Swimming of a sphere in a viscous incompressible fluid with inertia

TL;DR

This paper analyzes how a sphere swims in a viscous incompressible fluid with inertia, optimizing surface modulations to maximize efficiency based on the Navier-Stokes equations.

Contribution

It introduces a method to optimize sphere swimming efficiency by solving a generalized eigenvalue problem using quadratic forms derived from surface modulations.

Findings

Optimal swimming efficiency depends on a complex interplay of physical parameters.

Surface modulations of low multipole order are most effective for optimization.

Efficiency varies with a dimensionless scale number involving sphere size, cycle period, and fluid viscosity.

Abstract

The swimming of a sphere immersed in a viscous incompressible fluid with inertia is studied for surface modulations of small amplitude on the basis of the Navier-Stokes equations. The mean swimming velocity and the mean rate of dissipation are expressed as quadratic forms in term of the surface displacements. With a choice of a basis set of modes the quadratic forms correspond to two hermitian matrices. Optimization of the mean swimming velocity for given rate of dissipation requires the solution of a generalized eigenvalue problem involving the two matrices. It is found for surface modulations of low multipole order that the optimal swimming efficiency depends in intricate fashion on a dimensionless scale number involving the radius of the sphere, the period of the cycle, and the kinematic viscosity of the fluid.