Hydrodynamic mobility of a sphere moving on the centerline of an elastic tube

TL;DR

This paper derives analytical Green's functions for a sphere's hydrodynamic mobility in an elastic cylindrical channel, revealing shear resistance as the dominant factor affecting particle mobility and deformation.

Contribution

It provides the first analytical expressions for mobility functions in elastic channels, highlighting the role of shear versus bending resistance and validating results with boundary integral simulations.

Findings

Mobility is mainly influenced by membrane shear, with bending being negligible.

Particle self- and pair mobilities match no-slip cylinder results when shear rigidity is present.

Deformation is more significant along the axis than radially, depending on motion direction.

Abstract

Elastic channels are an important component of many soft matter systems, in which hydrodynamic interactions with confining membranes determine the behavior of particles in flow. In this work, we derive analytical expressions for the Green's functions associated to a point-force (Stokeslet) directed parallel or perpendicular to the axis of an elastic cylindrical channel exhibiting resistance against shear and bending. We then compute the leading order self- and pair mobility functions of particles on the cylinder axis, finding that the mobilities are primarily determined by membrane shear and that bending does not play a significant role. In the quasi-steady limit of vanishing frequency, the particle self- and pair mobilities near a no-slip hard cylinder are recovered only if the membrane possesses a non-vanishing shear rigidity. We further compute the membrane deformation, finding that deformation is generally more pronounced in the axial (radial) directions, for the motion along (perpendicular) to the cylinder centerline, respectively. Our analytical calculations for the Green's functions in an elastic cylinder can serve as a fundamental building block for future studies and are verified by fully resolved boundary integral simulations where a very good agreement is obtained.