Consider the flow of water at \(20^{\circ} \mathrm{C}\) between two very large horizontal plates. The lower plate is stationary and the upper plate moves to the right at a velocity of \(0.25 \mathrm{~m} / \mathrm{s}\). For a plate spacing of \(2 \mathrm{~mm}\), we want to determine the pressure gradient and its direction required to produce zero net flow at a cross-section.

a. What does the general situation look like?

b. Derive the differential equation governing the flow between the plates.

c. What are the boundary conditions?

d. Solve the equation for the velocity profile between the plates.

e. Determine an expression for the volumetric flow rate between the plates.

f. What is the pressure gradient and direction required to produce zero net volumetric flow?