$1.(7$ points$)$ In the lectures, we derived an expression for the velocity field for viscous flow between two parallel plates for two scenarios: With stationary plates and a pressure gradient $\backslash (\backslash$left$(\backslash$frac$\{\backslash$partial p$\}\{\backslash$partial x$\}\backslash$right$)\backslash )$ in the direction of flow, and with moving plates but no pressure gradient. Here we combine these two scenarios: Suppose now that we again have two plates that are a distance of $\backslash (2$ h $\backslash )$ apart, with the top plate is moving with a velocity of $\backslash ($ U$\_\{$U$\}\backslash ),$ and the bottom plate is moving with a velocity $\backslash ($ U$\_\{$L$\}\backslash ),$ and with a nonzero pressure gradient $\backslash (\backslash$frac$\{\backslash$partial p$\}\{\backslash$partial x$\}\backslash )($which we can assume to be constant$).$

$($a$)$ Determine the velocity profile of this flow.

$($b$)$ Determine the volumetric flow rate between the plates, per unit width.