## 1 Introduction
### Modeling of Cell Separation/Partitioning
#### 1.2.2 Conventional CFD Methods  
These methods are based on continuum assumption and include discretizing conservation equations coupled with boundary conditions. Widely accepted conventional CFD techniques include finite difference, finite volume, and finite element methods (FEM). There are other, yet infrequent methods, such as boundary element, spectral element, and other in-house developed software tools. There are two reviews covering these tools in the context of microfluid devices in general and, capturing rare cell systems. [PERSON] [69] provided an overview of traditional and emerging CFD techniques for integrated microfluidic devices covering thermal and chemical microscale flow and species transport simulation and cellular/particulate transport. On the other hand, the recent work of [PERSON] and [PERSON] [70] centers on overviewing these tools on cell sorting and rare cell capturing, such as circulating tumor cells. Herein, a brief introduction of some of the main traditional CFD tools is given. These techniques can be clustered by the particular meshing approach to discretize the governing equations such as finite difference, finite volume, FEM. In finite difference method, the governed partial differential equations are approximated with Taylor series based on the values of neighboring nodes. This, then, renders a system of algebraic equations to be solved using a variety of well-developed explicit and implicit solvers. This method has the advantage of generally being  
Figure 4: A few examples of past LBM simulations of blood cells in different domains. (**a**) Pressure and vector fields of three red blood cells (RBCs) flowing a white blood cell (WBC) in a blood vessel branch [65]. (**b**) Clusters distribution of RBCs in blood-like suspension [66]. (**c**) Simulation of the RBCs with platelets [67]. (**d**) RBCs simulated in a blood channel [68]. the easiest to implement, though, it is practically limited to well-structured grids and thus, only advantageous in simple geometries (e.g., capillaries or channel cross-sections) (refer to [PERSON] and [PERSON] [71] for more detail). Writing codes mostly use this method in generic programming software such as C, C++ FORTRAN, etc. In finite volume method, partial differential equations describing conservation laws are integrated and evaluated at the surfaces and finally discretized for very small, but finite-sized elements constituting the whole body. Unlike finite difference method, this method does not demand a structured grid and is thus, suitable for more complex and commonly encountered geometries (e.g., looping channels used in on-chip capillary electrophoresis). Since the conservation equations are to be applied on each irregular volume, the definition of derivatives and following Taylor series expansion to represent such is very challenging. It has been shown that this method is best suited for problems where the viscous terms are absent as opposed to the low Reynolds number flows encountered in microfluidics, in which these terms are dominant [71]. This method is incorporated in many commercialized software packages. For more detail on fundamentals, refer to [72]. In FEM, like finite volume method, the whole body is subsidized to smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then systematically recombined to form a more extensive system of equations that models the entire elements. Unlike the finite volume method, at this point, whereby the PDEs are to be discretized, in FEM variational methods from the calculus of variations are used to approximate a solution by minimizing an associated error function. FEM shares many of the same advantages of the finite volume method, most importantly, the ability to manage arbitrary grids and complex geometries.