Question:medium

Consider a flow with the following velocity field
\[ \vec{V} = (x+y)\hat{i} + (y+z)\hat{j} + (z+x)\hat{k} \]
where \(\hat{i}\), \(\hat{j}\) and \(\hat{k}\) are the unit vectors in the x, y and z directions, respectively. Which one of the following is CORRECT?

Show Hint

Compute the divergence of the field to check compressibility and the curl to check rotationality; neither comes out to be zero for this field.
Updated On: Aug 10, 2026
  • The flow is incompressible and irrotational.
  • The flow is incompressible and rotational.
  • The flow is compressible and rotational.
  • The flow is compressible and irrotational.
Show Solution

The Correct Option is C

Solution and Explanation

The velocity gradient tensor is $\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{pmatrix}$. The divergence is the trace, $1+1+1=3$, nonzero, so the flow is compressible. The vorticity vector from the antisymmetric part is $(-1,-1,-1)$, nonzero, so the flow is rotational.\[ \boxed{\text{Compressible and rotational}} \]
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