$(30$ pts$)$ 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 twodimensional in the $x-y$ plane. The velocity field is given by

vec$(V)=(u,v)=(V\frac{y}{h})vec(i)+0$vec$(j).$

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

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