GATE AE 2026 • Q19 MCQ • 1 Mark Aerodynamics & Fluid Mechanics

Aerodynamics & Fluid Mechanics

Basic Fluid Mechanics

Subtopic: Vorticity, Circulation & Kelvin's Theorem

Question Statement:

The velocity potential function ($\phi$) given below represents which one of the following? $\phi$ = 5x - 12y

Options:

  • (A) Doublet
  • (B) Irrotational vortex
  • (C) Source
  • (D) Uniform flow

Official Answer Key:

D

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2026 Q.19)** **1. Intuitive Physical Setup & Given Parameters**: We are given the 2D velocity potential function: $\phi(x, y) = 5x - 12y$. **2. Step-by-Step Mathematical Derivation**: 1. **Velocity Field Derivation**: In potential flow theory, velocity components are gradients of the potential function $\vec{V} = \nabla \phi$: u = \frac{\partial \phi}{\partial x} = \frac{\partial}{\partial x}(5x - 12y) = 5\text{ m/s} v = \frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y}(5x - 12y) = -12\text{ m/s} 2. **Analysis of the Velocity Vector**: The velocity vector is $\vec{V} = 5\hat{i} - 12\hat{j}$. Both $u$ and $v$ are independent of coordinates $(x, y)$, meaning magnitude and direction are constant everywhere: |\vec{V}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13\text{ m/s} \theta = \tan^{-1}\left(\frac{-12}{5}\right) = \text{constant} 3. **Conclusion**: A flow field with constant velocity magnitude and direction everywhere in space is by definition a **uniform flow**. **3. Comprehensive Options & Distractor Analysis**: - **(A) Doublet**: ❌ **INCORRECT**. Potential is $\phi = \frac{\kappa \cos\theta}{r}$, decaying with distance. - **(B) Irrotational vortex**: ❌ **INCORRECT**. Potential is $\phi = \frac{\Gamma}{2\pi}\theta$. - **(C) Source**: ❌ **INCORRECT**. Potential is $\phi = \frac{\Lambda}{2\pi}\ln r$. - **(D) Uniform flow**: ✅ **CORRECT**. Linear potential $\phi = Ax + By$ produces a spatially uniform velocity field. **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **D** - **Exam Takeaway**: Linear velocity potential $\phi = u_\infty x + v_\infty y$ always represents uniform parallel streamflow with velocity vector $\vec{V} = (u_\infty, v_\infty)$.
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