# Pressure-Driven Nitrogen Flow in Divergent Microchannels with Isothermal Walls
## 5 Results and Discussion
The heat generated in the Knudsen layer due to viscous dissipation diminishes with decreasing inlet-to-outlet pressure ratio \(\Pi\), decreasing the contribution of thermal gradients to the net heat flow close to the channel walls. Figure 6: The effect of rarefaction on the profiles of (**a**) gas temperature and (**b**) Mach number along the microchannel centreline (\(\phi=7.00\) and \(\Pi=2.5\)). The inlet temperature (\(T_{\rm i}\)) is used to non-dimensionalise the predicted temperatures. Figure 7: Heatlines (coloured by the magnitude of heat flux) overlaid on contours of dimensionless temperature (\(T/T_{\rm i}\)) predicted for different inlet Knudsen numbers \(\rm{Kn_{i}}\). \(\phi=7.00\) and \(\Pi=2.5\). The influence of the tangential accommodation coefficient (\(\alpha\)) on the profiles of temperature and velocity along divergent channels is shown in Figure 10 for \(\Pi=2.5\) and \(\mathrm{Kn_{i}}=0.05\). For all the cases considered in the present work, decreasing the tangential accommodation coefficient weakens the interactions between the gas molecules and the channel walls, leading to an increase in the gas velocity. A similar behaviour has also been observed both experimentally and numerically for rarefied gas flows in microchannels with uniform cross-sections [37, 50, 51, 69]. As expected, the augmentation of gas velocity in the channel results in an increase in the kinetic energy and a decrease in internal energy and thus the gas temperature. The results presented in Figure 11 show that the gas reaches supersonic velocities when \(\alpha=0\) (i.e., specular reflection); this happens for the case with \(\phi=7.0\) only when \(\alpha=0.5\) and does not happen when \(\alpha=1\) (i.e., fully-diffused reflection). Figure 8: The effect of the inlet-to-outlet pressure ratio \(\Pi\) on the profiles of **(a)** gas temperature and **(b)** Mach number along the microchannel centreline (\(\phi=7.00\) and \(\mathrm{Kn}=0.05\)). The inlet temperature (\(\mathrm{T_{i}}\)) is used to non-dimensionalise the predicted temperatures. Figure 9: The influence of the inlet-to-outlet pressure ratio \(\Pi\) on the thermal field (contours) and heat flow direction (coloured by the magnitude of heat flux). \(\mathrm{Kn_{i}}=0.05\) and \(\phi=7\). The results presented in Figure 11 show that heat and fluid flow patterns in the channel are sensitive to the value of the tangential accommodation coefficient \(\alpha\), particularly for low values of the Knudsen number. The number of gas molecules contributing to the adiabatic  
Figure 3:0 The influence of the tangential accommodation coefficient \(\alpha\) on the thermal field (contours) and heat flow direction (coloured by the magnitude of heat flux). \(\Pi=2.5\) and \(\phi=7\). Figure 3:1 The effect of the tangential accommodation coefficient \(\alpha\) on the profiles of (**a**) gas temperature and (**b**) Mach number along the microchannel centreline (\(\Pi=2.5\) and \(\text{Kn}_{\text{i}}=0.05\)). The inlet temperature (\(T_{\text{i}}\)) is used to non-dimensionalise the predicted temperatures. exchange of energy increases with decreasing values of the tangential accommodation coefficient \(\alpha\), increasing the contribution of thermal gradients to the net heat flow. Reducing the value of the tangential accommodation coefficient to \(\alpha=0\) as an extreme condition, it is observed that the heat flow converges to a sink point, where pressure gradients balance thermal gradients. Heat lines are directed toward the channel outlet in regions where thermal gradients dominate the net heat flow.