# Computational Study of Inertial Flows in Helical Microchannels
## 4 Results--Effects of the Simulation Parameters on the Cell's Trajectory
### Reference Solution  
The trajectory of solid cells is influenced by their size and, if they are small enough, by their initial position. In Figure 3, the cell with a radius of 3.91 \(\upmu\)m has only one stabilised position, which is eventually reached from any starting point within the cross-section. The cell with a radius of 2 \(\upmu\)m has, however, two different stabilised positions. The upper one is reached when the starting position of the cell is approximately in the upper 3/4 of the channel, while the lower one is reached from the lower quarter of the channel cross-section. In Figure 4, there is a dashed separation line that divides the cross-section into two parts. Smaller cells starting from the lower part converge to the lower stabilized position,  
Figure 2: (**a**) Streamlines of an inertial flow. Negative values on the x-axis are for the inner wall of the cross-section, and positive values are for the outer wall of the cross-section. Projection of the streamlines to a plane that is perpendicular to the local axis. (**b**) Projection of the stream-tracer trajectory to a plane that is perpendicular to the local axis. The initial positions of the tracers are marked with diamonds. whereas those starting from the upper part converge to the upper stabilized position. This line is defined by the trajectories of the smaller cells, and it is at around 15 \(\upmu\)m below the centre of the cross-section for this case. Separation of cells with two different sizes can be achieved by a proper installation of inlets and outlets to and from the helical structure. If the inlet of a cell mixture is located below the dashed line, smaller cells converge to lower stabilized position and large cells to the upper one resulting in their separation. Finally, two outlets can be installed so that one collects larger cells and one the smaller ones. We can see in this picture that the the stabilized position is not reached directly but rather after many concentric turns following a spiral path. Therefore, in the next section, we do not show the exact stabilised position. We show only the onset of the trajectory, from which it will be clear whether there are one or two different stabilised positions. For the clarity of the images, the trajectories of the reference solution are presented only by their onset. The position of the dashed separation line can also help us to define minimal length of the helical tube for sorting the cells. We need to ensure the helical tube is long enough, and thus the large cells will traverse through the separation line from any starting position in the lower part of the cross-section. The smaller cells will all remain below the separation line, as they will become stabilised in the lower part of the channel. The larger cell traverses the separation line relatively fast in this case. It crosses the line once, and then it stays in the upper part during its elliptical converging trajectory. This might not always be the case. For certain combination of the experimental parameters, the larger cell may re-descend to the lower part of the cross-section and then go back up again several times during its elliptical motion. However, in our case, the larger cell needs to travel a total distance of 370 \(\upmu\)m in the direction of the global helix axis, which is  
Figure 4: The separation line between the upper and lower stabilised position for cells with 2 \(\upmu\)m radius. Cells starting their trajectory below the separation line will reach the lower stabilised position, while the cells starting above this line will reach the upper stabilised position. Figure 3: The trajectory and stabilised positions for the reference case. The starting positions of cells are marked with diamonds. Negative values on the x-axis are for the inner wall of the cross-section, and positive values are for the outer wall of the cross-section. (**a**) The cell with a radius of 3.91 \(\upmu\)m has only one stabilised position. (**b**) The cell with a radius of 2 \(\upmu\)m has two different stabilised positions.