An engineer is attempting to measure the force needed to shear two plates past one another when the fluid inside is a slurry. The slurry is relatively dilute and obeys the equation:

\[\mu_{r}=\frac{\mu}{\mu_{o}}=\frac{1+0.5 \phi}{\left(\phi_{c}-\phi\right)^{2}} \quad\begin{aligned}& \mu_{o}=\text { viscosity of pure fluid } \\& \phi_{c}=\text { critical volume fraction }\end{aligned}\]

a. Derive an expression for the velocity profile between the plates assuming the lower plate is stationary, the upper plate moves with a velocity, \(v_{0}\), and the gap between the plates is \(d\).

b. Determine an expression for the shear stress between the plates.

c. What is the stress if \(d=1 \mathrm{~mm}, v_{0}=1 \mathrm{~cm} / \mathrm{s}, \mu_{0}=1 \mathrm{~Pa} \cdot \mathrm{s}, \phi=0.05\), and \(\phi_{c}=0.35\) ?

d. When \(\phi \rightarrow \phi_{c}\), particles can span the space between the plates leading to jamming. Plot the shear stress between the plates as a function of \(\phi\) for the conditions in (c). When does the shear stress become a problem, i.e., an order of magnitude larger than that for the pure fluid?