Stopping a Viscous Fluid by a Feedback Dissipative Field: Thermal Effects without Phase Changing

  • S.N. Antontsev
  • J.I. Díaz
  • H.B. de Oliveira
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 61)

Abstract

We show how the action on two simultaneous effects (a suitable coupling about velocity and temperature and a low range of temperature but upper that the phase changing one) may be responsible of stopping a viscous fluid without any changing phase. Our model involves a system, on an unbounded pipe, given by the planar stationary Navier-Stokes equation perturbed with a sublinear term f(x,θ, u) coupled with a stationary (and possibly nonlinear) advection diffusion equation for the temperature θ.

After proving some results on the existence and uniqueness of weak solutions we apply an energy method to show that the velocity u vanishes for x large enough.

Keywords

Non-Newtonian fluids nonlinear thermal diffusion equations feedback dissipative field energy method heat and mass transfer localization effect 

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References

  1. [1]
    S.N. Antontsev, J.I. Díaz, H.B. de Oliveira. Stopping a viscous fluid by a feedback dissipative external field: I. The stationary Stokes equations. Book of abstracts of NSEC8, Euler International Mathematical Institute, St. Petersburg, 2002.Google Scholar
  2. [2]
    S.N. Antontsev, J.I. Díaz, H.B. de Oliveira. On the confinement of a viscous fluid by means of a feedback external field. C.R. Mécanique 330 (2002), 797–802.MATHCrossRefGoogle Scholar
  3. [3]
    S.N. Antontsev, J.I. Díaz, H.B. de Oliveira. Stopping a viscous fluid by a feedback dissipative field: I. The stationary Stokes problem. To appear in J. Math. Fluid Mech.Google Scholar
  4. [4]
    S.N. Antontsev, J.I. Díaz, H.B. de Oliveira. Stopping a viscous fluid by a feedback dissipative field: I. The stationary Navier-Stokes problem. To appear in Rend. Lincei Mat. Appl.Google Scholar
  5. [5]
    S.N. Antontsev, J.I. Díaz, S.I. Shmarev. Energy Methods for Free Boundary Problems: Applications to Non-linear PDEs and Fluid Mechanics. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48, Birkhäuser, Boston, 2002.Google Scholar
  6. [6]
    J.R. Canon, E. DiBenedetto, G.H. Knightly. The bidimensional Stefan problem with convection: the time-dependent case. Comm. Partial Differential Equations, 8 (1983), 1549–1604.CrossRefMathSciNetGoogle Scholar
  7. [7]
    J. Carrillo, M. Chipot, On some nonlinear elliptic equations involving derivatives of the nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A 100(3–4) (1985), 281–294.MATHMathSciNetGoogle Scholar
  8. [8]
    E. DiBenedetto, M. O’Leary. Three-dimensional conduction-convection problems with change of phase. Arch. Rational Mech. Anal. 123 (1993), 99–117.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    G.P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Nonlinear Steady Problems. Springer-Verlag, New York, 1994.MATHGoogle Scholar
  10. [10]
    D. Gilbarg, N.S. Trudinger. Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin Heidelberg, 1998.Google Scholar
  11. [11]
    O.A. Ladyzhenskaya, N.N. Ural’tseva. Linear and Quasilinear Elliptic Equations. Academic Press, New York, 1968.MATHGoogle Scholar
  12. [12]
    O.A. Ladyzhenskaya. The Mathematical Theory of Viscous Incompressible Fluids. Gordon and Breach Science Publishers Inc., New York, 1969.Google Scholar
  13. [13]
    V.A. Solonnikov. On the solvability of boundary and initial boundary value problems for the Navier-Stokes systems in domains with noncompact boundaries. Pacific J. Math. 93(2) (1981), 443–458.MATHMathSciNetGoogle Scholar
  14. [14]
    X. Xu, M. Shillor. The Stefan problem with convection and Joule’s heating. Adv. Differential Equations 2 (1997), 667–691.MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • S.N. Antontsev
    • J.I. Díaz
      • H.B. de Oliveira
        1. 1.Universidade da Beira InteriorCovilhãPortugal
        2. 2.Universidad ComplutenseMadridSpain
        3. 3.Universidade do AlgarveFaroPortugal

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