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Vorticity

Vortex is a region in a fluid in which flow revolves around an axis line. Tornadoes form during thunderstorms when warm, humid air collides with colder air to form a vortex. A whirlpool is a body of swirling water produced by the meeting of opposite currents. Whirlpool having a downdraft is termed as vortex. 
Vorticity defines the dynamics of vortices, by a vector that describes the local rotary motion at a point in the fluid. In a velocity field, the curl of velocity is equal to vorticity\[\xi  = \nabla  \times V\] i) If \(\nabla  \times V = 0\) at every point in a flow, it is a irrotational flow.
ii) If \(\nabla  \times V \ne 0\) at every point in a flow, it is a rotational flow. 
Example: Consider a velocity field where x and y components of velocity are \(u = 4y/\left( {{x^2} + {y^2}} \right)\) and \(v =  - 4x/\left( {{x^2} + {y^2}} \right)\).Calculate the vorticity.
Solution: Vorticity is given as \[\xi  = \nabla  \times V\]
\( \Rightarrow \xi  = \left| {\begin{array}{*{20}{c}}i&j&k\\{\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\u&v&w\end{array}} \right|\)

\( \Rightarrow \xi \)\( = \left| {\begin{array}{*{20}{c}}i&j&k\\{\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}}\\{\frac{{4y}}{{{x^2} + {y^2}}}}&{\frac{{ - 4x}}{{{x^2} + {y^2}}}}&0\end{array}} \right|\)

\( \Rightarrow \xi \)\( = i\left[ {0 - 0} \right] - j\left[ {0 - 0} \right] + k\left[ {\frac{{\left( {{x^2} + {y^2}} \right)\left( { - 4} \right) + 4x\left( {2x} \right)}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}} - \frac{{\left( {{x^2} + {y^2}} \right)4 - 4y\left( {2y} \right)}}{{{{\left( {{x^2} + {y^2}} \right)}^2}}}} \right]\)
\( \Rightarrow \xi \)\( = 0i + 0j + 0k = 0\)
\( \Rightarrow \xi \)\( = 0\)
Therefore, it is a irrotational flow except at origin where \({x^2} + {y^2} = 0\).

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