Skip to main content

Steady States of Anisotropic Generalized Newtonian Fluids

Abstract.

We consider the stationary flow of a generalized Newtonian fluid which is modelled by an anisotropic dissipative potential f. More precisely, we are looking for a solution \(u:\Omega \to \mathbb{R}^n ,\Omega \subset \mathbb{R}^n ,\,n = 2,3,\) of the following system of nonlinear partial differential equations

$$ \left. \begin{aligned} - {\text{div}}\{ T(\varepsilon (u))\} + u^k \tfrac{{\partial u}} {{\partial x_k }} + \nabla \pi & = g{\text{ in}}\,\Omega , \\ {\text{div }}u = 0{\text{ in}}\,\Omega ,\quad u & = 0{\text{ on}}\,\partial \Omega . \\ \end{aligned} \right\} $$
((*))

Here \(\pi :\Omega \to \mathbb{R}\) denotes the pressure, g is a system of volume forces, and the tensor T is the gradient of the potential f. Our main hypothesis imposed on f is the existence of exponents 1 < p ≤ q0 < ∞ such that

$$ \lambda (1 + |\varepsilon |^2 )^{\tfrac{{p - 2}} {2}} |\sigma |^2 \leq D^2 f(\varepsilon )(\sigma ,\sigma ) \leq \Lambda (1 + |\varepsilon |^2 )^{\tfrac{{q_0 - 2}} {2}} |\sigma |^2 $$

holds with constants λ, Λ > 0. Under natural assumptions on p and q0 we prove the existence of a weak solution u to the problem (*), moreover we prove interior C1,α-regularity of u in the two-dimensional case. If n = 3, then interior partial regularity is established.

This is a preview of subscription content, access via your institution.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Darya Apushkinskaya.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Apushkinskaya, D., Bildhauer, M. & Fuchs, M. Steady States of Anisotropic Generalized Newtonian Fluids. J. math. fluid mech. 7, 261–297 (2005). https://doi.org/10.1007/s00021-004-0118-6

Download citation

Mathematics Subject Classification (2000).

  • 76M30
  • 49N60
  • 35J50
  • 35Q30

Keywords.

  • Generalized Newtonian fluids
  • anisotropic dissipative potentials
  • regularity