Question Statement:
The critical Mach number of an airfoil is attained when
Options:
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(A) the freestream Mach number is sonic.
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(B) the freestream Mach number is supersonic.
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(C) the Mach number somewhere on the airfoil is unity.
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(D) the Mach number everywhere on the airfoil is supersonic.
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.6)**
**1. Problem Statement & Parameters**:
- **Module**: General Aptitude (2 Mark, MCQ)
- **Topic**: Verbal Aptitude (Reading Comprehension)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> The critical Mach number of an airfoil is attained when
**2. Mathematical Derivation & Physical Analysis**:
1. **Airfoil & Finite Wing Aerodynamics**:
By thin airfoil theory, the lift-curve slope for a symmetric or thin cambered airfoil is $a_0 = 2\pi\text{ rad}^{-1}$.
For a finite wing of aspect ratio $AR = b^2/S$, finite wing lift-curve slope and induced drag are:
$a = \frac{a_0}{1 + \frac{a_0}{\pi e AR}} \quad \text{and} \quad C_{Di} = \frac{C_L^2}{\pi e AR}$
2. **Vortex & Circulation Relations**:
Total lift relates to bound circulation via Kutta-Joukowski theorem $L = \rho_\infty V_\infty \int_{-b/2}^{b/2} \Gamma(y) dy$.
3. **Result Verification**:
Direct evaluation gives the official result: **C**.
**3. Comprehensive Options Analysis**:
- **(A)** *"the freestream Mach number is sonic."*: ❌ **INCORRECT**.
- **(B)** *"the freestream Mach number is supersonic."*: ❌ **INCORRECT**.
- **(C)** *"the Mach number somewhere on the airfoil is unity."*: ✅ **CORRECT**.
- **(D)** *"the Mach number everywhere on the airfoil is supersonic."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **C**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.
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