## 1 Introduction
### Modeling of Cell Separation/Partitioning
#### 1.2.1 Lattice Boltzmann Method  
When LBM first appeared on the front page of _Washington Post_ on November 19, 1985, a dramatic enthusiasm was brought into computational methods, which was diminished later by revealed limitations, mostly because of convergence issues and computational expenses at the time, but these issues have been alleviated over the period since then. This CFD method is based on the kinetic theory, which correlates macroscopic properties of a medium to basic mechanical laws governing the behavior of its constituents as hard-sphere particles with elastic collision. State of each molecule is described by two variables, position(x(x,y,z)) and velocity (v(v,v,v,v,z)) in space. Since it is challenging, if possible, to go after each molecule, instead, a velocity distribution function, f(x,v,t), which is the velocity number density of molecules at position \(x\) and speed \(v\) at a time \(t\) is defined; and then is used to obtain macroscopic flow properties such as pressure, temperature, density, etc. Then Boltzmann equation, which expresses a balance between particle/molecule transport (streaming) and collision must be  
Figure 2: Different levels of numerical methods used in cellular flow manipulation. Adapted from [54]. analytically/numerically solved to find the mentioned velocity distribution function. This equation (in the absence of external forces) can be written as:  
\[\left(Df/\text{Dt}\right)_{\text{transport}}=\left(Df/\text{Dt}\right)_{\text{ collision}}\rightarrow\frac{\partial f(x,v,t)}{\partial t}+v.\
abla f(x,v,t)=C(x,v,t) \tag{1}\]  
where \(x\), \(v\), and \(t\) represent position, velocity of the particles, and the associated time during the simulation. In this equation, \(C\) models the pairwise collision among particles. There exist, several models, to describe the function \(C\): BGK, the classic and reliable method to solve problems with low Reynolds number (Re) (for more technical details, refer to [55]). MRT, enhances stability by presenting relaxing parameters to ensure stability [56]. A regularized model, renders better accuracy and stability compared to BGK by eliminating higher-order, non-hydrodynamic terms from the particle populations [57]; and Entropic model in which relaxation time is tailored locally to avoid an entropy decrease during the collision and ensuring unconditional numerical stability [58]. Another essential key input is the lattice types on which these models are operated. Several different lattices, both cubic and triangular, and with or without rest particles in the discrete distribution function exist. A typical way of clustering the various methods by lattice is the DnQm scheme. Here \"Dn\" stands for \"n dimensions,\" while \"Qm\" stands for \"m speeds.\" For example, D3Q15 is a 3-dimensional lattice Boltzmann model on a cubic grid, with rest particles present. Each node has a crystal shape and can deliver particles to 15 nodes: each of the 6 neighboring nodes that share a surface, the 8 neighboring nodes sharing a corner, and itself. It has been shown that the solution of the Boltzmann equation can represent the NS equations at low Mach number. Having this function, one can configure the 3D flow field of the system. Among many advantages, one can mention clear physical pictures with the detailed resolution, easy implementation of intricated boundary conditions with simple mechanical rules such as bounce-back and reflection and dynamic interfaces, and fully parallel algorithms for implementing high-performance computing. Nevertheless, as a natural-born dynamic scheme, the LBM is not a method of choice for steady-state analysis; and also is not well-versed for body-fitted coordinates and adaptive time-stepping [59]. As of now, there are several open-source software such as J-lattice-Boltzmann, SunlightLB, OpenLB, LIMBES, LBSim, LB2D, LB3D, Palabos, LBM-C, Taxila LBM, and Sanifish; and commercial ones like PowerFlow, XFlow, FlowKit, LBHydra and utraFluid taking advantage of this method.