# why LBM applicable to fluid

Hi All,
I have this stupid question: why lbm is applicable to dense fluids?
i am new to LBM however according to my understanding, the derivative of lattice Boltzmann equation is derived with a very basic assumption that the subject is rarefied gas with only binary collision. Then why we use lbm for fluid which is dense and may be binary collision no longer holds? Thank you very much if you can help me clear this problem.

Hi,

That’s not a stupid question at all, it is a very good question!

As you say, the lattice Boltzmann method is derived from the kinetic theory of gases, which does not hold hold for very dense gases or for fluids. However, according to the Chapman-Enskog analysis which you may have heard of, the Boltzmann equation of kinetic theory (the one that LB is a discretisation of) predicts the conservation equations of fluid mechanics. By that, I mean the conservation equation of mass, known as the continuity equation, and the conservation equation of momentum, known as the Navier-Stokes equation.

Now, these equations hold equally well for both gases and liquids. The difference between the fluids is manifested mainly by a difference in material constants. For instance, liquids tend to have a significantly higher speed of sound than gases. Even so, this doesn’t really matter for basic LB; in fact, the isothermal fluid that it directly simulates is already pretty far away from something you find in reality, but the results are still useful!

In short, then, LB works to simulate liquids because its behaviour corresponds to the equations of fluid mechanics, which hold for all fluids. Hope that helps!

Hi
I have a question in this relation,
is Pressure=Rho/3 relation true for all isotropic fluids like water in lattice boltzmann method? or it is true only for ideal gas?
im looking forward to ur expert guidance
thanks

Dear all
LBM connected parabolic density fuction to collision step
in fact it is easy to undrestand; population density; f; is chaneged via convective term or collision term
this fact can be applied to every fluids or convective media, the problem in dense media is to set collision term
this can be hold easily for very rarifeid flow, i mean we can set it to zero!. for more dense flow where collision between 2 particle is very dominated than between 3, … the collision term can be obtained via maxwel method who realated collision to f
for dense flow it is can be related to viscosity! viscisity is a key factor to showing collision strength but how we can relate it to viscosity?
the answer is that we set lbm in such away it recover navier stokes formula!
it can be done via multiscale expansion framework like chapman enskog that this expansion enforce lbm to equality its relaxation time to nu:
nu=(tau-0.5)/3
good luck

Hi 91126074,

This relation can be written as Pressure=Rho*c[sub]s[/sub][sup]2[/sup], where c[sub]s[/sub] related to sound speed. As far as I know, This relation is true for any fluid like water if you set the lattice sound speed equal to the fluid’s sound speed. And you also should .be care that the Mach number should be small so that you can get the right N-S equation. Under this condition, the fluid can be considered as imcompressible fluid, the sound speed has little influence on the simulation result, so in many simulations, the lattice sound speed needn’t be equal to the sound speed in fluid,

thank u very much anjie