Existence of the passage to the limit of an inviscid fluid

  • Denis S. GoldobinEmail author
Regular Article
Part of the following topical collections:
  1. Non-equilibrium processes in multicomponent and multiphase media


In the dynamics of a viscous fluid, the case of vanishing kinematic viscosity is actually equivalent to the Reynolds number tending to infinity. Hence, in the limit of vanishing viscosity the fluid flow is essentially turbulent. On the other hand, the Euler equation, which is conventionally adopted for the description of the flow of an inviscid fluid, does not possess proper turbulent behaviour. This raises the question of the existence of the passage to the limit of an inviscid fluid for real low-viscosity fluids. To address this question, one should employ the theory of turbulent boundary layer near an inflexible boundary (e.g., rigid wall). On the basis of this theory, one can see how the solutions to the Euler equation become relevant for the description of the flow of low-viscosity fluids, and obtain the small parameter quantifying accuracy of this description for real fluids.

Graphical abstract


Topical issue: Non-equilibrium processes in multicomponent and multiphase media 


  1. 1.
    H.S.G. Swann, Trans. Am. Math. Soc. 157, 373 (1971)Google Scholar
  2. 2.
    T. Kato, J. Funct. Anal. 9, 296 (1972)CrossRefGoogle Scholar
  3. 3.
    N. Masmoudi, Commun. Math. Phys. 270, 777 (2007)ADSCrossRefGoogle Scholar
  4. 4.
    M. Sammartino, R.E. Caflisch, Commun. Math. Phys. 192, 433 (1998)ADSCrossRefGoogle Scholar
  5. 5.
    M. Sammartino, R.E. Caflisch, Comm. Math. Phys. 192, 463 (1998)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    D. Iftimie, G. Planas, Nonlinearity 19, 899 (2006)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Constantin, I. Kukavica, V. Vicol, Proc. Am. Math. Soc. 143, 3075 (2015)CrossRefGoogle Scholar
  8. 8.
    Y. Maekawa, A. Mazzucato, The Inviscid Limit and Boundary Layers for Navier-Stokes Flows, in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, edited by Y. Giga, A. Novotny (Springer, 2016) pp. 1--48, e-Print: arXiv:1610.05372Google Scholar
  9. 9.
    Th. von Karman, Nachr. Ges. Wiss. Göttingen, Fachgruppe 1 (Mathematik) 5, 58 (1930) (English translation: Th. von Karman, Mechanical Similitude and TurbulenceGoogle Scholar
  10. 10.
    L. Prandtl, Z. Ver. Dtsch. Ing. 77, 105 (1933) (English translation: L. Prandtl, Recent results of turbulence researchGoogle Scholar
  11. 11.
    H. Schlichting, K. Gersten, Boundary-Layer Theory (Springer, 2000)Google Scholar
  12. 12.
    L.D. Landau, E.M. Lifshitz, Fluid Mechanics (Nauka, Moscow, 1986)Google Scholar
  13. 13.
    E. Buckingham, Phys. Rev. 4, 345 (1914)ADSCrossRefGoogle Scholar
  14. 14.
    T. Lyubimova, A. Lepikhin, V. Konovalov, Ya. Parshakova, A. Tiunov, J. Hydrol. 508, 328 (2014)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute of Continuous Media MechanicsUB RASPermRussia
  2. 2.Department of Theoretical PhysicsPerm State UniversityPermRussia

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