Hydrodynamic coupling and rotational mobilities near planar elastic membranes

TL;DR

This paper develops a theoretical and numerical framework to analyze how planar elastic membranes influence the hydrodynamic interactions and rotational behaviors of spherical particles, revealing effects of membrane properties on particle mobility and rotation.

Contribution

It introduces a combined analytical and numerical approach to quantify frequency-dependent hydrodynamic mobilities near elastic membranes, highlighting the effects of shear and bending resistance.

Findings

Shear and bending contributions can be additive or suppressive depending on membrane properties.

The induced rotation of particle doublets can reverse direction based on membrane elasticity.

Good agreement between analytical predictions and boundary integral simulations.

Abstract

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.