Swimming of a linear chain with a cargo in an incompressible viscous fluid with inertia

TL;DR

This paper develops an approximation for the added mass matrix of sphere assemblies to analyze swimming dynamics of a chain with a cargo in viscous fluids, covering both inertia and friction regimes.

Contribution

It introduces a new approximation for the added mass matrix considering a large sphere among small spheres, enabling comprehensive swimming analysis across flow regimes.

Findings

The approximation accurately predicts swimming velocity and dissipation.

Results agree with bilinear theory for small strokes.

The model covers the full range from Stokes to inertia-dominated flow.

Abstract

An approximation to the added mass matrix of an assembly of spheres is constructed on the basis of potential flow theory for situations where one sphere is much larger than the others. In the approximation the flow potential near a small sphere is assumed to be dipolar, but near the large sphere it involves all higher order multipoles. The analysis is based on an exact result for the potential of a magnetic dipole in the presence of a superconducting sphere. Subsequently, the approximate added mass hydrodynamic interactions are used in a calculation of the swimming velocity and rate of dissipation of linear chain structures consisting of a number of small spheres and a single large one, with account also of frictional hydrodynamic interactions. The results derived for periodic swimming on the basis of a kinematic approach are compared with bilinear theory, valid for small amplitude of stroke, and with the numerical solution of the approximate equations of motion. The calculations cover the whole range of scale number between the friction-dominated Stokes limit and the inertia-dominated regime.