$1.(15$ points$)$ Consider steady, laminar flow of a viscous incompressible fluid between two very long parallel plates with spacing $\backslash ($ b $\backslash ).$ Far from the channel entrance, the $\backslash ($ z $\backslash )-$ and $\backslash ($ y $\backslash )$ components of fluid velocity are zero. The upper wall is stationary. The flow is driven by the motion of the lower wall at constant velocity $\backslash ($ U $\backslash )$ and by a pressure gradient in the $\backslash ($ x $\backslash )$ direction $\backslash ($ d p $/$ d x $\backslash ).$ Effects of gravity can be neglected. NOTE: This is a combination of Couette and Poiseuille flows.

a$.(5$ points$)$ Draw velocity profile for this fluid flow, indicate the coordinates, distance $\backslash ($ b $\backslash ),$ and fluid flow direction between the moving and fixed plates.

b$.(10$ points$)$ Starting from equations for the fully developed and incompressible flow, i$.$e$.$ continuity and $\backslash ($ x $\backslash )-$momentum N$-$S equations, derive an expression for fluid velocity, $\backslash ($ u$($y$)\backslash ).$ Check lecture slides referring to these two types of flows. Clearly state all boundary conditions you will apply to solve the equations.