Dynamics of Purcell’s three-link microswimmer with a passive elastic tail

Regular Article
Part of the following topical collections:
  1. Active Matter

Abstract

One of the few possible mechanisms for self-propulsion at low Reynolds number is undulations of a passive elastic tail, as proposed in the classical work of Purcell (1977). This effect is studied here by investigating a variant of Purcell’s three-link swimmer model where the front joint angle is periodically actuated while the rear joint is driven by a passive torsional spring. The dynamic equations of motion are formulated and explicit expressions for the leading-order solution are derived by using perturbation expansion. The dependence of the motion on the actuation amplitude and frequency is analyzed, and optimization with respect to the swimmer’s geometry is conducted.

Keywords

Topical contribution 

Supplementary material

10189_2012_9755_MOESM1_ESM.zip (619 kb)
Supplementary material, approximately 619 MB.

References

  1. 1.
    C. Brennen, H. Winet, Annu. Rev. Fluid Mech. 9, 339 (1977).ADSCrossRefGoogle Scholar
  2. 2.
    J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, 1965).Google Scholar
  3. 3.
    E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009).MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    E.M. Purcell, Am. J. Phys. 45, 3 (1977).ADSCrossRefGoogle Scholar
  5. 5.
    J. Gray, H.W. Lissmann, J. Exp. Biol. 41, 135 (1964).Google Scholar
  6. 6.
    K.E. Machin, J. Exp. Biol. 35, 796 (1958).Google Scholar
  7. 7.
    C.J. Brokaw, Science 178, 455 (1972).ADSCrossRefGoogle Scholar
  8. 8.
    S. Camalet, F. Jülicher, J. Prost, Phys. Rev. Lett. 82, 1590 (1999).ADSCrossRefGoogle Scholar
  9. 9.
    E.A. Gaffney, H. Gadelha, D.J. Smith, J.R. Blake, J.C. Kirkman-Brown, Annu. Rev. Fluid Mech. 43, 501 (2011).MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    R. Dreyfus, J. Baudry, M.L. Roper, M. Fermigier, H.A. Stone, J. Bibette, Nature 437, 862 (2005).ADSCrossRefGoogle Scholar
  11. 11.
    B.J. Nelson, I.K. Kaliakatsos, J.J. Abbott, Rev. Biomed. Engin. 12, 55 (2010).CrossRefGoogle Scholar
  12. 12.
    G.I. Taylor, Proc. R. Soc. London, Ser. A 209, 447 (1951).ADSCrossRefMATHGoogle Scholar
  13. 13.
    J. Gray, G.J. Hancock, J. Exp. Biol. 32, 802 (1955).Google Scholar
  14. 14.
    M.J. Lighthill, Comm. Pure Appl. Math. 5, 109 (1952).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    J.R. Blake, J. Fluid Mech. 46, 199 (1971).ADSCrossRefMATHGoogle Scholar
  16. 16.
    A. Najafi, R. Golestanian, Phys. Rev. E 69, 062901 (2004).ADSCrossRefGoogle Scholar
  17. 17.
    J.E. Avron, O. Kenneth, D.H. Oakmin, New J. Phys. 7, 234 (2005).ADSCrossRefGoogle Scholar
  18. 18.
    A.M. Leshansky, O. Kenneth, Phys. Fluids 20, 063104 (2008).ADSCrossRefGoogle Scholar
  19. 19.
    L.E. Becker, S.A. Koehler, H.A. Stone, J. Fluid Mech. 490, 15 (2003).MathSciNetADSCrossRefMATHGoogle Scholar
  20. 20.
    R.G. Cox, J. Fluid Mech. 44, 791 (1970).ADSCrossRefMATHGoogle Scholar
  21. 21.
    D. Tam, A.E. Hosoi, Phys. Rev. Lett. 98, 068105 (2007).ADSCrossRefGoogle Scholar
  22. 22.
    J.E. Avron, O. Raz, New J. Phys. 10, 063016 (2008).ADSCrossRefGoogle Scholar
  23. 23.
    Y. Or, Phys Rev. Lett. 108, 258101 (2012).ADSCrossRefGoogle Scholar
  24. 24.
    C.H. Wiggins, R.E. Goldstein, Phys. Rev. Lett. 80, 3879 (1998).ADSCrossRefGoogle Scholar
  25. 25.
    M.L. Roper, R. Dreyfus, J. Baudry, M. Fermigier, J. Bibette, H.A. Stone, J. Fluid Mech. 554, 167 (2006).MathSciNetADSCrossRefMATHGoogle Scholar
  26. 26.
    E. Lauga, Phys. Rev. E 75, 041916 (2007).MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    A. Shapere, F. Wilczek, J. Fluid Mech. 198, 557 (1989).MathSciNetADSCrossRefMATHGoogle Scholar
  28. 28.
    A supplementary document supp1.pdf with more technical details is submitted online.Google Scholar
  29. 29.
    A.H. Nayfeh, Perturbation Methods (Willey-VCH, 2004).Google Scholar
  30. 30.
    J. Lighthill, Mathematical Biofluiddynamics (SIAM, Philadelphia, PA, 1975).Google Scholar
  31. 31.
    E.M. Purcell, Proc. Natl. Acad. Sci. U.S.A. 94, 11307 (1997).ADSCrossRefGoogle Scholar
  32. 32.
    J.E. Avron, O. Gat, O. Kenneth, Phys. Rev. Lett. 93, 186001 (2004).ADSCrossRefGoogle Scholar
  33. 33.
    S. Childress, J. Fluid Mech. 705, 77 (2012).MathSciNetADSCrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Faculty of Mechanical EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael

Personalised recommendations