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.

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    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 \(t=3\),
    \[a=\left ( 8\times\times 2\times 5 \right )\hat{j}+\left [ \left ( 5\times 2\times 3 \right ) \left ( 5\times 3 \right )\hat{i}+
    \left ( 8\times 3\times 5 \right )\hat{j}+5\hat{k}\right ]+\]\[\left [ \left ( 8\times 3\times 2\times 5 \right )\left ( 5\times 2 \right )\hat{i} \right ]+
    \left [ \left ( 2\times 5 \right )\left ( 8\times 3\times 2 \right )\hat{j}+2\hat{k} \right ]\]\[\Rightarrow a=80\hat{j}+450\hat{i}+120\hat{j}+5\hat{k}+2400\hat{i}+480\hat{j}+2\hat{k}\]\[\Rightarrow a = 2850\hat{i}+680\hat{j}+7\hat{k}\]

    Answered on 19th May 2021.
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