Advertisement

Archive for Rational Mechanics and Analysis

, Volume 211, Issue 1, pp 205–255 | Cite as

Weak Solutions for an Incompressible Newtonian Fluid Interacting with a Koiter Type Shell

  • Daniel Lengeler
  • Michael Růžička
Article

Abstract

In this paper we analyse the interaction of an incompressible Newtonian fluid with a linearly elastic Koiter shell whosemotion is restricted to transverse displacements. The middle surface of the shell constitutes the mathematical boundary of the three-dimensional fluid domain. We show that weak solutions exist as long as the magnitude of the displacement stays below some (possibly large) bound that rules out selfintersections of the shell.

Keywords

Weak Solution Newtonian Fluid Uniform Convergence Solution Operator Stokes System 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev spaces, 2nd edn. In: Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier, Amsterdam, 2003Google Scholar
  2. 2.
    Beiraoda Veiga H.: On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mech. 6, 21–52 (2004)ADSCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boulakia, M.: Existence of weak solutions for an interaction problem between an elastic structure and a compressible viscous fluid. J. Math. Pures Appl. (9) 84(11), 1515–1554 (2005)Google Scholar
  4. 4.
    Boyer, F., Fabrie, P.: Éléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 52. Springer, Berlin, 2006Google Scholar
  5. 5.
    Chambolle A., Desjardins B., Esteban M.J., Grandmont C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. J. Math. Fluid Mech. 7(3), 368–404 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cheng, C.H.A., Coutand, D., Shkoller, S.: Navier–Stokes equations interacting with a nonlinear elastic biofluid shell. SIAM J. Math. Anal. 39(3), 742–800 (electronic) (2007)Google Scholar
  7. 7.
    Cheng C.H.A., Shkoller S.: The interaction of the 3D Navier–Stokes equations with a moving nonlinear Koiter elastic shell. SIAM J. Math. Anal. 42(3), 1094–1155 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ciarlet, P.G.: Mathematical elasticity. Vol. II. Theory of plates. In: Studies in Mathematics and its Applications, vol. 27. North-Holland, Amsterdam, 1997Google Scholar
  9. 9.
    Ciarlet, P.G.: Mathematical elasticity, vol. III. Theory of shells. In: Studies in Mathematics and its Applications, vol. 29. North-Holland Publishing Co., Amsterdam, 2000Google Scholar
  10. 10.
    Ciarlet, P.G.: An Introduction to Differential Geometry with Applications to Elasticity. Springer, Dordrecht, 2005. Reprinted from J. Elasticity 78/79, no. 1–3 (2005)Google Scholar
  11. 11.
    Coutand D., Shkoller S.: Motion of an elastic solid inside an incompressible viscous fluid. Arch. Rational Mech. Anal. 176(1), 25–102 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Coutand D., Shkoller S.: The interaction between quasilinear elastodynamics and the Navier–Stokes equations. Arch. Rational Mech. Anal. 179(3), 303–352 (2006)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Diening L., Růžička M.: Strong solutions for generalized Newtonian fluids. J. Math. Fluid Mech. 7(3), 413–450 (2005)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Friesecke G., James R.D., Mora M.G., Müller S.: Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence. C. R. Math. Acad. Sci. Paris 336(8), 697–702 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations, vol. I. Linearized steady problems. In: Springer Tracts in Natural Philosophy, vol. 38. Springer, New York, 1994Google Scholar
  16. 16.
    Galdi G.P., Simader C.G., Sohr H.: A class of solutions to stationary Stokes and Navier–Stokes equations with boundary data in W −1/q,q. Math. Ann. 331(1), 41–74 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Granas A., Dugundji J.: Fixed point theory. Springer Monographs in Mathematics.. Springer, New York (2003)Google Scholar
  18. 18.
    Grandmont C.: Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate. SIAM J. Math. Anal. 40(2), 716–737 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Koiter, W.T.: A consistent first approximation in the general theory of thin elastic shells. In: Proc. Sympos. Thin Elastic Shells (Delft, 1959) (Amsterdam), pp. 12–33. North-Holland, Amsterdam, 1960Google Scholar
  20. 20.
    Koiter, W.T.: On the nonlinear theory of thin elastic shells. I, II, III. Nederl. Akad. Wetensch. Proc. Ser. B 69, 1–17, 18–32, 33–54 (1966)Google Scholar
  21. 21.
    Kufner A., John O., Fucík, S.: Function spaces. In: Monographs and Textbooks on Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Publishing, Leyden, 1977Google Scholar
  22. 22.
    Le Dret, H., Raoult, A.: The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. In: Mechanics: from theory to computation (New York), pp. 59–84. Springer, New York, 2000Google Scholar
  23. 23.
    Lee, J.M.L.: Introduction to smooth manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York, 2003Google Scholar
  24. 24.
    Lengeler, D.: Globale Existenz für die Interaktion eines Navier–Stokes-Fluids mit einer linear elastischen Schale. Ph.D. thesis, Universität Freiburg, 2011, FREIDOK ServerGoogle Scholar
  25. 25.
    Lequeurre J.: Existence of strong solutions to a fluid-structure system. SIAM J. Math. Anal. 43(1), 389–410 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Mathematisches InstitutUniversität FreiburgFreiburgGermany

Personalised recommendations