The quantities that are measured in a cup and bob viscometer are the rotation rate of the cup (rpm) and the corresponding torque transmitted to the bob. These quantities are then converted to corresponding values of shear rate $(\dot{\gamma})$ and shear stress $(\tau)$, which in turn can be converted to corresponding values of viscosity $(\eta)$.

(a) Show what a $\log -\log$ plot of $\tau$ versus $\dot{\gamma}$ and $\eta$ versus $\dot{\gamma}$ would look like for materials that follow the following models: (i) Newtonian, (ii) power law (shear thinning), (iii) power law (shear thickening), (iv) Bingham plastic, and (v) structural.

(b) Show how the values of the parameters for each of the models listed in (a) can be evaluated from the respective plot of $\eta$ versus $\dot{\gamma}$. That is, relate each model parameter to some characteristic or combination of characteristics of the plot such as the slope, specific values read from the plot, intersection of tangent lines, etc.