Journal of Mathematical Sciences

, Volume 93, Issue 5, pp 620–635 | Cite as

Stability and bifurcation in viscous incompressible fluids

  • H. Amann
Article
  • 38 Downloads

Abstract

We consider heat-conducting viscous incompressible (not necessarily Newtonian) fluids under the general Stokesian constitutive hypotheses. Given a natural and mild condition on the stress tensor at vanishing velocity, which is satisfied for Newtonian fluids, we discuss the stability behavior of stationary states at which the fluid is at rest and at constant temperature. In particular, we prove the existence of global small strong solutions for rather general isothermal non-Newtonian fluids. We also study bifurcation problems and show that subcritical bifurcations can occur. This effect can be seen only if the full energy equation is taken into consideration, that is, if the energy dissipation term is not dropped, as is done in the usual Boussinesq approximation. Bibliography: 29 titles.

Keywords

Stationary State Constant Temperature Stress Tensor Energy Dissipation Mild Condition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    H. Amann, “Dynamic theory of quasilinear parabolic equations — II. Reaction-diffusion systems”,Diff. Int. Equat.,3, 13–75 (1990).MATHMathSciNetGoogle Scholar
  2. 2.
    H. Amann,Ordinary Differential Equations. An Introduction to Nonlinear Analysis, W. de Gruyter, Berlin (1990).Google Scholar
  3. 3.
    H. Amann, “Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems”, in:Function Spaces, Differential Operators and Nonlinear Analysis, Teubner, Stuttgart-Leipzig, (1993), pp. 9–126.Google Scholar
  4. 4.
    H. Amann, “Stability of the rest state of a viscous incompressible fluid”,Arch. Rat. Mech. Anal.,126, 231–242 (1994).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    H. Amann, “Heat-conducting incompressible viscous fluids” in:Navier-Stokes Equations and Related Non-Linear Problems, Plenum Press, New York (1995), pp. 231–243.Google Scholar
  6. 6.
    H. Amann,Linear and Quasilinear Parabolic Problems, Birkhäuser, Basel (1995).Google Scholar
  7. 7.
    R. B. Bird, R. C. Armstrong, and O. Hassager,Dynamics of Polymeric Liquids, Vol. 1: Fluid Mechanics, Wiley, New York (1987).Google Scholar
  8. 8.
    D. Cioranescu, “Quelques examples de fluides Newtonien generalisés”, in:Mathematical Topics in Fluid Mechanics. Pitman Research Notes in Math., #274 (1992), pp. 1–31.Google Scholar
  9. 9.
    D. Fujiwara and H. Morimoto, “AnL r-theorem of the Helmholtz decomposition of vector fields”,J. Fac. Sci. Univ. Tokyo, Sec. IA Math.,24, 685–700 (1977).MathSciNetGoogle Scholar
  10. 10.
    G. P. Galdi,An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Springer-Verlag, New York (1994).Google Scholar
  11. 11.
    G. P. Galdi and M. Padula, “A new approach to energy theory in the stability of fluid motion”,Arch. Rat. Mech. Anal.,110, 187–286 (1990).CrossRefMathSciNetGoogle Scholar
  12. 12.
    Y. Giga, “Analyticity of the semigroup generated by the Stokes operator inL r spaces”,Math. Z.,178, 297–329 (1981).CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    D. D. Joseph,Stability of Fluid Motions, Springer-Verlag, Berlin (1976).Google Scholar
  14. 14.
    S. Kaniel, “On the initial value problem for an incompressible fluid with nonlinear viscosity”,J. Math. Mech.,19, 681–707 (1970).MATHMathSciNetGoogle Scholar
  15. 15.
    O. A. Ladyzhenskaja,Mathematical Theory of Viscous Incompressible Flow, Gordon & Breach, New York (1969).Google Scholar
  16. 16.
    O. A. Ladyzhenskaya, “New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them”,Zap. Nauchn. Semin. LOMI,102, 95–118 (1967).Google Scholar
  17. 17.
    A. Lunardi,Analytic Semigroups and Optimal Regularity in Parabolic Equations, Birkhäuser, Basel (1995).Google Scholar
  18. 18.
    J. Málek, J. Neĉas, and M. Růžiĉka, “On the non-Newtonian incompressible fluids”,Math. Meth. Models Appl. Sci.,3, 35–63 (1993).Google Scholar
  19. 19.
    J. Málek, K. R. Rajagopal, and M. Růžiĉka, “Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity”, Preprint (1994).Google Scholar
  20. 20.
    J. Málek, M. Růžiĉka, and G. Thäter, “Fractal dimension, attractors, and Boussinesq approximation in three dimensions”, Preprint (1993).Google Scholar
  21. 21.
    T. Miyakawa, “On the initial value problem for the Navier-Stokes equations inL p spaces”,Hiroshima Math. J.,11, 9–20 (1981).MATHMathSciNetGoogle Scholar
  22. 22.
    K. R. Rajagopal, “Mechanics of non-Newtonian fluids,”, in:Recent Developments in Theoretical Fluid Dynamics. Pitman Research Notes in Math., #291, (1993), pp. 129–162.Google Scholar
  23. 23.
    W. R. Schowalter,Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford (1978).Google Scholar
  24. 24.
    J. Serrin, “Mathematical principles of classical fluid dynamics,”, in:Handbuch der Physik,VIII/1, Springer-Verlag (1959), pp. 125–263.Google Scholar
  25. 25.
    V. A. Solonnikov, “Estimates for solutions of nonstationary Navier-Stokes equations”,J. Sov. Math.,8, 467–529 (1977).MATHGoogle Scholar
  26. 26.
    B. Straughan,The Energy Method, Stability, and Nonlinear Convection, Springer-Verlag, New York (1992).Google Scholar
  27. 27.
    C. Truesdell and W. Noll, “The nonlinear field theories of mechanics”, in:Handbuch der Physik,III/3, Springer-Verlag (1965).Google Scholar
  28. 28.
    W. von Wahl,The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg & Sohn, Braunschweig (1985).Google Scholar
  29. 29.
    W. von Wahl, “Necessary and sufficient conditions for the stability of flows in incompressible visous fluids”,Arch. Rat. Mech. Anal.,126, 103–129 (1994).CrossRefMATHGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • H. Amann

There are no affiliations available

Personalised recommendations