Skip to main content
SpringerLink
Log in

The Role of \(\mathop{\mathrm{SE}}\nolimits (d)\)-Reduction for Swimming in Stokes and Navier-Stokes Fluids

  • Chapter
Geometry, Mechanics, and Dynamics

Part of the book series: Fields Institute Communications ((FIC,volume 73))

Abstract

Steady swimming appears both periodic and stable. These characteristics are the very definition of limit cycles, and so we ask “Can we view swimming as a limit cycle?” In this paper we will not be able to answer this question in full. However, we shall find that reduction by \(\mathop{\mathrm{SE}}\nolimits (d)\)-symmetry brings us closer. Upon performing reduction by symmetry, we will find a stable fixed point which corresponds to a motionless body in stagnant water. We will then speculate on the existence of periodic orbits which are “approximately” limit cycles in the reduced system. When we lift these periodic orbits from the reduced phase space, we obtain dynamically robust relatively periodic orbits wherein each period is related to the previous by an \(\mathop{\mathrm{SE}}\nolimits (d)\) phase. Clearly, an \(\mathop{\mathrm{SE}}\nolimits (d)\) phase consisting of nonzero translation and identity rotation means directional swimming, while non-trivial rotations correspond to turning with a constant turning radius.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Central patter generators (CPGs) are neural networks which produce time-periodic signals.

  2. 2.

    We view \(\mathfrak{X}(M)\) as a Fréchet vector space.

  3. 3.

    This definition was taken from the introduction of [13] and is equivalent to the definition used in [19].

  4. 4.

    This is a pseudo Lie group. We will assume that all diffeomorphisms approach the identity as \(\|x\| \rightarrow \infty \) sufficiently rapidly for all computations to make sense. In particular, the existence of a Hodge-decomposition for our space is important. Sufficient conditions for our purposes are provided in [8] and [43].

  5. 5.

    The immersed boundary method [39] and smooth-particle hydrodynamics [17, 32] are both candidates.

References

  1. Abraham, R., Marsden, J.E.: Foundations of Mechanics, Benjamin/Cummings Publishing Co. Inc. Advanced Book Program, Reading, MA (1978). Second edition, revised and enlarged, with the assistance of Tudor Ratiu and Richard Cushman. Reprinted by AMS Chelsea (2008)

    Google Scholar 

  2. Alben, S., Shelley, M.J.: Coherent locomotion as an attracting state for a free flapping body. Proc. Natl. Acad. Sci. USA 102(32), 11163–11166 (2005)

    Article  Google Scholar 

  3. Arnold, V.I.: Sur la géométrie différentielle des groupes de lie de dimenision infinie et ses applications à l’hydrodynamique des fluides parfaits. Annales de l’Institute Fourier 16, 316–361 (1966)

    Google Scholar 

  4. Arnold, V.I., Khesin, B.A.: Topological Methods in Hydrodynamics. Applied Mathematical Sciences, vol. 24. Springer, New York (1992)

    Google Scholar 

  5. Bates, P.W., Lu, K.,  Zeng, C.: Existence and persistence of invariant manifolds for semiflows in Banach space. Mem. Am. Math. Soc. 135(645), viii+129 (1998)

    Google Scholar 

  6. Beal, D.N., Hover, F.S., Triantafyllou, M.S., Liao, J.C., Lauder, G.V.: Passive propulsion in vortex wakes. J. Fluid Mech. 549, 385–402 (2006)

    Article  Google Scholar 

  7. Bhalla, A.P.S., Griffith, B.E., Patankar, N.A.: A forced damped oscillation framework for undulatory swimming provides new insights into how propultion arises in active and passive swimming. PLOS Comput. Biol. 9(6), e1003097 (2013)

    Article  MathSciNet  Google Scholar 

  8. Cantor, M.: Perfect fluid flows over R n with asymptotic conditions. J. Funct. Anal. 18 (1975), 73–84.

    Article  MATH  MathSciNet  Google Scholar 

  9. Cendra, H., Marsden, J.E., Ratiu, T.S.: Lagrangian reduction by stages. Memoirs of the American Mathematical Society, vol. 152. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  10. Delcomyn, F.: Neural basis of rhythmic behavior in animals. Science 210(4469), 492–498 (1980)

    Article  Google Scholar 

  11. Ebin, D.G., Marsden, J.E.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ehlers, K.M.,  Koiller, J.: Micro-swimming without flagella: propulsion by internal structures. Regul. Chaotic Dynam. 16(6), 623–652 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Eldering, J.: Normally Hyperbolic Invariant Manifolds: The Noncompact Case. Atlantis Series in Dynamical Systems. Atlantis Press, Paris, France (2013)

