Question:
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?
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?
\[ \vec{V} = (x+y)\hat{i} + (y+z)\hat{j} + (z+x)\hat{k} \]
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: Jul 17, 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
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The Correct Option is C
Solution and Explanation
Step 1: Recall the tests for compressibility and rotationality.
A velocity field represents incompressible flow only if its divergence vanishes everywhere. It represents irrotational flow only if its curl vanishes everywhere.
Step 2: Write down the velocity components.
\(V_x = x+y\), \(V_y = y+z\), \(V_z = z+x\).
Step 3: Test for incompressibility.
\[ \nabla \cdot \vec{V} = 1+1+1 = 3 \]
Since this is nonzero, the flow is compressible.
Step 4: Test for irrotationality.
\[ \nabla \times \vec{V} = -\hat{i}-\hat{j}-\hat{k} \]
Nonzero, so the flow is rotational.
Step 5: Match with the options.
Only compressible and rotational is consistent with the divergence of 3 and the nonzero curl.
Step 6: Conclusion.
\[ \boxed{\text{The flow is compressible and rotational.}} \]
A velocity field represents incompressible flow only if its divergence vanishes everywhere. It represents irrotational flow only if its curl vanishes everywhere.
Step 2: Write down the velocity components.
\(V_x = x+y\), \(V_y = y+z\), \(V_z = z+x\).
Step 3: Test for incompressibility.
\[ \nabla \cdot \vec{V} = 1+1+1 = 3 \]
Since this is nonzero, the flow is compressible.
Step 4: Test for irrotationality.
\[ \nabla \times \vec{V} = -\hat{i}-\hat{j}-\hat{k} \]
Nonzero, so the flow is rotational.
Step 5: Match with the options.
Only compressible and rotational is consistent with the divergence of 3 and the nonzero curl.
Step 6: Conclusion.
\[ \boxed{\text{The flow is compressible and rotational.}} \]
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