# Collinear velocity relaxation of two spheres in a viscous incompressible fluid

## TL;DR

This paper investigates the velocity relaxation dynamics of two spheres in a viscous incompressible fluid, revealing long-time decay behaviors and providing simplified analytical expressions for the relaxation functions.

## Contribution

It introduces an approximate method for analyzing hydrodynamic interactions and long-time relaxation tails in collinear sphere systems.

## Key findings

- Velocities decay with a t^{-3/2} tail over time.
- Relative velocity decays with a t^{-5/2} tail.
- Relaxation functions are approximated by simple pole expressions.

## Abstract

Collinear velocity relaxation of two spheres immersed in a viscous incompressible fluid is studied on the basis of an approximate expression for the retarded hydrodynamic interaction. After a sudden impulse applied to one sphere, the other one instantaneously starts to move as well, with amplitude determined by the added mass effect. The velocities of both spheres eventually decay with a t^{-3/2} long-time tail, but the relative velocity decays with a t^{-5/2} long-time tail. The three relaxation functions are approximated by simple expressions involving only a small number of poles in the complex square root of frequency plane.