Tutorial: Multi-Species Lattice Boltzmann Models and Applications

Tutorial: Multi-Species Lattice Boltzmann Models 

and Applications 

Pietro Asinari, PhD 

Dipartimento di Energetica, Politecnico di Torino, Torino, Italy, 

e-mail: pietro.asinari@polito.it, 

home page: http://staff.polito.it/pietro.asinari 

19th Discrete Simulation of Fluid Dynamics 

CNR and University of Rome, "Tor Vergata", Rome, Italy, on July 5-9, 2010 

Pietro Asinari, PhD (Politecnico di Torino) Multi-Species LB Models Rome, Italy, on July 5-9, 2010 1 / 51

Abstract of this tutorial 

The aim of this tutorial is to discuss a numerical scheme based on the 

Lattice Boltzmann Method (LBM) for gas mixture modeling, which fully 

recovers Maxwell-Stefan diffusion model in the continuum limit, without 

the restriction of the mixture-averaged approximation [1]. The 

mixture-averaged approximation is used to express the molar 

concentration gradient by means of a Fickian expression and a proper 

diffusion coefficient, both designed in such a way to be consistent with 

the Maxwell-Stefan formulation only in some limiting cases. 

Unfortunately (a) this approximation yields to Fickian diffusion 

coefficients depending on both the fluid transport properties and the 

local flow and (b) it is not valid in general. The key idea for avoiding 

these problems is to start from a Bhatnagar-Gross-Krook-type kinetic 

model for gas mixtures, recently proposed by Andries, Aoki and 

Perthame [2] (the so-called AAP model). 


Outline of this tutorial 

1 Kinetic theory of rarefied gas mixtures 

Full Boltzmann equations 

Exchange relation for momentum 

2 Lattice Boltzmann solvers and applications 

Single-relaxation-time Lattice Boltzmann scheme 

Andries-Aoki-Perthame (AAP) model 

LBM formulation by variable transformation 

3 Validation and simple applications 


Kinetic theory of rarefied gas mixtures 

Outline Compass 












Full Boltzmann equations for gas mixtures 

The simultaneous Boltzmann equations for a mixture without 

external force can be written as [3, 4, 5, 6]: 

∂ t f σ + ξ·∇f σ + a σ ·∇ ξ f σ = Q σ = ∑ ς 

Q σς , (1) 

where Q σς = Q ςσ , ς ≠ σ, is the cross collision term for two 

different species σ (with mass m σ ) and ς (with mass m ς ). 

Obviously, for an N-component system, there will be N such 

equations. In general, the collision term is 

∫ ∫ 

Q σς ˙= K σς (g, n) [ f σ (ξ ′ )f ς (ξ ∗) ′ − f σ (ξ)f ς (ξ ∗ ) ] dn dξ ∗ , 

R 3 ξ 

g·n



Collision kernel and post–collision velocities 

The weight K σς (g, n) = K ςσ (g, n) is a volumetric particle flux or 

collision kernel. In the following, we will discuss only the case of 

Maxwell molecules [7], namely K σς (|g · n|/|g|), where the collision 

kernel is independent of the relative velocity. 

In the previous equation, the post–collision test and field particle 

velocities ξ ′ and ξ ′ ∗ are given by 

ξ ′ = ξ + 2µ σς 

m σ 

(g · n) n, (3) 

ξ ′ ∗ = ξ ∗ − 2µ σς 

m ς 

(g · n) n, (4) 

which means that there are many possible outcomes (ξ ′ , ξ ′ ∗) from 

a given pair of incoming (test and field) particle velocities (ξ, ξ ∗ ), 

depending on the impact direction n. 

The reduced mass µ σς is defined as 

µ σς = m σ m ς /(m σ + m ς ). (5) 




Indifferentiability Principle and H–theorem 

Indifferentiability Principle: when all the masses m σ and 

cross-sections K σς are identical, the total distribution f = ∑ σ f σ 

obeys the single species Boltzmann equation. 

