Hydrodynamic coupling and rotational mobilities near planar elastic membranes

We study theoretically and numerically, the coupling and rotational hydrodynamic interactions between spherical particles near a planar elastic membrane that exhibits resistance toward shear and bending. Using a combination of the multipole expansion and Faxén'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 mobilities 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 very good agreement is obtained over the whole range of applied frequencies.