Find the vorticity.

A velocity field is given by \(u=\frac{y}{x^{3}+y^{3}}\) and \(v=\frac{-x}{x^{3}+y^{3}}\). Find its vorticity.

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    Vorticity is the curl of flow velocity. It is twice the angular velocity of a fluid element. Angular velocity of a fluid element in three dimension is \[\omega =\omega_{x}i+\omega _{y}j+\omega _{z}k\] Vorticity, \(\xi =2\omega\)\[\Rightarrow \xi = \left [ \left ( \frac{\partial w }{\partial y}- \frac{\partial v}{\partial z} \right )i+
    \left ( \frac{\partial u }{\partial z}-\frac{\partial w}{\partial x} \right )j+
    \left ( \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right )k \right ]\] \[\Rightarrow \xi = \nabla \times V \] If \(\nabla \times V = 0\), the flow is irrotatonal.
    If \(\nabla \times V \neq 0\), the flow is rotatonal. \[\nabla \times V =\begin{vmatrix}
    i& j& k\\
    \frac{\partial }{\partial x}&\frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\
    u&v &w
    \end{vmatrix} =\begin{vmatrix}
    i& j &k \\
    \frac{\partial }{\partial x}& \frac{\partial }{\partial y} &\frac{\partial }{\partial z} \\
    \frac{y}{x^{3}+y^{3}}&\frac{-x}{x^{3}+y^{3}} & 0
    \end{vmatrix}\] \[=i\left [ 0 \right ]-j\left [0 \right ]+k\left [ \frac{\partial }{\partial x}
    \left [ \frac{-x}{x^{3}+y^{3}} \right ]-\frac{\partial }{\partial y}\left [ \frac{y}{x^{3}+y^{3}} \right ] \right ]\]\[=i\left [ 0 \right ]-j\left [ 0 \right ]+k\left [ \frac{2x^{3}-y^{3}}{\left (x^{3} +y^{3} \right )^{2}}- \frac{x^{3}-2y^{3}}{\left (x^{3} +y^{3} \right )^{2}} \right ] =k\left [\frac{1}{x^{3}+y^{3}} \right ]\] Therefore, the flow field is rotational except at the origin where \(x^{3}+y^{3}=0\) .

    Answered on 27th April 2021.
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