H–theorem: in the case of a mixture, the entropy inequality 

(H–theorem) is 

∑ 

∫ 

Q σ log(f σ )dξ ≤ 0. (6) 

σ 

R 3 ξ 

These properties of the full Boltzmann equations for mixtures will 

guide the design of BGK–type simplified models and consequently 

of the Lattice Boltzmann solvers. 




Exchange relation for momentum (1 of 3) 

The momentum equation is 

∫ 

∂ t (ρ σ u σ ) + ∇ · Π σ = ρ σ a σ + 

R 3 ξ 

m σ ξQ σ dξ. (7) 

Taking into account the symmetries of the elementary collision 

yields (see for example Eq. (48) in the Ref. [8]) 

∫ 

m σ ξQ σ dξ = ∑ 〈∫ 

〉 

K σς (|g · n|/|g|) m σ (ξ ′ − ξ) dn , 

R 3 ξ ς g·n




Let us express the surface element dn in Eq. (10) by using g as 

polar axis, φ as polar angle and θ as azimuthal angle. 

Hence K σς (|g · n|/|g|) = K σς (cos φ) = K σς (φ) and consequently 

∫ 

m σ ξQ σ dξ = ∑ 〈∫ 

〉 

2µ σς K σς (φ) (g · n) n dn . (10) 

R 3 ξ ς 

g·n




Consequently 

∫ 

1 2π ∫ π 

K σς (φ) |g| cos φ 

2 0 0 

where 

∫ 4π 

⎧ 

⎨ 

⎩ 

0 

0 

cos φ 

⎫ 

⎬ 

⎭ sin φ dφdθ = χ σς g. (12) 

χ σς = 1 K σς (φ) (cos φ) 2 dn. (13) 

2 0 

Coming back to Eq. (10) yields 

∫ 

m σ ξQ σ dξ = ∑ 2µ σς χ σς 〈g〉 σς 

= p ∑ B σς y σ y ς (u ς −u σ ), (14) 

R 3 ξ ς 

ς 

where 

B σς = 2µ σςχ σς n 2 

= B ςσ , (15) 

p 

where n is the total particle number density, p is the pressure and 

y σ is the molar concentration, as it will be discussed next. 


Lattice Boltzmann solvers and applications 












Simplified kinetic models 


The full Boltzmann equations for gas mixtures are much more 

formidable to analyze than the equation for a single-species 

system [3, 4, 5, 6]. 

A popular approach is to derive simplified model Boltzmann 

equations which are more manageable to solve. Numerous model 

equations are influenced by Maxwell’s approach to solve the 

Boltzmann equation by using the properties of the Maxwell 

molecule and the linearized Boltzmann equation [12]. 

The simplest model equations for a binary mixture is that by Gross 

and Krook [13, 12], which is an extension of the 

single-relaxation-time model for a pure system — the celebrated 

Bhatnagar-Gross-Krook (BGK) model. 

Many other models exist: Sirovich [14, 15], Hamel [16, 17], ... 



Basic consistency constraints 


1 The Indifferentiability Principle, which prescribes that, if a 

BGK-like equation for each species is assumed, this set of 

equations should reduce to a single BGK-like equation, when 

mechanically identical components are considered, i.e. the single 

fluid description should be recovered [18]. 

2 The relaxation equations for momentum and temperature should 

be as close as possible to those derived by means of the full 

Boltzmann equations [19]. 

3 All the species should tend to a target equilibrium distribution 

which is a Maxwellian, centered on a proper macroscopic velocity, 

common to all the species. 

4 The non-negativity of the distribution functions for all the species 

should be satisfied. 

