Has any work been done to describe the transient behaviour of a non-uniform ideal gas in an adiabatic process?

Question

Recently, I've come across a discussion where a person believed classical thermodynamics predicted loss of energy in free expansion. After discussing for a while, I found out that that person made a mistake by applying the ordinary adiabatic equation that requires pressure and temperature equilibrium to free expansion, a situation where such equilibrium cannot be assumed. The problem can be solved easily by acknowledging that no collision happens during the expansion and the wall stays stationary after that. However, I also want to study the more general case where adiabatic expansion is not instantly but it happens within $\Delta t > 0$. This requires calculating diffusion and the work that the gas does on the moving wall of the container. Unfortunately, this approach faces a lot difficulties:

1. Molecules with different kinetic energy diffuse at different speed.
2. When the wall is moving, it reacts with molecules at different velocity differently. For example, low speed particles won't collide, and particles are more likely to collide and also lose more energy.

As a result, this seems to break every single commonly used equilibrium, including Maxwell-Boltzmann distribution. Well if I let $\Delta t = 0$ (the free expansion), the second problem can be resolved, and I keep the Maxwell-Boltzmann distribution, with the caveat that I have to keep track of volumetric concentration distribution at each kinetic energy level.

I've also tried to search for online articles but most do not consider the transient behaviour at all, perhaps due to its sheer complexity. So my question is: Is there any work done on giving a general description of the transient properties of this kind of adiabatic process?

Answer 1

Free expansion does not occur instantaneously, and that assumption is not needed to find the final equilibrium state using classical thermodynamics.

For determining what takes place during an irreversible process, one must consider transport processes such as viscous momentum transfer and conductive heat transfer within the gas. You end up with a set of partial differential equations in space and time involving conservation of mass (the continuity equation), conservation of momentum (the Navier Stokes equation), and conservation of energy (the thermal energy equation). The derivation of the thermal energy equation involves the assumption of local thermodynamic equilibrium. This is all laid out in the book Transport Phenomena by Bird, Stewart, and Lightfoot.