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.

Asked on 15th April 2021 in
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$$ .