Variational theorem for the generalized diffusion matrix

摘要

A variational theorem is developed for a generalized diffusion matrix that describes the motion of one or more objects in a continuous solvent. The solvent motion obeys the linearized Navierndash;Stokes equation. The objects interact with the solvent through frictional forces. The magnitude of the frictional force density may vary from point to point in the object, and is controlled by a spatially varying friction tensor. If the friction tensors have nonhyphen;negative eigenvalues at all spatial points, the variational principle becomes a minimization principle. The approach is illustrated for the two sphere problem in the nondraining limit, for which several exact calculations are available for comparison.