## 2 Numerical Procedure
### Data Evaluation  
After the numerical computation, the flow rate in each microchannel was calculated by velocity integration. In addition, the standardized flow rate, that is, the ratio between the flow rate at channel iqjand the mean flow rateqmed, was calculated, enabling the comparison of the flow rate distribution's uniformity within different geometries. An example can be seen in Figure 5, where a higher flow is responsible for the concentration of the flow in the central microchannels, which are aligned with the inlet, showing an effect of the inertia of the fluid on the FDU coefficient S.  
\begin{table}
\begin{tabular}{c c c} \hline
**Surface** & **Boundary Conditions** & **Mathematical Expression** \\ \hline Inlet & Flow Rate Inlet & \(q_{IN}\to t,\ t\in[1,\ 100]\), Tmean = 20 \({}^{\circ}\)C \\ \hline Outlet & Pressure Outlet & \(\text{P}_{\text{OUT}}\) = 0 Pa (free atmosphere) and \\  & & \(\text{\Delta T}_{\text{OUT}}\) = 0 \\ \hline Fluid–Solid Interface & No-slip Interface & \(K_{f}\times\
abla T_{f}\ =\ K_{PDMS}\times\
abla T_{PDMS}\) and T\({}_{f}\) = TPDMS \\ \hline Sided Walls & Adiabatic & \(-K_{i}\times\left(\frac{\partial T_{i}}{\partial x}\right)=-K_{i}\left(\frac{ \partial T_{i}}{\partial y}\right)=0\) \\ \hline Interface between the Heater and PDMS & Thermal Coupled & \(K_{H}\times\
abla T_{H}\ =\ K_{PDMS}\times\
abla T_{PDMS}\) and \\  & & T\({}_{H}\) = TPDMS \\ \hline Bottom Surface of the PDMS & Mixed Radiation and Convection & \(-K_{PDMS}\times\left(\frac{\partial T_{PDMS}}{\partial z}\right)\) & \(q_{\text{{\it mrad}}\ JDMS}\) \\  & & \(\to s+q_{\text{{\it cov}},\ PDMS}\) \\  & & \(\to a\) \\ \hline \end{tabular}
\end{table}
Table 4: Boundary conditions used in the current simulation for solid and fluid domains.  
The FDUcoefficient, S, can be calculated by Equation (6):  
\[\text{S}\ =\ \sqrt{\frac{\sum_{i=\ 1}^{\text{N}}\left(q_{i}-\overline{q}\right)^{2}} {\text{N}}} \tag{6}\]  
where \(\overline{q}\) is the mean flow rate, \(q_{i}\) the flow rate in the microchannel i, and N is the total number of microchannels of the device. A higher S means a lower flow distribution uniformity, and vice-versa. The temperature was determined in the fluid and solid domains. The temperature evolution throughout the complete pathway of the working fluid was analyzed, from the collectors to the individual channels, allowing for the evaluation of the impact of the FDU on temperature distribution.  
Figure 6 shows two cases that enable the analysis of the effect of the flow rate on the temperature field of the device. As can be seen, for lower flow rates (Figure 6a), higher temperatures are globally attained, resulting in a lower heat removal. This type of representation improves the understanding of the heat transport phenomena in the device. For example, a study of the evolution of the temperature in the different channels can also be useful for understanding how FDU affects temperature.  
The results presented in Figure 7 highlight a variation in the registered temperatures of the channels. They show some symmetry, with a smaller temperature difference for the central channels and a greater difference with distance from the centralaxis.  
Figure 5: Flow distribution in the microchannels for two different flow rates.  
Figure 6: View of the temperature profile of a fluid flow at different flow rates:(**a**) 1 mL/min; (**b**) 30 mL/min.  
This profile can be explained with the help of the flow trajectories, depicted in Figure 8, where it can be seen thatthe fluid coming from the inlet disperses in all the directions. Therefore, the fluid that crosses the central channels originates directly from the jet, while the external channels receive fluid that has covered a greater distance since it comes from the rear of the collector. The temperature profile will possibly be influencedby the same phenomena that occur in Figure 6, only on a smaller scale.