Subtopic: Tsiolkovsky Rocket Equation & Specific Impulse
Question Statement:
A flow is steady, inviscid and one-dimensional, with no shaft work or body forces.
Which of the following is/are possible under the given conditions?
Options:
(A) Oblique shocks
(B) Sound propagation
(C) Rayleigh flow
(D) Fanno flow
Official Answer Key:
B, C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.30)**
**1. Intuitive Physical Setup & Given Parameters**:
We are given the following flow conditions:
- Steady flow ($\partial / \partial t = 0$)
- Inviscid fluid ($\mu = 0$, frictionless)
- One-dimensional flow ($1\text{-D}$)
- No shaft work ($w_s = 0$)
- No body forces ($f_b = 0$)
We need to determine which phenomena are **possible** under these specific constraints (MSQ).
**2. Step-by-Step Gas Dynamic Derivation**:
1. **Evaluation of Given Phenomena**:
- **Oblique Shocks**: An oblique shock wave requires a two-dimensional deflection of flow (turning angle $ heta$ and shock angle $eta$). A purely 1-D flow can only support normal shocks, not oblique shocks. Thus, oblique shocks are **NOT possible** in 1-D flow.
- **Sound Propagation**: Acoustic wave propagation is governed by small, isentropic, 1-D inviscid perturbations where the speed of sound $a = \sqrt{\gamma R T}$ propagates along the flow axis. In an inviscid 1-D duct, sound wave propagation is fully admitted. Statement B is **POSSIBLE**.
- **Rayleigh Flow**: Rayleigh flow is defined specifically as 1-D, steady, frictionless (inviscid) flow in a constant-area duct with **heat addition or rejection**, with no shaft work or body forces. This matches the problem's criteria identically! Statement C is **POSSIBLE**.
- **Fanno Flow**: Fanno flow is defined as 1-D, steady, adiabatic flow with **friction** ($\mu
eq 0$, wall shear stress $ au_w
eq 0$). The problem explicitly states that the flow is *inviscid* (frictionless), which completely rules out Fanno flow. Thus, Fanno flow is **NOT possible**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) Oblique shocks**: ❌ **INCORRECT**. Oblique shocks are fundamentally 2D/3D phenomena involving streamline deflection; 1D flow only allows normal shocks.
- **(B) Sound propagation**: ✅ **CORRECT**. One-dimensional acoustic disturbance propagation is inviscid and 1D.
- **(C) Rayleigh flow**: ✅ **CORRECT**. Textbook definition of Rayleigh flow: 1D, steady, frictionless with heat transfer.
- **(D) Fanno flow**: ❌ **INCORRECT**. Fanno flow requires viscous friction, contradicting the inviscid condition.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **B, C**
- **Exam Takeaway**: Remember the twin 1D duct flows: Rayleigh = Frictionless + Heat Transfer; Fanno = Adiabatic + Friction.
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