Discussion
The displacement thickness (in mm) when \(\delta = 6\) mm, is
Answer: B
The displacement thickness is 2 mm.
With \(\eta = y/\delta\) and the profile \(u/U = 2\eta - \eta^{2}\):
\[ \begin{aligned} \delta^{*} &= \delta\int_{0}^{1}\left(1 - \frac{u}{U}\right)d\eta \\ &= \delta\int_{0}^{1}(1 - 2\eta + \eta^{2})\,d\eta \\ &= \delta\left(1 - 1 + \tfrac{1}{3}\right) = \frac{6}{3} = 2\ \text{mm} \end{aligned} \]
Displacement thickness measures the outward shift of the streamlines caused by the slowed fluid, so it must be positive and smaller than \(\delta\); the negative options are impossible. 2.25 mm would come from a different (cubic) profile.
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