Advertisement

Translations and rotations at low Reynolds number: A study of simple model swimmers with finite amplitude strokes

  • M. Leoni
  • T. B. Liverpool
Regular Article
Part of the following topical collections:
  1. Active Matter

Abstract

We present a simple dynamical model of self-propeller at low Reynolds number in which self-propulsion is achieved via rotary elements (rotors). In this model by changing the sense of rotation of the rotors, the self-propeller can switch between a “linear swimming” phase where it swims in a straight line and a “tumbling” phase in which it can change direction in a controllable way via a global rotation of its body. We study the dynamics of this propeller in detail. To do this we provide an analytic framework within which the non-perturbative aspects of the internal dynamics can be treated allowing us to study the swimming process for arbitrary values of the swimmer deformations. Using it, we compute the averages (over a deformation cycle) of a number of characteristic properties of the swimmer such as its self-propulsion velocity, the dissipated power, its efficiency and the fluid flow patterns it generates. We compare these results to the corresponding average quantities for another class of model swimmers, where self-propulsion is achieved via periodic translations. Finally, we provide an explanation of why non-perturbative results can be obtained for these models using the geometrical language of gauge theory.

Graphical abstract

Keywords

Regular Article - Topical issue: Active Matter 

References

  1. 1.
    G. Taylor, Proc. R. Soc. London, Ser. A 209, 447 (1951)CrossRefADSGoogle Scholar
  2. 2.
    E.M. Purcell, Am. J. Phys. 45, 3 (1977)CrossRefADSGoogle Scholar
  3. 3.
    M.J. Lighthill, Mathematical Biofluiddynamics (SIAM, Philadelphia, 1975)Google Scholar
  4. 4.
    S. Childress, Mechanics of Swimming and Flying (Cambridge University Press, 1981)Google Scholar
  5. 5.
    H.C. Berg, E. coli in Motion (Springer, New York, 2003)Google Scholar
  6. 6.
    C. Brennen, H. Winet, Annu. Rev. Fluid Mech. 9, 339 (1977)CrossRefADSGoogle Scholar
  7. 7.
    E. Lauga, T.R. Powers, Rep. Progr. Phys. 72, 096601 (2009)MathSciNetCrossRefADSGoogle Scholar
  8. 8.
    S.J. Ebbens, J.R. Howse, Soft Matter 6, 726 (2010)CrossRefADSGoogle Scholar
  9. 9.
    E.A. Gaffney et al., Annu. Rev. Fluid Mech. 46, 501 (2011)CrossRefADSGoogle Scholar
  10. 10.
    M.J. Lighthill, Comm. Pure Appl. Math. 5, 109 (1952)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J.R. Blake, Bull. Austral. Math. Soc. 3, 255 (1971)CrossRefADSGoogle Scholar
  12. 12.
    G.I. Taylor, Proc. R. Soc. London, Ser. A 211, 225 (1952)CrossRefADSGoogle Scholar
  13. 13.
    A.T. Chwang, T.Y. Wu, Proc. R. Soc. London, Ser. B 178, 327 (1971)CrossRefADSGoogle Scholar
  14. 14.
    J. Garcia de la Torre, V.A. Bloomfield, Biophys. J. 20, 49 (1977)CrossRefGoogle Scholar
  15. 15.
    G.J. Hancock, Proc. R. Soc. London, Ser. A 217, 96 (1953)CrossRefADSGoogle Scholar
  16. 16.
    J. Lighthill, SIAM Rev. 18, 161 (1975)CrossRefGoogle Scholar
  17. 17.
    A. Najafi, R. Golestanian, Phys. Rev. E 69, 062901 (2004)CrossRefADSGoogle Scholar
  18. 18.
    R. Golestanian, A. Ajdari, Phys. Rev. E 77, 036308 (2008)CrossRefADSGoogle Scholar
  19. 19.
    A. Najafi, R. Zargar, Phys. Rev. E 81, 067301 (2010)CrossRefADSGoogle Scholar
  20. 20.
    M. Leoni, T.B. Liverpool, Europhys. Lett. 92, 64004 (2010)CrossRefADSGoogle Scholar
  21. 21.
    Y. Fily, A. Baskaran, M.C. Marchetti, Soft Matter 8, 3002 (2012)CrossRefADSGoogle Scholar
  22. 22.
    R. Dreyfus et al., Nature 437, 862 (2005)CrossRefADSGoogle Scholar
  23. 23.
    P. Tierno et al., Phys. Rev. Lett. 101, 218304 (2008)CrossRefADSGoogle Scholar
  24. 24.
    A. Ghosh, P. Fischer, Nano Lett. 9, 2243 (2009)CrossRefADSGoogle Scholar
  25. 25.
    L. Zhang et al., Nano Lett. 9, 3663 (2009)CrossRefADSGoogle Scholar
  26. 26.
    P. Tierno et al., Phys. Rev. E 81, 011402 (2010)CrossRefADSGoogle Scholar
  27. 27.
    P. Dhar et al., Nano Lett. 6, 66 (2006)CrossRefADSGoogle Scholar
  28. 28.
    S. Ebbens et al., Phys. Rev. E 82, 015304 (2010)CrossRefADSGoogle Scholar
  29. 29.
    M. Leoni et al., Soft Matter 5, 472 (2009)CrossRefADSGoogle Scholar
  30. 30.
    M. Leoni et al., Phys. Rev. E 81, 036304 (2010)CrossRefADSGoogle Scholar
  31. 31.
    J.S. Guasto, K.A. Johnson, J.P. Gollub, Phys. Rev. Lett. 105, 168102 (2010)CrossRefADSGoogle Scholar
  32. 32.
    K. Drescher et al., Phys. Rev. Lett. 105, 168101 (2010)CrossRefADSGoogle Scholar
  33. 33.
    A. Shapere, F. Wilczek, Phys. Rev. Lett. 58, 2051 (1987)CrossRefADSGoogle Scholar
  34. 34.
    A. Shapere, F. Wilczek, J. Fluid Mech. 198, 557 (1989)MathSciNetCrossRefADSGoogle Scholar
  35. 35.
    M. Doi, S. Edwards, The Theory of Polymer Dynamics (Oxford University Press, 1986)Google Scholar
  36. 36.
    P.G. Saffman, J. Fluid Mech. 73, 593 (1976)MathSciNetCrossRefADSGoogle Scholar
  37. 37.
    B.A. Hughes, B.A. Pailthorpe, L.R. White, J. Fluid Mech. 310, 349 (1981)CrossRefADSGoogle Scholar
  38. 38.
    A.J. Levine, T.B. Liverpool, F.C. MacKintosh, Phys. Rev. Lett. 93, 038102 (2004)CrossRefADSGoogle Scholar
  39. 39.
    H.C. Berg, D.A. Brown, Nature 239, 500 (1972)CrossRefADSGoogle Scholar
  40. 40.
    M. Leoni, T.B. Liverpool, Phys. Rev. Lett. 105, 238102 (2010)CrossRefADSGoogle Scholar
  41. 41.
    J.E. Avron, O. Raz, New J. Phys. 10, 063016 (2008)CrossRefADSGoogle Scholar
  42. 42.
    I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, seventh edition (Academic Press, 2007)Google Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • M. Leoni
    • 1
  • T. B. Liverpool
    • 1
  1. 1.Department of MathematicsUniversity of BristolBristolUK

Personalised recommendations