$3.(40$ pts$)$ Consider steady, incompressible, laminar flow that is two$-$dimensional in the $\backslash ($ x$-$y $\backslash )$ plane. Ignore gravity effects and units. The $\backslash ($ x $\backslash )-$component of velocity is $\backslash ($ u$=-5$ x$+3\backslash ).$ Do not be concerned with units.

$($a$)$ Calculate the $\backslash ($ y $\backslash )-$component of velocity, $\backslash ($ v $\backslash ).$

$($b$)$ Regardless of your answer to Part $($a$),$ set $\backslash ($ v$=5$ y$+2\backslash ),$ and calculate the stream function $\backslash (\backslash$psi$($x$,$ y$)\backslash ).$

$($c$)$ Generate an expression for the pressure field associated with this velocity field, i$.$e$.,$ calculate $\backslash ($ P$($x$,$ y$)\backslash ).$ Is this a valid steady, two$-$dimensional, incompressible, laminar flow field? Explain. Hint: To calculate the pressure field, you will need to write out both the $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ components of the Navier$-$Stokes equation, reduce them as much as possible, and then integrate them to calculate $\backslash ($ P$($x$,$ y$)\backslash ),$ noting that the pressure derivatives are partial derivatives, so the integration must be handled carefully, as discussed in class.