Oseen tensor and is obtained from Eqs. (18), (20) and (21). At large enough distances, appears sufficiently uniform that a sphere at is simply advected according to Faxén's law:

(24) | ||

(25) |

where is a component of a mobility tensor describing sphere 's motion in the direction due to a force applied to sphere in the direction. More to the point, . The diagonal elements, , describe the sphere's own response to an external force.

Any number of sources may contribute to the flow past sphere , with each contributing linearly in the Stokes approximation. The mobility tensor therefore may be factored into a self-mobility and a mobility, , due to all external sources,

The results in this section all follow from this equation, once the necessary flow fields have been calculated. In particular, we will consider contributions to a sphere's mobility from (1) neighboring spheres and (2) bounding planar surfaces.