Existence of stationary solutions of the Navier–Stokes equations in two dimensions in the presence of a wall

Article

Abstract

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.

Mathematics Subject Classification (2000)

76D05 76D25 76M10 41A60 35Q35 

Keywords

Navier–Stokes equations Stationary solutions Fluid structure interaction 

References

  1. 1.
    Bönisch S., Heuveline V., and Wittwer P. Adaptive boundary conditions for exterior flow problems. Journal of Mathematical Fluid Mechanics 7: (2005), 85–107.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Bönisch, V. Heuveline, and P. Wittwer, Second order adaptive boundary conditions for exterior flow problems: non-symmetric stationary flows in two dimensions, Journal of mathematical fluid fechanics 8 (2006), 1–26.CrossRefGoogle Scholar
  3. 3.
    T. Z. Boulmezaoud and U. Razafison, On the steady Oseen problem in the whole space, Hiroshima Math. J. 35 (2005), no. 3, 371–401. MR MR2210715 (2006m:35290)Google Scholar
  4. 4.
    B. Bunner and G. Tryggvason, Direct numerical simulations of three-dimensional bubbly flows, Physics of Fluids 11 (1999), 1967–1969.MATHCrossRefGoogle Scholar
  5. 5.
    A. Esmaeeli and G. Tryggvason, A direct numerical simulation study of the buoyant rise of bubbles at O(100) Reynolds number, Phys. Fluids 17 (2005), 093303–093322.CrossRefGoogle Scholar
  6. 6.
    J. Magnaudet F. Takemura, The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number, Journal of Fluid Mechanics 495 (2003), 235–253.MATHCrossRefGoogle Scholar
  7. 7.
    R. Farwig, The stationary exterior 3 d-problem of oseen and Navier–Stokes equations in anisotropically weighted sobolev spaces, Mathematische Zeitschrift 211 (1992), no. 1, 409–447.Google Scholar
  8. 8.
    G.P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations: Linearized steady problems, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, Heidelberg, 1998.Google Scholar
  9. 9.
    G.P. Galdi, An introduction to the mathematical theory of the Navier–Stokes equations: Nonlinear steady problems, Springer Tracts in Natural Philosophy, Vol. 39, Springer-Verlag, Heidelberg, 1998.Google Scholar
  10. 10.
    V. Heuveline and P. Wittwer, Exterior flows at low Reynolds numbers: concepts, solutions and applications, 2007.Google Scholar
  11. 11.
    M. Hillairet, Interactive features in fluid mechanics, Ph.D. thesis, Ecole normale supérieure de Lyon, 2005.Google Scholar
  12. 12.
    M. Hillairet and D. Serre, Chute stationnaire d’un solide dans un fluide visqueux incompressible le long d’un plan incliné, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 5, 779–803. MR 2004h:35174Google Scholar
  13. 13.
    J. Hua and J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 222 (2007), no. 2, 769–795.Google Scholar
  14. 14.
    R. S. Gorelik L. S. Timkin and P. D. Lobanov, Rise of a single bubble in ascending laminar flow: Slip velocity and wall friction, Journal of Engineering Physics and Thermophysics 78 (2005), 762–768.Google Scholar
  15. 15.
    J. Lu and G. Tryggvason, Numerical study of turbulent bubbly downflows in a vertical channel, Phys. Fluids 18 (2006), 103302–103312.CrossRefGoogle Scholar
  16. 16.
    P. Wittwer and M. Hillairet, On a universal asymptotic description for solutions of the stationary Navier–Stokes equations, In preparation (2009).Google Scholar
  17. 17.
    D. Serre, Chute libre d’un solide dans un fluide visqueux incompressible. Existence, Japan J. Appl. Math. 4 (1987), no. 1, 99–110. MR 89m:76032Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.IMT, UMR CNRS 5219Université Paul Sabatier (Toulouse 3)Toulouse Cedex 09France
  2. 2.Département de Physique ThéoriqueUniversité de GenèveGenève 4Switzerland

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