A 2D source of strength $\Lambda = 10\text{ m}^2/\text{s}$ located at $(-1\text{ m}, 0)$ and a sink of equal strength $-\Lambda = -10\text{ m}^2/\text{s}$ located at $(+1\text{ m}, 0)$ are placed in an undisturbed quiescent fluid. The induced velocity $u$ along the $x$-axis at the origin $(0, 0)$ is:

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### 1. Strategy & Theoretical Principles For a 2D point source of strength $\Lambda$ located at $(x_0, y_0)$, the velocity induced at $(x, y)$ is purely radial: $$V_r = \frac{\Lambda}{2\pi r}$$ Along the $x$-axis ($y = 0$): - Source at $x = -1$: induces velocity directed in $+x$ direction at $x = 0$ with distance $r_1 = 1\text{ m}$: $$u_1 = +\frac{\Lambda}{2\pi r_1} = +\frac{10}{2\pi (1)} = +\frac{5}{\pi}$$ - Sink at $x = +1$: pulls fluid towards itself (in $+x$ direction) at $x = 0$ with distance $r_2 = 1\text{ m}$: $$u_2 = +\frac{\Lambda}{2\pi r_2} = +\frac{10}{2\pi (1)} = +\frac{5}{\pi}$$ ### 2. Analytical Formulation & Calculation Total velocity $u$ at the origin is the sum of both contributions: $$u = u_1 + u_2 = \frac{5}{\pi} + \frac{5}{\pi} = \frac{10}{\pi}$$ Evaluating numerically: $$u = \frac{10}{3.14159265} \approx 3.183\text{ m/s} \approx 3.18\text{ m/s}$$ ### 3. Distractor Forensics - Candidate believing source and sink cancel at the midpoint gets $u = 0$, forgetting that fluid flows *from* source *towards* sink, so both velocities point in the $+x$ direction. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Source and sink velocities reinforce each other between the singularities; they do NOT cancel. - **Standard Reference:** Anderson, J.D., *Fundamentals of Aerodynamics*, Chapter 3.

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