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Acoustic propulsion of a small, bottom-heavy sphere

Published online by Cambridge University Press:  25 June 2020

François Nadal Open the ORCID record for François Nadal [Opens in a new window]  and
Sébastien Michelin Open the ORCID record for Sébastien Michelin [Opens in a new window]
François Nadal*
Affiliation:
Wolfson School of Mechanical Electrical and Manufacturing Engineering, Loughborough University, LoughboroughLE11 3TU, UK
Sébastien Michelin
Affiliation:
LadHyX – Département de Mécanique, CNRS – École Polytechnique, Institut Polytechnique de Paris, 91128Palaiseau, France
*
† Email address for correspondence: F.R.Nadal@lboro.ac.uk
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Abstract

We present here a comprehensive derivation for the speed of a small bottom-heavy sphere forced by a transverse acoustic field and thereby establish how density inhomogeneities may play a critical role in acoustic propulsion. The sphere is trapped at the pressure node of a standing wave whose wavelength is much larger than the sphere diameter. Due to its inhomogeneous density, the sphere oscillates in translation and rotation relative to the surrounding fluid. The perturbative flows induced by the sphere’s rotation and translation are shown to generate a rectified inertial flow responsible for a net mean force on the sphere that is able to propel the particle within the zero-pressure plane. To avoid an explicit derivation of the streaming flow, the propulsion speed is computed exactly using a suitable version of the Lorentz reciprocal theorem. The propulsion speed is shown to scale as the inverse of the viscosity, the cube of the amplitude of the acoustic field and is a non-trivial function of the acoustic frequency. Interestingly, for some combinations of the constitutive parameters (fluid-to-solid density ratio, moment of inertia and centroid to centre of mass distance), the direction of propulsion is reversed as soon as the frequency of the forcing acoustic field becomes larger than a certain threshold. The results produced by the model are compatible with both the observed phenomenology and the orders of magnitude of the measured velocities.

JFM classification

Biological Fluid Dynamics: Propulsion Biological Fluid Dynamics: Swimming/flying
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 898 , 10 September 2020 , A10
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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