Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T08:35:29.576Z Has data issue: false hasContentIssue false
  • English
  • Français

On the self-propulsion of an $N$-sphere micro-robot

Published online by Cambridge University Press:  28 January 2013

V. A. Vladimirov
V. A. Vladimirov*
Affiliation:
Department of Mathematics, University of York, Heslington, York YO10 5DD, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The aim of this paper is to describe the self-propulsion of a micro-robot (or micro-swimmer) consisting of $N$ spheres moving along a fixed line. The spheres are linked to each other by arms with their lengths changing periodically. We use the asymptotic procedure containing the two-timing method and a distinguished limit. We show that self-propulsion velocity appears (in the main approximation) as a linear combination of velocities of all possible triplets of spheres. Velocities and efficiencies of three-, four- and five-sphere swimmers are calculated.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Low-Reynolds-number flows: Low-Reynolds-number flows Micro-/Nano-fluid dynamics: Micro-/Nano-fluid dynamics Biological Fluid Dynamics: Propulsion
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 716 , 10 February 2013 , R1
Copyright
©2013 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

Alexander, G. P., Pooley, C. M. & Yeomans, J. M. 2009 Hydrodynamics of linked sphere model swimmers. J. Phys.: Condens. Matter 21, 204108.Google ScholarPubMed
Alouges, F., DeSimone, A. & Lefebvre, A. 2008 Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18, 277–302.CrossRefGoogle Scholar
Avron, J. E., Kenneth, O. & Oaknin, D. H. 2005 Pushmepullyou: an efficient micro-swimmer. New J. Phys. 7, 234.CrossRefGoogle Scholar
Becker, D. J., Koelher, C. M. & Ryder, Stone 2003 On self-propulsion of micro-machines at low Reynolds number: Purcell’s three-link swimmer. J. Fluid Mech. 490, 15–35.CrossRefGoogle Scholar
Blake, J. R. 1971 Infinite models for ciliary propulsion. J. Fluid Mech. 49, 209–227.CrossRefGoogle Scholar
Chang, S. T., Paunov, V. N., Petsev, D. N. & Orlin, D. V. 2007 Remotely powered self-propelling particles and micropumps based on miniature diodes. Nat. Mater. 6, 235–240.CrossRefGoogle ScholarPubMed
Childress, S. 1981 Mechanics of Swimming and Flying. Cambridge University Press.CrossRefGoogle Scholar
Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. & Bibette, J. 2005 Microscopic artificial swimmers. Nature 437 (6), 862–865.CrossRefGoogle ScholarPubMed
Earl, D. J., Pooley, C. M., Ryder, J. F., Bredberg, I. & Yeomans, J. M. 2007 Modelling microscopic swimmers at low Reynolds number. J. Chem. Phys. 126, 064703.CrossRefGoogle ScholarPubMed
Gilbert, A. D., Ogrin, F. Y., Petrov, P. G. & Wimlove, C. P. 2010 Theory of ferromagnetic microswimmers. Q. J. Mech. Appl. Maths 64, 239–263.CrossRefGoogle Scholar
Golestanian, R. & Ajdari, A. 2008 Analytic results for the three-sphere swimmer at low Reynolds number. Phys. Rev. E 77, 036308.CrossRefGoogle ScholarPubMed
Golestanian, R. & Ajdari, A. 2009 Stochastic low Reynolds number swimmers. J. Phys.: Condens. Matter 21, 204104.Google ScholarPubMed
Kanso, E. & Newton, P. K. 2009 Passive locomotion via normal-mode coupling in a submerged spring-mass system. J. Fluid Mech. 641, 205–215.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics. Butterworth-Heinemann.Google Scholar
Lauga, E. 2011 Life around the scallop theorem. Soft Matt. 7, 3060–3065.CrossRefGoogle Scholar
Leoni, M., Kotar, J., Bassetti, B., Cicuta, P. & Lagomarsino, M. C. 2009 A basic swimmer at low Reynolds number. Soft Matt. 5, 472–476.CrossRefGoogle Scholar
Moffatt, H. K. 1996 Dynamique des Fluides, Tome 1, Microhydrodynamics. Ecole Polytechnique.Google Scholar
Najafi, A. & Golestanian, R. 2004 Simple swimmer at low Reynolds number: three linked spheres. Phys. Rev. E 69, 062901.CrossRefGoogle ScholarPubMed
Pedley, T. J. 2009 Biomechanics of aquatic micro-organisms. New Trends Fluid Mech. Res. 1, 1–6.Google Scholar
Pedley, T. J. & Kessler, J. O. 1987 The orientation of spheroidal microorganisms swimming in a flow field. Proc. R. Soc. Lond. B 231, 47–70.Google Scholar
Polin, M., Tuval, I., Dresher, K., Gollub, J. P. & Goldstein, R. E. 2009 Chlamydomonas swims with two gears in a eukaryotic version of run-and-tumble locomotion. Science 325, 487–490.CrossRefGoogle Scholar
Purcell, E. M. 1977 Life at low Reynolds number. Am. J. Phys. 45 (1), 3–11.CrossRefGoogle Scholar
Saffman, P. G. 1967 The self-propulsion of a deformable body in a perfect fluid. J. Fluid Mech. 28 (2), 385–389.CrossRefGoogle Scholar
Shapere, A. & Wilczek, F. 1989 Efficiencies of self-propulsion at low Reynolds number. J. Fluid Mech. 198, 587–599.CrossRefGoogle Scholar
Taylor, G. I. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461.Google Scholar
Vladimirov, V. A. 2005 Vibrodynamics of pendulum and submerged solid. J. Math. Fluid Mech. 7, S397–412.CrossRefGoogle Scholar
Vladimirov, V. A. 2008 Viscous flows in a half-space caused by tangential vibrations on its boundary. Stud. Appl. Maths 121 (4), 337–367.CrossRefGoogle Scholar
Vladimirov, V. A. 2010 Admixture and drift in oscillating fluid flows. E-print: ArXiv:1009.4085v1, (physics, flu-dyn).Google Scholar
Vladimirov, V. A. 2011 Theory of non-degenerate oscillatory flows. E-print: ArXiv:1110.3633v2, (physics, flu-dyn).Google Scholar
Vladimirov, V. A. 2012a Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo?. J. Fluid Mech. 698, 51–61.CrossRefGoogle Scholar
Vladimirov, V. A. 2012b Self-propulsion of V-shape micro-robot. Phys. Fluids (submitted), E-print: ArXiv:1209.2835v1 (physics, flu-dyn).Google Scholar
Vladimirov, V. A. 2012c Theory of a triangular micro-robot. J. Fluid Mech. (submitted), E-print: ArXiv:1210.0747v1 (physics, flu-dyn).Google Scholar
Vladimirov, V. A. 2012d Acoustic-drift equation. E-print: ArXiv:1206.1297v1 (physics, flu-dyn). An upgraded version is submitted to the J. Acoust. Soc. Am. (jointly with Prof. K.I. Ilin).Google Scholar
Vladimirov, V. A., Wu, M. S., Pedley, T. J., Denissenko, P. V. & Zakhidova, S. G. 2004 Measurements of cell velocity distributions in populations of motile algae. J. Expl Biol. 207, 1203–1216.CrossRefGoogle ScholarPubMed