Effect of fluid inertia on swimming of a sphere in a viscous incompressible fluid

TL;DR

This paper investigates how fluid inertia influences the swimming velocity of a sphere in a viscous incompressible fluid, revealing complex interactions between surface distortions, flow patterns, and fluid viscosity across different regimes.

Contribution

It provides a comprehensive analysis of the effects of fluid inertia on spherical swimming, bridging Stokes and inertia-dominated regimes using the Navier-Stokes equations.

Findings

Mean swimming velocity depends on the balance of flow generated by surface distortions and Reynolds forces.

Flow patterns vary significantly with kinematic viscosity and scale number.

The model covers the entire range from viscous to inertial regimes, offering broad insights.

Abstract

Swimming of a sphere in a viscous incompressible fluid is studied on the basis of the Navier-Stokes equations for wave-type distortions of the spherical shape. At sizable values of the dimensionless scale number the mean swimming velocity is the result of a delicate balance between the net time-averaged flow generated directly by the surface distortions and the flow generated by the mean Reynolds force density. Depending on the stroke, this can lead to a surprising dependence of the mean swimming velocity on the kinematic viscosity of the fluid. The net flow pattern is calculated as a function of kinematic viscosity for axisymmetric strokes of the swimming sphere. The calculation covers the full range of scale number, from the friction-dominated Stokes regime in the limit of vanishing scale number to the inertia-dominated regime at large scale number. The model therefore provides paradigmatic insight into the fluid dynamics of swimming or flying of a wide range of organisms.