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GATE 2026 CH – Question 37

Fluid Mechanics and Mechanical Operations · Fluid properties, fluid statics and fluid kinematics · 2 marks · Multiple choice

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?

  1. The flow is incompressible and irrotational.
  2. The flow is incompressible and rotational.
  3. The flow is compressible and rotational.
  4. The flow is compressible and irrotational.

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Correct answer: (C) The flow is compressible and rotational.

Explanation

The divergence is $\nabla \cdot \vec{V} = 1 + 1 + 1 = 3 \neq 0$, so the flow is not incompressible. The curl is $\left(\frac{\partial V_z}{\partial y} - \frac{\partial V_y}{\partial z},\ \frac{\partial V_x}{\partial z} - \frac{\partial V_z}{\partial x},\ \frac{\partial V_y}{\partial x} - \frac{\partial V_x}{\partial y}\right) = (0 - 1,\ 0 - 1,\ 0 - 1) = (-1, -1, -1) \neq 0$, so the flow is rotational.