A Newtonian fluid with constant density $\backslash (\backslash$rho $\backslash )$ flows in a slot of width $\backslash ($ b $\backslash ).$ The slot is modeled as having infinite depth. The slot is at an angle of $\backslash (\backslash$theta $\backslash )$ to the horizontal, and gravity acts vertically downward with the gravitational acceleration of $\backslash ($ g $\backslash ).$ There is no driving pressure gradient in the flow direction. Construct a coordinate system rotated at an angle $\backslash (\backslash$theta $\backslash )$ to the horizontal, with $\backslash ($ x $\backslash )$ in the flow direction and $\backslash ($ y $\backslash )$ normal to the flow direction.

$($a$)$ If the viscosity $\backslash (\backslash$mu $\backslash )$ is a constant, find the steady state velocity profile.

$($b$)$ Find the difference in pressure between the top and bottom of the channel, measured normal to the channel walls.

$($c$)$ If the viscosity $\backslash (\backslash$mu $\backslash )$ varies with distance as $\backslash (\backslash$mu$($y$)=\backslash$mu$\_\{$o$\}+\backslash$mu$\_\{1\}($y $/$ b$)\backslash ),$ find the steady state velocity profile.