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The Giesekus model adds the nonlinear anisotropic-drag term $\alpha\lambda\boldsymbol\tau^2/\eta$ to the <Upper-convected Maxwell model>. Positive dimensionless mobility factor $\alpha$ produces shear thinning and a nonzero second normal-stress difference.

Giesekus model

Setting $\alpha=0$ gives the <Upper-convected Maxwell model>
$$
\boldsymbol\tau+\lambda
\overset{\triangledown}{\boldsymbol\tau}
=\eta\dot{\boldsymbol\gamma}.
$$
In steady incompressible uniaxial extension of rate $\dot\epsilon$, its tensile and transverse stresses give
$$
\eta_E
=\frac{\tau_{zz}-\tau_{xx}}{\dot\epsilon}
=\frac{2\eta}{1-2\lambda\dot\epsilon}
+\frac{\eta}{1+\lambda\dot\epsilon}.
$$
The <extensional viscosity> rises above the <Trouton ratio> value $3\eta$ and diverges at $\lambda\dot\epsilon=1/2$, the ideal model's extensional catastrophe. An extensional rheometer can locate that rapid growth and estimate $\lambda\simeq1/(2\dot\epsilon_c)$. More robustly, impose a small deformation, stop the flow, and fit the exponential stress decay $\tau\propto e^{-t/\lambda}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 352 / 2 / b / Solution

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The upper-convected Maxwell model evolves viscoelastic stress according to
$$
\boldsymbol\tau+\lambda\overset{\triangledown}{\boldsymbol\tau}
=\eta\dot{\boldsymbol\gamma}.
$$
It has constant shear viscosity and predicts a divergent extensional viscosity at a finite extension rate.


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 Incoming links: Upper-convected Maxwell model