Advertisement

Elliptic Regularization and the Solvability of Self-Propelled Locomotion Problems

  • Adam BoucherEmail author
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 155)

Abstract

Humans have been fascinated by the wonders of animal locomotion since antiquity. The effortless grace of birds in flight, and the silent glide of fish through water exemplify the elegance possible with control over the local fluid flow. While we have been free to observe the poetry in these motions for centuries, the ability to predict and mimic these motions with any degree of accuracy is a much more recent development. Advances in computing hardware and our numerical simulation capabilities have opened the door to the analysis and prediction of the motion of complex shapes in fluid environments. As we continue to develop more depth and sophistication in our ability to simulate the mathematical models of fluid mechanics, we must be sure to maintain an awareness of the existence and smoothness of the exact solutions to our mathematical models. In this paper we present a very brief overview of some techniques from partial differential equations which can be used to analyze physically relevant problems in animal locomotion. The ideas used here give important insight into situations where control systems for ordinary differential equations may be used to control the trajectories of vehicles through fluid environments.

Keywords

Fluid Environment Boundary Velocity Elliptic Regularization Ambient Flow Fluid Problem 
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.

Notes

Acknowledgements

I would like to thank the workshop organizers for their hard work in preparation, and the workshop participants for their presentations and ideas. I would like to thank the IMA for hosting the Natural Locomotion in Fluids workshop. I would also like to thank my colleagues at the University of New Hampshire for their support and insight.

References

  1. [1]
    Agarwal RP, Meehan M, O’Regan D (2001) Fixed point theory and applications. Cambridge University Press, Cambridge/New YorkzbMATHCrossRefGoogle Scholar
  2. [2]
    Childress S (1981) Mechanics of swimming and flying. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  3. [3]
    Evans C (1998) Partial differential equations. Graduate studies in mathematics. AMS, ProvidencezbMATHGoogle Scholar
  4. [4]
    Galdi GP (1999) On the steady self-propelled motion of a body in a viscous incompressible fluid. Arch Ration Mech Anal 148:53-88MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge/New YorkzbMATHCrossRefGoogle Scholar
  6. [6]
    Lighthill J (1975) Mathematical biofluiddynamics. CBMS 17. SIAM, PhiladelphiaGoogle Scholar
  7. [7]
    Martin JS, Scheid J-F, Takahashi T, Tucsnak M (2008) An initial boundary value problem modeling of fish-like swimming. Arch Ration Mech Anal 188:429–455MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Martin JS, Takahashi T, Tucsnak M A (2007) control theoretic approach to the swimming of microscopic organisms Quart Appl Math 65: 405–424Google Scholar
  9. [9]
    Medeiros LA, Ferrel JL (1997) Elliptic regularization and Navier-stokes system. Mem Differ Equ Math Phys 12:165–177MathSciNetzbMATHGoogle Scholar
  10. [10]
    Salvi R (1985) On the existence of weak solutions of a nonlinear mixed problem for the Navier-Stokes equations in a time dependent domain. J Fac Sci Univ Tokyo Sect IA Math 32:213–221MathSciNetzbMATHGoogle Scholar
  11. [11]
    Salvi R (1988) On the Navier-stokes equations in non-cylindrical domains: one the existence and regularity. Math Z 199:153–170MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Salvi R (1994) On the existence of periodic weak solutions of Navier-stokes equations in regions with periodically moving boundaries. Acta Appl Math 37:169–179MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Simon J (1986) Compact Sets in the space L p(0, T, B). Annali Di Matematica Pura ed Applicata 146(1):65–96CrossRefGoogle Scholar
  14. [14]
    Silvestre AL (2002) On the slow motion of a self-propelled rigid body in a viscous incompressible fluid. J Math Anal Appl 274:203-227MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Takahashi T (2003) Analysis of strong solutions for the equations modeling the motion of a rigid-fluid system in a bounded domain. Adv Differ Equ 8: 1499–1532zbMATHGoogle Scholar
  16. [16]
    Temam R (1994) Navier Stokes equations: theory and numerical analysis. AMS ChelseaGoogle Scholar
  17. [17]
    Wang ZJ (2000) Vortex shedding and frequency selection in flapping flight. J Fluid Mech 410:323–341zbMATHCrossRefGoogle Scholar
  18. [18]
    Wang ZJ (2005) Dissecting insect flight. Annu Rev Fluid Mech 37:183–210CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of New HampshireDurhamUSA

Personalised recommendations