Discussion
For a laminar, incompressible, and fully-developed flow through a circular pipe, the ratio of the maximum velocity to the average velocity of the flow is
Answer: B
The ratio is 2.
Fully developed laminar pipe flow (Hagen-Poiseuille flow) has a parabolic profile:
\[ u(r) = u_{max}\left(1-\frac{r^{2}}{R^{2}}\right) \]
Averaging over the circular cross-section:
\[ \begin{aligned} \bar{u} &= \frac{1}{\pi R^{2}}\int_{0}^{R}u_{max}\left(1-\frac{r^{2}}{R^{2}}\right)2\pi r\,dr \\ &= \frac{u_{max}}{2} \end{aligned} \]
So \(u_{max}/\bar{u} = 2\). The value 1.5 belongs to laminar flow between parallel plates, where the profile is parabolic across a slot rather than a circle. 3 and 4 have no basis for laminar flow.
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