    Book  Google Scholar 

  14. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21, 193–226 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  15. Friesen, W.O.: Reciprocal inhibition: a mechanism underlying oscillatory animal movements. Neurosci. Biobehav. Rev. 18(4), 547–553 (1994)

    Article  Google Scholar 

  16. Gadelha, H., Gaffney, E.A.,  Goriely, A.: The counterbend phenomena in flagellar axonemes and cross-linked filament bundles. Proc. Natl. Acad. Sci. 110(30), 12180–12185 (2013)

    Article  Google Scholar 

  17. Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181, 375–389 (1977)

    Article  MATH  Google Scholar 

  18. Grillner, S., Wallen, P.: Central pattern generators for locomotion, with special reference to vertebrates. Annu. Rev. Neurosci. 8, 233–261 (1985)

    Article  Google Scholar 

  19. Hirsch, M.W., Pugh, C.C.,  Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. Springer, New York (1977)

    Google Scholar 

  20. Hobbelen, D.G.E.,  Wisse, M.: Limit-cycle walking. In: Humanoid Robots: Human-Like Machines, pp. 277–294. Itech, Vienna (2007)

    Google Scholar 

  21. Jacobs, H.O.,  Eldering, J.: Limit cycle walking on a regularized ground. Preprint. March 2013. arXiv:1212.1978[math.DS]

    Google Scholar 

  22. Jacobs, H.,  Vankerschaver, J.: Fluid-structure interaction in the Lagrange-Poincaré formalism: the Navier-Stokes and inviscid regimes. J. Geom. Mech. 6(1), 39–66 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jones D.A., Shkoller, S.: Persistence of invariant manifolds for nonlinear PDEs. Stud. Appl. Math. 102(1), 27–67 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kanso, E., Marsden, J.E., Rowley, C.W., Melli-Huber, J.B.: Locomotion of articulated bodies in a perfect fluid. J. Nonlinear Sci. 15(4), 255–289 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kelly, S.D., Murray, R.M.: The geometry and control of dissipative systems. In: Proceedings of the 35th Conference on Decisions and Control, December (1996)

    Google Scholar 

  26. Kelly, S.D., Murray, R.M.: Modelling efficient pisciform swimming for control. Int. J. Robust Nonlinear Control 10(4), 217–241 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Koiller, J.,  Ehlers, K.,  Montgomery, R.: Problems and progress in microswimming. J. Nonlinear Sci. 6(6), 507–541 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Lewis, D.,  Marsden, J.,  Montgomery, R.,  Ratiu, T.: The Hamiltonian structure for dynamic free boundary problems. Physica D (Nonlinear Phenomena) 18(1–3), 391–404 (1986)

    Google Scholar 

  29. Liao, J.C., Beal, D.N., Lauder, G.V., Triantafyllou, M.S..: Fish exploiting vortices decrease muscle activity. Science 302, 1566–1569 (2003)

    Article  Google Scholar 

  30. Liao, J.C., Beal, D.N., Lauder, G.V., Triantafyllou, M.S.: The Karman gait: novel body kinematics of rainbow trout swimming in a vortex street. J. Exp. Biol. 206(6), 1059–1073 (2003)

    Article  Google Scholar 

  31. Liu, B.,  Ristroph, L.,  Weathers, A., Childress, S.,  Zhang, J.: Intrinsic stability of a body hovering in an oscillating airflow. Phys. Rev. Lett. 108, 068103 (2012)

    Article  Google Scholar 

  32. Lucy, B.L.: A numerical approach to testing the fission hypothesis. Astron. J. 82, 1013–1924 (1977)

    Article  Google Scholar 

  33. Mariano, G.,  Chatterjee, A.,  Ruina, A.,  Coleman, M.: The simplest walking model: stability, complexity, and scaling. J. Biomech. Eng. 120(2), 281–288 (1998)

    Article  Google Scholar 

  34. Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, Columbia (1983)

    MATH  Google Scholar 

  35. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry. Texts in Applied Mathematics, vol. 17, 2nd edn. Springer, New York (1999)

    Google Scholar 

  36. Marsden, J.E.,  Montgomery, R., Ratiu, T.S.: Reduction, Symmetry, and Phases in Mechanics. Memoirs of the American Mathematical Society, no. 436, vol. 88. American Mathematical Society, Providence, RI (1990)

