# Find the angular velocity about the $$z$$-axis, acceleration and the vorticity at point $$(2,3,5)$$ at $$t = 3$$ seconds.

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.

Asked on 5th May 2021 in

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 $$\omega = \omega _{x}\hat{i}+\omega _{y}\hat{j}+\omega _{z}\hat{k}$$
$\omega =\frac{1}{2}\left [ \left ( \frac{\partial w }{\partial y}-\frac{\partial v}{\partial z} \right )\hat{i}+ \left ( \frac{\partial u}{\partial z}-\frac{\partial w}{\partial x} \right )\hat{j}+ \left ( \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right )\hat{k} \right ]$
$\omega_{z}=\frac{1}{2}\left ( \frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \right )=\frac{1}{2}\left ( \frac{\partial }{\partial x}\left ( 8txz \right )-\frac{\partial }{\partial y}\left ( 5xy \right ) \right )=\frac{1}{2}\left ( 8tz-5x \right )$
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}$