Presets
Playback
Gas and plates
Kn = mean free path / plate gap. Small Kn = continuum, large Kn = free molecular.
Knudsen sweep
Runs eight Kn values at the current plate settings and plots the measured heat flux.
Model checks
The science
Knudsen number and regimes
The mean free path of hard disks of diameter d at number density n is λ = 1/(√2 n d) in 2D (1/(√2 π n d²) in 3D). The Knudsen number Kn = λ/H compares it with the gap H. Roughly: Kn < 0.01 continuum, 0.01-0.1 slip, 0.1-10 transition, > 10 free molecular.
Direct Simulation Monte Carlo
Each step the disks move freely, are reflected by the plates, then are sorted into cells. In a cell with Nc disks the no-time-counter scheme tests M = ½ Nc(Nc-1) Fnum (d cr)max Δt / Vc random pairs and accepts a pair with probability d cr/(d cr)max. An accepted hard-disk collision keeps the centre-of-mass velocity and the relative speed and turns the relative velocity to a random angle, so momentum and energy are conserved exactly. This samples the Boltzmann collision integral (Bird, 1994).
Walls and heat flux
A diffuse wall re-emits a disk with a flux-weighted Maxwell distribution at the wall temperature: normal speed from a Rayleigh law, tangential speed Gaussian. With accommodation coefficient α a fraction 1-α is reflected mirror-like. In the free-molecular limit the net heat flux is q = [α/(2-α)] · (3/2) k (Th-Tc) Γ, where Γ = n c̄ / π is the one-sided particle flux (2D). As Kn falls, collisions block the flight from plate to plate and q drops roughly as 1/(1 + b/Kn); the sweep fits b to your own data. Near a wall the gas temperature does not reach the wall value (temperature jump, of order Kn), which shows in the left graph.
Honest simplifications: a flat 2D gas of hard disks (no rotation or vibration), constant diameter instead of VHS temperature dependence, periodic sides, finite particle count so the profile is noisy, cells are twice as wide as tall, and the Kn=0.05 end is under-resolved unless N is large. Reduced units: k = m = 1, so the speed unit is sqrt(k Tref/m) = about 298 m/s for nitrogen at 300 K. The pressure readout assumes a real 3D nitrogen gas in a gap of 1 micrometre.
What to try
- Press the Continuum preset and then Free molecular: the temperature profile changes from a straight ramp to a flat plateau with large jumps at both plates.
- Lower the accommodation coefficient: the jump grows and the flux falls.
- Set the temperature difference to zero and watch the speed histogram match the Maxwell curve.
- Run the Kn sweep and compare the dots with the fitted 1/(1+b/Kn) curve.