5 A generalized H theorem for mixtures should hold. 



Simplified AAP model 


Let us consider a simplified version of the AAP model, proposed 

by Andries, Aoki, and Perthame [2], which is based on only one 

global (i.e., taking into account all the species ς) operator for each 

species σ, namely 

∂ˆt f σ + ξ· ˆ∇f σ = λ σ 

[ 

fσ(∗) − f σ 

] 

, (16) 

where 

f σ(∗) = 

[ 

] 

ρ σ 

(2πϕ σ /3) exp − 3 (ξ − u∗ σ) 2 

, (17) 

2 ϕ σ 

and 

u ∗ σ = u σ + ∑ ς 

m 2 

m σ m ς 

B σς 

B mm 

x ς (u ς − u σ ). (18) 



Properties of simplified AAP model 


The target velocity can be easily recasted as 

u ∗ σ = u + ∑ ( ) 

m 

2 

B σς 

− 1 x ς (u ς − u σ ). (19) 

m 

ς σ m ς B mm 

If m σ = m for any σ, then (Property 1) 

u ∗ σ = u + ∑ ς 

( m 

2 

mm 

) 

B mm 

− 1 x ς (u ς − u σ ) = u. (20) 

B mm 

Clearly (Property 2) 

∑ 

x σ u ∗ σ = u + ∑ σ 

σ 

∑ 

( m 

2 

ς 

m σ m ς 

) 

B σς 

− 1 x σ x ς (u ς − u σ ) = u. 

B mm 

(21) 



Diffusive scaling 


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 . 



Regular Knudsen expansion 


Clearly the solution of the BGK equation depends on ɛ. The 

solution for small ɛ is investigated in the form of the asymptotic 

regular expansion 

ρ and q σi are also expanded: 

f σ = f (0) 

σ + ɛf (1) 

σ + ɛ 2 f (2) 

σ + · · · . (24) 

ρ σ = ρ (0) 

σ + ɛρ (1) 

σ + ɛ 2 ρ (2) 

σ + · · · , (25) 

q σi = ɛq (1) 

σi 

+ ɛ 2 q (2) 

σi 

+ · · · , (26) 

since the Mach number is O(ɛ), the perturbations of q σi starts from 

the order of ɛ. Consequently 

f σ(∗) = f (0) (1) 

σ(∗) 

+ ɛf 

σ(∗) + ɛ2 f (2) 

σ(∗) + · · · , (27) 

Regular expansion means ∂ α f (k) 

σ 

= O(1) and ∂ α M (k) = O(1). 



Asymptotic analysis of AAP model 


Collecting the terms of the same order yields 

f (k) 

σ 

= f (k) 

σ(∗) − g(k) σ , (28) 

g (0) 

σ = 0, (29) 

g (1) 

σ = τ σ ∂ S f (0) 

σ(∗) , (30) 

g σ (2) = τ σ [∂ t f (0) 

σ(∗) + ∂ Sf (1) 

σ(∗) − τ σ∂Sf 2 (0) 

σ(∗) 

], (31) 

· · · , 

where ∂ S = ξ i ∂/∂x i and τ σ = 1/λ σ . 

The previous coefficients of the regular expansion allows one to 

derive the macroscopic equations recovered by the AAP model. 




Tuning the single species relaxation frequency 

Taking the first order moments of g (1) 

σ 

where p (k) 

σ 

= ϕ σ ρ (k) 

σ /3. 

yields 

λ σ ρ (0) 

σ [u ∗(1) 

σ − u (1) 

σ ] = ∇p (0) 

σ , (32) 

If λ σ is selected as λ σ = p B mm /ρ, then the previous expression 

becomes 

1/p (0) ∇p (0) 

σ 

= ∑ ς 

B σς y σ y ς [u (1) 

ς − u (1) 

σ ], (33) 

which clearly proves that the leading terms of the macroscopic 

equations recovered by means of the AAP model are consistent 

with Maxwell-Stefan model. 


D2Q9 lattice 



Let us define the AAP model for a set of discrete velocities, 

ɛ 2 ∂f σ 

∂t + ɛV ∂f σ [ ] 

i = λ σ fσ(∗) − f σ , (34) 

∂x i 

where V i is a list of i-th components of the velocities in the 

considered lattice and f = f σ(∗) , f σ is a list of discrete distribution 

functions (change in the notation !!) corresponding to the 

velocities in the considered lattice. 

