Problem 1.334

A tube of length $l$ and radius $R$ carries a steady flow of fluid whose density is $\rho$ and viscosity $\eta .$ The fluid flow velocity depends on the distance $r$ from the axis of the tube as $v=v_{0}\left(1-r^{2} / R^{2}\right)$. Find: (a) the volume of the fluid flowing across the section of the tube per unit time; (b) the kinetic energy of the fluid within the tube's volume; (c) the friction force exerted on the tube by the fluid; (d) the pressure difference at the ends of the tube.