All Problems Hydrodynamics
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.
1.334. (a) \(Q=1 / 2 \pi v_{0} R^{2}\) (b) \(T=1 /_{6} \pi l R^{2} \rho v_{0}^{2}\) (c) \(F_{f r}=\) \(=4 \pi \eta l v_{0} ;\) (d) \(\overline{\Delta p}=4 \eta l v_{0} / R^{2}\)