A Low-Reynolds-Number Treadmilling Swimmer Near a Semi-infinite Wall

  • Kiori Obuse
  • Jean-Luc ThiffeaultEmail author
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 155)


We investigate the behavior of a treadmilling microswimmer in a two-dimensional unbounded domain with a semi-infinite no-slip wall. The wall can also be regarded as a probe or pipette inserted into the flow. We solve the governing evolution equations in an analytical form and numerically calculate trajectories of the swimmer for several different initial positions and orientations. We then compute the probability that the treadmilling organism can escape the vicinity of the wall. We find that many trajectories in a ‘wedge’ around the wall are likely to escape. This suggests that inserting a probe or pipette in a suspension of organism may push away treadmilling swimmers.


Hydrodynamic Interaction Escape Probability Symmetric Periodic Orbit Conformal Mapping Technique Tangential Velocity Profile 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors are grateful for the hospitality of the Geophysical Fluid Dynamics Program at the Woods Hole Oceanographic Institution (supported by NSF), and thank Matthew D. Finn for his helpful advice and suggestions. Some of the numerical calculations for this project were performed at the Institute for Information Management and Communication of Kyoto University.


  1. [1]
    Avron JE, Kenneth O, Oakmin DH (2005) Pushmepullyou: an efficient micro-swimmer. New J Phys 7:234CrossRefGoogle Scholar
  2. [2]
    Berke AP, Turner L, Berg HC, Lauga E (2008) Hydrodynamic attraction of swimming microorganisms by surfaces. Phys Rev E 101:038102Google Scholar
  3. [3]
    Cosson J, Huitorel P, Gagnon C (2003) How spermatozoa come to be confined to surfaces. Cell Motil Cytoskel 54:56–63CrossRefGoogle Scholar
  4. [4]
    Crowdy DG, Or Y (2010) Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. Phys Rev E 81:036313CrossRefGoogle Scholar
  5. [5]
    Crowdy DG, Samson O (2011) Hydrodynamic bound states of a low-Reynolds-number swimmer near a gap in a wall. J Fluid Mech 667:309–335MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Drescher K, Leptos K, Tuval I, Ishikawa T, Pedley TJ, Goldstein RE (2009) Dancing volvox: hydrodynamic bound states of swimming algae. Phys Rev Lett 102:168101CrossRefGoogle Scholar
  7. [7]
    Hernandez-Ortiz JP, Dtolz CG, Graham MD (2005) Transport and collective dynamics in suspensions of confined swimming particles. Phys Rev Lett 95:204501CrossRefGoogle Scholar
  8. [8]
    Lauga E, DiLuzio WR, Whitesides GM, Stone HA (2006) Swimming in circles: motion of bacteria near solid boundaries. Biophys J 90:400–412CrossRefGoogle Scholar
  9. [9]
    Lauga E, Powers TR (2009) The hydrodynamics of swimming micro-organisms. Rep Prog Phys 72:096601MathSciNetCrossRefGoogle Scholar
  10. [10]
    Leshansky AM, Kenneth O, Gat O, Avron JE (2007) A frictionless microswimmer. New J Phys 9:145CrossRefGoogle Scholar
  11. [11]
    Najafi A, Golestanian R (2004) Simple swimmer at low Reynolds number: three linked spheres. Phys Rev E 69:062901CrossRefGoogle Scholar
  12. [12]
    Obuse K (2010) Trajectories of a low Reynolds number treadmilling organism near a half-infinite no-slip wall. In: Proceedings of the 2010 summer program in geophysical fluid dynamics, Woods Hole Oceanographic Institute, Woods HoleGoogle Scholar
  13. [13]
    Or Y, Murray RM (2009) Dynamics and stability of a class of low Reynolds number swimmers near a wall. Phys Rev E 79:045302CrossRefGoogle Scholar
  14. [14]
    Pedley TJ, Kessler JO (1992) Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu Rev Fluid Mech 24:313–358MathSciNetCrossRefGoogle Scholar
  15. [15]
    Ramia M, Tullock DL, Phan-Thien N (1993) The role of hydrodynamics interaction in the locomotion of microorganism. Biophys J 65:755–778CrossRefGoogle Scholar
  16. [16]
    Rothschild AJ (1963) Non-random distribution of bull spermatozoa in a drop of sperm suspension. Nature 198:1221–1222CrossRefGoogle Scholar
  17. [17]
    Shapere A, Wilczek F (1989) Geometry of self-propulsion at low Reynolds numbers. J Fluid Mech 198:557–585MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Winet H, Bernstein GS, Head J (1984) Observations on the response of human spermatozoa to gravity, boundaries and fluid shear. J Reprod Fert 70:511–523CrossRefGoogle Scholar
  19. [19]
    Zhang S, Or Y, Murray RM (2010) Experimental demonstration of the dynamics and stability of a low Reynolds number swimmer near a plane wall. In: Proceedings of american control conference, Baltimore, pp 4205–4210Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA

Personalised recommendations