# Pressure-Driven Nitrogen Flow in Divergent Microchannels with Isothermal Walls
## 5 Results and Discussion  
The influence of the microchannel divergence angle on variationsinof temperature and Mach number along the microchannel centreline is shown in Figure 4. For \(\phi<2.5\), the gas temperature decreases along the channel because of the augmentation of the gas velocity toward the channel outlet. The increase in kinetic energy of the gas results in a reduction in its internal energy and thus its temperature. The gas velocity remains almost constant along the channel with \(\phi=2.5\), and the gas temperature hardly changes along the channel, except in regions close to the inlet and outlet boundaries. Further increase in the divergence angle \(\phi\) results in a decrease in gas velocity along the channel, which is attributed to the increased cross-sectional area of the channel as well as the increased shear stress at the channel walls [31, 39]. The kinetic energy of the gas converts into internal energy as the fluid velocity decreases toward the channel outlet for the cases with \(\phi>2.5\), leading to an increase in gas temperature. Additionally, the bulk gas temperature reduces with an increase in divergence angle of the channel because of the increased bulk gas velocity and the reduced rate of molecular collision. Because of the development of a viscous boundary layer close to the channel inlet [39, 64], the gas flow accelerates in the entrance region; however, the flow decelerates after passing this region moving toward the channel outlet to conform to the pressure prescribed at the outlet. Close to the channel outlet (\(x/L>0.9\)), the gas temperature drops, which is attributed to the rapid gas expansion in that region, which is known as the expansion cooling phenomenon [20, 65]. A similar behaviour close to the channel outlet has been reported by others [28, 43, 44, 45] for Poiseuille micro-flows in microchannels with uniform cross-sections. Figure 3: The effects of (**a,b**) the computational grid size, (**c,d**) number of simulator particles per cell, and (**e,f**) time-step size on the predicted profiles of temperature and Mach number along the channel centreline. \(\text{Kn}_{\text{i}}=0.1\), \(\phi=1\) and \(\Pi=2.5\). Curves shown with symbols indicate the chosen values. The overall heat flow direction for the problem considered in the present work is determined by both the thermal and pressure gradients in the microchannel [66]. The heat flow direction and contours of dimensionless temperature (\(T/T_{\rm i}\)) obtained from the DSMC simulations are shown in Figure 5 for microchannels with different divergence angles at \(\text{Kn}_{\rm i}=0.05\) and \(\Pi=2.5\). Heat flows generally toward the channel inlet, except in regions adjacent to the walls in the Knudsen layer, where the heat flow is dominated by viscous heating. For the cases with \(\phi>2.5\), the amount of heat induced by viscous dissipation decreases along the microchannel due to the reductions in wall shear stress and slip velocity [39], decreasing the influence of viscous dissipation on the net heat flow direction in regions close to the walls. In contrast, the amount of heat generated by viscous dissipation increases as the gas moves along the channels with \(\phi\leq 2.5\), increasing its influence on the net heat flow direction in regions adjacent to the walls. Accordingly, it can be argued that the heat flow is dominated by the pressure gradient (known as a rarefaction effect [66]) in microchannels with a divergence angle of \(\phi\leq 2.5\). Due to the rarefaction effects, heat flow from cold to hot regions is observed, which is not predictable using the NSF equations because of the neglect of high-order rarefaction terms [67]. Heat flow in the central region of the channel is dominated by the mass flow driven by the pressure gradient. However, the magnitude of the thermal gradient increases with an increase in divergence angle \(\phi\), affecting the net heat flow direction.