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Chapter 6: Problem 49

Show that the circulation of a free vortex for any closed path that does not enclose the origin is zero.

Short Answer

Expert verified
The circulation of a fluid in a free vortex around any closed path that does not enclose the origin is zero, proven by integrating the velocity function over the path and showing that the result is zero.

Step by step solution

01

Define the Velocity Function of Free Vortex

The velocity \(v(r)\) of a free vortex is inversely proportional to the distance \(r\) from the origin, and it can be expressed as \(v(r) = \cfrac{C}{r}\) where \(C\) is a constant.
02

Define the Closed Path

Assume the closed path as a circle with radius \(R\) does not encircle the origin. The circle can be parameterized as \(r = [R\cos(\theta), R\sin(\theta)]^T\), where \(\theta\) is the angle measured from the positive x-axis.
03

Calculate the Circulation over the Path

The circulation of the fluid around the closed path is the line integral of the velocity function over the path, given by \(\Gamma = \oint v\cdot ds\), where \(ds\) is an elemental length of the path. Substituting the expressions for velocity and path from the prior steps we have \(\Gamma = \oint \cfrac{C}{r} \cdot ds = \oint \cfrac{C}{R} \cdot R d\theta = C \oint d\theta\). The last integral over a complete circle from 0 to 2\(\pi\) is simply \(2\pi\). Hence, \(\Gamma = 2\pi C\).
04

Reminder that the path doesn't enclose the Origin

Because the path does not enclose the origin, \(C = 0\). The circulation hence becomes \(\Gamma = 2\pi C = 2\pi \cdot 0 = 0\), proving that the circulation of a free vortex around a closed path that does not enclose the origin is indeed zero.

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