$(20)$ An airplane flying at a flight speed $V_f$ through air with density $$ requires a propulsive force $F$ and propulsive power $P.$ The airplane has a propeller with diameter $D_p.$ Assume the flow to be incompressible and inviscid, and neglect gravity effects.

a$)$ For the control volume a$-$b$-$c$-$d$,$ obtain an expression for $F.$

b$)$ Obtain an expression for the pressure change across the propeller $\frac{p_3-p_2}{}$ in terms of the flight and wake velocities $(V_f$&$V_w).$

c$)$ For control volume e$-$f$-$g$-$h$,$ obtain an expression for air velocity through the propeller $V_p$

d$)$ What is the expression for propulsive power $P?$ Solve fully and show working $($please dont use AI$)$