Journal of Mathematical Fluid Mechanics

, Volume 14, Issue 2, pp 295–309 | Cite as

On the 3D Steady Flow of a Second Grade Fluid Past an Obstacle

  • Paweł Konieczny
  • Ondřej KremlEmail author


We study steady flow of a second grade fluid past an obstacle in three space dimensions. We prove existence of solution in weighted Lebesgue spaces with anisotropic weights and thus existence of the wake region behind the obstacle. We use properties of the fundamental Oseen tensor together with results achieved in Koch (Quad Mat 15:59–122, 2004) and properties of solutions to steady transport equation to get up to arbitrarily small ε the same decay as the Oseen fundamental solution.

Mathematics Subject Classification (2010)

35Q35 76D03 


Second grade fluid Steady flow past an obstacle Asymptotic behavior Weighted estimates 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Coscia V., Galdi G.P.: Existence, uniqueness and stability of regular steady motions of a second-grade fluid. Int. J. Non-Linear Mech. 29(4), 493–506 (1994)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Dunn J.E., Fosdick R.L.: Thermodynamics, stability and boundedness of fluids of complexity 2 and fluids of second grade. Arch. Ration. Mech. Anal. 56, 191–252 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Finn R.: Estimates at infinity for stationary solution of the Navier–Stokes equations. Bult. Math. de la Soc. Sci. Math. de la R. P. R., Tome 3 51(4), 387–418 (1959)Google Scholar
  4. 4.
    Galdi G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations I, Springer Tracts in Natural Philosophy, vol. 38. Springer, New York (1994)Google Scholar
  5. 5.
    Galdi G.P., Padula M., Rajagopal K.R.: On the conditional stability of the rest state of a fluid of second grade in unbounded domains. Arch. Ration, Mech. Anal. 109(2), 173–182 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Koch H.: Partial differential equations and singular integrals, Dispersive nonlinear problems in mathematical physics, Department of Mathematics, Seconda University, Napoli. Caserta. Quad. Mat. 15, 59–122 (2004)Google Scholar
  7. 7.
    Kračmar, S., Novotný, A., Pokorný, M.: Estimates of three-dimensional Oseen kernels in weighted L p spaces. In: da Veiga, B., Sequeira, A., Videman, J. (eds.) Applied Functional Analysis, 281–316 (1999)Google Scholar
  8. 8.
    Mogilevskii I.Sh., Solonnikov V.A.: Problem on a stationary flow of a second-grade fluid in Hölder classes of functions. Zapisky Nauc. Sem. LOMI 243, 154–165 (1997)Google Scholar
  9. 9.
    Novotný, A.: About the steady transport equation. In: Proceedings of the Fifth Winter School at Paseky, Pitman Research Notes in Mathematics (1998)Google Scholar
  10. 10.
    Novotný A., Pokorný M.: Three-dimensional steady flow of viscoelastic fluid past an obstacle. J. Math. Fluid Mech. 2, 294–314 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Pokorný, M.: Asymptotic behaviour of solutions to certain PDE’s describing the flow of fluids in unbounded domains, Ph.D. thesis, Charles University in Prague and University of Toulon and Var, Toulon-La Garde (1999)Google Scholar
  12. 12.
    Smith D.R.: Estimates at infinity for stationary solutions of the N. S. equations in two dimensions. Arch. Rat. Mech. Anal. 20, 341–372 (1965)zbMATHCrossRefGoogle Scholar
  13. 13.
    Truesdell C., Noll W.: The Nonlinear Field Theories of Mechanics, Handbuch der Physik, III/3. Springer, Heidelberg (1965)Google Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institute for Mathematics and its ApplicationsUniversity of MinnesotaMinneapolisUSA
  2. 2.Mathematical Institute of Charles UniversityPraha 8Czech Republic

Personalised recommendations