# Help on Chapman Enskog

Hi,

I’m very new to LBM and i’m trying to understand the Chapman Enskog expansion. Please, could anyone point me to a good, graduate level description/introduction on that topic? I’m trying to get it with the details in Guo’s paper on body-force, but i’m really lost!

Fred

Hi Fred,

You will find details of the Chapman-Enskog expansion in all good books on kinetic theory, the most famous is probably the one by Chapman and Cowling. This is for “true” kinetic theory, rather than discrete kinetic theory of LBE. In terms of the application of the expansion to LBE, I think a lot of thesis include details, and there is a paper by He and Luo) on “Incompressible LBE” from about 1997. You may also find these slides useful https://www.maths.ox.ac.uk/system/files/attachments/Oxford_2010_Intro.pdf.

Good luck!

It depends on what you want. If you are going the pure kinetic theory route that Tim talks about (starting from the Boltzmann equation to the fluid conservation equations), then I would heartily recommend Paul Dellar’s 2001 paper [1], which goes through the Chapman-Enskog expansion first before it goes into LBM. In my opinion, that is a very easily digestible and well-written text on Chapman-Enskog.

If you want to start from the lattice Boltzmann equation and include the forcing terms, I suggest Jonas Lätt’s PhD thesis [2]. It goes through that derivation with a forcing term that I think is the same as Guo’s. (Forgive me if I’m wrong, I have never really looked into LBM forcing terms.) It’s quite concisely written, so if you find it tough going you might want to supplement it with my own master’s thesis [3], which goes through the same derivation as Lätt’s in more detail, but with no forcing term.

Hope that helps!

It is perhaps worth noting that these forms of the force terms are designed to ensure you get the correct viscous stress. This dates back to Luo 1998, where it was derived from discrete Hermite Polynomials.

To All,

I think it’s not the basics of kinetic theory that makes me stuggling, but more the mathematical details of the CE expansion. I’m not used to such math, and I’ll have to go through it step by step. Since I would really like to undertsand, I’m trying to comprehend every step.

So far, I’d like to thank you very much! I will have a look at the references you suggested, and - if necessary - come back to you … if that is tolerable in the subject of this forum?

Kind Regards,

Fred

Of course, feel free to ask.