## 4 Results and Discussion
### Skin Friction Coefficient and the Nusselt Number  
The effects of various parameters on the skin friction coefficient and the Nusselt number are plotted in Figures 28-35. Both stretching and shrinking cases are taken into account. For the case of a divergent channel, Figure 28 gives a description of the varying angle and the nanoparticle volume fraction on the skin friction coefficient. Clearly, with an increase in the nanoparticle volume fraction, the skin friction is seen to be declining for both stretching and shrinking divergent channels. Furthermore, the greater the channel opening for the divergent channel, the lower are the skin friction values. It can be observed that the stretching channel has lower values of the skin friction as compared to the shrinking case. Due to the higher density values, silver nanoparticles have lower values as compared to copper nanoparticles. For the case of the convergent channel, the behavior of the skin friction coefficient can be observed from Figure 29. The skin friction is seen to be increasing with the increase in the angle for the convergent channel. Here, the stretching convergent channel has lower values of the skin friction as compared to the shrinking convergent channel. The manners in which skin friction behaves for increasing values of Re are plotted in Figures 30-31. For the divergent channel, the increase in Re results in declining values of the skin friction coefficient, as well as for the nanoparticle volume fraction. Stretching divergent channels have lower skin friction values than shrinking divergent channels. Almost the reverse behavior is observed for the convergent channel, _i.e._, higher Re values result in higher skin friction values for both stretching and shrinking convergent channels. Figure 28: Variation in the skin friction coefficient with the angle and the nanoparticle volume fraction (divergent channel). Figure 29: Variation in the skin friction coefficient with the angle and the nanoparticle volume fraction (convergent channel). Figure 31: Variation in the skin friction coefficient with the Reynold number and the nanoparticle volume fraction (convergent channel). Figure 30: Variation in the skin friction coefficient with the Reynold number and the nanoparticle volume fraction (divergent channel). Figure 32: Variation in the Nusselt number with the angle and the nanoparticle volume fraction (divergent channel). Figure 34: Variation in the Nusselt number with the Eckert number and the nanoparticle volume fraction (divergent channel). Figure 35: Variation in the Nusselt number with the Eckert number and the nanoparticle volume fraction (convergent channel). Figure 33: Variation in the Nusselt number with the angle and the nanoparticle volume fraction (convergent channel). Changes in the Nusselt number with varying parameters are plotted in Figures 32-35 both for divergent and convergent channels. Figure 32 gives a description of the change in the Nusselt number with the varying nanoparticle volume fraction and the angle for the divergent channel case. The heat transfer rate at the wall is seen to be an increasing function of both the angle and the nanoparticle volume fraction. This behavior is quite fast for a shrinking divergent channel. A similar pattern is seen for the convergent channel, the difference being only that, for the convergent channel case, the values are slightly on the higher side as compared to the same values for the divergent channel, as depicted in Figure 33. Apparently, there is no difference in the values for copper and silver nanoparticles. The increase in the Eckert number results in a higher heat transfer rate at the wall for both divergent and convergent channels. The stretching of walls has a dominant effect as compared to the shrinking of the walls. The nanoparticle volume fraction increases the heat transfer rate and, thus, can be useful in many industrial processes. The problem under consideration is solved using two methods: the variational iteration method and the Runge-Kutta-Fehlberg method.