A velocity field is given by $V= 5xyi+8txzj+xzk$. Find the angular velocity about the$z$-axis, acceleration and the vorticity at point $(2,3,5)$ at $t = 3$. Units of $x,y,z$ are in meters and time – $t$ is in seconds.

Velocity field,$V=u\hat{i}+v\hat{i}+w\hat{k}$
Here, $V=5xy\hat{i}+8txz\hat{j}+xz\hat{k}$
Angular velocity of a fluid particle in a fluid flow is $\omega = \omega _{x}\hat{i}+\omega _{y}\hat{j}+\omega _{z}\hat{k}$
$$\omega =\frac{1}{2}\left [ \left ( \frac{\partial w }{\partial y}-\frac{\partial v}{\partial z} \right )\hat{i}+ \left ( \frac{\partial u}{\partial z}-\frac{\partial w}{\partial x} \right )\hat{j}+ \left ( \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right )\hat{k} \right ]$$
Angular velocity about the z-axis
$$\omega_{z}=\frac{1}{2}\left ( \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right )=\frac{1}{2}\left ( \frac{\partial }{\partial x}\left ( 8txz \right )-\frac{\partial }{\partial y}\left ( 5xy \right ) \right )=\frac{1}{2}\left ( 8tz-5x \right )$$
At point $(2,3,5)$ and $t=3$, angular velocity is $$\omega _{z}=\frac{1}{2}\left ( 8tz-5x \right )=\frac{1}{2}\left ( \left ( 8\times 3\times 5 \right )-\left ( 5\times 2 \right ) \right )=\frac{1}{2}\left ( 120-10 \right )=55$$
Vorticity of fluid particle in a fluid flow is $\xi =2\omega$
$$\Rightarrow \xi = \left ( \frac{\partial w}{\partial y}-\frac{\partial v}{\partial z} \right )\hat{i} +\left ( \frac{\partial u}{\partial z}-\frac{\partial w}{\partial x} \right )\hat{j} +\left ( \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right )\hat{k}$$ $$\Rightarrow \xi = \left ( \frac{\partial }{\partial y}\left ( xz \right ) -\frac{\partial }{\partial z} \left ( 8txz \right )\right )\hat{i} +\left ( \frac{\partial }{\partial z}\left ( 5xy \right ) -\frac{\partial }{\partial x}\left ( xz \right ) \right )\hat{j}+\left ( \frac{\partial }{\partial x}\left ( 8txz \right ) -\frac{\partial }{\partial y}\left ( 5xy \right ) \right )\hat{k}$$ $$\Rightarrow \xi =\left ( 0-8tx \right )\hat{i}+\left ( 0-z \right )\hat{j}+\left ( 8tz -5x \right )\hat{k}$$
At point $(2,3,5)$ and at $t=3$,
$$\xi =\left ( -8\times 3\times 2 \right )\hat{i}+\left ( -5 \right )\hat{j} +\left ( \left ( 8\times 3\times 5 \right ) -\left ( 5\times 2 \right ) \right )\hat{k}$$ $$\Rightarrow \xi =-48\hat{i}-5\hat{j}+110\hat{k}$$
Acceleration of a fluid particle in a fluid flow is
$$a=\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x} +v\frac{\partial v}{\partial y}+w\frac{\partial v }{\partial z}$$ $$\Rightarrow a=\frac{\partial }{\partial t}\left ( 5xy\hat{i} +8txz\hat{j}+xz\hat{k}\right )+\left ( 5xy \right ) \frac{\partial }{\partial x}\left ( 5xy\hat{i}+ 8txz\hat{j}+xz\hat{k}\right )+$$ $$\left ( 8txz \right )\frac{\partial } {\partial y}\left ( 5xy\hat{i} +8txz\hat{j}+xz\hat{k}\right )+\left ( xz \right )\frac{\partial } {\partial z}\left ( 5xy\hat{i} +8txz\hat{j}+xz\hat{k}\right )$$ $$\Rightarrow a=8xz\hat{j}+\left [ \left ( 5xy \right )\left ( 5y\hat{i}+8tz\hat{j}+z\hat{k} \right ) \right ]+\left [ \left ( 8txz \right )\left ( 5x\hat{i} \right ) \right ]+\left ( xz \right )\left [ \left ( 8tx\hat{j}+x\hat{k} \right ) \right ]$$
At point $(2,3,5)$ and at $t=3$,
$$a=\left ( 8\times\times 2\times 5 \right )\hat{j}+\left [ \left ( 5\times 2\times 3 \right ) \left ( 5\times 3 \right )\hat{i}+ \left ( 8\times 3\times 5 \right )\hat{j}+5\hat{k}\right ]+$$ $$\left [ \left ( 8\times 3\times 2\times 5 \right )\left ( 5\times 2 \right )\hat{i} \right ]+ \left [ \left ( 2\times 5 \right )\left ( 8\times 3\times 2 \right )\hat{j}+2\hat{k} \right ]$$ $$\Rightarrow a=80\hat{j}+450\hat{i}+120\hat{j}+5\hat{k}+2400\hat{i}+480\hat{j}+2\hat{k}$$ $$\Rightarrow a = 2850\hat{i}+680\hat{j}+7\hat{k}$$