# Computational Study of Inertial Flows in Helical Microchannels
## 3 Setup of the Numerical Simulation  
The geometry of the baseline reference solution is depicted in Figure 1. We have chosen this geometry because we observed different type of trajectories for two different cells in this helical channel, and consequently different stabilised positions. One such trajectory is indicated in the figure on the left as a blue helical line. Its projection onto the cross-section (cross-section = a plane that is perpendicular to the local axis of the helix) is indicated as a dense blue spiral on right of the figure.  
The simulation box contained only one loop of the helix, in order to reduce the computational cost. The cyclic boundary conditions ensure the continuity of the simulation on both ends of the channel, and thus the helical channel is modelled as a channel with infinite length.  
Both cells used in simulations were spherical and almost rigid, they differed only by their size. We chose only two different cell diameters of 4 and 7.82 \(\upmu\)m as two use-cases. The actual values were motivated by the size of red blood cells and smaller cells. For sorting larger cells, the dimensions of the helix could be scaled up with similar qualitative results. The discretization of a cell's inner volume was used in order to ensure their rigidity. To maintain a similar density of discretization points with respect to the discretization of the fluid, the larger cell was discretized with more nodes. The sizes of cells and their discretization are listed in Table 1. The spatial discretization of the fluid was 1 \(\upmu\)m, and all the simulations were run with a simulation timestep of 0.01 \(\upmu\)s.  
Starting from this reference simulation, we modified several parameters linked to the fluid as well as parameters linked to the geometry of the helical channel. The examined parameters and their values are listed in Table 2. As the maximum fluid velocity, we took the maximal velocity magnitude found in the velocity field. The maximum velocity was oriented parallel with the local axis of the helix curvature. Throughout all the simulations, we used a fixed fluid density of water.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Cell Radius [\(\upmu\)m]** & **N. of Surface Nodes** & **N. of Inner Nodes** \\ \hline Smaller cell & 2 & 113 & 13 \\ Bigger cell & 3.91 & 303 & 134 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The size and discretization of the cells present in the set of simulations.  
Figure 1: The geometry of the helix used in the reference simulation. The blue curve at the right side stands for the projection of a cell trajectory to a cross-section that is perpendicular to the local helix axis.  
The \"Reference\" line in the table stands for the reference simulation. The results of all the other simulations were compared with the reference simulation to establish the stability analysis. In each simulation, we modified only one parameter among the considered parameters listed in Table 2. This means that each field in rows Mod 1-Mod 4 corresponds to one simulation setting, in which we kept all the parameters from the reference simulation, except the one parameter, which was modified as it is specified in the field's value. The majority of these sensibility studies were run twice--once with the smaller cell and a second time with the larger cell. Only in the case of the pitch and the tube radius, we chose different parameter values for different cell sizes.  
The range of Reynolds numbers used in simulations was from 2.5 to 10, and the reference simulation had a Reynolds number of 5. The Dean numbers ranged from 1 to 6, and the reference simulation had Dean number of 3.