# Computational Study of Inertial Flows in Helical Microchannels
## 2 Theoretical Background
### Lift Force  
The lift force is composed from the wall-induced force and shear-gradient-induced force. The combination of these two forces lies in a Segre-Silberberg effect in straight channels [12]. The stabilised positions of particles in straight channels depend on the cross-section of the channel, the physical parameters of the particles, and the physical parameters of the fluid [5,13].  
When a cell is sufficiently close to a wall, the wall-induced force arises from different pressures in the fluid that flow around the cell. In the constriction between the wall and the particle, the pressure will not be changed. On the other side of the particle, the streamlines will become denser, and thus the fluid velocity will be larger; therefore, the pressure will decrease. This will cause a force directed away from the walls of the channel. The formula that describes the magnitude of the wall-induced force is  
\[F_{LW}=C_{W}\rho U_{max}^{2}a^{6}/D_{h}^{4}, \tag{1}\]  
where \(C_{W}\) is a lift coefficient dependent on the particle position and on the Reynolds number [14], \(\rho\) is the fluid density, \(U_{max}\) is the maximal fluid velocity, \(a\) is the particle diameter, and \(D_{h}\) is the hydraulic diameter of the channel.  
The shear-gradient-induced force arises also from a different velocity profile around the cell; however, this one takes its origins in the parabolic velocity profile. The magnitudes of fluid velocities on the two sides of the cell are different. The force that is acting on cell due to this difference pushes the cell closer to the wall. The formula that describes the magnitude of the shear-gradient-induced force is  
\[F_{LS}=C_{S}\rho U_{max}^{2}a^{3}/D_{h}, \tag{2}\]  
where \(C_{S}\) is a shear coefficient dependent on particle position and on Reynolds number [14].  
The final lift force is sum of the wall-induced force and shear-gradient-induced force. The formula for the total lift force is not the same across the literature. In [14], where a squared cross-section was examined, the final lift force is either the wall-induced force or the shear-induced force, as a function of the particle position. For particles closer to the centre of the cross-section, the shear-induced force is dominant, whereas for particles that are closer to the channel walls, the wall-induced force is dominant.  
Several authors agree on the following formula for the lift force, e.g., [7],  
\[F_{L}=C_{L}\rho U_{max}^{2}a^{4}/D_{h}^{2}, \tag{3}\]  
where \(C_{L}\) is the lift coefficient as a function of the particle position within the channel cross-section, considered to be 0.5 for simplification. This formula is taken from a study where a spherical particle in a straight channel with plane Poiseuille flow at large channel Reynolds number was examined [15].  
In our study, the geometry and the range of Reynolds numbers is more similar to [14]; therefore, we consider the same expressions for the lift force as used in this study.  
The final formula, however, is tricky: As we have seen, even in straight channels, it is impossible to write down an explicit dependence of the forces depending on the flow parameters. Even the well-accepted formula (3) appearing in [15] contains a lift coefficient depending on the position of the particle in a channel and the channel Reynolds number. This dependence is not listed in the original works. Moreover, in [14], the authors showed that the term \(a^{4}/D_{h}^{2}\) changes to \(a^{3}/D_{h}\) in near-centre regions and to \(a^{6}/D_{h}^{4}\) in near-wall regions without further specifying what \"near\" means. Not to mention that the \(C_{L}\) might change as well. Therefore, the actual simulations of scenarios with curvilinear channels are crucial for the correct determination of stable positions in channels.