Journal of Nonlinear Science

, Volume 18, Issue 3, pp 277–302 | Cite as

Optimal Strokes for Low Reynolds Number Swimmers: An Example

  • François Alouges
  • Antonio DeSimoneEmail author
  • Aline Lefebvre


Swimming, i.e., being able to advance in the absence of external forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers. This is the regime of interest for micro-organisms and micro- or nano-robots. We focus in this paper on a simple yet representative example: the three-sphere swimmer of Najafi and Golestanian (Phys. Rev. E, 69, 062901–062904, 2004). For this system, we show how to cast the problem of swimming in the language of control theory, prove global controllability (which implies that the three-sphere swimmer can indeed swim), and propose a numerical algorithm to compute optimal strokes (which turn out to be suitably defined sub-Riemannian geodesics).


Biological and artificial micro-swimmers Optimal control Optimal gait Propulsion efficiency Movement and locomotion Low-Reynolds-number (creeping) flow 

AMS Subject Classifications

76Z10 49J20 92C10 93B05 


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  1. Agrachev, A., Sachkov, Y.: Control theory from the geometric viewpoint. In: Control Theory and Optimization. Encyclopaedia of Mathematical Sciences, vol. 87. Springer, New York (2004) Google Scholar
  2. Avron, J.E., Kenneth, O., Oakmin, D.H.: Pushmepullyou: An efficient micro-swimmer. New J. Phys. 7, 234–238 (2005) CrossRefGoogle Scholar
  3. Batchelor, G.K.: Brownian diffusion of particles with hydrodynamic interactions. J. Fluid Mech. 74, 1–29 (1976) zbMATHCrossRefMathSciNetGoogle Scholar
  4. Becker, L.E., Koehler, S.A., Stone, H.A.: On self-propulsion of micro-machines at low Reynolds numbers: Purcell’s three-link swimmer. J. Fluid Mech. 490, 15–35 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  5. Berg, H.C., Anderson, R.: Bacteria swim by rotating their flagellar filaments. Nature 245, 380–382 (1973) CrossRefGoogle Scholar
  6. Bressan, A.: Impulsive control of Lagrangian systems and locomotion in fluids. Preprint (2006) Google Scholar
  7. Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 4. Springer, Berlin (1990) Google Scholar
  8. Golestanian, R.: personal communication to ADS (2006) Google Scholar
  9. Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.B.: Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15, 255–289 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  10. Koiller, J., Ehlers, K., Montgomery, R.: Problems and progress in microswimming. J. Nonlinear Sci. 6, 507–541 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  11. Leshansky, A.M., Kenneth, O., Gat, O., Avron, J.E.: A frictionless microswimmer. Preprint (2007) Google Scholar
  12. Lighthill, M.J.: On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Math. 5, 109–118 (1952) zbMATHCrossRefMathSciNetGoogle Scholar
  13. Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS Mathematical Surveys and Monographs, vol. 91. AMS, Providence (2002) Google Scholar
  14. Morrey, C.B., Niremberg, L.: On the analyticity of the solutions of linear elliptic systems of partial differential equations. Commun. Pure Appl. Math. 10, 271–290 (1957) zbMATHCrossRefGoogle Scholar
  15. Najafi, A., Golestanian, R.: Simple swimmer at low Reynolds numbers: Three linked spheres. Phys. Rev. E 69, 062901–062904 (2004) CrossRefGoogle Scholar
  16. Purcell, E.M.: Life at low Reynolds numbers. Am. J. Phys. 45, 3–11 (1977) CrossRefGoogle Scholar
  17. Taylor, G.I.: Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461 (1951) zbMATHCrossRefGoogle Scholar
  18. Wilkening, J., Hosoi, A.E.: Shape optimization of swimming sheets. Preprint (2007) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • François Alouges
    • 1
  • Antonio DeSimone
    • 2
    Email author
  • Aline Lefebvre
    • 1
  1. 1.Laboratoire de MathématiquesUniversité Paris-SudOrsay cedexFrance
  2. 2.SISSA-International School for Advanced StudiesTriesteItaly

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