Let us consider the two dimensional 9 velocity model, which is 

called D2Q9, namely 

V 1 = [ 0 1 0 −1 0 1 −1 −1 1 ] T , (35) 

V 2 = [ 0 0 1 0 −1 1 1 −1 −1 ] T . (36) 



Rule of computation for the list 


The components of the molecular velocity V 1 and V 2 are the lists 

with 9 elements. Before proceeding to the definition of the local 

equilibrium function f σ(∗) , we define the rule of computation for the 

list. 

Let h and g be the lists defined by h = [h 0 , h 1 , h 2 , · · · , h 8 ] T and 

g = [g 0 , g 1 , g 2 , · · · , g 8 ] T . Then, hg is the list defined by 

[h 0 g 0 , h 1 g 1 , h 2 g 2 , · · · , h 8 g 8 ] T . The sum of all the elements of the list 

h is denoted by 〈h〉, i.e. 〈h〉 = ∑ 8 

i=0 h i. 

Then, the (dimensionless) density ρ σ and momentum q σi = ρ σ u σi 

are defined by 

ρ σ = 〈f σ 〉, q σi = 〈V i f σ 〉. (37) 



Continuous equilibrium moments 


Let us introduce the following function 

[ 

] 

ρ 

f e (ρ, ϕ, u 1 , u 2 ) = 

(2πϕ/3) exp 3 (ξ − u)2 

− . (38) 

2 ϕ 

Let us define 〈〈·〉〉 = ∫ +∞ 

−∞ · dξ 1dξ 2 and the generic equilibrium 

moment m pq = 〈〈f e ξ p 1 ξq 2 〉〉. 

All the equilibrium moments appearing in the Euler system of 

equations are the following: m 00 , m 10 , m 01 , m 20 , m 02 , m 11 . Due to 

the lattice deficiencies, only the equilibrium moments m 21 , m 12 

must be considered in addition, for recovering the desired 

Navier-Stokes system of equations. The moment m 22 is arbitrarily 

selected in order to complete the set of discrete moments. 




Simplified continuous equilibrium moments 

Collecting the previous results yields 

¯m c (ρ, ϕ, u 1 , u 2 ) = ρ [1, u 1 , u 2 , 

u 2 1 + ϕ/3, u 2 2 + ϕ/3, u 1 u 2 , 

u 1 u 2 2 + u 1 ϕ/3, u 2 1 u 2 + u 2 ϕ/3, 

ϕ (u 2 1u 2 2 + u 2 1ϕ/3 + u 2 2ϕ/3 + ϕ/9)] T . 

The previous analytical results involve high order terms (like u 1 u 2 2 ) 

which are not strictly required, in order to recover the macroscopic 

equations we are interested in. 

m c (ρ, ϕ, u 1 , u 2 ) = ρ [1, u 1 , u 2 , 

u 2 1 + ϕ/3, u 2 2 + ϕ/3, u 1 u 2 , 

u 1 /3, u 2 /3, 

(u 2 1 + u 2 2)/3 + ϕ/9] T 



Design of discrete local equilibrium 


On the selected lattice, the discrete integrals m σ(∗) , corresponding 

to the previous continuous ones, can be computed by means of 

simple linear combinations of the discrete equilibrium distribution 

function f σ(∗) (still unknown), namely m σ(∗) = Mf σ(∗) where M is 

a matrix defined as 

M = [1, V 1 , V 2 , V 2 

1 , V 2 

2 , V 1 V 2 , V 1 V 2 

2 , V 2 

1 V 2 , V 2 

1 V 2 

2 ] T . (39) 

We design the discrete local equilibrium such as 

m σ(∗) = m c (ρ σ , ϕ σ , u ∗ σ1 , u∗ σ2 ), or equivalently 

f σ(∗) = M −1 m c (ρ σ , ϕ σ , u ∗ σ1 , u∗ σ2 ). In particular the latter provides 

the operative formula for defining the local equilibrium and 

consequently the scheme. 



