Water flows vertically upward in a circular cross-sectional pipe. At section (1), the velocity profile over the cross-sectional area is uniform. At section (2), the velocity profile is

\[ \mathbf{V}=w_{c}\left(\frac{R-r}{R}\right)^{1 / 7} \hat{\mathbf{k}} \]

where \(\mathbf{V}=\) local velocity vector, \(w_{c}=\) centerline velocity in the axial direction, \(R=\) pipe inside radius, and \(r=\) radius from pipe axis. Develop an expression for the loss in available energy between sections (1) and (2).