Consider a velocity field  $\overrightarrow{V}=3z\hat{i}+0\hat{j}+Cx\hat{k}$  , where C is a constant. If the flow is irrotational, the value of C is _________ (rounded off to 1 decimal place).

The correct answer is 3.

To find the value of the constant C for which the given velocity field is irrotational, we start by identifying the components of the velocity field. The velocity vector is given by:
$\overrightarrow{V}=3z\hat{i}+0\hat{j}+Cx\hat{k}$

From this expression, we can extract the individual velocity components in the x, y, and z directions:
u = 3z
v = 0
w = Cx

For a fluid flow to be irrotational, the curl of the velocity field must be zero everywhere:
$\nabla \times \overrightarrow{V}=\overrightarrow{0}$

The curl of a three-dimensional velocity field is defined as:
$\nabla \times \overrightarrow{V}=\left(\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}\right)\hat{i}+\left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\right)\hat{j}+\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right)\hat{k}$

Let us compute each component of the curl:

1. The x-component:
$\frac{\partial w}{\partial y}-\frac{\partial v}{\partial z}=\frac{\partial }{\partial y}\left(Cx\right)-\frac{\partial }{\partial z}\left(0\right)=0-0=0$

2. The z-component:
$\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}=\frac{\partial }{\partial x}\left(0\right)-\frac{\partial }{\partial y}\left(3z\right)=0-0=0$

3. The y-component:
$\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}=\frac{\partial }{\partial z}\left(3z\right)-\frac{\partial }{\partial x}\left(Cx\right)=3-C$

For the entire curl to be equal to zero, each individual component must vanish. Setting the y-component to zero gives:
3 - C = 0
C = 3

Thus, the value of the constant C for an irrotational flow field is 3.