Advertisement

Czechoslovak Mathematical Journal

, Volume 54, Issue 3, pp 637–656 | Cite as

Density-Dependent Incompressible Fluids with Non-Newtonian Viscosity

  • F. Guillén-González
Article

Abstract

We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of p-coercivity and (p−1)-growth, for a given parameter p > 1. The existence of Dirichlet weak solutions was obtained in [2], in the cases p ≥ 12/5 if d = 3 or p ≥ 2 if d = 2, d being the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all p ≥ 2. In addition, we obtain regularity properties of weak solutions whenever p ≥ 20/9 (if d = 3) or p ≥ 2 (if d = 2). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained.

variable density shear-dependent viscosity power law Carreau's laws weak solution strong solution periodic boundary conditions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. A. Antonzev and A. V. Kazhikhov: Mathematical Study of Flows of Non-Homogeneous Fluids. Lectures at the University of Novosibirsk, U.S.S.R, 1973. (In Russian.)Google Scholar
  2. [2]
    E. Fernández-Cara, F. Guillén and R. R. Ortega: Some theoretical results for visco-plastic and dilatant uids with variable density. Nonlinear Anal. 28 (1997), 1079–1100.Google Scholar
  3. [3]
    J. Frehse, J. Málek and M. Steinhauer: On existence results for uids with shear depen-dent viscosity—unsteady ows. Partial Differential Equations, Praha 1998 Chapman & Hall/CRC, Res. Notes Math., 406, Boca Raton, FL, 2000, pp. 121–129.Google Scholar
  4. [4]
    A. V. Kazhikhov: Resolution of boundary value problems for nonhomogeneous viscous uids. Dokl. Akad. Nauk 216 (1974), 1008–1010.Google Scholar
  5. [5]
    O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow. Gor-don and Breach, 1969.Google Scholar
  6. [6]
    J. L. Lions: Quelques méthodes de résolution des problemes aux limites non linéaires. Dunod, Gauthier-Villars, 1969.Google Scholar
  7. [7]
    P. L. Lions: Mathematical Topics in Fluid Mechanics. Volume 1, Incompressible models. Clarendon Press, 1996.Google Scholar
  8. [8]
    J. Málek, K. R. Rajagopal and M. Ružička: Existence and regularity of solutions and the stability of the rest state for uids with shear dependent viscosity. Math. Models and Methods in Applied Sciences 5 (1995), 789–812.Google Scholar
  9. [9]
    J. Málek, J. Nečas, M. Rokyta and M. Ružička: Weak and Measure-Valued Solutions to Evolutionary PDEs. Chapman & Hall, 1996.Google Scholar
  10. [10]
    J. Málek, J. Nečas and M. Ružička: On weak solutions of non-Newtonian incompressible uids in bounded three-dimensional domains. The case p ⩾ 2. Advances in Differential Equations 6 (2001), 257–302.Google Scholar
  11. [11]
    J. Simon: Compact sets in L p (0; T; B). Ann. Mat. Pura Appl. 4 (1987), 65-96.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2004

Authors and Affiliations

  • F. Guillén-González
    • 1
  1. 1.Depto. Ecuaciones Diferenciales y Análisis NuméricoUniversidad de SevillaSevillaSpain

Personalised recommendations