# Hydrodynamic mobility of a solid particle nearby a spherical elastic membrane. II. Asymmetric motion

## TL;DR

This paper derives analytical expressions for the hydrodynamic mobility of a small particle near a spherical elastic membrane, revealing how membrane resistance influences particle motion and diffusion, with validation through boundary integral simulations.

## Contribution

It extends previous work by providing the analytical solution for asymmetric tangential motion of particles near elastic capsules, highlighting the effects of shearing and bending resistance.

## Key findings

- Shearing resistance causes a low-frequency peak in self-mobility.
- Membrane induces long-lasting subdiffusive behavior in particle diffusion.
- Analytical results agree well with boundary integral simulations.

## Abstract

In this paper, we derive analytical expressions for the leading-order hydrodynamic mobility of a small solid particle undergoing motion tangential to a nearby large spherical capsule whose membrane possesses resistance towards shearing and bending. Together with the results obtained in the first part (Daddi-Moussa-Ider and Gekle, Phys. Rev. E {\bfseries 95}, 013108 (2017)) where the axisymmetric motion perpendicular to the capsule membrane is considered, the solution of the general mobility problem is thus determined. We find that shearing resistance induces a low-frequency peak in the particle self-mobility, resulting from the membrane normal displacement in the same way, although less pronounced, to what has been observed for the axisymmetric motion. In the zero frequency limit, the self-mobility correction near a hard sphere is recovered only if the membrane has a non-vanishing resistance towards shearing. We further compute the particle in-plane mean-square displacement of a nearby diffusing particle, finding that the membrane induces a long-lasting subdiffusive regime. Considering capsule motion, we find that the correction to the pair-mobility function is solely determined by membrane shearing properties. Our analytical calculations are compared and validated with fully resolved boundary integral simulations where a very good agreement is obtained.