Mathematics of Complexity and Dynamical Systems

2011 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Biological Fluid Dynamics, Non-linear Partial Differential Equations

  • Antonio DeSimone
  • François Alouges
  • Aline Lefebvre
Reference work entry
DOI: https://doi.org/10.1007/978-1-4614-1806-1_3

Article Outline

Glossary

Definition of the Subject

Introduction

The Mathematics of Swimming

The Scallop Theorem Proved

Optimal Swimming

The Three-Sphere Swimmer

Future Directions

Bibliography

Keywords

Translational Invariance Hydrodynamic Interaction Small Length Scale Power Phase Adjacent Link 
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.
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Bibliography

Primary Literature

  1. 1.
    Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for low Reynolds number swimmers: an example. J Nonlinear Sci 18:277–302MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for low Reynolds number axisymmetric swimmers. Preprint SISSA 61/2008/MGoogle Scholar
  3. 3.
    Avron JE, Kenneth O, Oakmin DH (2005) Pushmepullyou: an efficient micro‐swimmer. New J Phys 7:234-1–8CrossRefGoogle Scholar
  4. 4.
    Becker LE, Koehler SA, Stone HA (2003) On self-propulsion of micro‐machines at low Reynolds numbers: Purcell's three-link swimmer. J Fluid Mechanics 490:15–35MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Berg HC, Anderson R (1973) Bacteria swim by rotating their flagellar filaments. Nature 245:380–382CrossRefGoogle Scholar
  6. 6.
    Lighthill MJ (1952) On the Squirming Motion of Nearly Spherical Deformable Bodies through Liquids at Very Small Reynolds Numbers. Comm Pure Appl Math 5:109–118MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Najafi A, Golestanian R (2004) Simple swimmer at low Reynolds numbers: Three linked spheres. Phys Rev E 69:062901-1–4CrossRefGoogle Scholar
  8. 8.
    Purcell EM (1977) Life at low Reynolds numbers. Am J Phys 45:3–11CrossRefGoogle Scholar
  9. 9.
    Tan D, Hosoi AE (2007) Optimal stroke patterns for Purcell's three-link swimmer. Phys Rev Lett 98:068105-1–4Google Scholar
  10. 10.
    Taylor GI (1951) Analysis of the swimming of microscopic organisms. Proc Roy Soc Lond A 209:447–461MATHCrossRefGoogle Scholar

Books and Reviews

  1. 11.
    Agrachev A, Sachkov Y (2004) Control Theory from the Geometric Viewpoint. In: Encyclopaedia of Mathematical Sciences, vol 87, Control Theory and Optimization. Springer, BerlinGoogle Scholar
  2. 12.
    Childress S (1981) Mechanics of swimming and flying. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  3. 13.
    Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Nijhoff, The HagueGoogle Scholar
  4. 14.
    Kanso E, Marsden JE, Rowley CW, Melli-Huber JB (2005) Locomotion of Articulated Bodies in a Perfect Fluid. J Nonlinear Sci 15:255–289MathSciNetMATHCrossRefGoogle Scholar
  5. 15.
    Koiller J, Ehlers K, Montgomery R (1996) Problems and Progress in Microswimming. J Nonlinear Sci 6:507–541MathSciNetMATHCrossRefGoogle Scholar
  6. 16.
    Montgomery R (2002) A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS Mathematical Surveys and Monographs, vol 91. American Mathematical Society, ProvidenceGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

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