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Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid

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Abstract

We consider the three-dimensional motion of a self-propelled deformable structure into a viscous incompressible fluid. The deformation of the solid is given whereas its position is unknown. Such a system could model the propulsion of fish-like swimmers. The equations of motion of the fluid are the Navier-Stokes equations and the equations for the structure are deduced from Newton’s laws. The corresponding system is a free boundary problem and the main result of the paper is the existence of weak solutions for this problem.

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References

  1. Alouges, F., DeSimone, A., Lefebvre, A.: Optimal strokes for low Reynolds number swimmers: an example. J. Nonlinear Sci. 18, 277–302 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Borelli, G.: On the Movement of Animals. Springer, New York (1989)

    Google Scholar 

  3. Bost, C., Cottet, G.-H., Maitre, E.: Convergence analysis of a penalization method for the three-dimensional motion of a rigid body in an incompressible viscous fluid. SIAM J. Numer. Anal. 48, 1313–1337 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boulakia, M.: Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid. J. Math. Fluid Mech. 9, 262–294 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bressan, A.: Impulsive control of Lagrangian systems and locomotion in fluids. Discrete Contin. Dyn. Syst. 20, 1–35 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chambrion, T., Munnier, A.: On the locomotion and control of a self-propelled shape-changing body in a fluid. J. Nonlinear Sci. (to appear)

  7. Childress, S.: Mechanics of Swimming and Flying. Cambridge Studies in Mathematical Biology, vol. 2. Cambridge University Press, Cambridge (1981)

    Book  MATH  Google Scholar 

  8. Conca, C., San Martín, J.H., Tucsnak, M.: Existence of solutions for the equations modelling the motion of a rigid body in a viscous fluid. Commun. Partial Differ. Equ. 25, 1019–1042 (2000)

    Article  MATH  Google Scholar 

  9. Desjardins, B., Esteban, M.J., Grandmont, C., Le Tallec, P.: Weak solutions for a fluid-elastic structure interaction model. Rev. Mat. Complut. 14, 523–538 (2001)

    MathSciNet  MATH  Google Scholar 

  10. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Springer Tracts in Natural Philosophy, vol. 38. Springer, New York (1994). Linearized steady problems

    Book  Google Scholar 

  12. Grandmont, C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 40, 716–737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gray, J.: Study in animal locomotion IV—the propulsive powers of the dolphin. J. Exp. Biol. 10, 192–199 (1932)

    Google Scholar 

  14. Gunzburger, M.D., Lee, H.-C., Seregin, G.A.: Global existence of weak solutions for viscous incompressible flows around a moving rigid body in three dimensions. J. Math. Fluid Mech. 2, 219–266 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hillairet, M., Takahashi, T.: Collisions in three-dimensional fluid structure interaction problems. SIAM J. Math. Anal. 40, 2451–2477 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ladyzhenskaya, O.: The Mathematical Theory of Viscous Incomprehensible Flow. Gordon and Breach, New York (1969)

    Google Scholar 

  17. Legendre, G., Takahashi, T.: Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation. M2AN Math. Model. Numer. Anal. 42, 609–644 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lighthill, J.: Mathematical Biofluiddynamics. SIAM, Philadelphia (1975)

    Book  MATH  Google Scholar 

  19. Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 1. Oxford Lecture Series in Mathematics and Its Applications, vol. 3. Clarendon, Oxford University Press, New York (1996). Incompressible models, Oxford Science Publications

    MATH  Google Scholar 

  20. San Martín, J., Scheid, J.-F., Takahashi, T., Tucsnak, M.: Convergence of the Lagrange-Galerkin method for the equations modelling the motion of a fluid-rigid system. SIAM J. Numer. Anal. 43, 1536–1571 (2005) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  21. San Martín, J., Scheid, J.-F., Takahashi, T., Tucsnak, M.: An initial and boundary value problem modeling of fish-like swimming. Arch. Ration. Mech. Anal. 188, 429–455 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. San Martín, J., Takahashi, T., Tucsnak, M.: A control theoretic approach to the swimming of microscopic organisms. Q. Appl. Math. 65, 405–424 (2007)

    MATH  Google Scholar 

  23. San Martín, J.A., Starovoitov, V., Tucsnak, M.: Global weak solutions for the two-dimensional motion of several rigid bodies in an incompressible viscous fluid. Arch. Ration. Mech. Anal. 161, 113–147 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sigalotti, M., Vivalda, J.-C.: Controllability properties of a class of systems modeling swimming microscopic organisms. ESAIM Control Optim. Calc. Var. 16, 1053–1076 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Silvestre, A.L.: On the slow motion of a self-propelled rigid body in a viscous incompressible fluid. J. Math. Anal. Appl. 274, 203–227 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sparenberg, J.A.: Survey of the mathematical theory of fish locomotion. J. Eng. Math. 44, 395–448 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Starovoitov, V.N.: Solvability of the problem of the self-propelled motion of several rigid bodies in a viscous incompressible fluid. Comput. Math. Appl. 53, 413–435 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Taylor, G.: Analysis of the swimming of long and narrow animals. Proc. R. Soc. London A 214 (1952)

  29. Temam, R.: Problèmes Mathématiques en Plasticité. Gauthier-Villars, Montrouge (1983)

    MATH  Google Scholar 

  30. Wu, T.Y.: Mathematical biofluiddynamics and mechanophysiology of fish locomotion. Math. Methods Appl. Sci. 24, 1541–1564 (2001). Biofluiddynamics

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematical Institute, Academy of Sciences, Zitna 25, Prague 1, 11567, Czech Republic

    Šárka Nečasová

  2. Institut Élie Cartan, UMR 7502, INRIA, Nancy-Université, CNRS, BP239, 54506, Vandœuvre-lès-Nancy Cedex, France

    Takéo Takahashi & Marius Tucsnak

  3. Team-Project CORIDA, INRIA Nancy - Grand Est, 615, rue du Jardin Botanique, 54600, Villers-lès-Nancy, France

    Takéo Takahashi & Marius Tucsnak

Authors
  1. Šárka Nečasová
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  2. Takéo Takahashi
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  3. Marius Tucsnak
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Corresponding author

Correspondence to Takéo Takahashi.

Additional information

The research of S.N. was supported by Grant Agency of Czech Republic n. P201/11/1304 and by Institutional Reasearch Plan n. AVOZ10190503, of the Academy of Sciences of the Czech Republic.

The work of the second and third authors was partially supported by ANR Grant BLAN07-2 202879 and by ANR Grant 09-BLAN-0213-02.

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Nečasová, Š., Takahashi, T. & Tucsnak, M. Weak Solutions for the Motion of a Self-propelled Deformable Structure in a Viscous Incompressible Fluid. Acta Appl Math 116, 329–352 (2011). https://doi.org/10.1007/s10440-011-9646-2

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  • DOI: https://doi.org/10.1007/s10440-011-9646-2

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