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A. Bourgeat
France

 

Mathematical modeling of a non-Newtonian viscous flow through a thin filter
Alain Bourgeat
Faculté des Sciences, 23 rue du Dr Paul Michelon, 42023 Saint-Etienne Cedex 02, France
e-mail : bourgeat@univ-st-etienne.fr


Abstract. We consider a non-Newtonian flow, like polymer melt or solution, pushed through a thin periodic filter with a period and thickness $\varepsilon\ll 1$. Starting from the Stokes system with a nonlinear viscosity obeying the Carreau's law (with possibly a high rate viscosity) we study the asymptotic behavior of the flow as $\varepsilon \to 0$. The main goal of this work is to study the global convergence of the pressure in the whole domain and not separately in the upper and in the lower part as in previous papers.

The flow is described by a quasi-Newtonian model with a nonlinear viscosity $\eta$ depending on the rate-of-strain tensor $\;e(u)$:

\begin{displaymath} \eta^r (e(u)) = (\eta_0 -\eta_\infty ) (1+\lambda \vert e(u)\vert^2)^{r/2 -1} +\eta_\infty \end{displaymath} (1)

with the flow index $r\in ]1,2[ $ , $\lambda >0 \;,\;\eta_0>\eta_\infty \geq 0$. We consider both cases $\eta_\infty >0$ (corresponding to a polymer solution with the solvent's viscosity equal to $\eta_\infty $) and $\eta_\infty =0$ (corresponding to a polymer melt). For $\eta_\infty >0$, and for a Newtonian fluid, i.e. $r=2$ we find the effective behavior of the velocity in the whole domain $\Omega$ and the effective behavior of the pressure in two separated domains, the upper one $\Omega^+$ and the lower one $\Omega^-$. In addition we prove that, globally, the pressure in the whole domain $\Omega$ behaves like $\varepsilon^\gamma C^{+-}$ in $\Omega^{+-}$ (with $\gamma =-1$ if $\eta_\infty >0$ and $\gamma =1-r$ if $\eta_\infty =0$) where the two constants $C^+$ and $C^-$ are different. Moreover the difference $C^+ -C^- $ is equal to $\frac{1}{F}E_{bl}$ , with $F$ the flux through the filter and $E_{bl}$ the boundary layer dissipated energy.

 

 

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Last modified: December 20, 1999