Swimming at small Reynolds number of a planar assembly of spheres in an incompressible viscous fluid with inertia

TL;DR

This paper investigates the swimming behavior of a planar assembly of spheres at low Reynolds number, incorporating fluid inertia effects, and derives formulas for swimming velocities and power consumption based on hydrodynamic interactions.

Contribution

It introduces a detailed model including inertia and hydrodynamic interactions for sphere assemblies, providing explicit formulas for swimming velocities and power at small amplitudes.

Findings

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

Explicit calculations for a three-sphere equilateral triangle configuration.

Quantified effects of inertia and hydrodynamic interactions on swimming performance.

Abstract

Translational and rotational swimming at small Reynolds number of a planar assembly of identical spheres immersed in an incompressible 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 translational and rotational swimming velocities of the assembly are deduced from momentum and angular momentum balance equations. The mean power required during a period is calculated from an instantaneous power equation. Expressions are derived for the mean swimming velocities and the power, valid to second order in the amplitude of displacements from the relative equilibrium positions. Hence these quantities can be evaluated for prescribed periodic displacements. Explicit calculations are performed for three spheres interacting such that they form an equilateral triangle in the rest configuration.