Transport momentum and energy between neighboring layers

Suppose mean flow in the x direction increases along y. Molecules arriving from larger y tend to carry excess x momentum into a slower layer, while molecules arriving from smaller y carry a deficit. Random motion therefore transports x momentum down its gradient. In the kinetic pressure tensor convention, the off-diagonal momentum flux is minus viscosity times the velocity gradient. The viscous part of the Cauchy stress uses the opposite sign because stress is traction, while the pressure tensor appears as an outward momentum flux.

A dimensional estimate gives dynamic viscosity of order mass density times thermal speed times mean free path. At fixed temperature in a dilute gas, increasing number density increases the carriers but shortens their mean free path approximately inversely. These effects roughly cancel, explaining weak density dependence in the dilute regime. This cancellation fails near dense-fluid correlations, where the simple binary transport picture and collision frequency no longer suffice.

Molecules crossing a temperature gradient carry excess or deficient random energy, yielding heat flux down the gradient. Thermal conductivity therefore involves not only the flight scale but also energy per temperature carried by each molecule. Fourier’s law relates a heat flux in watts per square metre to a temperature gradient in kelvin per metre; its coefficient has units watts per metre kelvin. It must not be inserted directly as the diffusivity of a temperature equation.

To obtain a thermal diffusivity, divide conductivity by the relevant heat capacity per volume. For an ideal gas at approximately constant pressure, that capacity is mass density times specific heat at constant pressure. In a constrained fixed-density setting, a different caloric factor may appear. The dynamical and thermodynamic constraints matter just as much as the material coefficient. Calling conductivity and diffusivity by the same symbol can conceal this physically important distinction.

First-order BGK gives viscosity equal to pressure times relaxation time. Its conductivity is five halves the gas constant per mass times that viscosity. Their dimensionless Prandtl ratio is therefore one. A more faithful elastic collision calculation can produce the near-two-thirds value characteristic of dilute monatomic gases. This is a model limitation with a concrete consequence: fitting the momentum diffusion time does not independently fit the thermal diffusion time.

The heat lab represents a diffusivity after such conversion. Reset and set mode to 2 and diffusivity to 0.25, then Step twice; predict amplitude exp(−1). Reset and set mode to 2 and diffusivity to 0.5, then Step twice; predict exp(−2). Read the decay rate as diffusivity times squared mode number. The lab has no mass density, viscosity, or heat-capacity control, so it cannot derive gas coefficients or compare Prandtl numbers.