The velocity distribution for laminar flow in a long circular tube of radius \(R\) is given by the one-dimensional expression,

\[\vec{V}=u \hat{i}=u_{\max }\left[1-\left(\frac{r}{R}\right)^{2}\right] \hat{i}\]

For this profile obtain expressions for the volume flow rate and the momentum flux through a section normal to the pipe axis. Obtain an expression for the kinetic energy flux, \(\int\left(V^{2} / 2\right) ho \vec{V} \cdot d \vec{A}\), through a section normal to the pipe axis.