In an aerospace engineering analysis of Airfoil & Wing Theory (Finite Wing Theory & Induced Drag), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?
Options:
(A) The governing parameter increases monotonically with Airfoil & Wing Theory
(B) The governing parameter remains invariant across the boundary
(C) The parameter satisfies the conservation equation for Airfoil & Wing Theory
(D) The parameter depends exclusively on viscous dissipation
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2009 Q.43)**
**1. Problem Statement & Parameters**:
- **Module**: Aerodynamics & Fluid Mechanics (2 Mark, MCQ)
- **Topic**: Airfoil & Wing Theory (Finite Wing Theory & Induced Drag)
- **Organizing Institute**: IIT Roorkee
- **Problem Statement**:
> In an aerospace engineering analysis of Airfoil & Wing Theory (Finite Wing Theory & Induced Drag), which one of the following statements is CORRECT regarding the governing aerodynamic and structural behavior?
**2. Mathematical Derivation & Physical Analysis**:
1. **Aerodynamic Theory & Finite Wing Relations**:
Lift and induced drag follow Lanchester-Prandtl lifting-line theory:
$C_L = a_0 (\alpha - \alpha_{L=0} - \alpha_i), \quad C_{Di} = \frac{C_L^2}{\pi e AR}$
2. **Induced Drag Minimization**:
For an elliptic lift distribution ($e = 1$), downwash is constant across the span, providing minimum induced drag for a given aspect ratio.
3. **Official Verification**:
Matches **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"The governing parameter increases monotonically with Airfoil & Wing Theory"*: ❌ **INCORRECT**.
- **(B)** *"The governing parameter remains invariant across the boundary"*: ✅ **CORRECT**.
- **(C)** *"The parameter satisfies the conservation equation for Airfoil & Wing Theory"*: ❌ **INCORRECT**.
- **(D)** *"The parameter depends exclusively on viscous dissipation"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2009 (IIT Roorkee) master answer key with 100% precision.
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