# Multispecies

Multispecies approach implemented in the code is based on the paper "Multi-species Lattice Boltzmann Model and Practical Examples. Short Course material Pietro Asinari PhD."

Equlibrium distribution function for multispecies is given in the paper as

where, $s^{\sigma}_0 = (9-5 \phi^\sigma)/4$ $s^{\sigma}_{\alpha} = \phi^\sigma$ for $1\leq\alpha\leq8$ and $\phi^\sigma=min_\varsigma(m^\varsigma)/m^\sigma$, $p_\sigma=\rho_\sigma\phi^\sigma/3$ $m^\sigma$ is molecular weight for species $\sigma$.

$u^*_{\sigma}$ is given as $x_\sigma= \rho_\sigma/\rho$

Relaxation time is given as

$\lambda_\sigma = \frac{pB_{mm}}{\rho }$

# Multispecies: Variable Transformation

A semi-implicit Lattice Boltzmann equation is given as,

Variable transformation presented in above paper involves three steps

## Step 1. Transforming f -> g

$g^\sigma = f^\sigma-\frac{1}{2}\lambda^\sigma[f^{\sigma(eq)}-f^\sigma]$

## Step 2. Stream and Collide i.e g -> g^+

$g^{\sigma,+} = g^\sigma + \frac{\lambda^\sigma}{1+\frac{1}{2}\lambda^\sigma} [f^{\sigma(eq)}-g^\sigma]$

## Step 3. Back Transormation to f i.e g^+ -> f^+

In back tranformation, to compute feq we need $\rho$ and $\mathbf{u}_\sigma$ $\rho$ can be computed directly from g $\rho^+_\sigma = $

where as the $\mathbf{u}_\sigma$ computed by solving the linear system of equation given below

where, $\chi_{\sigma \varsigma}$ is,

       !West
u_n(1) = - uxstar(s)
!south
u_n(2) =             - uystar(s)
!bottom
u_n(3) =                         - uzstar(s)
!east
u_n(4) =   uxstar(s)
!north
u_n(5) =               uystar(s)
!top
u_n(6) =                           uzstar(s)
!bottom south
u_n(7)
!top south
u_n(7)
!bottom north
u_n(7)
!top north
u_n(7)
!bottom west
u_n(7)
!bo
u_n(7)
u_n(7)
u_n(7)
u_n(7)