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Vortex flow

In a vortex flow streamlines are concentric circles about a given point.The velocity along any given streamline is constant and varies from one streamline to another inversely with distance from the common centre.The velocity components in radial and tangential directions are\[\begin{array}{l}{V_r} = 0\,{\rm{and}}\,\,{V_\theta } = \frac{{{\rm{constant}}}}{r}\\{V_\theta } = \frac{{ - \tau }}{{2\pi r}}\end{array}\]\(\tau \) = strength of vortex flow.
Example: For a vortex flow,\(u = \frac{{4x}}{{{x^2} + {y^2}}}\,and\,v = \frac{{ - 4y}}{{{x^2} + {y^2}}}\),calculate(a) The time rate of change of the volume of a fluid element per unit volume.(b) The vorticity.
Solution: On changing the equation to polar co-ordinates \[\begin{array}{l}x = r\cos \theta \\y = r\sin \theta \\{V_r} = u\cos \theta  + v\sin \theta \\{V_\theta } =  - u\sin \theta  + v\cos \theta \end{array}\]\[\begin{array}{l}u = \frac{{4y}}{{({x^2} + {y^2})}} = \frac{{4r\sin \theta }}{{{r^2}}} = \frac{{4\sin \theta }}{r}\\v = \frac{{ - 4x}}{{({x^2} + {y^2})}} = \frac{{4r\cos \theta }}{{{r^2}}} = \frac{{ - 4\cos \theta }}{r}\\{V_r} = \frac{4}{r}\cos \theta \sin \theta  - \frac{4}{r}\cos \theta \sin \theta  = 0\\{V_\theta } = \frac{{ - 4}}{r}{\sin ^2}\theta  - \frac{4}{r}{\cos ^2}\theta  = \frac{{ - 4}}{r}\end{array}\]
Time rate of change of volume of a fluid element per unit volume is given as\[\begin{array}{l}\nabla .\mathop V\limits^ \to   = \frac{1}{r}\frac{{\partial r{V_r}}}{{\partial r}} + \frac{1}{r}\frac{{\partial {V_\theta }}}{{\partial \theta }}\\\nabla .\mathop V\limits^ \to   = \frac{1}{r}\frac{{\partial (0)}}{{\partial r}} + \frac{1}{r}\frac{{\partial \left( { - c/r} \right)}}{{\partial \theta }} = 0 + 0 = 0\end{array}\] (b) Vorticity = \[\begin{array}{l}\nabla  \times \mathop V\limits^ \to   = {e_z}\left[ {\frac{{\partial \left( { - c/r} \right)}}{{\partial r}} - \frac{c}{{{r^2}}} - \frac{1}{r}\frac{{\partial \left( 0 \right)}}{{\partial \theta }}} \right]\\ = {e_z}\left[ {\frac{c}{{{r^2}}} - \frac{c}{{{r^2}}} - 0} \right]\\ = 0\end{array}\]\(\nabla  \times \mathop V\limits^ \to   = 0\) except at origin, since it is singular at origin.

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