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Hindered mobility of a particle near a soft interface

The translational motion of a solid sphere near a deformable fluid interface is studied in the low Reynolds number regime. In this problem, the fluid flow driven by the sphere is dynamically coupled the instantaneous conformation of the interface. Using a two-dimensional Fourier transform technique, we are able to account for the multiple backflows scattered from the interface. The mobility tensor is then obtained from the matrix elements of the relevant Green function. This analysis allows us to express the explicit position and frequency dependence of the mobility. We recover in the steady limit the result for a sphere near a perfectly flat interface. At intermediate time scales, the mobility exhibits an imaginary part, which is a signature of the elastic response of the interface. In the short time limit, we find the intriguing feature that the perpendicular mobility may, under some circumstances, become lower than the bulk value. All those results can be explained from the definition of the relaxation time of the soft interface.