A student expects that hotter gas should be less viscous — molecules move faster, so flow should be easier, just as heating a liquid reduces its viscosity. What does Chapman-Enskog theory actually predict?
AHotter gas has lower viscosity — faster molecules encounter less resistance from neighboring layers
BGas viscosity is temperature-independent in the dilute limit
CHotter gas has higher viscosity (η ∝ √T) — faster molecules carry more momentum across streamlines, increasing viscous stress
DViscosity depends on pressure, not temperature, in kinetic theory
This counterintuitive result is one of Chapman-Enskog theory's key predictions, confirmed experimentally. Unlike liquids (where viscosity is dominated by intermolecular attraction that weakens with heat), gas viscosity arises from momentum transport by molecules crossing streamlines. Hotter molecules move faster, so they carry more momentum per crossing — viscous stress increases. Liquids and gases have opposite temperature dependences of viscosity, and kinetic theory explains why.
Question 2 Multiple Choice
Why does the zeroth-order (ε = 0) term in the Chapman-Enskog expansion give the Euler equations rather than the Navier-Stokes equations?
AThe Euler equations are an approximation of Navier-Stokes valid at low Reynolds numbers
BAt zeroth order the distribution is exactly the local Maxwell-Boltzmann — perfect local equilibrium with no gradients — so there are no irreversible transport processes; viscosity and conductivity only emerge from the first-order correction
CThe Boltzmann equation is only valid for ideal fluids, which the Euler equations describe
DTransport coefficients were not included in Euler's original formulation and must be added separately
At zeroth order (ε = 0), the system is assumed to be in perfect local thermodynamic equilibrium everywhere — the distribution function is a local Maxwellian with no deviation. No gradients means no momentum flux from velocity shear (no viscosity) and no energy flux from temperature gradient (no heat conduction). These irreversible transport processes only appear when ε > 0 introduces first-order corrections that encode the response of the distribution to spatial gradients.
Question 3 True / False
Chapman-Enskog theory derives transport coefficients (viscosity, thermal conductivity) from the Boltzmann equation without requiring empirically fitted parameters for dilute monatomic gases.
TTrue
FFalse
Answer: True
This is the central achievement of the theory. For hard-sphere or specified interaction-potential gases, the transport coefficients are expressed as explicit integrals over the collision operator — no free parameters. The theory predicts, for instance, η ∝ √T for hard spheres, and this prediction is confirmed by experiment. The derivation realizes the kinetic theory program of connecting atomic-level interactions to macroscopic fluid behavior.
Question 4 True / False
The Chapman-Enskog expansion is valid when the Knudsen number ε = λ/L is much greater than 1, meaning the mean free path is large compared to the macroscopic scale.
TTrue
FFalse
Answer: False
The expansion requires ε ≪ 1 — the mean free path must be small compared to macroscopic length scales. This condition ensures that collisions are frequent enough to keep the distribution function close to local equilibrium, so that a perturbative expansion around f^(0) is valid. When ε ≫ 1 (rarefied gas or shock waves), the gas is far from local equilibrium and the Chapman-Enskog hierarchy breaks down — higher-order terms become large rather than small corrections.
Question 5 Short Answer
What is the physical meaning of the Knudsen number, and why must it be small for the Chapman-Enskog expansion to be valid?
Think about your answer, then reveal below.
Model answer: The Knudsen number ε = λ/L is the ratio of the mean free path (average distance between molecular collisions) to the macroscopic length scale over which temperature or velocity vary. When ε ≪ 1, collisions occur frequently relative to macroscopic gradients, keeping the distribution function close to local equilibrium. The Chapman-Enskog expansion treats deviations from equilibrium as small perturbations proportional to ε; if ε is not small, the perturbative assumption fails and the expansion diverges.
Physically, small ε means the gas 'forgets' its non-equilibrium initial conditions over a distance much shorter than the macroscopic gradient scale — it relaxes to local equilibrium between macroscopic events. This justifies writing f as a small perturbation around f^(0). High Knudsen number situations (spacecraft re-entry, microfluidics, rarefied gas dynamics) require the full Boltzmann equation or other approaches that do not assume near-equilibrium.