Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-20T03:46:53.316Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Ahmed, S., Gentekos, D. T., Fink, C. A. & Mallouk, T. E. 2014 Self-assembly of nanorod motors into geometrically regular multimers and their propulsion by ultrasound. ACS Nano 8 (11), 11053–11060.CrossRefGoogle ScholarPubMed
Ahmed, S., Wang, W., Bai, L., Gentekos, D. T., Hoyos, M. & Mallouk, T. E. 2016 Density and shape effects in the acoustic propulsion of bimetallic nanorod motors. ACS Nano 10 (4), 4763–4769.CrossRefGoogle ScholarPubMed
Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21, 61–99.CrossRefGoogle Scholar
Baraban, L., Streubel, R., Makarov, D., Han, L., Karnaushenko, D., Schmidt, O. G. & Cuniberti, G. 2012 Fuel-free locomotion of janus motors: magnetically induced thermophoresis. ACS Nano 7, 1360–1367.CrossRefGoogle Scholar
Bruss, H. 2012 Acoustofluidics 2: perturbation theory and ultrasound resonance modes. Lab on a Chip 12, 20–28.CrossRefGoogle Scholar
Burdick, J., Laocharoenshuk, R., Wheat, P. M., Posner, J. D. & Wang, J. 2008 Synthetic nanomotors in microchannel networks: directional microchip motion and controlled manipulation of cargo. J. Am. Chem. Soc. 130, 8164–8165.CrossRefGoogle ScholarPubMed
Campuzano, S., Kagan, D., Orozco, J. & Wang, J. 2011 Motion-driven sensing and biosensing using electrochemically propelled nanomotors. Analyst 136, 4621–4630.CrossRefGoogle ScholarPubMed
Collis, J., Jesse, F., Chakraborty, D. & Sader, J. E. 2017 Autonomous propulsion of nanorods trapped in an acoustic field. J. Fluid Mech. 825, 29–48.CrossRefGoogle Scholar
Cordova-Figueroa, U. M. & Brady, J. F. 2008 Osmotic propulsion: the osmotic motor. Phys. Rev. Lett. 100 (15), 158303.CrossRefGoogle ScholarPubMed
Ebbens, S. J. & Howse, J. R. 2010 In pursuit of propulsion at the nanoscale. Soft Matt. 6, 726–738.CrossRefGoogle Scholar
Ebbens, S. J. & Howse, J. R. 2011 Direct observation of the direction of motion for spherical catalytic swimmers. Langmuir 27, 12293–12296.CrossRefGoogle ScholarPubMed
Golestanian, R., Liverpool, T. B. & Ajdari, A. 2007 Designing phoretic micro- and nano-swimmers. New J. Phys. 9, 126.CrossRefGoogle Scholar
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65 (2), 365–400.CrossRefGoogle Scholar
Ibele, M. E., Wang, Y., Kline, T. R., Mallouk, T. E. & Sen, A. 2007 Hydrazine fuels for bimetallic catalytic microfluic pumping. J. Am. Chem. Soc. 129 (25), 7762–7763.CrossRefGoogle Scholar
Jiang, H. R., Yoshinaga, N. & Sano, M. 2010 Active motion of a janus particle by self-thermophoresis in a defocused laser beam. Phys. Rev. Lett. 105, 268302.CrossRefGoogle Scholar
Kaynak, M., Ozcelik, A., Nourhani, A., Lammert, P. E., Crespi, V. H. & Huang, T. J. 2017 Acoustic actuation of bioinspired microswimmers. Lab on a Chip 17, 395–400.CrossRefGoogle ScholarPubMed
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics. Dover Publication, Inc.Google Scholar
Lauga, E. & Powers, T. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Lippera, K., Dauchot, O., Michelin, S. & Benzaquen, M. 2019 No net motion for oscillating near-spheres at low Re numbers. J. Fluid Mech. 866, R1.CrossRefGoogle Scholar
Mazur, P. & Bedeaux, D. 1974 A generalization of Faxén’s theorem to nonsteady motion of a sphere through an incompressible fluid in arbitrary flow. Physica D 76, 235–246.CrossRefGoogle Scholar
Nadal, F. & Lauga, E. 2014 Asymmetric steady streaming as a mechanism for acoustic propulsion of rigid bodies. Phys. Fluids 26, 082001.CrossRefGoogle Scholar
Nelson, B. J., Kaliakastos, I. K. & Abbott, J. J. 2010 Microrobots for minimally invasive medicine. Ann. Rev. Biomed. Eng. 12 (1), 041916.CrossRefGoogle ScholarPubMed
Pavlick, R. A., Dey, K. K., Sirjoosingh, A., Benesi, A. & Sen, A. 2013 A catalytically driven organometallic molecular motor. Nanoscale 5, 1301–1304.CrossRefGoogle ScholarPubMed
Pavlick, R. A., Sengupta, S., McFadden, T., Zhang, H. & Sen, A. 2011 A polymerization powered-motor. Angew. Chem. Intl Ed. 50, 9374–9377.CrossRefGoogle ScholarPubMed
Paxton, W. F., Kistler, K. C., C.c., O., Sen, A., Angelo, S. K. S., Mallouk, Y., Thomas, E., Lammert, P. E. & Crespi, V. H. 2004 Catalytic nanomotors: autonomous movement of stripped nanorods. J. Am. Chem. Soc. 126 (41), 13424–13431.CrossRefGoogle Scholar
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45, 3–11.CrossRefGoogle Scholar
Qian, B., Montiel, D., Bregulla, A., Cichos, F. & Yang, H. 2013 Harnessing thermal fluctuations for purposeful activities: the manipulation of single microswimmers by adaptative photon nudging. Chem. Sci. 4, 1420–1429.CrossRefGoogle Scholar
Riley, N. 1966 On a sphere oscillating in a viscous fluid. Q. J. Mech. Appl. Maths XIX (4), 461–472.CrossRefGoogle Scholar
Sabrina, S., Tasinkevych, M., Ahmed, S., Brooks, A. M., Olvera de la Cruz, M., Mallouk, T. E. & Bishop, K. J. M. 2018 Shape-directed microspinners powered by ultrasound. ACS Nano 12 (3), 2939–2947.CrossRefGoogle ScholarPubMed
Smoluchowsky, M. 1921 Handbuch der Electrizitat und des Magnetismus. Graetz (ed.), Leipzig.Google Scholar
Stokes, Sir G. G. 1850 On the effects of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. IX, 8.Google Scholar
Sundararajan, S., Lammert, P. E., Zudans, A. W., Crespi, V. H. & Sen, A. 2008 Catalytic motors for transport of colloidal cargo. Nano Lett. 8, 1271–1276.CrossRefGoogle ScholarPubMed
Wang, W., Castro, L. A., Hoyos, M. & Mallouk, T. E. 2012 Autonomous motion of metallic microrods propelled by ultrasound. ACS Nano 6 (7), 6122–6132.CrossRefGoogle ScholarPubMed
Wang, W., Duan., W., Ahmed, S., Mallouk, T. E. & Sen, A. 2013 Small power: autonomous nano- and micromotors propelled by self-generated gradients. Nano Today 8, 531–554.CrossRefGoogle Scholar
Wu, J., Kagan, S. B. D., Manesh, K. M., Campuzano, S. & Wang, J. 2010 Motion-based DNA detection using catalytic nanomotors. Nat. Commun. 1 (3), 36.CrossRefGoogle ScholarPubMed
Zhang, W. & Stone, H. A. 1998 Oscillatory motions of circular disks and nearly spherical particles in viscous flows. J. Fluid Mech. 367, 329–358.CrossRefGoogle Scholar