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.
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.
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.
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 an irrotational flow except at origin where \({x^2} + {y^2} = 0\).
