Consider a flow of liquid in a horizontal circular tube which is mobile. The flow in the tube is assumed to be fully developed and pressures at two ends are $P_o$ and $P_L.$ The cylindrical tube has radius of $R$ and length of $L.$ The tube itself is moving along the z$-$direction with a velocity of $V_z(R)=-{}_{rz}($ R $),$ where $>0.$

a$)$ Obtain the radial velocity profile in the circular tube.

b$)$ Find the average velocity

Hint $($Answer$)$:

$U_z=\frac{P}{4L}{r}^2+\frac{P}{2L}(R-\frac{R^2}{2}),U_{\text{a}\text{v}\text{g}}=[\frac{P}{8L}{R}^2+\frac{P}{2L}(R-\frac{R^2}{2})]$