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The general problems of particle motion in the vicinity of a flat, non-deforming fluid interface is studied. The approximate singularity method used by previous workers in this research group has been generalized to consider the motion of a sphere in any linear velocity field compatible with the existence of the undisturbed flat interface, and the motion of slender rod-like particles which undergo an arbitrary translation or rotation in either a quiescent fluid or in a linear flow. The theory yields the hydrodynamic mobility tensors which are necessary to describe Brownian movement near a phase boundary, as well as general trajectory equations for sedimenting particles near a fluid interface with an arbitrary viscosity ratio. These approximate solution results are in good agreement with both exact-solutions where they are available and experimental data for motion of a sphere near a rigid plane wall. Among the most interesting results for motion of slender bodies is the generalization of Jeffery orbit equations for linear simple shear flow.

The Brownian motion of a sphere in the presence of a deformable fluid interface is also examined. First, the fluctuation-dissipation theorem is derived for the random distortions of interface shape that are caused by spontaneous thermal impulses from the surrounding fluids. This analysis is carried out using the method of normal modes in conjunction with a Langevim type equation for the Brownian particle, and results in the prediction of autocorrelation functions for the location of the interface, for the random force acting on the particle (evaluated by a generalization of the Faxen's law), and for the particle velocity. The particle velocity correlation, in turn, yields the effective diffusion coefficient due to random fluctuations of the interface shape. Finally, we investigate the effects of interface deformation that are induced by the impulsive motion of a sphere that is undergoing Brownian motion. In this phase of our study, we consider both the spatially modified hydrodynamic mobility which occurs as a consequence of hydrodynamic interactions, and influence on the mean-square displacement of the Brownian particle of the interface relaxation back towards the flat equilibrium configuration after an initial deformation that is caused by the particle motion.

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