Skip to main content
SpringerLink
Log in

Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We tackle the issue of the inviscid limit of the incompressible Navier–Stokes equations when the Navier slip-with-friction conditions are prescribed on impermeable boundaries. We justify an asymptotic expansion which involves a weak amplitude boundary layer, with the same thickness as in Prandtl’s theory and a linear behavior. This analysis holds for general regular domains, in both dimensions two and three.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achdou Y., Pironneau O., Valentin F.: Effective boundary conditions for laminar flows over periodic rough boundaries. J. Comput. Phys. 147(1), 187–218 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Alekseenko S.N.: The existence and asymptotic representation of weak solutions of the flow problem with the condition of regular slippage along rigid wall. Sibirsk. Mat. Zh. 35(2), 235–257 (1994)

    MathSciNet  Google Scholar 

  3. Barrenechea G.R., Le Tallec P., Valentin F.: New wall laws for the unsteady incompressible Navier–Stokes equations on rough domains. Math. Model. Numer. Anal. 36(2), 177–203 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beavers G.S., Joseph D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30(01), 197–207 (1967)

    Article  ADS  Google Scholar 

  5. Berselli L.C., Romito M.: On the existence and uniqueness of weak solutions for a vorticity seeding model. SIAM J. Math. Anal. 37(6), 1780–1799 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bourguignon J.P., Brezis H.: Remarks on the Euler equation. J. Funct. Anal. 15, 341–363 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boyer, F., Fabrie, P.: Eléments d’analyse pour l’étude de quelques modèles d’écoulements de fluides visqueux incompressibles. Mathématiques et Applications 52, 405 p. Springer, Berlin (2006)

  8. Bresch, D., Milisic, V.: Higher order boundary layer correctors and wall laws derivation: a unified approach. http://arxiv.org/abs/math.AP/0611083

  9. Carbou G., Fabrie P., Guès O.: Couche limite dans un modèle de ferromagnétisme. Commun Partial Differ. Equ. 27(7-8), 1467–1495 (2002)

    Article  MATH  Google Scholar 

  10. Clopeau T., Mikelić A., Robert R.: On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions. Nonlinearity 11(6), 1625–1636 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  11. Ebin D.G., Marsden J.: Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid. Bull. Am. Math. Soc. 75, 962–967 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  12. Grenier, E.: Boundary layers. Handbook of Mathematical Fluid Dynamics, Vol. III, 245–309. North-Holland, Amsterdam, 2004

  13. Grenier E., Guès O.: Boundary layers for viscous perturbations of noncharacteristic quasilinear hyperbolic problems. J. Differ. Equ. 143(1), 110–146 (1998)

    Article  MATH  Google Scholar 

  14. Grubb G.: Nonhomogeneous Navier–Stokes problems in L p Sobolev spaces over exterior and interior domains. Ser. Adv. Math. Appl. Sci. 47, 46–63 (1998)

    MathSciNet  Google Scholar 

  15. Grubb G., Solonnikov V.A.: Boundary value problems for the nonstationary Navier–Stokes equations treated by pseudo-differential methods. Math. Scand. 69(2), 217–290 (1991)

    MATH  MathSciNet  Google Scholar 

  16. Grubb, G., Solonnikov, V.A.: A pseudo-differential treatment of general inhomogeneous initial-boundary value problems for the Navier–Stokes equation. Journées “Équations aux Dérivées Partielles” (Saint Jean de Monts, 1988), Exp. No. III, 8

  17. Iftimie D., Planas G.: Inviscid limits for the Navier–Stokes equations with Navier friction boundary conditions. Nonlinearity 19, 899–918 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Itoh S., Tanaka N., Tani A.: The initial value problem for the Navier–Stokes equations with general slip boundary condition in Hölder spaces. J. Math. Fluid Mech. 5(3), 275–301 (2003)

    MATH  MathSciNet  ADS  Google Scholar 

  19. Jäger W., Mikelić A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170(1), 96–122 (2001)

    Article  MATH  Google Scholar 

  20. Jäger W., Mikelić A.: Couette flows over a rough boundary and drag reduction. Comm. Math. Phys. 232(3), 429–455 (2003)

