Effect of slip velocity and knudsen number in B.C

Dear all

as you know many articles and papers that have published in LBM assumed that flow is continuum. then they imply no slip boundary condition Bounce Back to their walls. this method was good in macroscopic and microscopic scales. But if I want to simulated rare flow in nano channel I should imply effect of slip Boundary condition and Kn (knudsen number) in my code.
also i search in many journal in internet . I found that many researcher work on but their methods is very different. someone imply no slip B.C but study effect of kn(knudsen number) like" Lattice Boltzmann versus Molecular Dynamics Simulation of Nanoscale Hydrodynamic Flows Ju rgen Horbach and Sauro Succi" and someone suggest their different methods like related relaxation time with kn and etc.

I really confused when I have study this papers. if i wanted to study effect of kn number in my channel (single component or multiphase model ) what papers or book should i read ?
i see many articles with many method but i dont know which one is better and can be apply in shan-chen model. who can help me in this case? can you guided me for this problem?
any book ,articles, papers
I really so sorry
Many thanks in advance and with kind regards


You need to be aware that bounce-back boundary conditions introduce a non-zero velocity at the walls. This has often been called Knudsen slip but it is not. It is numerical slip. That is, it is simply a numerical error and does not represent any physical slip (microscopic or otherwise).

Moreover, the LBE can be solved exactly for Poiseuille flow and simple shear (see see He, Zou, Luo and Dembo, 1997; and Ginzburg and Adler, 1994). These papers are normally cited to show that bounce back doesn’t get no-slip correct but the He at al paper says something else too: the analytic solution shows that the LBE in Poiseuille flow is a parabolic profile shifted by an amount U_s. This U_s is the numerical slip. Since the profile is simply a shifted parabola then the LBE can not possibly capture what is called the Knudsen layer (a boundary layer near the wall), at least not in a consistent way. Therefore the best that we can hope for (with the standard D2Q9 LBE) is Navier-slip (or Maxwell-Navier slip) boundary conditions. See Verhaeghe et al (2009) for a good review.