In the Falkner-Skan boundary layer family where external potential velocity varies as $U_e(x) = C x^m$, boundary layer separation occurs when the pressure gradient parameter $\beta = \frac{2m}{m+1}$ reaches approximately:

(A) $\beta \approx -0.1988$ (Adverse pressure gradient condition)

(B) $\beta = 0$ (Zero pressure gradient Blasius flow)

(C) $\beta = +1.0$ (Favorable stagnation point flow)

(D) $\beta = +2.0$ (Extreme accelerating flow)

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Falkner-Skan Equation:** $f''' + f f'' + \beta [1 - (f')^2] = 0$. 2. **Stage 2: Boundary Conditions:** $f(0) = 0$, $f'(0) = 0$, $\lim_{\eta\to\infty} f'(\eta) = 1$. 3. **Stage 3: Wall Shear Stress:** $\tau_w = \mu \left(\frac{\partial u}{\partial y}\right)_{y=0} \propto f''(0)$. 4. **Stage 4: Separation Threshold:** As $\beta$ becomes increasingly negative (adverse pressure gradient), the velocity profile develops an inflection point and wall slope decreases. At $\beta = -0.1988$, $f''(0) = 0$, indicating boundary layer detachment. 5. **Stage 5: Official Reference:** Hermann Schlichting, *Boundary-Layer Theory*, Chapter 9.

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