# Pressure-Driven Nitrogen Flow in Divergent Microchannels with Isothermal Walls
## 5 Results and Discussion
For cases with \(\phi>2.5\), a thermal separation occurs at a certain location in the channel beyond which heat flows toward the channel outlet. Thermal separation for pressure-driven gas flow in microchannels of variable cross-sections is attributed to the enhanced contribution of thermal gradient to the total heat transfer [68]. The position of the thermal separation point moves toward the inlet with an increase in the divergence angle of the channel. Notably, no hydrodynamic separation was observed for the cases considered in the present study [38; 39]. These observations can also be explained through the asymptotic analysis of the Boltzmann equation [66] for small Knudsen numbers (i.e., slip and early-transition regimes). However, this type of analysis, which is based on the continuum theory, fails to accurately predict heat transfer at high Knudsen numbers [38]. Figure 4: Distribution of (**a**) gas temperature and (**b**) Mach number along the microchannel centreline for different divergence angles \(\phi\) (\(\text{Kn}_{\rm i}=0.05\) and \(\Pi=2.5\)). The inlet temperature (\(T_{\rm i}\)) is used to non-dimensionalise the predicted temperatures. Numerical predictions of the gas temperature and Mach number along the channel centreline are presented in Figure 6 for different inlet Knudsen numbers \(\text{Kn}_{\text{i}}\) at \(\phi=7\) and \(\Pi=2.5\). The bulk gas velocity along the channel and variations in the gas velocity decreases with an increase in the Knudsen number. However, this reduction in the velocity magnitude and its variations becomes insignificant for \(\text{Kn}_{\text{i}}\geq 1\). Variation in gas velocity along the channel and the molecular collision frequency decreases with an increase in Knudsen number, limiting the variation in gas temperature along the channel. The magnitude of temperature drop close to the channel outlet also decreases with increasing inlet Knudsen number. Moreover, the predicted gas temperatures close to the channel inlet are slightly higher than the inlet gas temperature \(T_{\text{i}}\) because of the augmentation of the temperature jump with increasing Knudsen number [37]. The influence of the Knudsen number on the thermal field and heat flow direction in a divergent channel with \(\phi=7\) and \(\Pi=2.5\) is shown in Figure 7. The temperature gradient in the channel decreases with an increase in Knudsen number, decreasing the contribution of the Fourier heat transfer to the net heat flow. According to Figure 7c,d, at sufficiently large \(\text{Kn}_{\text{i}}\), the heat flow in the channel is merely toward the channel inlet, which demonstrates the domination of the pressure gradient in the net heat flow direction, intensifying the counter-gradient cold-to-hot heat transfer (also known as anti-Fourier heat transfer). Moreover, the total heat flow rate decreases with increasing Knudsen number. The distance between the thermal separation point and the channel outlet increases with a decrease in \(\text{Kn}_{\text{i}}\). Figure 5: Contours of temperature overlaid with heatlines (coloured by the magnitude of heat flux) obtained from the simulations for microchannels with different divergence angles \(\phi\). (\(\text{Kn}_{\text{i}}=0.05\) and \(\Pi=2.5\)). The bulk gas velocity and temperature in the channel decrease with a decrease in the inlet-to-outlet pressure ratio \(\Pi\), as shown in Figure 8. Increasing the inlet-to-outlet pressure ratio increases the gas kinetic energy and decreases its internal energy. The thermal field as well as the heat lines are shown in Figure 9 for channels with a divergence angle of \(\phi=7\) and Knudsen number of 0.05. The magnitude of the thermal gradient close to the channel outlet increases with decreasing the pressure ratio \(\Pi\), which enhances the contribution of the thermal gradient to the net heat flow; this affects the net heat flow direction and leads to the occurrence of thermal separation farther away from the channel outlet.