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GATE 2021 AE – Question 13

Aerodynamics · Basic fluid mechanics: kinematics, conservation laws, dimensional analysis, incompressibility and Newtonian fluids · 1 mark · Multiple choice

The velocity field $\mathbf V=(2x+3y)\hat i+(3x+2y)\hat j$ represents

  1. divergence-free and curl-free
  2. curl-free but not divergence-free
  3. divergence-free but not curl-free
  4. neither divergence-free nor curl-free

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Correct answer: (B) curl-free but not divergence-free

Explanation

For $\mathbf V=u\hat i+v\hat j$ with $u=2x+3y$ and $v=3x+2y$:

**Divergence:**
$$\nabla\cdot\mathbf V=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=2+2=4\neq0 .$$
So the field is **not** divergence-free.

**Curl (the $z$ component):**
$$\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}=3-3=0 .$$
So the field is **curl-free** (irrotational).

Answer: curl-free but not divergence-free (B).