Consider fully developed, laminar flow between infinite parallel plates separated by a gap width, \(h\). The upper plate \((y=h)\) moves to the right (positive \(x\)-direction) with a velocity of magnitude, \(v_{u p}\); the lower plate \((y=0)\) moves to the left (negative \(x\)-direction) with a velocity of magnitude, \(v_{\text {low }}\). The net volumetric flow rate of fluid in the positive \(x\)-direction, per unit depth into the page, is:

\[\dot{V}=\frac{v_{\text {low }} h}{2} \text { in units of }\left(\mathrm{m}^{3} / \mathrm{s}\right) / \mathrm{m}\]

a. Sketch the flow situation.

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

c. Specify the boundary conditions.

d. Solve the equation for the velocity profile.

e. Use the velocity profile to determine the volumetric flow rate.

f. Find the speed ratio, \(v_{u p} / v_{\text {low }}\) required to match the observed net volumetric flow rate.