## 4 Results and Discussion
### Velocity Profile  
This section is dedicated to exploring the behavior of the velocity profile for both convergent and divergent channels under the variations in the parameters, like the channel opening \(\alpha\), the Reynolds number \(Re\), the volume fraction of nanoparticles \(\phi\), the Hartmann number \(Ha\) and the stretching/shrinking parameter S. For the said purpose, Figures 2-11 are plotted. Figure 2 gives a \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\boldsymbol{\rho(kg/m^{3})}\) & \(\boldsymbol{C_{p}(J/KgK)}\) & \(\boldsymbol{k(W/mK)}\) & **Electrical Conductivity** \(\sigma(\Omega m)^{-1}\) \\ \hline Pure Water & 997.1 & 4179 & 0.613 & 0.05 \\ Copper (Cu) & 8933 & 385 & 401 & \(5.96\times 10^{7}\) \\ Silver & 10,500 & 235 & 429 & \(6.3\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical properties of water, copper and silver nanoparticles [22]. graphical description of the variations in the velocity profile for a divergent channel for different values of channel opening \(\alpha\). Both the stretching and shrinking cases are discussed in the same figure. A clear drop in the velocity profile is evident for increasing values of \(\alpha\). Here, clearly, for the stretching walls, the change in velocity is seen to be quite rapid at the central portion; there comes a point at which the backflow phenomena are observed particularly for \(\alpha=5^{\circ}\). This backflow behavior may result in separation at some point. In Figure 3, the same effects of \(\alpha\) for the convergent channel are portrayed. A quite opposite behavior is observed for this case, and increasing values of velocity are observed. For the stretching walls, the change in velocity is on the lower side as compared to the shrinking walls. This means that for a shrinking, convergent channel, more fluid is moved towards the walls as compared to the same for a diverging channel. Copper nanoparticles are seen to have a slightly higher velocity for divergent channels as compared to the silver nanoparticles. The behavior is reversed for a convergent channel. Figure 3: Change in velocity for varying \(\alpha\) (convergent channel). Figure 2: Change in velocity for varying \(\alpha\) (divergent channel). Figure 4: Change in velocity for varying \(\,\mathrm{Re}\,\) (divergent channel). Figure 5: Change in velocity for varying \(\,\mathrm{Re}\,\) (convergent channel). Figure 6: Change in velocity for varying \(\,\mathit{Ha}\,\) (divergent channel). Figure 8: Change in velocity for varying \(\varphi\) (divergent channel). Figure 7: Change in velocity for varying \(Ha\) (convergent channel). Figure 9: Change in velocity for varying \(\varphi\) (convergent channel). The velocity profile for the increasing values of the Reynolds number Re for the case of a divergent channel is plotted in Figure 4. The higher the values of Re, the lower the velocity becomes in the diverging channel. This results in a backflow near the walls of the channel for stretching walls. A steep decrease in velocity is seen for the stretching/diverging channel as compared to the shrinking one. A quite opposite scenario is seen for the convergent channel, and velocity is an increasing function of Re. This entire discussion shows that the upsurge in viscous forces results in lower values of the Reynolds number, and an increase in the mainstream velocity of the channel increases the values of Re that result in the increasing of the velocity for convergent channels and a decrease in velocity for the divergent channel. Again, for increasing Re, the copper nanoparticles have slightly higher values for the velocity of the divergent channel. This is due to the lower density of copper as compared to silver. The Hartmann number that represents the strength of the magnetic field results in a growth in the velocity profile when the channel is divergent, as is clear from Figure 3:0 The behavior is similar for both the stretching and shrinking of walls.