Question

A free vortex filament (oriented along Z–axis) of strength \(K = 5 \, \text{m}^2/\text{s}\) is placed at the origin. The circulation around the closed loop ABCDEFA is \(\underline{\hspace{1cm}}\). 

Hint:

Circulation due to a free vortex is non-zero only if the integration contour encloses the vortex core. If the contour does not enclose the origin, circulation is zero.

Answer 1

Correct Answer: 0

For a free vortex of strength \(K\), the tangential velocity field is:
\[ V_\theta = \frac{K}{2\pi r} \] and the circulation around any closed contour encircling the vortex once is:
\[ \Gamma = K \] Now we check whether the path ABCDEFA encloses the origin \((0,0)\).
The polygon passes through points A(1,0), B(2,0), C(2,-2), D(0,-2), E(0,-1), F(1,-1).
Clearly, the entire loop lies in the region \(x \ge 0\). The origin \((0,0)\) is not inside the contour.
Since the vortex is not enclosed by the loop:
\[ \Gamma = 0 \] Thus the circulation around ABCDEFA is \(0\).