Question

In potential flow, a uniform stream of strength $U$ flows along x-axis. Line sources of strength $\pi/2, -\pi/3, \pi/4, -\pi/5$ are placed at $x=0,1,2,3$ respectively. Find strength of an additional line source at $x=4$ such that a closed streamline encircles all five sources. (round off to two decimal places)

Hint:

In 2D potential flow, a finite closed streamline exists only if algebraic sum of source/sink strengths equals zero.

Answer 1

Step 1: Condition for closed streamline.
In source/sink flow superimposed with uniform stream, a closed streamline exists if net source strength = 0.

Step 2: Sum of given sources.
\[ q_{total} = \frac{\pi}{2} - \frac{\pi}{3} + \frac{\pi}{4} - \frac{\pi}{5} \]

Step 3: Simplify.
Common denominator $60$: \[ = \frac{30\pi - 20\pi + 15\pi - 12\pi}{60} = \frac{13\pi}{60} \]

Step 4: Required additional source.
\[ q_{5} = - \frac{13\pi}{60} \approx -0.681 \] Wait — but statement asks positive magnitude such that streamline closes. If sign convention preserved, answer = $-0.68$. Magnitude = $0.68$. \[ \boxed{-0.68 \;\; (\text{sink})} \]