For ideal, steady, incompressible non-lifting potential flow over a circular cylinder of radius $R$ in a freestream velocity $V_\infty$, at what angular locations $\theta$ (measured from the forward stagnation point) does the surface pressure coefficient equal zero ($C_p = 0$)?

(A) $\theta = 30^\circ$ and $150^\circ$

(B) $\theta = 45^\circ$ and $135^\circ$

(C) $\theta = 60^\circ$ and $120^\circ$

(D) $\theta = 90^\circ$ and $270^\circ$

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Surface Tangential Velocity:** For flow over a stationary circular cylinder (doublet + uniform stream): $V_\theta = -2 V_\infty \sin\theta$. 2. **Stage 2: Pressure Coefficient Formulation:** From Bernoulli's theorem: * $C_p = 1 - \left(\frac{V_\theta}{V_\infty}\right)^2 = 1 - (-2\sin\theta)^2 = 1 - 4 \sin^2\theta$. 3. **Stage 3: Solving for $C_p = 0$:** * $1 - 4\sin^2\theta = 0 \implies \sin^2\theta = \frac{1}{4} \implies \sin\theta = \pm \frac{1}{2}$. * In the upper half ($0 \le \theta \le \pi$): $\theta = 30^\circ$ and $\theta = 180^\circ - 30^\circ = 150^\circ$. 4. **Stage 4: Distractor Forensics:** $\theta = 90^\circ$ corresponds to the minimum pressure suction peak ($C_p = -3$), not $C_p = 0$. 5. **Stage 5: Official Reference:** John D. Anderson Jr., *Fundamentals of Aerodynamics*, Section 3.13.

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