# Research on the Inertial Migration Characteristics of Bi-Disperse Particles in Channel Flow
## 2 Materials and Methods  
### Lattice Boltzmann Method  
The lattice Boltzmann method (LBM) comes from lattice gas automata, which is a numerical simulation method based on the discrete Boltzmann equation. LBM is considered to be an effective numerical simulation method for studying multiphase flow. It is a kind of mesoscopic model that has the advantages of both macroscopic and microscopic models.  
The main idea of LBM is to numerically solve the discrete Boltzmann equation to simulate the macroscopic motion of fluids. LBM models usually include discrete velocity models, equilibrium distribution function, and evolution equation for the distribution function. The discrete velocity model adopted in this paper is the D2Q9 model ([PERSON] et al., 2010), which applies to two-dimensional space and contains nine discrete velocity directions ([PERSON] et al., 2010):  
\[e_{i}=\left\{\begin{array}{l}(0,0)\quad\quad i=0\\ c\left(\cos\left[(i-1)\frac{\pi}{4}\right],\sin\left[(i-1)\frac{\pi}{4}\right] \right)\,i=1,2,3,4\\ \sqrt{2}c\left(\cos\left[(2i-1)\frac{\pi}{4}\right],\sin\left[(2i-1)\frac{\pi }{4}\right]\right)i=5,6,7,8\end{array}\right., \tag{1}\]  
where \(i\) represents the component of the function in the \(i\) direction; \(e_{i}\) represents the direction vector of the discrete velocity in the \(i\) direction; and \(c=\Delta x/\Delta t\), where \(\Delta x\), and \(\Delta t\) are the grid step size and the unit lattice time, respectively, in the standard LBM \(\Delta x=1\), \(\Delta t=1\).  
The equilibrium distribution function is as follows:  
\[f_{i}^{eq}=\rho w_{i}\bigg{[}1+\frac{e_{i}\cdot u}{c_{s}^{2}}+ \frac{\left(e_{i}\cdot u\right)^{2}}{2c_{s}^{4}}-\frac{u^{2}}{2c_{s}^{2}}\bigg{]} \tag{2}\] \[c_{s}=\frac{1}{\sqrt{3}},\,w_{i}=\left\{\begin{array}{l}4/9\,i =0\\ 1/9\,i=1,2,3,4\\ 1/36i=5,6,7,8\end{array}\right.,\]where \(c_{s}\) is the speed of sound; \(w_{i}\) is the weight factor; and \(\rho\) and \(u\) represent the fluid density and velocity, respectively. The evolution equation of the distribution function with the external force term is as follows:  
\[f_{i}(\mathbf{x}+\Delta t\mathbf{e}_{i},t+\Delta t)=f_{i}(\mathbf{x},t)+\frac{1}{\tau}\Big{[} f_{i}^{eq}(\mathbf{x},t)-f_{i}(\mathbf{x},t)\Big{]}+\Delta t\cdot F_{p}, \tag{3}\]  
where \(\tau\) is the relaxation time, \(f_{i}(x,t)\) is the distribution function at the position \(x\) and time \(t\), and \(F_{p}\) is the external force term that drives the Poiseuille flow. Using the external force model with good stability proposed by [PERSON] et al. [17],  
\[F_{p}=\left(1-\frac{1}{2\tau}\right)\frac{(\mathbf{e}_{i}-\mathbf{u})\cdot\mathbf{F}_{b}}{ c_{s}^{2}}f_{i}^{eq}(\mathbf{r},t), \tag{4}\]  
where \(\mathbf{F}_{b}\) is the body force. The fluid density and velocity are calculated as follows:  
\[\rho=\sum f_{i}\mathbf{u}=\frac{1}{\rho}\sum f_{i}\mathbf{e}_{i}+\frac{\Delta t}{2\rho }\mathbf{F}_{b}. \tag{5}\]  
Navier-Stokes equations can be derived through the Chapman-Enskog expansion and have second-order accuracy in both time and space [18].  
The hydrodynamic force experienced by the particles is calculated by means of momentum exchange, and the specific process is as described in [10].