## 3 Results
### Many-Cell Experiments  
Many-cell cases are employed to simulate the collective transport dynamics of blood in vessel-like structures. The verification of the added viscosity contrast to the HemoCell framework consists of simulations with a large number of RBCs in a straight circular channel flow where the channel has a diameter of 70 \(\upmu\)m and a length of 140 \(\upmu\)m. The openings of the channel are periodic and the flow is driven by body force. For an overview of the simulation setup see Figure 11.  
Generating the homogeneous random initial positions of RBCs is done using a short kinetic simulation of enclosing ellipsoids which is described in [25]. With this method, we generated three different initial configurations of RBCs. These configurations differ in haematocrit, with \(H_{d}=\) 20%, \(H_{d}=\) 28% and \(H_{d}=\) 33%. The only present cells are the RBCs as they dictate the bulk mechanical behaviour of blood flows and the simulations are performed with and without the internal viscosity contrast and for \(\text{Re}=\) 0.5 and \(\text{Re}=\) 1.3. The first fundamental transport property that is investigated is the relative apparent viscosity of the vessel section. [PERSON] et al. [12] collected the available experimental data and derived an empirical formula for the relative apparent viscosity of blood with respect to the pipe diameter for different haematocrit values. We measured the relative apparent viscosity, \(\mu_{r}\), by finding the ratio between the  
Figure 10: \(E_{yy}\) component of the rate of strain tensor just before a collision event in bulk shear rate of 1000 \(\text{s}^{-1}\): (**Left**) with viscosity contrast; and (**Right**) without viscosity contrast. Note that the elongational strain is higher in the case of added viscosity contrast. Figure 9: The average platelet displacement after a collision with a RBC depending on the initial collision distance between the cells, \(h\): (**Left**) \(\dot{\gamma}=4000\)\(\text{s}^{-1}\); and (**Right**) \(\dot{\gamma}=400\)\(\text{s}^{-1}\). observed average fluid velocity and the theoretical average velocity of Poiseuille flow based on the driving body force:  
\[\mu_{r}=\frac{\overline{u}_{Poiseuille}}{\overline{u}_{Cellular}}, \tag{6}\]  
where \(\overline{u}_{Poiseuille}\) is the theoretical average velocity of the channel flow of pure plasma and no cells and \(\overline{u}_{Cellular}\) is the average velocity of the channel flows with RBCs using the same driving body force. The top left and bottom images in Figure 12 show that the differences in \(\mu_{r}\) that are introduced by the viscosity contrast are larger when the average flow velocity is increased. This is caused by the higher shear rates which are present in these flows. They have similar effects on the bulk flow to what we observed for single cell collisions. Due to the short time scales of collisions, the increase in cytoplasmic viscosity is decreasing the deformability of the cells which results in a higher apparent viscosity. As before, this suggests a higher importance of modelling the viscosity contrast for higher shear rate flows. As the Reynolds number is increasing, the shear rate is increasing accordingly in these simulations, which is decreasing the relative apparent viscosity of blood as it is a shear-thinning fluid. It has been captured in numerous experiments (e.g., [57; 58]) and several empirical models have been proposed to describe this behaviour on the continuum level (such as the Carreau-Yasuda and Casson models [59]). The top right image in Figure 12 shows a qualitatively good agreement with the experimental results. Please note that the saturation of the relative apparent viscosity for higher Reynolds numbers is not reached in these simulations due to the numerical limitations that arise as the Reynolds number passes a certain threshold.