# Displacement field around a rigid sphere in a compressible elastic environment, corresponding higher-order Fax\'en relations, as well as higher-order displaceability and rotateability matrices

## TL;DR

This paper presents an efficient method for calculating the displacement field around a rigid sphere in a compressible elastic medium, derives higher-order Faxén relations, and computes interaction matrices up to sixth order, with applications to hydrodynamics.

## Contribution

It introduces a novel approach for displacement field calculation, explicitly derives higher-order Faxén relations, and computes displaceability and rotateability matrices up to sixth order.

## Key findings

- Higher-order Faxén relations are explicitly derived for compressible media.
- Displaceability and rotateability matrices are calculated up to sixth order.
- The methods are applicable to both elastic media and low-Reynolds-number hydrodynamics.

## Abstract

An efficient route to the displacement field around a rigid spherical inclusion in an infinitely extended homogeneous elastic medium is presented in a slightly alternative way when compared to some common textbook methods. Moreover, two Fax\'en relations of next-higher order beyond the stresslet are calculated explicitly for compressible media. They quantify higher-order moments involving the force distribution on rigid particles in a deformed elastic medium. Additionally, the displaceability and rotateability matrices are calculated up to (including) sixth order in inverse particle separation distance. These matrices describe the interactions mediated between the rigid embedded particles by the elastic environment. All methods and results can formally be transferred to the corresponding case of incompressible hydrodynamic low-Reynolds-number Stokes flow by considering the limit of an incompressible environment. The roles of compressibility of the embedding medium and of the here additionally derived higher-order contributions are highlighted by some selected example configurations.