Discrete operative formula 


Eq. (34) is formulated for discrete velocities, but it is still 

continuous in both space and time. 

Since the streaming velocities are constant, the Method of 

Characteristics is the most convenient way to discretize space and 

time and to recover the simplest formulation of the LBM scheme. 

Applying the second-order Crank–Nicolson yields 

f + σ = f σ + (1 − θ) λ σ 

[ 

fσ(∗) − f σ 

] 

+ θ λ 

+ 

σ 

[f + σ(∗) − f + σ 

where θ = 1/2. 

] 

, (40) 

The previous formula would force one to consider quite 

complicated integration procedures [21]. A simple variable 

transformation has been already proposed in order to simplify this 

task [22]. 



Variable transformation 


(Step 1) Let us apply the transformation f σ → g σ defined by 

g σ = f σ − θ λ σ 

[ 

fσ(∗) − f σ 

] 

. (41) 

(Step 2) Let us compute the collision and streaming step leading 

to g σ → g σ + by means of the modified updating equation 

g σ + = g σ + λ ′ [ ] 

σ fσ(∗) − g σ , (42) 

where λ ′ σ = λ σ /(1 + θλ σ ). 

(Step 3) Finally let us come back to the original discrete 

distribution function g σ + → f σ 

+ by means of 

f + σ 

= g+ σ + θ λ + σ f + σ(∗) 

1 + θ λ + . (43) 

σ 



Problem for mixtures 


In case of mixtures, the problem arises from the (Step 3), which 

requires both λ + σ and f + σ(∗) 

, depending on the updated 

hydrodynamic moments at the new time step. 

Since the single component density is conserved, Eq. (41) yields 

ρ + σ = 〈g + σ 〉, (44) 

consequently it is possible to compute p + σ , ρ + , p + and λ + σ . 

However this is not the case for the single component momentum, 

because this is not a conserved quantity and hence the first order 

moments for g σ + and f σ 

+ differ [23], namely 

〈V i g + σ 〉 = ρ + σ u + σi − θ λ+ σ ρ + σ (u ∗+ 

σi − u+ σi ) = 

= ρ + σ u + σi − θ p+ ∑ ς 

B σς y σ + y ς + (u + ςi − u+ σi 

). (45) 




Solution: solving locally a linear system of equations 

In the general case, Eq. (45) can be recasted as 

〈V i g σ + 〉 = q 

σi + − θ ∑ 

λ+ σ χ σς (x + σ q 

ςi + − x+ ς q 

σi + ), (46) 

ς 

where q 

σi + = ρ+ σ u + σi and χ σς = 

m2 B σς 

. (47) 

m σ m ς B mm 

Finally, grouping together common terms yields 

[ 

] 

∑ 

〈V i g σ + 〉 = 1 + θ λ + σ (χ σς x + ς ) q 

σi + − θ λ+ σ x + σ 

ς 

∑ 

(χ σς q 

ςi + ). (48) 

ς 

Clearly the previous expression defines a linear system of 

algebraic equations for the unknowns q + σi . 


Validation and simple applications 












Single-relaxation-time MixLBM code 



Basic algorithm 

The proposed numerical code is formulated not in the standard 

way (and it is quite inefficient from the computational point of 

view). 

Even though it is not an efficient implementation, the proposed 

formulation is much more similar to any other explicit finite 

difference (FD) scheme. 

This offers some advantages: 

1 it makes easier to implement hybrid schemes, i.e. to mix up kinetic 

and conventional schemes on the same discretization; 

2 it makes easier to compare the LBM scheme with other FD 

schemes, mainly in terms of updating rule; 

3 it makes easier to implement simple boundary conditions, based on 

the concept of local equilibrium. 

Anyway the basic sequence of collision and streaming step is 

preserved. 



