  We study theoretically and numerically the coupling and rotational
hydrodynamic interactions between spherical particles near a planar elastic
membrane that exhibits resistance towards shear and bending. Using a
combination of the multipole expansion and Faxen's theorems, we express the
frequency-dependent hydrodynamic mobility functions as a power series of the
ratio of the particle radius to the distance from the membrane for the self
mobilities, and as a power series of the ratio of the radius to the
interparticle distance for the pair mobilities. In the quasi-steady limit of
zero frequency, we find that the shear- and bending-related contributions to
the particle mobilities may have additive or suppressive effects depending on
the membrane properties in addition to the geometric configuration of the
interacting particles relative to the confining membrane. To elucidate the
effect and role of the change of sign observed in the particle self and pair
mobilities, we consider an example involving a torque-free doublet of
counterrotating particles near an elastic membrane. We find that the induced
rotation rate of the doublet around its center of mass may differ in magnitude
and direction depending on the membrane shear and bending properties. Near a
membrane of only energetic resistance toward shear deformation, such as that of
a certain type of elastic capsules, the doublet undergoes rotation of the same
sense as observed near a no-slip wall. Near a membrane of only energetic
resistance toward bending, such as that of a fluid vesicle, we find a reversed
sense of rotation. Our analytical predictions are supplemented and compared
with fully resolved boundary integral simulations where a very good agreement
is obtained over the whole range of applied frequencies.
