Question

Velocity vector of a flow field is given as
$\overrightarrow{V}=2xy\hat{i}-x^{2}z\hat{j}$. The vorticity vector at (1, 1, 1) is

• A. $4\hat{i}-\hat{j}$
• B. $4\hat{i}-\hat{k}$
• C. $\hat{i}-4\hat{j}$
• D. $\hat{i}-4\hat{k}$

Answer 1

The correct option is D $\hat{i}-4\hat{k}$

$Vorticityvector=\left|\begin{array}{ccc}i& j& k\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ 2xy& -x^{2}z& 0\end{array}\right|$

$=\hat{i}[0-\frac{\partial }{\partial z}(-x^{2}z\left)\right]-\hat{j}[0-\frac{\partial }{\partial z}(2xy\left)\right]+\hat{k}\left[\frac{\partial }{\partial x}\right(-x^{2}z)-\frac{\partial }{\partial y}(2xy\left)\right]$

$=\hat{i}\left(x^2\right)+0+\hat{k}(-2xz-2x)$

$\therefore$ At ( 1, 1, 1)

Vorticity vector = $\hat{i}-4\hat{k}$