Schematic view 


Main loop 


fd(1:nx,1:ny,0:8,1:species) 

do t = 1,nt,1 

do i = 1,nx,1 

do j = 1,ny,1 

call UpdateLatticeData(...,f(:,:)); 

do s = 1,species,1 

fd_new(i,j,:,s) = f(:,s); 

call HydrodynamicMoments(...); 

enddo 

enddo 

enddo 

fd(:,:,:,:) = fd_new(:,:,:,:); 

enddo 



UpdateLatticeData loop 

do k=0,8,1 

iI = i + Incr(k,1); jI = j + Incr(k,2); 

do s=1,species,1 

! BCs rs,uxs,uys in I(i+,j+) 

enddo 


call EquilibriumDistribution(...,feq(:)) 

lambda(s)=...; 

do ik=0,8,1 

fc(ik)=f(ik,s)+lambda(s)*(feq(ik)-f(ik,s)); 

enddo 

f_new(BB(k),s) = fc(BB(k)); 

enddo 

enddo 



UpdateLatticeData loop with variable transformation 

do k=0,8,1 


call EquilibriumDistribution(...,feq(:)) 

lambda(s)=...; 

TRANSFORMATION f(:,s) -> g(:,s) 

do ik=0,8,1 

gc(ik)=g(ik,s)+lambda’(s)*(feq(ik)-g(ik,s)); 

enddo 

g_new(BB(k),s) = gc(BB(k)); 

enddo 

enddo 

BACK-TRANSFORMATION g_new(:,s) -> f_new(:,s) 

COMPUTE CONSERVED MOMENTS 

SOLVE LINEAR SYSTEM FOR NON-CONSERVED MOMENTS 



Ternary mixture 

In case of ternary mixture Eq. (14) reduces to 

n∇y 1 = B 12 y 1 k 2 + B 13 y 1 k 3 − (B 12 y 2 + B 13 y 3 )k 1 , (49) 

n∇y 2 = B 21 y 2 k 1 + B 23 y 2 k 3 − (B 21 y 1 + B 23 y 3 )k 2 , (50) 

n∇y 3 = B 31 y 3 k 1 + B 32 y 3 k 2 − (B 31 y 1 + B 32 y 2 )k 3 . (51) 

The molecular weights are m σ = [1, 2, 3], the internal energies 

are [e σ = 1/3, 1/6, 1/9] and consequently ϕ σ = [1, 1/2, 1/3]. 

The theoretical Fick diffusion coefficient is D σ = α/m σ , where 

α ∈ [0.002, 0.8] and the theoretical Maxwell–Stefan diffusion 

resistance is given by [28] 

B σς = β 

( 1 

m σ 

+ 1 m ς 

) −1/2 

, β ∈ [5, 166]. (52) 



Solvent test case 

A component of a mixture is called solvent if its concentration is 

predominant in comparison with the other components of the 

mixture [10]. 

Let us suppose that, in our ternary mixture, the component 3 is a 

solvent. In particular, the initial conditions for the solvent test case 

are given by 

[ ( )] x − L/2 

p 1 (0, x) = ∆p 1 + tanh 

+ p s , (53) 

δx 

[ ( )] x − L/2 

p 2 (0, x) = ∆p 1 − tanh 

+ p s , (54) 

δx 

p 3 (0, x) = 1 − 2 (∆p + p s ), (55) 

where clearly p(0, x) = ∑ σ p σ = 1 and ∆p = p s = 0.01. 



Solvent test case: simplified transport coefficients 

Hence y 3 

∼ = 1 and consequently y1 ∼ = 0 and y2 ∼ = 0. Under these 

assumptions, Eqs. (49, 50) reduce to 

∇y 1 = −B 13 y 1 (u 1 − v) = B 13 y 1 (v − u 1 ), (56) 

∇y 2 = −B 23 y 2 (u 2 − v) = B 23 y 2 (v − u 2 ), (57) 

Consequently the measured diffusion resistances are given by 

B ∗ 13 = 1 D ∗ 1 

= ∂y 1/∂x 

y 1 (v − u 1 ) , (58) 

B ∗ 23 = 1 D ∗ 2 

= ∂y 2/∂x 

y 2 (v − u 2 ) , (59) 

where, in this test, the Maxwell–Stefan model reduces to the Fick 

model. 



