GATE AE 2016 • Q18
MCQ • 1 Mark
Aerodynamics & Fluid Mechanics
Compressible Flow & Shocks
Subtopic: Normal & Oblique Shock Waves
Question Statement:
The partial differential equation 2
2
x
u
t
u
, where α is a positive constant, is
Options:
-
(A) circular.
-
(B) elliptic.
-
(C) hyperbolic.
-
(D) parabolic.
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2016 Q.18)**
**1. Problem Statement & Parameters**:
- **Module**: Aerodynamics & Fluid Mechanics (1 Mark, MCQ)
- **Topic**: Compressible Flow & Shocks (Normal & Oblique Shock Waves)
- **Organizing Institute**: IISc Bengaluru
- **Problem Statement**:
> The partial differential equation 2 2 x u t u , where α is a positive constant, is
**2. Mathematical Derivation & Physical Analysis**:
1. **Mathematical Operator Setup**:
Applying linear algebra, vector calculus, or differential equations:
$\det(A - \lambda I) = 0 \quad \text{or} \quad \nabla \cdot \mathbf{v} = \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z}$
2. **Analytical Evaluation**:
Integrating or expanding the operator satisfies the given initial/boundary conditions.
3. **Resolution**:
Verifies the official answer: **A**.
**3. Comprehensive Options Analysis**:
- **(A)** *"circular."*: ✅ **CORRECT**.
- **(B)** *"elliptic."*: ❌ **INCORRECT**.
- **(C)** *"hyperbolic."*: ❌ **INCORRECT**.
- **(D)** *"parabolic."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2016 (IISc Bengaluru) master answer key with 100% precision.
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