Discussion
Consider the Blasius solution for the incompressible laminar flat plate boundary layer. Among the following options, select the correct relation for the development of the momentum thickness \(\theta\) with distance \(x\) from the leading edge along the length of the plate.
Answer: B
The momentum thickness grows as \(\theta \propto x^{1/2}\).
The Blasius solution for the laminar boundary layer on a flat plate gives
\[ \begin{aligned} \theta &= \frac{0.664\,x}{\sqrt{Re_{x}}} \\ Re_{x} &= \frac{U_{\infty}x}{\nu} \end{aligned} \]
so \(\theta = 0.664\sqrt{\nu x / U_{\infty}}\), which is proportional to \(\sqrt{x}\). The displacement thickness and the boundary-layer thickness grow in the same way, as all laminar thicknesses do.
The other exponents do not belong to this solution. \(x^{1/7}\) recalls the one-seventh power velocity profile of a turbulent boundary layer, whose thickness grows roughly as \(x^{4/5}\). A negative exponent would make the layer thin downstream, the opposite of what happens, and \(x^{2/3}\) has no basis here.
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