$2.(60$ pts$)$ Consider steady, incompressible, parallel, laminar flow of a viscous fluid falling between two infinite vertical walls $($see sketch$).$ The distance between the walls is $\backslash ($ h $\backslash ),$ and gravity acts in the negative $\backslash ($ z $\backslash )$ direction $($downward in the figure$).$ There is no applied $($forced$)$ pressure driving the flow $-$ the fluid falls by gravity alone. The pressure is constant everywhere in the flow field.

$($a$)$ Calculate the velocity field. For full credit, follow the step$-$by$-$step procedure for solving the Navier$-$Stokes equation, as discussed in class, showing all your steps and all your work.

$($b$)$ Sketch the velocity profile using appropriate nondimensionalized variables.

$($c$)$ Generate an expression for volume flow rate per unit depth through a cross$-$sectional area at some $\backslash ($ z $\backslash )$ location.