# Computational Study of Inertial Flows in Helical Microchannels
## Appendix A Dean Force Significance over Lift Force  
In [8], the dimensions of the cross-section were approximately 10-times larger than the dimensions of the treated particles, and the curvature radius was 200-times larger than the dimension of the cross-section. The helical loop was relatively flat in this case, the value of the helical pitch was small in comparison with the curvature radius; therefore, the local geometry of the loops was not very different from an annulus geometry. An advantage of such geometry is its relatively good symmetry in Dean vortices due to its similarity to the annulus. Another advantage is the stability of the Dean and lift forces acting on articles all along the microfludic channel due to its helical structure: the geometry of the loops does not change over the particle trajectory. The Reynolds number in this study varied from 8 to 50. To estimate the ratio of the lift force over the average Dean force in our computational studies, we can use (1), (2) and (7). The ratio is in the range of 0.2-40 if we consider the wall lift force with coefficient 10 or in the range of 0.6-4.6 if we consider the shear lift force with coefficient 0.01. The coefficients were estimated from [14]. The ratio is, in both cases, not far from 1 if we take into consideration that the lift force coefficients and the Dean force are only estimated. In several experiments of this work, the equilibrium was obtained and the particles become focused, in several experiments the particles did not focused. Figure 14: The proposed shape for inlets and outlets for the helical channel. We can estimate the ratio of the lift force over the Dean force in experiments of this work also by using (1), (2) and (9). The ratio is in the range of 0.007-1.28 for case of wall-induced lift force, and 0.02-0.16 for case of shear-induced lift force. These ranges should be close to 1, in the case we would take into account also a missing parameter mentioned previously for (8) and (9). In [9], the dimension of the squared channel cross-section was 4 to 20 times larger than the dimension of the particles, and the curvature of the channel was 20 to 70 times larger than the cross-section dimension. The helical loop was not negligible, it was comparable with the curvature radius of helical loops. The Reynolds numbers were in the range of 20 to 200. To evaluate the lift force over the Dean force ratio, (7) is less suitable as in previous case. The curvature geometry is different: The curvature radius is approximately 10 times smaller. Therefore, the formula does not necessarily fit the Dean force. Let us estimate the ratio, while keeping this potential error in mind. With using (1), (2) and (7), the ratio of the wall-induced lift force over the average Dean force is in the range of 0.001 to 0.1 for cases where the equilibrium position of the particles was not reached. For particles that are stabilised, the wall-induced lift force over the average Dean force was in the range of 0.002 to 15.7. The wall lift coefficient was estimated to be 10. The ratio of the shear-induced lift force over the average Dean force was in the range of 0.14 to 1.9 for the cases where the equilibrium was not reached, and in the range of 0.4 to 13.1 in the cases with particles that reached an equilibrium position. The shear lift coefficient was estimated to be between 0.05 and 0.14 in the function of the Reynolds number [14]. These lift force over Dean force ratios are, similarly to the previously described study, not far from 1. Therefore, this is consistent with the fact that the particles reached their stabilised position in some cases. By using (1), (2) and (9) to evaluate the lift force over Dean force ratio, we obtained values of 0.00002-0.002 for the case of wall-induced lift force in experiments where the particles did not focus, and 0.0001-0.3 for experiments where the particles did focus. For the shear-induced lift force, the ranges were 0.002-0.08 for experiments without successful focusing and 0.008-0.55 for experiments with successful focusing.