Tutorial: Multi-Species Lattice Boltzmann Models and Applications

Lattice Boltzmann solvers and applications
Diffusive scaling
Andries-Aoki-Perthame (AAP) model
In the following asymptotic analysis [20], we introduce the
dimensionless variables, defined by
x i = (l c /L) ˆx i , t = (UT c /L) ˆt. (22)
Defining the small parameter ɛ as ɛ = l c /L, which corresponds to
the Knudsen number, we have x i = ɛ ˆx i .
Furthermore, assuming U/c = ɛ, which is the key of derivation of
the incompressible limit [20], we have t = ɛ 2 ˆt. Then, AAP model is
rewritten as
ɛ 2 ∂f σ
∂t + ɛ ξ ∂f σ [ ]
i = λ σ fσ(∗) − f σ . (23)
∂x i
In this new scaling, we can assume ∂ α f σ = ∂f σ /∂α = O(f σ ) and
∂ α M = ∂M/∂α = O(M), where α = t, x i and M = ρ σ , q σi where
q σi = ρ σ u σi .
Pietro Asinari, PhD (Politecnico di Torino) Multi-Species LB Models Rome, Italy, on July 5-9, 2010 16 / 51