Discussion
For an incompressible boundary layer flow over a flat plate shown in the figure, the momentum thickness is expressed as
Answer: C
The momentum thickness is \(\int_{0}^{\infty} \frac{u}{U_{\infty}}\left(1 - \frac{u}{U_{\infty}}\right) dy\).
Momentum thickness \(\theta\) is the thickness of a layer of free-stream fluid whose momentum flux equals the momentum flux lost because the boundary layer slows the fluid. The mass actually passing through a strip \(dy\) is \(\rho u\,dy\), and each unit of it has lost velocity \(U_{\infty} - u\), so
\[ \begin{aligned} \rho U_{\infty}^{2}\theta \\ = \int_{0}^{\infty} \rho u (U_{\infty} - u)\,dy \end{aligned} \]
which gives the expression above.
\(\int (1 - u/U_{\infty})\,dy\) is the displacement thickness, the mass-flux deficit. \(\int (1 - u^{2}/U_{\infty}^{2})\,dy\) is the energy-type form wrongly built from \(u^{2}\) alone, and \(\int u/U_{\infty}\,dy\) diverges, because \(u \to U_{\infty}\) outside the layer.
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