GATE AE 2021 • Q13
MCQ • 1 Mark
Engineering Mathematics
Calculus
Subtopic: Limits, Continuity & Differentiability
Question Statement:
Consider the 2D velocity field $\vec{V} = (2x + 3y)\hat{i} + (3x + 2y)\hat{j}$. The field $\vec{V}$ is:
Options:
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(A) divergence-free and curl-free
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(B) curl-free but not divergence-free
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(C) divergence-free but not curl-free
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(D) neither divergence-free nor curl-free
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2021 Q.13)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the 2D planar velocity field: $\vec{V} = (2x + 3y)\hat{i} + (3x + 2y)\hat{j}$, where $u = 2x + 3y$ and $v = 3x + 2y$.
**2. Step-by-Step Mathematical Derivation**:
1. **Divergence Evaluation (Mass Conservation / Incompressibility)**:
$\nabla \cdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \frac{\partial}{\partial x}(2x + 3y) + \frac{\partial}{\partial y}(3x + 2y) = 2 + 2 = 4$
Since $\nabla \cdot \vec{V} = 4 \ne 0$, the flow field is **NOT divergence-free** (it has a uniform volumetric dilation rate of 4).
2. **Curl Evaluation (Vorticity / Irrotationality)**:
$\nabla \times \vec{V} = \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right)\hat{k} = \left( \frac{\partial}{\partial x}(3x + 2y) - \frac{\partial}{\partial y}(2x + 3y) \right)\hat{k} = (3 - 3)\hat{k} = 0$
Since $\nabla \times \vec{V} = 0$, the flow is strictly **curl-free (irrotational)**.
3. **Conclusion**:
The field is curl-free but not divergence-free.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) divergence-free and curl-free**: ❌ **INCORRECT**. Fails divergence-free condition since $\nabla \cdot \vec{V} = 4$.
- **(B) curl-free but not divergence-free**: ✅ **CORRECT**. Curl is identically $3 - 3 = 0$, while divergence is $2 + 2 = 4 \ne 0$.
- **(C) divergence-free but not curl-free**: ❌ **INCORRECT**. Inverts the two properties.
- **(D) neither divergence-free nor curl-free**: ❌ **INCORRECT**. Fails to recognize that the cross-derivatives $\partial v/\partial x = \partial u/\partial y = 3$ cancel.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B**
- **Exam Takeaway**: Divergence is the trace of the velocity gradient tensor ($\partial u/\partial x + \partial v/\partial y$). Curl in 2D is the anti-symmetric component ($\partial v/\partial x - \partial u/\partial y$).
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