$1.$ Figure $1$ shows two infinite parallel plates separated by a distance $2$h where the top plate is moving at a velocity $2$U

to the right and the bottom plate is moving to the left at a speed U$.$ Noting that the coordinate system starts at the

midpoint between the plates find the following:

$($a$)$ Use continuity and boundary conditions to show v$=0$ everywhere.

$($b$)$ Derive the velocity profile u$($y$)$ using Navier$-$Stokes Equation.

$($c$)$ At what position between the two plates is u$($y$)=0?$

$2.$ Figure $2$ shows an infinite $2$ D plate which is tilted at an angle theta$=30$ with a thin sheet of water running down it which

has a thickness of h$.$

$($a$)$ Utilize the continuity equation and boundary conditions to show v$=0$ everywhere.

$($b$)$ Derive the velocity profile u$($y$)$ using the Navier$-$Stokes Equation.

$($c$)$ Using this velocity profile, determine if there is anywhere that the flow is irrotational.