Two horizontal, infinite, parallel plates are spaced a distance $\backslash ($ b $\backslash )$ apart. A viscous liquid is contained between the plates. The bottom plate is fixed, and the upper plate moves parallel to the bottom plate with a velocity $\backslash ($ U $\backslash ).$ Because of the no$-$slip boundary condition $($see the Video$),$ the liquid motion is caused by the liquid being dragged along by the moving boundary. There is no pressure gradient in the direction of flow. Note that this is a so$-$called simple Couette flow discussed in Section $6\mathrm{.}9\mathrm{.}2.$

$($a$)$ Start with the Navier$-$Stokes equations and determine the velocity distribution between the plates.

$($b$)$ Determine an expression for the flowrate passing between the plates $($for a unit width$).$ Express your answer in terms of $\backslash ($ y$,$ b $\backslash )$ and $\backslash ($ U $\backslash ).$ A viscous fluid $($specific weight $\backslash (=80\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$ft$\}^\{3\}$ ; $\backslash )$ viscosity $\backslash (=0\mathrm{.}03\backslash$mathrm$\{$lb$\}\backslash$cdot $\backslash$mathrm$\{$s$\}/\backslash$mathrm$\{$ft$\}^\{2\}\backslash ))$ is contained between two infinite, horizontal parallel plates as shown in the figure below. The fluid moves between the plates under the action of a pressure gradient, and the upper plate moves with a velocity $\backslash ($ U $\backslash )$ while the bottom plate is fixed. A $\backslash ($ U $\backslash )-$tube manometer connected between two points along the bottom indicates a differential reading of $0\mathrm{.}1$ in $.$ If the upper plate moves with a velocity of $\backslash (0\mathrm{.}03\backslash$mathrm$\{$ft$\}/\backslash$mathrm$\{$s$\}\backslash ),$ at what distance from the bottom plate does the maximum velocity in the gap between the two plates occur? Assume laminar flow.