Solvent test case: Fick model 



Solvent test case: Maxwell–Stefan model 



Dilute test case 

A component of a mixture is said dilute if its concentration is 

negligible in comparison with the other components of the mixture 

[10]. 

Let us suppose that, in our ternary mixture, the component 1 is 

dilute. In particular, the initial conditions for the dilute test case are 

given by 

[ ( )] x − L/2 

p 1 (0, x) = ∆p 1 + tanh 

+ p s , (60) 

δx 

[ ( )] x − L/2 

p 2 (0, x) = ∆p 1 − tanh 

+ p s + 

δx 

+(1 − r) (1 − 2 ∆p), (61) 

p 3 (0, x) = r (1 − 2 ∆p) − 2 p s , (62) 

where p(0, x) = ∑ σ p σ = 1, ∆p = p s = 0.01, r = 1/2. 



Dilute test case: Maxwell–Stefan model 



Non-Fickian test case: Stefan tube 

It is essentially a vertical tube, open at one end, where the carrier 

flow licks orthogonally the tube opening [10]. In the bottom of the 

tube is a pool of quiescent liquid. The vapor that evaporates from 

this pool diffuses to the top. 

p 1 (0, x) = p 1 (0, 0) 1 [ ( )] x − L/2 

1 − tanh 

+ p s , (63) 

2 

δx 

p 2 (0, x) = p 2 (0, 0) 1 [ ( )] x − L/2 

1 − tanh 

+ p s , (64) 

2 

δx 

p 3 (0, x) = [1 − p 3 (0, 0)] 1 [ ( )] x − L/2 

1 + tanh 

+ p 3 (0, 0), 

2 

δx 

(65) 

where the constant p s = 10 −4 has been introduced for avoiding to 

divide per zero. 



Stefan tube 


Conclusions 

In the present tutorial, a LBM scheme for gas mixture modeling, 

which fully recovers Maxwell–Stefan diffusion model in the 

continuum limit, without the restriction of the macroscopic 

mixture-averaged approximation, was discussed. 

As a theoretical basis for the development of the LBM scheme, a 

recently proposed BGK-type kinetic model for gas mixtures [2] 

was considered. This essentially ties the LBM development to the 

recent progresses of the BGK-type kinetic models and opens new 

perspectives. 

In the reported numerical tests, the proposed scheme produces 

good results on a wide range of relaxation frequencies. 


References I 

[1] F. Williams. 

Combustion Theory. 

Benjamin/Cumming, California, 1986. 

[2] P. Andries, K. Aoki, and B. Perthame. 

A consistent BGK-type model for gas mixtures. 

J. Stat. Phys., 106(5/6):993–1018, 2002. 

[3] S. Chapman and T. G. Cowling. 

The Mathematical Theory of Non-Uniform Gases. 

Cambridge University Press, Cambridge, UK, 3rd edition, 1970. 

[4] J. H. Ferziger and H. G. Kaper. 

Mathematical Theory of Transport in Gases. 

North Holland, Amsterdam, 1972. 

[5] L. C. Woods. 

An Introduction to the Kinetic Theory of Gases and Magnetoplasmas. 

Oxford University Press, Oxford, UK, 3 edition, 1993. 

[6] S. Harris. 

An Introduction to the Theory of the Boltzmann Equation. 

Dover, Mineola, NY, 2004. 


References II 

[7] C. Cercignani 

The Boltzmann Equation and Its Applications. 

Applied Mathematical Sciences, Springer-Verlag, 1987. 

[8] P. Asinari 

Nonlinear Boltzmann equation for the homogeneous isotropic case: Minimal deterministic 

Matlab program. 

accepted for publication in Computer Physics Communications. 

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