# Computational Study of Inertial Flows in Helical Microchannels
## 1 Introduction  
The precise control of the cell motion is essential in microfluidics. One method to do this is to exploit inertial effects. Inertial microfluidics are utilized to sort different types of cells, as a function of their size and their physical properties, such as deformability and shape. There are different channel shapes that are suitable for cell separation or focusing--for example, a 2D spiral channel [1, 2], a laminar vortex technology [3], or zig-zag curved channels [4]. A general overview of the available techniques and their physical background can be found in [5], and a description of inertial techniques with respect to clinical use can be found in [6].  
In this study, we consider almost rigid spherical cells in a helical channel. Inertial sorting in helical channels was also used in [7] where the authors considered a helical channel with a trapezoid cross-section. Their aim was to detect _E. coli_ bacteria in a real food matrix. In [8], a commercially available capillary tube with a squared cross-section was used to focus micro-beads of different sizes. Other aspects of particle focusing within a helical micro-channel were studied in [9, 10].  
The advantage of a helical channel is that the particle separation efficiency can be adjusted by modifying the geometry of the channel--by adjusting the 3D coiling radius of a single device rather than using multiple devices with a different designs. This idea was exploited by using flexible pipes used to create the helical channel [8, 9]. The basic principle of inertial sorting is the balance of the Dean force and the lift force acting on a particle. The particle finds its stabilized position within the cross-section in a place where the magnitudes of these two forces are almost equal and the directions are the opposite.  
This principle is often difficult to achieve since both the Dean force and the lift force partially depend on the same quantities with different orders of magnitude. It is delicate to find a balance so that one force does not dominate the other [11]. In this study, weconsidered a different approach. We considered Dean-force-dominated flows where the stabilized position is not the result of an equilibrium between the Dean and the lift force.  
The Dean force is dominant over the lift force in the examined geometries. Therefore, the stabilized position is mainly defined by a place where the Dean force vanishes, which is close to the centre of the Dean vortices. In this paper, we show that smaller cells tend to stabilise in the lower part of the channel and larger cells in the upper part. This finding has a practical implication: By suitable design of the channel output with two branches, one can collect larger cells and smaller cells in different outlets. In our work, we show that this is possible, and we suggest concrete dimensions of the helix to achieve this behaviour.  
In the following sections, we first present the theoretical background for our approach in sorting cells in a helical channel. Then, we present a setup for the numerical simulation where the sorting of microbeads was observed. After that, we perform a sensibility analysis--we examined the sensibility of the results to the modification of specific physical parameters linked to fluid flowing through the channel, or parameters linked to channel geometry. In the end, we present also a stability analysis--a sensibility analysis of the results with modification of the numerical parameters.