The velocity profile in laminar flow in a pipe is a parabolic function of \(r\) and can be represented as

\[v_{z}=v_{\mathrm{c}}\left[1-(r / R)^{2}\right]\]

The parameter \(v_{\mathrm{c}}\) appearing in this equation can be interpreted as the center-line velocity.

(a) Using the Hagen-Poiseuille equation, how is the center-line velocity related to other parameters of the problem.

(b) Calculate the following for the above velocity profile: \(\langle vangle,\left\langle v^{2}\rightangle\), and \(\left\langle v^{3}\rightangle\).

(c) What error would be introduced if the rate of momentum flow were based on \(\langle vangle^{2}\) instead of \(\left\langle v^{2}\rightangle\) ?

(d) What error would be introduced if the rate of kinetic energy flow were based on \(\langle vangle^{3}\) instead of \(\left\langle v^{3}\rightangle\) ?

(e) What is the value for the momentum correction factor for this velocity profile? (Answer: 4/3.)

(f) What is the value for the kinetic-energy correction factor for this velocity profile? (Answer: 2.)