    Google Scholar 

  37. McGreer, T.: Passive dynamic walking. Int. J. Robot. 9(2), 62–82 (1990)

    Article  Google Scholar 

  38. Munnier, A.: Passive and self-propelled locomotion for an elastic swimmer in a perfect fluid. SIAM J. Appl. Dyn. Syst. 10(4), 1363–1403 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. Peskin, C.: The immersed boundary method. Acta Numerica 11, 479–513 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  40. Shadwick, R.E.: Muscle dynamics in fish during steady swimming. Am. Zool. 38, 755–70 (1998)

    Google Scholar 

  41. Shapere, A.,  Wilczek, F.: Geometry of self-propulsion at low Reynolds number. J. Fluid Mech. 198, 557–585 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  42. Shelley, M. J.,  Vandenberghe, N.,  Zhang, J.: Heavy flags undergo spontaneous oscillations in flowing water. Phys. Rev. Lett. 94, 094302 (2005)

    Article  Google Scholar 

  43. Troyanov, M.: On the Hodge decomposition in \(\mathbb{R}^{n}\). Mosc. Math. J. 9(4), 899–926, 936 (2009)

    Google Scholar 

  44. Tytell, E.D.,  Hsu, C., Williams, T.L., Cohen, A.H., Fauci, L.J.: Interactions between internal forces, body stiffness, and fluid environment in a neuromechanical model of lamprey swimming. Proc. Natl. Acad. Sci. 107(46), 19832–19837 (2010)

    Article  Google Scholar 

  45. Vogel, S.: Comparative Biomechanics. Princeton University Press, Princeton (2003)

    MATH  Google Scholar 

  46. Wallen, P., Williams, T.L.: Fictive locomotion in the lamprey spinal chord in vitro compared with swimming in the intact and spinal animal. J. Physiol. 347, 225–239 (1984)

    Article  Google Scholar 

  47. Wilson, M.W., Eldredge, J.D.: Performance improvement through passive mechanics in jellyfish-inspired swimming. Int. J. Non-Linear Mech. 46(4), 557–567 (2011)

    Article  Google Scholar 

  48. Zhang, J.,  Liu, N.,  Lu, X.: Locomotion of a passively flapping flat plate. J. Fluid Mech. 659, 43–68 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  49. Zhou, K., Doyle, J.C.: Essentials of Robust Control, 1st edn. Prentice Hall, New Jersey (1997)

    MATH  Google Scholar 

Download references

Acknowledgements

The notion of swimming as a limit cycle was initially introduced to me by Erica J. Kim while she was studying hummingbirds. Additionally, Sam Burden, Ram Vasudevan, and Humberto Gonzales provided much insight into how to frame this work for engineers. I would also like to thank Professor Shankar Sastry for allowing me to stay in his lab for a year and meet people who are outside of my normal research circle. I would like to thank Eric Tytell for suggesting relevant articles in neurobiology, Amneet Pal Singh Bhalla for allowing me to reproduce figures from [7], and Peter Wallen for allowing me to reproduce figures from [46]. An early version of this paper was written in the context of Lie groupoid theory, where the guidance of Alan Weinstein was invaluable. Jaap Eldering and Joris Vankerschaver have given me more patience than I may deserve by reading my papers and checking my claims. Major contributions to the bibliography and the overall presentation of the paper were provided by Jair Koiller. Finally, the writing of this paper was solidified with the help of Darryl Holm. This research has been supported by the European Research Council Advanced Grant 267382 FCCA and NSF grant CCF-1011944.

Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College London, Huxley Building, 180 Queen’s Gate Road, London, SW7 2AZ, UK

    Henry O. Jacobs

Authors
  1. Henry O. Jacobs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Henry O. Jacobs .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

    Dong Eui Chang

  2. Department of Mathematics, Imperial College London, London, United Kingdom

    Darryl D. Holm

  3. Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada

    George Patrick

  4. Section de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Tudor Ratiu

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media New York

About this chapter

Cite this chapter

Jacobs, H.O. (2015). The Role of \(\mathop{\mathrm{SE}}\nolimits (d)\)-Reduction for Swimming in Stokes and Navier-Stokes Fluids. In: Chang, D., Holm, D., Patrick, G., Ratiu, T. (eds) Geometry, Mechanics, and Dynamics. Fields Institute Communications, vol 73. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2441-7_8

Download citation

Publish with us

Policies and ethics

Search

Navigation