Advertisement

Manuscripta Mathematica

, 136:315 | Cite as

Asymptotic profile of steady Stokes flow around a rotating obstacle

  • Reinhard FarwigEmail author
  • Toshiaki Hishida
Article

Abstract

We analyze the spatial anisotropic profile at infinity of steady Stokes flow around a rotating obstacle. It is shown that the flow is largely concentrated along the axis of rotation in the leading term and that a rotating profile can be found in the second term. The proof relies upon a detailed analysis of the associated fundamental solution tensor.

Mathematics Subject Classification (2000)

35Q30 35Q35 35B40 76D07 

References

  1. 1.
    Borchers W.: Zur Stabilität und Faktorisierungsmethode für die Navier–Stokes-Gleichungen inkompressibler viskoser Flüssigkeiten. Habilitationsschrift, Universität Paderborn (1992)Google Scholar
  2. 2.
    Deuring P., Galdi G.P.: On the asymptotic behavior of physically reasonable solutions to the stationary Navier–Stokes system in three-dimensional exterior domains with zero velocity at infinity. J. Math. Fluid Mech. 2, 353–364 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Farwig R.: The stationary exterior 3D-problem of Oseen and Navier–Stokes equations in anisotropically weighted Sobolev spaces. Math. Z. 211, 409–447 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Farwig, R.: The stationary Navier–Stokes equations in a 3D-exterior domain. In: Kozono, H., Shibata, Y. (eds.) Recent Topics on Mathematical Theory of Viscous Incompressible Fluid. Lecture Notes in Num. Appl. Anal., vol. 16. pp. 53–115. Kinokuniya, Tokyo (1998)Google Scholar
  5. 5.
    Farwig R.: An L q-analysis of viscous fluid flow past a rotating obstacle. Tohoku Math. J. 58, 129–147 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Farwig R., Hishida T.: Stationary Navier–Stokes flow around a rotating obstacle. Funkcial. Ekvac. 50, 371–403 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Farwig, R., Hishida, T.: Leading term at infinity of steady Navier-stokes flow around a rotating obstacle. Math. Nachr. (to appear)Google Scholar
  8. 8.
    Farwig R., Hishida T., Müller D.: L q-theory of a singular “winding” integral operator arising from fluid dynamics. Pac. J. Math. 215, 297–312 (2004)zbMATHCrossRefGoogle Scholar
  9. 9.
    Farwig R., Neustupa J.: On the spectrum of a Stokes-type operator arising from flow around a rotating body. Manuscr. Math. 122, 419–437 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Finn R.: On the exterior stationary problem for the Navier–Stokes equations, and associated perturbation problems. Arch. Ration. Mech. Anal. 19, 363–406 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Finn R.: Mathematical questions relating to viscous fluid flow in an exterior domain. Rocky Mt. J. Math. 3, 107–140 (1973)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. I, II, revised edition, Springer, New York (1998)Google Scholar
  13. 13.
    Galdi, G.P.: On the motion of a rigid body in a viscous liquid: a mathematical analysis with applications. In: Handbook of Mathematical Fluid Dynamics, vol. I, pp. 653–791, North-Holland, Amsterdam (2002)Google Scholar
  14. 14.
    Galdi G.P.: Steady flow of a Navier–Stokes fluid around a rotating obstacle. J. Elast. 71, 1–31 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Galdi G.P., Silvestre A.L.: Strong solutions to the Navier–Stokes equations around a rotating obstacle. Arch. Ration. Mech. Anal. 176, 331–350 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Galdi G.P., Silvestre A.L.: Further results on steady-state flow of a Navier–Stokes liquid around a rigid body. Existence of the wake, Kyoto Conference on the Navier–Stokes equations and their applications. RIMS Kôkyûroku Bessatsu B 1, 127–143 (2007)MathSciNetGoogle Scholar
  17. 17.
    Galdi G.P., Silvestre A.L.: The steady motion of a Navier–Stokes liquid around a rigid body. Arch. Ration. Mech. Anal. 184, 371–400 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Geissert M., Heck H., Hieber M.: L p-theory of the Navier–Stokes flow in the exterior of a moving or rotating obstacle. J. Reine Angew. Math. 596, 45–62 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Heywood J.G.: On uniqueness questions in the theory of viscous flow. Acta Math. 136, 61–102 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Hishida T.: The Stokes operator with rotation effect in exterior domains. Analysis 19, 51–67 (1999)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Hishida T.: An existence theorem for the Navier–Stokes flow in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 150, 307–348 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hishida T.: L q estimates of weak solutions to the stationary Stokes equations around a rotating body. J. Math. Soc. Jpn. 58, 743–767 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hishida T., Shibata Y.: L p-L q estimate of the Stokes operator and Navier–Stokes flows in the exterior of a rotating obstacle. Arch. Ration. Mech. Anal. 193, 339–421 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Korolev, A., Šverák, V.: On the large-distance asymptotics of steady state solutions of the Navier–Stokes equations in 3D exterior domains. arXiv:math/07110560, preprint (2007)Google Scholar
  25. 25.
    Kozono H., Sohr H., Yamazaki M.: Representation formula, net force and energy relation to the stationary Navier–Stokes equations in 3-dimensional exterior domains. Kyushu J. Math. 51, 239–260 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Kračmar S., Nečasová Š., Penel P.: Anisotropic L 2-estimates of weak solutions to the stationary Oseen-type equations in 3D-exterior domain for a rotating body. J. Math. Soc. Jpn. 62, 239–268 (2010)zbMATHCrossRefGoogle Scholar
  27. 27.
    Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow, 2nd ed. Gordon and Breach, New York (1969)zbMATHGoogle Scholar
  28. 28.
    Nazarov S.A., Pileckas K.: On steady Stokes and Navier–Stokes problems with zero velocity at infinity in a three-dimensional exterior domain. J. Math. Kyoto Univ. 40, 475–492 (2000)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Shibata Y.: On an exterior initial boundary value problem for Navier–Stokes equations. Quart. Appl. Math. 57, 117–155 (1999)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Silvestre A.L.: On the existence of steady flows of a Navier–Stokes liquid around a moving rigid body. Math. Methods Appl. Sci. 27, 1399–1409 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.FB MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Graduate School of MathematicsNagoya UniversityNagoyaJapan

Personalised recommendations