$($a$)(20$pt$)$ Derive a flow model for the boundary layer flow, starting with the full $3$D Navier$-$Stokes equations and the following assumptions:

$-2$D flow in an $\backslash ($ x y $\backslash )$ plane. $\backslash ($ y$=0\backslash )$ is the wall with no$-$slip condition $($zero velocity$)$ at wall. The flow is in the $\backslash ($ x $\backslash )$ direction.

$-$ Wall$-$normal velocity $\backslash (($v$)\backslash )$ is small compared to the streamwise velocity $\backslash (($u$)\backslash ).$

$-$ Spatial gradients in the wall$-$normal direction $\backslash ((\backslash$partial $/\backslash$partial y$)\backslash )$ are much larger compared to the streamwise gradients $\backslash ((\backslash$partial $/\backslash$partial x$)\backslash ).$

$-$ Write the momentum and energy equations in an ODE $($ordinary differential equation$)$ form, commenting on the continuity equation.