In 1D frictionless compressible flow through a constant-area duct with heat addition (Rayleigh flow), if heat is added to an initially subsonic flow ($M_1 < 1$), how do the static pressure $p$ and Mach number $M$ change along the duct?

(A) Static pressure decreases ($p \downarrow$) and Mach number increases ($M \uparrow$)

(B) Static pressure increases ($p \uparrow$) and Mach number increases ($M \uparrow$)

(C) Static pressure decreases ($p \downarrow$) and Mach number decreases ($M \downarrow$)

(D) Static pressure increases ($p \uparrow$) and Mach number decreases ($M \downarrow$)

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Conservation:** For Rayleigh flow in a constant-area duct: $p + \rho u^2 = \text{const}$. 2. **Stage 2: Differential Form:** $\frac{dp}{p} = -\frac{\gamma M^2}{1 - M^2} \frac{dT_0}{T_0}$ and $\frac{dM^2}{M^2} = \frac{1 + \gamma M^2}{1 - M^2} \frac{dT_0}{T_0}$. 3. **Stage 3: For Subsonic Flow ($M < 1$) with Heat Addition ($dT_0 > 0$):** * Denominator $(1 - M^2) > 0$. * Therefore $\frac{dp}{p} < 0 \implies p$ decreases. * $\frac{dM^2}{M^2} > 0 \implies M$ increases towards 1 (Thermal Choking). 4. **Stage 4: Distractor Elimination:** Thermal acceleration decreases static pressure due to momentum conservation. 5. **Stage 5: Official Reference:** Ascher H. Shapiro, *The Dynamics and Thermodynamics of Compressible Fluid Flow*, Vol. 1.

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