Motion of a sphere in the presence of a deformable interface I. Perturbation of the interface from flat: the effects on drag and torque

We investigate the effect of small deformations of an initially flat fluid interface on the force and torque experienced by a nearby translating and rotating solid sphere. The small deformation problem is reformulated in terms of equivalent boundary conditions on a flat interface; this allows a separation of the rotation and translation problems, along with their respective components parallel and normal to the undeformed interface. Results for the force and torque corrections due to interface deformation are thus calculated for the four fundamental cases of translation normal and parallel to the undeformed interface, and rotation with the axis of rotation normal and parallel to the undeformed interface. These results can be superposed to obtain the force and torque on a sphere which is undergoing an arbitrary translational and/or rotational motion near the interface.