(Solved): 6. (a) Show that the Cross-Arrhenius (5-constant) viscosity model equation can be reduced to a Powe ...

6. (a) Show that the Cross-Arrhenius (5-constant) viscosity model equation can be reduced to a Power-law model like $\eta (T,\dot{\gamma })=m(T){\dot{\gamma }}^{n-1}$ at very high shear rates and at $T>T_{\text{trans. }}$. (3 points) $\begin{array}{l}\eta (T,\dot{\gamma },p)=\frac{{\eta }_0(T,p)}{1+{\left[\frac{{\eta }_0(T,p)\dot{\gamma }}{{\tau }^{\ast }}\right]}^{(1-n)}}\\ T>T_{\text{trans }},{\eta }_0(T,p)=B\times \exp \left(\frac{T_b}{T}\right)\times \exp (\beta \times p)\\ T<T_{\text{trans }},{\eta }_0(T,p)=\infty \end{array}$ ${\eta }_0$ is the zero-shear-rate viscosity $\dot{\gamma }$ is the shear rate $T$ is the temperature $p$ is the pressure $n,{\tau }^{\ast },B,T_b$, and $\beta$ are model constants