Question $2[20$ marks$]$

Consider steady, incompressible, laminar flow of a viscous fluid falling between two infinite vertical walls as shown in Figure Q$2.$ The distance between the walls is $\backslash ($ h $\backslash ),$ and gravity acts in the negative $\backslash ($ z $\backslash )$ direction $($downward in the figure$).$ The fluid falls by gravity alone. The fluid pressure is constant everywhere in the flow field. The fluid density is $\backslash (\backslash$rho $\backslash )$ and the fluid dynamic viscosity is $\backslash (\backslash$mu $\backslash ).$ Assume the $\backslash ($ x $\backslash )$ and $\backslash ($ y $\backslash )$ components of fluid velocity are zero, i$.$e$.\backslash ($ u$=0\backslash )$ and $\backslash ($ v$=0\backslash ).$ The component of fluid velocity in the $\backslash ($ z $\backslash )$ direction $($i$.$e$.\backslash ($ w $\backslash ))$ is only a function of $\backslash ($ x $\backslash ).$

$($a$)$ Derive the equation that formulates the fluid velocity profile, i$.$e$.\backslash ($ w $\backslash )$ as a function of $\backslash ($ x $\backslash ),$ by using $\backslash (\backslash$rho$,\backslash$mu$,$ h $\backslash )$ and $\backslash ($ g $\backslash )($gravitational acceleration$)$ as input parameters.

$($b$)$ Graphically sketch the fluid velocity profile in a diagram.

$($c$)$ Derive the equation that formulates the average fluid velocity by using $\backslash (\backslash$rho$,\backslash$mu$,$ h $\backslash )$ and $\backslash ($ g $\backslash )$ as input parameters.

Figure Q$2$

Please Focus on Part $($c$)$ on getting the average velocity