A Newtonian fluid $($density $,$ viscosity $)$ is steadily flowing down with a flow velocity $u_0$ through a DECLINED circular pipe. The diameter, the angle to the ground, the length and the frictional factor of the pipe are $D,,L$ and $f_F,$ respectively.

$($a$)$ Start from the momentum & energy balances, derive an analytical expression for the frictional pressure loss $(p_f)$ and the frictional heating loss $(F)$ in terms of $f_F.c.f.$

In addition to $($a$),$ please also solve $($b$)$ and $($c$).$