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 ${x^2} + {y^2} = 0$.