GATE 2023 AE – Question 57
Uniform flow U is superposed with line sources $\pi/2,-\pi/3,\pi/4,-\pi/5$ at x=0,1,2,3. The additional source strength at x=4 for a closed streamline enclosing all five (two decimal places) is _____.
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Correct answer: -0.69 to -0.67
Explanation
For a flow to have a **closed streamline** (a closed body) around a set of sources and sinks in a uniform stream, the total source strength must be **zero**. Otherwise there would be a net outflow and the dividing streamline would not close.
**Sum of the given strengths** (per unit length):
$$\frac\pi2-\frac\pi3+\frac\pi4-\frac\pi5=\pi\left(\frac12-\frac13+\frac14-\frac15\right)=\pi\left(\frac{30-20+15-12}{60}\right)=\frac{13\pi}{60}=0.681 .$$
**Source needed at $x=4$:** the strength that cancels the sum:
$$Q_5=-\frac{13\pi}{60}=\mathbf{-0.68}.$$
(A negative strength is a sink.)