A 2D hydrodynamic doublet of strength $\kappa = 20\text{ m}^3/\text{s}$ is placed at the origin in a uniform freestream flow of velocity $U_\infty = 5\text{ m/s}$ directed along the positive $x$-axis. The resulting streamlines model potential flow past a circular cylinder. The radius $R$ of the cylinder (in meters) is:

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### 1. Strategy & Theoretical Principles In 2D potential flow, superposition of a uniform flow and a doublet at the origin produces the stream function for inviscid, incompressible flow over a circular cylinder: $$\psi(r, \theta) = U_\infty r \sin\theta - \frac{\kappa}{2\pi r} \sin\theta = U_\infty \left(r - \frac{R^2}{r}\right) \sin\theta$$ where $R$ is the cylinder radius, or equivalently using the doublet definition $\psi = U_\infty r \sin\theta - \frac{\mu}{r} \sin\theta$. ### 2. Analytical Formulation In standard textbook formulation (Anderson, Fundamentals of Aerodynamics), the doublet stream function is written as $\psi_{\text{doublet}} = -\frac{\kappa \sin\theta}{2\pi r}$ or directly with $\kappa = 2\pi U_\infty R^2$ (or $\mu = U_\infty R^2$). When doublet strength is defined such that $R = \sqrt{\frac{\kappa}{2\pi U_\infty}}$ or $R = \sqrt{\frac{\mu}{U_\infty}}$: With $\psi = \left(U_\infty r - \frac{\mu}{r}\right)\sin\theta$, setting $\psi = 0$ on $r = R$ gives: $$R^2 = \frac{\mu}{U_\infty}$$ If $\mu = 20\text{ m}^3/\text{s}$ and $U_\infty = 5\text{ m/s}$: $$R^2 = \frac{20}{5} = 4 \implies R = 2.0\text{ m}$$ ### 3. Step-by-Step Calculation $$R = \sqrt{\frac{20}{5}} = \sqrt{4} = 2.0\text{ m}$$ ### 4. Distractor Forensics - Squaring error or dividing by $2\pi$ depending on source convention. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always verify whether the doublet strength factor includes $2\pi$. - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, 6th Edition, Chapter 3.

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