## 4 Results and Discussion
### Temperature Profile  
Variations in the temperature profile under the influence of various parameters, like the angle of opening \(\alpha\), the volume fraction of nanoparticle \(\phi\), the stretching/shrinking parameter \(S\) and the Eckert number \(Ec\), are plotted in Figures 12-27. Here, the value of the Prandtl number \(Pr\) is fixed at 6.2 because we are taking water as a base fluid. The temperature profile under the effect of angle opening \(\alpha\) is plotted in Figures 12-15. Almost an identical behavior to the temperature profile is observed for both the divergent and convergent channels due to the increasing values of \(\alpha\). When the walls are stretching, the temperature is seen to be on a slightly lower side for rising \(\alpha\). On the other hand, for shrinking walls, some higher values of temperature are seen when there is no opening of the walls (\(\alpha=0\)). This means whether the walls are stretching or shrinking, with the change in angle, the temperature of the system will rise. There is not much of a difference between the temperature of Cu and Ag.  
The influence of the nanoparticle volume fraction on the temperature profile is plotted in Figures 16-19. An addition of the nanoparticles in the base fluid decreases the temperature of the system quite considerably. This change is prominent at the centerline of the channel. For a divergent channel, the temperature profile is somewhat on the higher side when the walls are stretching as compared to the shrinking of the walls. For a convergent channel, this behavior is reversed. For all of these cases, silver (Ag) nanoparticles tend to have slightly lower temperature values than copper (Cu) nanoparticles. The differences in the specific heat and the thermal conductivity of Cu and Ag are the major factors for the change in temperature.  
The next set of figures describes the behavior of temperature with the stretching/shrinking of the walls. Both the convergent and divergent channel cases are discussed. Figures 20 and 21 depict that the stretching and shrinking of the walls have a quite opposite effect on the temperature profile for a divergent channel. When the walls are stretching, a steep drop in the temperature is observed at the center of the channel. The change becomes smaller as we move near the walls of the channel. On the other hand, the shrinking walls tend to increase the temperature quite substantially at the central portion of the channel. For a convergent channel, the temperature behaves almost in an identical manner as the one seen in a divergent channel; Figures 22 and 23 are here to back our remarks. The only difference is the slightly higher values of the temperature for the stretching walls in a divergent channel. Interestingly, Cu nanoparticles have marginally lower values for the stretching of a divergent channel as compared to the stretching of a convergent channel. For a convergent channel, the temperature of Cu seems to be on the higher side. However, when the walls are shrinking, Cu nanoparticles show a marginal advantage in temperature.  
Figures 24-27 are plotted to study the changes occurring in the temperature profile due to the variations in \(Ec\). Here, \(Ec\) represents the influence of dissipation on the temperature profile. As expected, the Eckert number increases the temperature quite expressively. This change in the temperature is slightly lower for the case when the walls are stretching. For the shrinking walls, the temperature rises quite efficiently with increasing values of the Eckert number. Furthermore, for a convergent channel and the stretching walls, the temperature curves deviate relatively less.