A fluid of density, \(ho\), and viscosity, \(\mu\), drains from between very long parallel flat plates oriented vertically with respect to gravity. One plate is held stationary and the other travels vertically, opposite to the direction of gravity, with a velocity, \(v_{o}\).

a. What is the differential equation governing the flow of fluid between the plates?

b. What are the boundary conditions for the problem?

c. Solve the equation for the velocity profile \(v_{z}(x)\).

d. If we are to have no net volumetric flow \(\left(\mathrm{m}^{3} / \mathrm{s}\right)\) between the plates, what value should \(v_{o}\) have and what is its direction? Assume the plates extend \(1 \mathrm{~m}\) deep in the \(y\)-direction.