Mathematical and Numerical Analysis of Thermally Coupled Quasi-Newtonian Flow Obeying a Power Law

  • Jiang Zhu
  • Xijun Yu
Conference paper


In this paper, we consider an incompressible quasi-Newtonian flow with a temperature dependent viscosity obeying a power law, and the thermal balance includes viscous heating. The corresponding mathematical model can be written as:
$$ \left\{ \begin{gathered} - 2\nabla \cdot \left( {\mu \left( \theta \right)\left| {D\left( u \right)} \right|^{r - 2} D\left( u \right)} \right) + \nabla p = f in \Omega \hfill \\ \nabla \cdot u = 0 in \Omega \hfill \\ - \Delta \theta = \mu \left( \theta \right)\left| {D\left( u \right)} \right|^r in \Omega \hfill \\ u = 0 on \Gamma \hfill \\ \theta = 0 on \Gamma \hfill \\ \end{gathered} \right. $$
where u : Ω → d is the velocity, p : Ω → is the pressure, θ : Ω → is the temperature, Ω is a bounded open subset of d , d 2 or 3 , Γ its boundary. The viscosity µ is a function of θ , µ = µ(θ). D is the strain rate tensor, D (u) (∇u + ∇u T ) , |D(u)|2 is the second invariant of D(u) , and 1< r < ∞ . Some mathematical results such as existence and uniqueness are established, finite element approximation is proposed, and convergence analysis is presented.

Copyright information

© Tsinghua University Press & Springer 2007

Authors and Affiliations

  • Jiang Zhu
    • 1
  • Xijun Yu
    • 2
  1. 1.National Laboratory for Scientific ComputingMCTPetrópolisBrazil
  2. 2.National Key Laboratory of Computational PhysicsInstitute of Applied Physics and Computational MathematicsBeijingChina

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