## 3 Results
### Periodic Cell Stretching  
A possible way to measure the characteristic time-scale of deformation for a RBC is by investigating the mechanical response of the RBC to a periodically varying force. A similar investigation was carried out by [PERSON] et al. [52] in optical-tweezers experiments. They attached silica beads to the opposing ends of the long axis of the RBC. One of these beads is kept stationary, whilst the other bead is moved with a constant velocity farther and then back. In our setup, we applied a periodic sawtooth-like force function over time (see Figure 5). We matched the magnitude of the force to the values reported in [52]. To imitate the attachment surface of the silica beads, the forcing was applied on five per cent of the membrane area on opposing ends of the RBC.  
Figure 5 shows the mechanical response of an RBC to the periodic stretching. There is a good qualitative agreement in the behaviour, however, the quantitative characteristic time-scale of the results are significantly different from those reported by [PERSON] et al. [52] since they prescribed the movement and inferred the forces from the deformation of the beads. In our case, the magnitude of the stretching force is prescribed and we measure the deformation versus time. A similar hysteresis loop is present,  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(z_{0}\) & \(3U\) & \(b\) & \(R^{2}\)-Value & \(UR_{eff}^{3}\) & \(\Lambda\) \\ \hline
5.000 & 0.328 & \(-0.143\) & 0.990 & 10.627 & 1 \\
9.000 & 0.389 & 27.845 & 0.996 & 12.6036 & 1 \\
5.000 & 0.215 & 1.997 & 0.991 & 6.966 & 5 \\
9.000 & 0.265 & 29.580 & 0.998 & 8.586 & 5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Fitting parameters for the non-dimensional data represented in the right image of Figure 4.  
Figure 4: **(Left)** Migration of two RBCs starting at 5 \(\upmu\)m and 9 \(\upmu\)m away from the channel wall, respectively. **(Right)** Non-dimensional trajectories and their linear fit to approximate [PERSON]’s prediction (Equation (4)).  
as can be seen in the right side of Figure 5, though the shape and the magnitude of the elongation (as expected) do not reproduce the experimental results. With the increase of the frequency of the forcing, the viscous effects are increasing as well and the hysteresis loop is widening, as depicted in Figure 6.  
Mechanical effects that can contribute to such a loop might originate from inertial fluid effects or the viscoelastic behaviour of the cell. For high frequency forcing, the pure inertial effects of the fluid might play a role, however, the stretchings in Figure 5 were done with low velocity (on the time-scale of 0.35 s), which resulted in viscosity dominated Stokes flow around the cell membrane. This is also  
Figure 5: Stretching results with a period of 0.35 s: (**Top left**) The elongation of the longest axis of the RBC (\(\Delta L\)) over time. (**Bottom left**) The magnitude of the force applied on opposing membrane areas over time. (**Right**) The applied force magnitude versus the corresponding elongation of the cell.  
Figure 6: Stretching results with a period of 10 ms: (**Top left**) The elongation of the longest axis of the RBC (\(\Delta L\)) over time. (**Bottom left**) The magnitude of the force applied on opposing membrane areas over time. (**Right**) The applied force magnitude versus the corresponding elongation of the cell.  
supported by the observation that the increased inner viscosity had little effect and produced almost the same curve in Figure 5 as the case with \(\Lambda=1\). Therefore, the viscous effects on deformations on the time-scale of several hundreds of milliseconds are governed by the membrane viscosity rather than the internal cytoplasmic viscosity. In Figure 6, the period of the stretching cycle is closer to the typical time-scales arising during cell-cell collisions and the two curves (\(\Lambda=1\) and 5) are starting to deviate. This effect strengthens as the frequency of the forcing period increases.