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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Brennen, H. Winet, Annu. Rev. Fluid Mech. 9, 339 (1977).
J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, 1965).
E. Lauga, T.R. Powers, Rep. Prog. Phys. 72, 096601 (2009).
E.M. Purcell, Am. J. Phys. 45, 3 (1977).
J. Gray, H.W. Lissmann, J. Exp. Biol. 41, 135 (1964).
K.E. Machin, J. Exp. Biol. 35, 796 (1958).
C.J. Brokaw, Science 178, 455 (1972).
S. Camalet, F. Jülicher, J. Prost, Phys. Rev. Lett. 82, 1590 (1999).
E.A. Gaffney, H. Gadelha, D.J. Smith, J.R. Blake, J.C. Kirkman-Brown, Annu. Rev. Fluid Mech. 43, 501 (2011).
R. Dreyfus, J. Baudry, M.L. Roper, M. Fermigier, H.A. Stone, J. Bibette, Nature 437, 862 (2005).
B.J. Nelson, I.K. Kaliakatsos, J.J. Abbott, Rev. Biomed. Engin. 12, 55 (2010).
G.I. Taylor, Proc. R. Soc. London, Ser. A 209, 447 (1951).
J. Gray, G.J. Hancock, J. Exp. Biol. 32, 802 (1955).
M.J. Lighthill, Comm. Pure Appl. Math. 5, 109 (1952).
J.R. Blake, J. Fluid Mech. 46, 199 (1971).
A. Najafi, R. Golestanian, Phys. Rev. E 69, 062901 (2004).
J.E. Avron, O. Kenneth, D.H. Oakmin, New J. Phys. 7, 234 (2005).
A.M. Leshansky, O. Kenneth, Phys. Fluids 20, 063104 (2008).
L.E. Becker, S.A. Koehler, H.A. Stone, J. Fluid Mech. 490, 15 (2003).
R.G. Cox, J. Fluid Mech. 44, 791 (1970).
D. Tam, A.E. Hosoi, Phys. Rev. Lett. 98, 068105 (2007).
J.E. Avron, O. Raz, New J. Phys. 10, 063016 (2008).
Y. Or, Phys Rev. Lett. 108, 258101 (2012).
C.H. Wiggins, R.E. Goldstein, Phys. Rev. Lett. 80, 3879 (1998).
M.L. Roper, R. Dreyfus, J. Baudry, M. Fermigier, J. Bibette, H.A. Stone, J. Fluid Mech. 554, 167 (2006).
E. Lauga, Phys. Rev. E 75, 041916 (2007).
A. Shapere, F. Wilczek, J. Fluid Mech. 198, 557 (1989).
A supplementary document supp1.pdf with more technical details is submitted online.
A.H. Nayfeh, Perturbation Methods (Willey-VCH, 2004).
J. Lighthill, Mathematical Biofluiddynamics (SIAM, Philadelphia, PA, 1975).
E.M. Purcell, Proc. Natl. Acad. Sci. U.S.A. 94, 11307 (1997).
J.E. Avron, O. Gat, O. Kenneth, Phys. Rev. Lett. 93, 186001 (2004).
S. Childress, J. Fluid Mech. 705, 77 (2012).
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Passov, E., Or, Y. Dynamics of Purcell’s three-link microswimmer with a passive elastic tail. Eur. Phys. J. E 35, 78 (2012). https://doi.org/10.1140/epje/i2012-12078-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1140/epje/i2012-12078-9