From microscopic to macroscopic step

Dear All,

I’m trying to think to the relationship between micro and macroscopic variables. I’m not expert so corrections and comments are welcome.

Starting from the lattice-boltzmann equation, we have streaming and collision parts:

Fi (x + ci dt, t + dt) - Fi (x, t) = -(dt/tau) (Fi (x, t) - Fi[sup]0[/sup] (x, t))

streaming = collision

The streaming acts for the non-steady and the non-uniform operator. It means that when we reach the steady in the simulation, that part simulates non uniformity in the domain. As a consequence when we have periodic boundary condition (as in poiseuille test case), it is zero. In this case we have only the collision operator. If we impose pressure drop between the inlet and the outlet of the poisuille case, the force term add to the Fi[sup]0[/sup]:

-(dt/tau) (Fi (x, t) - Fi[sup]0[/sup] (x, t)) + Fi_ext

Because the Fi_ext is in direct ratio to dt[sup]2[/sup]/dx, We add something that is proportional to dt/dx:

-(dt/tau) (Fi (x, t) - Fi[sup]0[/sup] (x, t) + Fi_ext/dt)

Hence, the macroscopic velocity:

v=sum[sub]i[/sub] (ci*Fi (x, t))

is independent on the ratio dx/dt. This is true for periodic, steady simulation.

Let me know your comments, thanks.

Stefano