Consider fully developed Couette flow $-$ flow between two infinite parallel plates separated by distance h$,$ with the top plate moving and the bottom plate stationary as sketched. The flow is steady, incompressible, and two$-$dimensional in the x$-$y plane. The velocity field is given by vec$($V$)=($u$,$v$)=($Vy$/$h$)$ vec$($i$)+0$ vec$($j$).$

$($a$)$ Verify that this velocity field satisfies the incompressible continuity equation.

$($b$)$ Generate an expression for stream function psi along the vertical dashed line in the figure. For convenience, let psi $=0$ along the bottom wall of the channel. What is the value of psi along the top wall?

$($c$)$ Calculate the volume flow rate per unit width $($into the page$)$ from first principles $($integration of the velocity field$).$ Compare your result to that obtained directly from the stream function. Discuss.

Solve each part of this question and show all work.