GATE AE 2026 • Q19
MCQ • 1 Mark
Aerodynamics & Fluid Mechanics
Compressible Flow & Shocks
Subtopic: 1D Isentropic Flow & Area-Mach Relations
Question Statement:
The velocity potential function (𝜑) given below represents which one of the
following?
𝜑 = 5x - 12y
Options:
-
(A) Doublet
-
(B) Irrotational vortex
-
(C) Source
-
(D) Uniform flow
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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