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: a. Derive an...

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:

a. Derive an expression for the velocity profile between the plates assuming the lower plate is stationary, the upper plate moves with a velocity, v0, 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 mm, v0 = 1 cm/s, μ0 = 1 Pa ∙s, ϕ = 0.05, and ϕc = 0.35?

d. When ϕ → ϕc, particles can span the space between the plates leading to jamming. Plot the shear stress between the plates as a function of ϕ 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?