    MATH  MathSciNet  ADS  Google Scholar 

  21. Kato T.: On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Ration. Mech. Anal. 25, 188–200 (1967)

    Article  MATH  Google Scholar 

  22. Kato T., Lai C.Y.: Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56(1), 15–28 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kelliher J.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kelliher, J.: The vanishing viscosity limit for incompressible fluids in two dimensions. Ph. D Thesis

  25. Kiselev A.A., Ladyzhenskaya O.A.: On existence and uniqueness of the solution of the nonstationary problem for a viscous incompressible fluid. Izv. Akad. Nauk SSSR 21, 655 (1957)

    MATH  Google Scholar 

  26. Lauga, E., Brenner, M.P., Stone, H.A.: Microfluidics: the no-slip boundary condition. Handbook of Experimental Fluid Dynamics (Chapter 19). (Eds. Tropea C., Yarin A., Foss J.F.), Springer, Berlin, 2007

  27. Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, 1969

  28. Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 1: incompressible models. Oxford Lecture Series in Mathematics and its Applications, Vol. 3. Clarendon Press, Oxford, 1996

  29. Lopes Filho M.C., Nussenzveig Lopes H.J., Planas G.: On the inviscid limit for two-dimensional incompressible flow with Navier friction condition. SIAM J. Math. Anal. 36(4), 1130–1141 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Masmoudi N., Saint-Raymond L.: From the Boltzmann equation to the Stokes-Fourier system in a bounded domain. Comm. Pure Appl. Math. 56(9), 1263–1293 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Navier C.-L.-M.-H.: Mémoire sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Paris 6, 389–416 (1823)

    Google Scholar 

  32. Neto C., Evans D.R., Bonaccurso E., Butt H.J., Craig V.S.J.: Boundary slip in Newtonian liquids: a review of experimental studies. Rep. Prog. Phys. 68, 2859–2897 (2005)

    Article  ADS  Google Scholar 

  33. Parés C.: Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids. Appl. Anal. 43(3-4), 245–296 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  34. Serrin, J.: Mathematical principles of classical fluid mechanics. Handbuch der Physik (herausgegeben von S. Flügge), Bd. 8/1, Strömungsmechanik I (Mitherausgeber C. Truesdell), 125–263, Springer, Berlin, 1959

  35. Steiger O.: Navier–Stokes equations with first Order boundary conditions. J Math. Fluid Mech. 8(4), 456–481 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Sueur F.: Couches limites: un problème inverse. Commun. Partial Differ. Equ. 31(1–3), 123–194 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Tani A., Itoh S., Tanaka N.: The initial value problem for the Navier–Stokes equations with general slip boundary condition. Adv. Math. Sci. Appl. 4(1), 51–69 (1994)

    MATH  MathSciNet  Google Scholar 

  38. Temam R., Wang X.: Boundary layers associated with incompressible Navier–Stokes equations: the noncharacteristic boundary case. J. Differ. Equ. 179(2), 647–686 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Wolibner W.: Un théorème sur l’existence du mouvement plan d’un fluide parfait, homogène, incompressible, pendant un temps infiniment long. Math. Z. 37(1), 698–726 (1933)

    Article  MathSciNet  Google Scholar 

  40. Xiao, Y., Xin, Z.: On the vanishing viscosity limit for the 3d Navier–Stokes equations with a slip boundary condition. The Institute of Mathematical Sciences. The Chinese University of Hong Kong, Hong Kong

  41. Ziane M.: On the two-dimensional Navier–Stokes equations with the free boundary condition. Appl. Math. Optim. Int. J. Appl. Stoch. 38(1), 1–19 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut Camille Jordan, Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Bâtiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Dragoş Iftimie

  2. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Université Paris 6, CNRS, UMR 7598, 175 rue du Chevaleret, 75252, Paris, France

    Franck Sueur

Authors
  1. Dragoş Iftimie
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Franck Sueur
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dragoş Iftimie.

Additional information

Communicated by Y. Brenier

Rights and permissions

Reprints and permissions

About this article

Cite this article

Iftimie, D., Sueur, F. Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions. Arch Rational Mech Anal 199, 145–175 (2011). https://doi.org/10.1007/s00205-010-0320-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-010-0320-z

Keywords

Search

Navigation