Question

Consider the Blasius solution for the incompressible laminar flat–plate boundary layer. Among the options, select the correct relation for the development of the momentum thickness \(\theta\) with distance \(x\) from the leading edge along the plate.

• A. \(\theta \propto x^{2/3}\)
• B. \(\theta \propto x^{1/2}\)
• C. \(\theta \propto x^{1/7}\)
• D. \(\theta \propto x^{-2/3}\)

Hint:

Always remember: for laminar flat–plate, displacement and momentum thickness scale with \(x^{1/2}\). For turbulent boundary layers, the 1/7th power law often applies.

Answer 1

The Correct Option is B

Step 1: Recall boundary layer theory (Blasius).
For a laminar flat–plate boundary layer, the thickness scales as \[ \delta \sim \sqrt{\frac{\nu x}{U_\infty}} \;\;\Rightarrow\;\; \delta \propto x^{1/2}. \]

Step 2: Momentum thickness.
The momentum thickness is defined as \[ \theta(x) = \int_0^\infty \frac{u}{U_\infty}\left(1-\frac{u}{U_\infty}\right)dy. \] From the Blasius similarity solution, one obtains \[ \theta(x) = \frac{0.664}{\sqrt{Re_x}}\, x \] where \(Re_x = \frac{U_\infty x}{\nu}\).

Step 3: Simplify scaling.
\[ \theta(x) \sim \frac{x}{\sqrt{x}} \sim x^{1/2}. \]

Final Answer:
\[ \boxed{\theta \propto x^{1/2}} \]