Effect of inertia on laminar swimming and flying of an assembly of rigid spheres in an incompressible viscous fluid

TL;DR

This paper develops a comprehensive mechanical model to analyze how inertia influences the swimming and flying of rigid sphere assemblies in viscous fluids, covering a broad viscosity range applicable to animals and microorganisms.

Contribution

It introduces a new theoretical framework that incorporates hydrodynamic interactions and inertia effects for rigid sphere assemblies in viscous fluids.

Findings

Model applies to a wide viscosity range, from bacteria to birds.

Derived equations describe the velocity of the assembly's center.

Example with three spheres illustrates the theory's application.

Abstract

A mechanical model of swimming and flying in an incompressible viscous fluid in the absence of gravity is studied on the basis of assumed equations of motion. The system is modeled as an assembly of rigid spheres subject to elastic direct interactions and to periodic actuating forces which sum to zero. Hydrodynamic interactions are taken into account in the virtual mass matrix and in the friction matrix of the assembly. An equation of motion is derived for the velocity of the geometric center of the assembly. The mean power is calculated as the mean rate of dissipation. The full range of viscosity is covered, so that the theory can be applied to the flying of birds, as well as to the swimming of fish or bacteria. As an example a system of three equal spheres moving along a common axis is studied.