Swimming at small Reynolds number of a collinear assembly of spheres in an incompressible viscous fluid

TL;DR

This paper models the swimming behavior of a linear chain of spheres at low Reynolds number, deriving formulas for velocity and power, and performs explicit calculations for a three-sphere chain.

Contribution

It introduces a detailed theoretical framework for analyzing swimming at small Reynolds number using a chain of spheres, including explicit formulas and calculations.

Findings

Derived expressions for mean swimming velocity and power to second order in displacement amplitude.

Validated the theoretical model with explicit calculations for a three-sphere chain.

Provided a basis for analyzing microscale swimming mechanisms.

Abstract

Swimming at small Reynolds number of a linear assembly of identical spheres immersed in a viscous fluid is studied on the basis of a set of equations of motion for the individual spheres. The motion of the spheres is caused by actuating forces and forces derived from a direct interaction potential, as well as hydrodynamic forces exerted by the fluid as frictional and added mass hydrodynamic interactions. The swimming velocity is deduced from the momentum balance equation for the assembly of spheres, and the mean power required during a period is calculated from an instantaneous power equation. Expressions are derived for the mean swimming velocity and the mean power, valid to second order in the amplitude of displacements from the relative equilibrium positions. Hence these quantities can be evaluated in terms of prescribed periodic displacements. Explicit calculations are performed for a linear chain of three identical spheres.