Content and numerical checks: 2026-10-09
Two independent modeling questions
An incompressible approximation asks whether density variation can be neglected. A continuum approximation asks whether molecular travel distances are small relative to the relevant physical length. Low Mach number does not establish continuum behavior, and small Knudsen number does not establish constant density.
The acoustic and stagnation-state relations below assume a calorically perfect ideal gas, constant specific-heat ratio, and adiabatic, isentropic deceleration. They do not describe an entire device with shocks, significant friction, heat input or strong rarefaction.
State variables and length scales
| Variable | SI unit | Meaning |
|---|---|---|
| γ | Dimensionless | Specific-heat ratio cₚ/cᵥ |
| R | J/(kg·K) | Specific gas constant |
| T, T₀ | K | Static and isentropic stagnation temperature |
| v, c | m/s | Local speed and local sound speed |
| ρ, ρ₀ | kg/m³ | Static and stagnation density |
| p, p₀ | Pa | Static and stagnation pressure |
| λ, L | m | Mean free path and relevant length |
Use kelvins in c = √(γRT), and a mass-specific R rather than the molar universal gas constant. Obtain λ for the gas at the relevant pressure and temperature. A microchannel width or gradient length may matter more than the overall device length.
Deriving the stagnation-state comparison
For this ideal gas c² = γRT and Ma = v/c. Reversible adiabatic deceleration conserves energy: cₚT₀ = cₚT + v²/2. Insert cₚ = γR/(γ−1) to obtain the first ratio below. Combine p/ρᵞ = constant with the ideal-gas equation for the other two ratios.
ρ₀/ρ = (T₀/T)^(1/(γ−1))
p₀/p = (T₀/T)^(γ/(γ−1))
These ratios compare a local state with the hypothetical state reached by ideal deceleration. They are not a prediction of actual inlet-to-outlet changes. Compare the resulting density difference with the accuracy required by your application.
A constructed numerical comparison
For illustrative γ = 1.4, R = 287 J/(kg·K) and T = 300 K, c = √120540 = 347.1887 m/s. Speeds below are constructed from that value; they are not certified atmospheric data.
| Ma / v | T₀/T | ρ₀/ρ | p₀/p |
|---|---|---|---|
| 0.3 / 104.1566 m/s | 1.018 | 1.04561 | 1.06443 |
| 0.8 / 277.7510 m/s | 1.128 | 1.35137 | 1.52434 |
Even Ma = 0.3 gives a density ratio about 4.56% above unity in this comparison. A convenient low-speed rule still needs to be checked against the required precision.
Check molecular scales after changing device size
Hold an illustrative mean free path λ = 6.8×10⁻⁸ m fixed. At L = 0.001 m, Kn = 6.8×10⁻⁵; at L = 10⁻⁶ m, Kn = 0.068. Holding speed and sound speed fixed leaves Ma unchanged. Thus the size change affects a separate assumption.
At Kn = 0.068, assess rarefaction effects including wall slip and temperature jump. The appropriate boundary model depends on surfaces and required accuracy. Reusing the same no-slip continuum model needs justification. This example changes length at fixed λ; it does not model the change in mean free path caused by changing pressure.
What to record when selecting a model
- State local temperature and the acoustic assumptions behind Ma.
- Choose an acceptable density error for an incompressible approximation.
- Explain which gas state and physical length λ and L represent.
- Do not apply isentropic ratios across a shock.
- Strong rarefaction or thermal nonequilibrium requires more than this ideal stagnation comparison.
Formula and concept references: NASA Glenn — Isentropic Flow Equations · NASA — Knudsen-number and slip discussion, pp. 106–107. Derivations and worked input sets were constructed for this article; they are not measured datasets or copied source examples.