## 5 Results and Discussion
### The Results of Microflows Hydrodynamics Simulations  
In Figure 15, the normalized velocity distributions, velocity vectors, and velocity streamlines in the middle of the height of a channel for three channels' configurations are shown.  
Figure 14: Comparison of normalized velocity profiles at the middle of the channel’s height \(u/u_{m}\) and y-mean velocities \(u_{m\gamma}/u_{m}\). The highlighted values were calculated for the S&M model. \(l\)–dimensionless distance from the center of profile (\(-0.5a<l<0.5a\)).  
Figure 13: Comparison of Spiga & Morini model with [PERSON]’s model: (**a**) Normalized velocity profiles–surface plot of S&M model and red dots for PM model; (**b**) Distribution of divergence between the two models.  
For velocity normalization, we used the mean values of \(u_{m}\) calculated for the whole channel's segment. As the channel's cross-section area changes in hexagonal and circular geometries, we used averaged values of \(u_{m}\) for these configurations.  
\[u_{m}=\frac{u_{m_{min}}+u_{m_{min}}\frac{w_{max}}{w_{min}}}{2} \tag{9}\]  
Figure 15: The normalized velocity distributions in the middle of the channel’s height: (**a**) Serpentine microchannel; (**b**) Hexagonal microchannel; (**c**) Circular microchannel. From the left: surface plot, surface plot with stream lines, and velocity vectors. The cross-sections 1–4 are marked by the black lines with arrows indicating the direction of the profile.  
where \(u_{m,min}\) is the mean velocity at the cross-section 3 (Figure 15) and \(w_{min}\) and \(w_{max}\) are the channel's widths, respectively, at inlet and cross-section 3. The values of \(u_{m}\) [m/s] calculated from simulated data, used for normalization, were \(1.664\times 10^{-3},13.11\times 10^{-3}\), and \(15.50\times 10^{-3}\), respectively, for serpentine, hexagonal, and circular channels.  
In all of the channel configurations, the regions with high-velocity magnitudes exist red to yellow color marked in Figure 15, regardless of whether the contractions occur in the channel geometry, like in circular or hexagonal configurations, or not, like in the serpentine channel. It is quite surprising that, in the case of the serpentine channel, fluid velocities four times higher than the average value occur in the bend. Additionally, the stagnant region occurs in all of the channels (navy blue marked). The circular-shaped channel appears to be the worst configuration, with the greatest observed velocity gradients across the channel. In Figure 16, the velocity profiles across the channels are compared. The courses of the profile lines are shown in Figure 15.  
Figure 16: Results of the simulation—the normalized velocity profiles within the microchannel at the half of the channel’s height: (**a**) Serpentine configuration; (**b**) Hexagonal configuration; (**c**) Circular configuration. \(l\)–dimensionless distance from the midpoint of profile.  
For the hexagonal and circular channels (Figure 16b,c), we observe similar axisymmetric profile plots with inlet \"buffalo horns-like\" shapes caused by channel contractions. The velocity gradients are higher for circular configuration, while, for both cases, the maximal values of \(u/u_{m}\) are about two times higher than predicted by the S&M model for fully developed duct flow. For the serpentine duct (Figure 16a) the velocity distributions are much more uniform, and the velocities, excluding the small part of the bend, are close to those predicted by the S&M model values (profiles 3 and 4).