Dyadics particle boundary

There exist few particle-boundary combinations for which the particleboundary resistance dyadics in Eqs. (38) and (39) are known for all physically possible particle-to-wall dimensions. They are known, for example, for the trivial case of a spherical particle at the center of a concentric spherical boundary, the space between them being filled with fluid.The translation and rotation dyadics for this case are clearly isotropic, white the coupling dyadic obviously vanishes at this common center. 

The sphere-plane wall configuration represents one of the few nontrivial cases for which the particle-boundary resistance dyadics in Eqs. (38) and (39) are completely known. If e, is a unit vector normal to the plane then the translation and rotation dyadics are given by equations of the form (123) and... 

Here, the vector constants V and 1 are to be determined so as to satisfy the respective pair of dyadic and triadic particle-surface boundary conditions... 

Ko are the translation and coupling dyadics for the particle in the unbounded fluid. Also, Wo is the wall-effect dyadic (B20, C20) evaluated at the point in the fluid presently occupied by O. In general, is a constant, symmetric dyadic which depends only upon the size and shape of the boundaries and upon the location of O relative to the bounding walls. In particular, Wp is independent of yt, Uq, co, of the size and shape of the particle, and of the orientation of the particle relative to the boundaries. At each point O in space it is, therefore, an intrinsic geometric property of the container boundaries... 

Stokes equations [cf. Eqs. (17) and (18)] associated with the motion of the yth particle while all other particles are at rest. Explicitly, let the translational solutions satisfy the dyadic boundary conditions... 

---