Skip to main content
SpringerLink
Log in

Existence and Regularity of Steady Flows for Shear-Thinning Liquids in Exterior Two-Dimensional

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

An Erratum to this article was published on 05 January 2011

Abstract

We show that the two-dimensional exterior boundary-value problem (flow past a cylinder) associated with a class of shear-thinning liquid models possesses at least one solution for data of arbitrary “size”. This result must be contrasted with its counterpart for the Navier–Stokes model, where a similar result is known to hold, to date, only if the size of the data is sufficiently restricted.

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. Adams R.A.: Sobolev Spaces. Academic Press, New York (1975)

    MATH  Google Scholar 

  2. Amick C.J.: On Leray’s problem of steady Navier–Stokes flow past a body in the plane. Acta Math. 161, 71–130 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Batchelor G.K.: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  4. Bellout H., Nečas J.: The exterior problem in the plane for a non-Newtonian incompressible bipolar viscous fluid. Rocky Mountain J. Math. 26, 1245–1260 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bird R.B., Armstrong R.C., Hassager O.: Dynamics of Polymeric Liquids, Vol. 1. Wiley, New York (1987)

    Google Scholar 

  6. Finn R., Smith D.R.: On the stationary solution of the Navier-Stokes equations in two dimensions. Arch. Ration. Mech. Anal. 25, 26–39 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fujita H.: On the existence and regularity of the steady-state solutions of the Navier–Stokes equation. J. Fac. Sci. Univ. Tokyo Sect. I 9, 59–102 (1961)

    MathSciNet  MATH  Google Scholar 

  8. Galdi, G.P.: Existence and uniqueness at low Reynolds number of stationary plane flow of a viscous fluid in exterior domains. Recent Developments in Theoretical Fluid Mechanics, Paseky, 1992. Pitman Research Notes in Mathematical Series, Vol. 291. Longman Sci. Tech., Harlow, 1–33, 1993

  9. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations. Vol I. Linearized Steady Problems. Springer Tracts in Natural Philosophy, Vol. 38. Springer, New York, 1994

  10. Galdi, G.P.: An introduction to the mathematical theory of the Navier–Stokes equations. Vol II. Nonlinear Steady Problems. Springer Tracts in Natural Philosophy, Vol. 39. Springer, New York, 1994

  11. Galdi G.P.: On the existence of symmetric steady-state solutions to the plane exterior Navier–Stokes problem for arbitrary large Reynolds number. Advances in fluid dynamics. Quad. Mat. 4, 1–25 (1999)

    MathSciNet  MATH  Google Scholar 

  12. Galdi, G.P.: Mathematical problems in classical and non-Newtonian fluid mechanics. In: Galdi, G.P., Robertson, A.M., Rannacher, R., Turek, S.: Hemodynamical Flows: Modeling, Analysis and Simulation (Oberwolfach Seminars) Vol. 37. Birkhaeuser, Basel, 2008

  13. Gilbarg D., Weinberger H.F.: Asymptotic properties of Leray’s solution of the stationary two-dimensional Navier–Stokes equations. Russ. Math. Surv. 29, 109–123 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gilbarg D., Weinberger H.F.: Asymptotic properties of steady plane solutions of the Navier–Stokes equations with bounded Dirichlet integral. Ann. Scuola Norm. Sup. Pisa 5, 381–404 (1978)

    MathSciNet  MATH  Google Scholar 

  15. Kondrat’ev V.A., Olenik O.A.: Boundary value problems for a system in elasticity theory in unbounded domains. Korn inequalities. Russ. Math. Surv. 43, 65–119 (1988)

    Article  MathSciNet  Google Scholar 

  16. Ladyzhenskaya O.A.: On some new equations describing dynamics of incompressible fluids and on global solvability of boundary value problems to these equations. Trudy Steklov Math. Inst. 102, 85–104 (1967)

    Google Scholar 

  17. Ladyzhenskaya O.A.: On some modifications of the Navier–Stokes equations for large gradients of velocity. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov (LOMI) 7, 126–154 (1968)

    Google Scholar 

  18. Leray J.: Etude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’ hydrodynamique. J. Math. Pures Appl. 12, 1–82 (1933)

    MATH  Google Scholar 

  19. Lions J.-L.: Quelques méthodes de résolution des problémes aux limites non linaires. Dunod, Gauthier-Villars, Paris (1969)

    MATH  Google Scholar 

  20. Marušic-Paloka E.: On the Stokes paradox for power-law fluids. ZAMM Z. Angew. Math. Mech. 81, 31–36 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Naumann J., Wolf J.: Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids. J. Math. Fluid Mech. 7, 298–313 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Pittsburgh, 630 Benedum Engineering Hall, Pittsburgh, PA, 15621, USA

    Giovanni P. Galdi

  2. Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa, Via Buonarroti, 1/c, 56127, Pisa, Italy

    Carlo R. Grisanti

Authors
  1. Giovanni P. Galdi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlo R. Grisanti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni P. Galdi.

Additional information

Communicated by C.M. Dafermos

An erratum to this article can be found at http://dx.doi.org/10.1007/s00205-010-0380-0

Rights and permissions

Reprints and permissions

About this article

Cite this article

Galdi, G.P., Grisanti, C.R. Existence and Regularity of Steady Flows for Shear-Thinning Liquids in Exterior Two-Dimensional. Arch Rational Mech Anal 200, 533–559 (2011). https://doi.org/10.1007/s00205-010-0364-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-010-0364-0

Keywords

Search

Navigation