Finite Time Localized Solutions of Fluid Problems with Anisotropic Dissipation

  • S. Antontsev
  • H. B. de Oliveira
Part of the International Series of Numerical Mathematics book series (ISNM, volume 154)

Abstract

In this work we consider an incompressible, non-homogeneous, dilatant and viscous fluid for which the stress tensor satisfies a general non-Newtonian law. The new contribution of this work is the consideration of an anisotropic dissipative forces field which depends nonlinearly on the own velocity. We prove that, if the flow of such a fluid is generated by the initial data, then in a finite time the fluid becomes immobile. We, also, prove that, if the flow is stirred by a forces term which vanishes at some instant of time, then the fluid is still for all time grater than that and provided the intensity of the force is suitably small.

Keywords

Non-Newtonian Fluids anisotropic dissipative field finite time localization effect 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Antontsev, J.I. Díaz and H.B. de Oliveira, On the confinement of a viscous fluid by means of a feedback external field. C.R. Mecanique, 330 (2002), 797–802.CrossRefGoogle Scholar
  2. [2]
    S. Antontsev, J.I. Díaz and H.B. de Oliveira, Stopping a viscous fluid by a feedback dissipative field. I. The stationary Stokes problem., J. Math. Fluid Mech., no. 4, 6 (2004), 439–461.CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    S. Antontsev, J.I. Díaz and H.B. de Oliveira, Stopping a viscous fluid by a feedback dissipative field. II. The stationary Navier-Stokes problem. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., no. 3–4, 15 (2004), 257–270.MathSciNetMATHGoogle Scholar
  4. [4]
    S. Antontsev, J.I. Díaz and H.B. de Oliveira, Stopping a viscous fluid by a feedback dissipative field: thermal effects without phase changing, Progress in Nonlinear Differential Equations and their Applications 61, Birkhäuser (2005), 1–14.CrossRefGoogle Scholar
  5. [5]
    S. Antontsev, J.I. Díaz and S.I. Shmarev, Energy methods for free boundary problems, Progress in Nonlinear Differential Equations and their Applications 48, Birkhäuser, 2002.Google Scholar
  6. [6]
    S. Antontsev, A.V. Kazhikhov and V.N. Monakhov, Boundary value problems in mechanics of nonhomogeneous fluids. Studies in Mathematics and its Applications 22, North-Holland, 1990.Google Scholar
  7. [7]
    E. Fernández-Cara, F. Guillén and R.R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable density., Nonlinear Anal., no. 6, 28 (1997), 1079–1100.CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    E. Fernández-Cara, F. Guillén and R.R. Ortega, Some theoretical results concerning non-Newtonian fluids of the Oldroyd kind., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), no. 1, 26 (1998), 1–29.MathSciNetMATHGoogle Scholar
  9. [9]
    O.A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Mathematics and its Applications 2, Gordon and Breach, 1969.Google Scholar
  10. [10]
    J. Simon, Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal., no. 5, 21 (1990), 1093–1117.CrossRefMathSciNetMATHGoogle Scholar
  11. [11]
    H. Sohr, The Navier-Stokes equations., Birkhäuser, Basel, 2001.MATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • S. Antontsev
    • 1
  • H. B. de Oliveira
    • 2
  1. 1.Departamento de MatemáticaUniversidade da Beira InteriorCovilhãPortugal
  2. 2.Faculdade de Ciências e TecnologiaUniversidade do AlgarveFaroPortugal

Personalised recommendations