GATE AE 2013 • Q13
MCQ • 1 Mark
Engineering Mathematics
Calculus
Subtopic: Limits, Continuity & Differentiability
Question Statement:
Consider a thin flat plate airfoil at a small angle to an oncoming supersonic stream of air.
Assuming the flow to be inviscid, 2
d
l
C
C
is
Options:
-
(A) zero
-
(B) independent of
-
(C) proportional to
-
(D) proportional to 2
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2013 Q.13)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Calculus (Limits, Continuity & Differentiability)
- **Organizing Institute**: IIT Bombay
- **Problem Statement**:
> Consider a thin flat plate airfoil at a small angle to an oncoming supersonic stream of air. Assuming the flow to be inviscid, 2 d l C C is
**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: **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"zero"*: ❌ **INCORRECT**.
- **(B)** *"independent of "*: ✅ **CORRECT**.
- **(C)** *"proportional to "*: ❌ **INCORRECT**.
- **(D)** *"proportional to 2"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2013 (IIT Bombay) master answer key with 100% precision.
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