Question 8.7: In the laminar flow of a fluid in a circular pipe, the veloc...
In the laminar flow of a fluid in a circular pipe, the velocity profile is exactly a parabola. The rate of discharge is then represented by volume of a paraboloid. Prove that for this case the ratio of the maximum velocity to mean velocity is 2.
See Fig. 8.17. For a paraboloid,
v_{z}=v_{z_{\max }}\left[1-\left(\frac{r}{R}\right)^{2}\right]
Q=\int v_{z} d A=\int_{0}^{R} v_{z_{\max }}\left[1-\left(\frac{r}{R}\right)^{2}\right](2 \pi r d r)
=2 \pi v_{z_{\max }}\left[\frac{r^{2}}{2}-\frac{r^{4}}{4 R^{2}}\right]_{0}^{R}=2 \pi v_{z_{\max }}\left[\frac{R^{2}}{2}-\frac{R^{2}}{4}\right]
=v_{z_{\max }}\left(\frac{\pi R^{2}}{2}\right)
v_{z_{\text {mean }}}=\frac{Q}{A}=\frac{v_{z_{\max }}\left(\pi R^{2} / 2\right)}{\left(\pi R^{2}\right)}=\frac{v_{z_{\max }}}{2}
Thus, \frac{v_{z_{\max }}}{v_{z_{\text {mean }}}}=2
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