In the potential flow past a stationary circular cylinder of radius $R$ in a uniform stream $U_\infty$, the surface pressure distribution is $C_p(\theta) = 1 - 4 \sin^2\theta$, where $\theta$ is measured from the front stagnation point. The surface angle $\theta$ at which the pressure gradient transitions from favorable $\left(\frac{dp}{d\theta} < 0\right)$ to adverse $\left(\frac{dp}{d\theta} > 0\right)$ is:

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### 1. Strategy & Theoretical Principles The static pressure on the cylinder surface is $p(\theta) = p_\infty + \frac{1}{2} \rho U_\infty^2 C_p(\theta)$. The pressure gradient along the circumference is: $$\frac{dp}{d\theta} \propto \frac{d C_p}{d\theta}$$ ### 2. Analytical Formulation & Calculation Given $C_p(\theta) = 1 - 4 \sin^2\theta$: $$\frac{d C_p}{d\theta} = -4 (2 \sin\theta \cos\theta) = -8 \sin\theta \cos\theta = -4 \sin(2\theta)$$ - For $0 < \theta < 90^\circ$: $\sin(2\theta) > 0 \implies \frac{d C_p}{d\theta} < 0$ (favorable pressure gradient, flow accelerates towards maximum velocity $2 U_\infty$). - At $\theta = 90^\circ$: $\sin(180^\circ) = 0 \implies \frac{d C_p}{d\theta} = 0$ (minimum pressure point $C_p = -3$). - For $90^\circ < \theta < 180^\circ$: $\sin(2\theta) < 0 \implies \frac{d C_p}{d\theta} > 0$ (adverse pressure gradient). Therefore, the transition occurs at exactly $\theta = 90^\circ$. ### 3. Distractor Forensics - Zero-pressure crossing ($C_p = 0$) occurs at $\sin\theta = 0.5 \implies \theta = 30^\circ$, but the gradient transition occurs at $\theta = 90^\circ$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Adverse pressure gradient begins at the shoulder ($\theta = 90^\circ$), causing the real viscous boundary layer to separate shortly thereafter (typically around $\theta \approx 82^\circ$). - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 3.

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