# A Velocity persistence in Chapman-Enskog theory

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1. Dec 21, 2017

### Dank matter

Hi guys!

My question is about the diffusion coefficient of the mixture of gases. Consider two gases in thermal equilibrium (Maxwell velocity distribution) having different densities n1 and n2. Let's suppose that the molecules are rigid spheres with diameters d1 and d2 and masses M and m respectively. As far as I am concerned, the most exact expression for the diffusion coefficient can be obtained via Chapman-Enskog approach and for the case described above the first approximation gives [Chapman, Cowling, 1970] $$D_{12} = \frac{3\pi}{32(n_1 + n_2)\sigma_{12}} \sqrt{\frac{8kT}{\pi\mu}}\, ,$$ where $\sigma_{12} = \pi(d_1+d_2)^2/4$ is the cross section of the collision and $\mu = mM/(m+M)$ is the reduced mass.

If $M \gg m$ the diffusion coefficient becomes independent of M. However, in Meyer's approach the diffusion coefficient depends on the persistence factor $$D_{12} \sim \frac{1}{1-\theta} \frac{1}{(n_1+n_2)\sigma_{12}}\sqrt{\frac{8kT}{\pi M}}.$$ For rigid spheres the average persistence factor is [Jeans, 1967] $$\frac{1}{1-\theta} = \frac{3}{4}\frac{M}{\mu},$$ so the diffusion coefficient grows as $\sqrt{M}$ with the increase of M.

I would (naively) expect that Meyer's corrected formula yields a better description, because particles with an enormous mass will travel through the second gas almost without any interruption. On the other hand, of course, their average thermal velocity goes to zero with the growth of M, but I am not sure whether those two effects compensate each other according to Chapman-Enskog formula. Thus, I wonder if the effect of velocity persistence is accounted for in Chapman-Enskog approach? I know that there are next-order corrections to this formula, but at least the second correction for the case $n_1 \ll n_2$ is also independent of M when $M \gg m$ and according to [Chapman,Cowling, 1970] the third correction is very hard to compute (and I do not expect it to influence the result much). I tried to follow the derivation of the Chapman-Enskog formula for the diffusion coefficient, but this is not that easy

I hope my question is clear. I would be very grateful if you could help me with that.

2. Dec 27, 2017

### PF_Help_Bot

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