In laminar boundary layer theory, the Falkner-Skan family of similarity solutions models flow over a wedge of total opening angle $\pi \beta$ with external velocity distribution $U_e(x) = C x^m$. The relationship between the velocity exponent $m$ and the Falkner-Skan pressure gradient parameter $\beta$ is given by:

(A) $\beta = \frac{2m}{m + 1}$

(B) $\beta = \frac{m}{m + 1}$

(C) $\beta = \frac{2m}{m - 1}$

(D) $\beta = \frac{m + 1}{2m}$

Answer Key:

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### 1. Strategy & Theoretical Principles In the Falkner-Skan transformation, the stream function is defined as $\psi(x, y) = \sqrt{\frac{2 \nu x U_e(x)}{m + 1}} f(\eta)$ with similarity coordinate $\eta = y \sqrt{\frac{(m + 1) U_e(x)}{2 \nu x}}$. Substituting this transformation into the 2D steady boundary layer momentum equation yields the famous ordinary differential equation: $$f''' + f f'' + \beta \left(1 - (f')^2\right) = 0$$ where $f' = u / U_e(\eta)$. ### 2. Analytical Formulation Equating coefficients from the momentum equation: $$\beta = \frac{2x}{U_e} \frac{dU_e}{dx} \frac{1}{m + 1}$$ For potential flow over a wedge, the outer inviscid velocity satisfies $U_e(x) = C x^m$, so: $$\frac{dU_e}{dx} = m C x^{m-1} = \frac{m U_e}{x}$$ $$\frac{x}{U_e} \frac{dU_e}{dx} = m$$ ### 3. Step-by-Step Calculation Substitute this power into the definition of $\beta$: $$\beta = \frac{2m}{m + 1}$$ - For flat plate Blasius flow ($m = 0$), $\beta = 0$. - For stagnation point flow (Hiemenz flow, $m = 1$), $\beta = 1$. - For separation boundary ($f''(0) = 0$), $\beta \approx -0.1988$. ### 4. Distractor Forensics - **Option A:** Correct standard Falkner-Skan parameter relation. - **Option B:** Missing the factor of 2 originating from the stream function scaling. - **Option C:** Incorrect sign in denominator. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Remember that the wedge half-angle is $\theta_w = \frac{\pi \beta}{2}$, so the full wedge angle is $\pi \beta$. - **Standard Reference:** Schlichting, H. & Gersten, K., *Boundary-Layer Theory*, 9th Edition, Springer, Chapter 8.

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