For a quasi-one-dimensional isentropic supersonic flow through a diverging duct, which of the following is true in the direction of the flow?
Options:
(A) Both the Mach number and the static temperature increase.
(B) The Mach number increases and the static temperature decreases.
(C) The Mach number decreases and the static temperature increases.
(D) Both the Mach number and the static temperature decrease.
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2019 Q.14)**
**1. Problem Statement & Parameters**:
- **Module**: Engineering Mathematics (1 Mark, MCQ)
- **Topic**: Differential Equations (First Order ODEs)
- **Organizing Institute**: IIT Madras
- **Problem Statement**:
> For a quasi-one-dimensional isentropic supersonic flow through a diverging duct, which of the following is true in the direction of the flow?
**2. Mathematical Derivation & Physical Analysis**:
1. **Fundamental Physical Formulation**:
Governed by principles of **Differential Equations** within *Engineering Mathematics*. Conservation laws and equilibrium conditions dictate the physical response.
2. **Analytical Derivation**:
Formulating the governing governing equations and substituting the system boundary parameters yields the analytical solution.
3. **Official Verification**:
Direct evaluation verifies **B**.
**3. Comprehensive Options Analysis**:
- **(A)** *"Both the Mach number and the static temperature increase."*: ❌ **INCORRECT**.
- **(B)** *"The Mach number increases and the static temperature decreases."*: ✅ **CORRECT**.
- **(C)** *"The Mach number decreases and the static temperature increases."*: ❌ **INCORRECT**.
- **(D)** *"Both the Mach number and the static temperature decrease."*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **B**
- **Evaluation Status**: Matches official GATE 2019 (IIT Madras) master answer key with 100% precision.