Journal of Statistical Physics

, Volume 127, Issue 1, pp 133–170 | Cite as

Stationary Flow Past a Semi-Infinite Flat Plate: Analytical and Numerical Evidence for a Symmetry-Breaking Solution

Article

Abstract

We consider the question of the existence of stationary solutions for the Navier Stokes equations describing the flow of a incompressible fluid past a semi-infinite flat plate at zero incidence angle. By using ideas from the theory of dynamical systems we analyze the vorticity equation for this problem and show that a symmetry-breaking term fits naturally into the downstream asymptotic expansion of a solution. Finally, in order to check that our asymptotic expressions can be completed to a symmetry-breaking solution of the Navier–Stokes equations we solve the problem numerically by using our asymptotic results to prescribe artificial boundary conditions for a sequence of truncated domains. The results of these numerical computations a clearly compatible with the existence of such a solution.

Keywords

Navier-Stokes equations semi-infinite plate symmetry-breaking 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. L. Alden, Second approximation to the laminar boundary layer flow over a flat plate. J. Math. Phys. XVII:91–104 (1948).MathSciNetGoogle Scholar
  2. 2.
    G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 1967).Google Scholar
  3. 3.
    H. Blasius. Zeitschrift für Mathematik und Physik 56:4–13 (1908).Google Scholar
  4. 4.
    S. Bönisch, V. Heuveline and P. Wittwer, Second order adaptive boundary conditions for exterior flow problems: non-symmetric stationary flows in two dimensions. to appear in the Journal of Mathematical Fluid Mechanics (2004).Google Scholar
  5. 5.
    S. Bönisch, V. Heuveline and P. Wittwer, Adaptive boundary conditions for exterior flow problems. J. Math. Fluid Mech. 7:85–107 (2005).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    J. F. Douglas, J. M. Gasiorek and J. A. Swaffield, Fluid Mechanics (Pearson, Prentice Hall).Google Scholar
  7. 7.
    J. H. Ferziger and M. Péric, Computational Methods in Fluid Dynamics (Springer-Verlag).Google Scholar
  8. 8.
    Th. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R2. Arch. Rat. Mech. Anal. 163:209–258 (2002).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Th. Gallay and C. E. Wayne, Long-time asymptotics of the Navier—Stokes and vorticity equations on R3. Phil. Trans. Roy. Soc. Lond. 360:2155–2188 (2002).MATHCrossRefADSMathSciNetGoogle Scholar
  10. 10.
    S. Goldstein, Lectures on Fluid Dynamics (Interscience Publishers, Ltd., London, 1960).Google Scholar
  11. 11.
    M. Griebel, T. Dornseifer and T. Neunhoeffer, Numerical Simulation in Fluid Dynamics. A Practical Introduction (SIAM).Google Scholar
  12. 12.
    F. Haldi and P. Wittwer, Leading order down-stream asymptotics of non-symmetric stationary Navier—Stokes flows in two dimensions. J. Math. Fluid Mech. 7:611–648 (2005).MATHCrossRefADSMathSciNetGoogle Scholar
  13. 13.
    I. Imai, Second approximation to the laminar boundary-layer flow over a flat plate. J. Aeronaut. Sci. 24:155–156 (1957).MATHGoogle Scholar
  14. 14.
    L. Landau and E. Lifchitz, Mécanique Des Fluides (éditions MIR, 1989).Google Scholar
  15. 15.
    H. Ockendon and J. R. Ockendon, Viscous Flow. (Cabridge texts in applied mathematics).Google Scholar
  16. 16.
    W. E. Olmstead. A homogeneous solution for viscous flow around a half-plane. Quart. Appl. Math 33:165–169 (1975).ADSMATHGoogle Scholar
  17. 17.
    W. E. Olmstead and D. L. Hector, On the nonuniqueness of oseen flow past a half plane. J. Math. Phys. 45:408–417 (1966).MATHMathSciNetGoogle Scholar
  18. 18.
    H. Schlichting and K. Gersten, Boundary Layer Theory (Springer, 1999).Google Scholar
  19. 19.
    F. T. Smith, Non-uniqueness in wakes and boundary layers. Proc. R. Soc. London, Ser. A 391:1–26 (1984).MATHADSCrossRefGoogle Scholar
  20. 20.
    I. J. Sobey, Introduction to Interactive Boundary Layer Theory (Oxford University Press, 2000).Google Scholar
  21. 21.
    K. Stewartson, On asymptotic expansions in the theory of boundary layers. J. Math. Phys. 36:173–191 (1957).MathSciNetMATHGoogle Scholar
  22. 22.
    K. Stüben and U. Trottenberg, Multigrid Methods: Fundamental Algorithms, Model Problem Analysis and Applications, volume 960 (Springer-Verlag).Google Scholar
  23. 23.
    G. v. Baalen, Stationary solutions of the Navier—Stokes equations in a half-plane down-stream of an object: Universality of the wake. Nonlinearity 15:315–366 (2002).MATHCrossRefADSMathSciNetGoogle Scholar
  24. 24.
    A. I. van de Vooren and D. Dijkstra, The Navier—Stokes solution for laminar flow past a semi-infinite flat plate. J. Eng. Math. 4:9–27 (1970).MATHCrossRefGoogle Scholar
  25. 25.
    M. van Dyke, Perturbation Methods in Fluid Mechanics (The Parabolic Press, Stanford, California, 1975).MATHGoogle Scholar
  26. 26.
    E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains. Arch. Rational Mech. Anal. 138:279–306 (1997).MATHCrossRefADSMathSciNetGoogle Scholar
  27. 27.
    P. Wittwer, On the structure of stationary solutions of the Navier—Stokes equations. Commun. Math. Phys. 226:455–474 (2002).MATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    P. Wittwer, Supplement: On the structure of stationary solutions of the Navier—Stokes equations. Commun. Math. Phys. 234:557–565 (2003).MATHCrossRefADSMathSciNetGoogle Scholar
  29. 29.
    P. Wittwer, Leading order down-stream asymptotics of stationary Navier-Stokes flows in three dimensions. J. Math. Fluid Mech. 8:147–186 (2006).MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Laboratoire de modélisation et simulationHESSOEcole d’Ingénieurs de GenèveSwitzerland
  2. 2.Département de Physique ThéoriqueUniversité de GenèveGenèveSwitzerland

Personalised recommendations