## 5 Results and Discussion
### Streamfunction  
_Case-1_: regardless of the value of \(\widetilde{\text{Ri}}\), the walls' movement creates two symmetric elliptical vortices in opposite directions, controlling the flow and heat inside the enclosure. The difference in density along the Y-direction causes the flow to move upward, where it interacts with the upper wall and then changes its direction. This process creates vortices. Additionally, the symmetrical boundary conditions along the Y-axis result in the formation of two symmetrical vortices. As presented in Figure 3, most of the flow is concentrated near the sliding walls due to the walls' movement, making the center of the elliptical streamfunction move and located near these walls. As \(\widetilde{\text{Ha}}\) increases, the core center of the elliptical streamfunction moves towards the moving isothermal walls for all \(\widetilde{\text{Ri}}\) values. The shape of the streamfunctions does not change significantly as \(\widetilde{\Phi}\) increases for various values of \(\widetilde{\text{Ha}}\), but the intensity of the flow increases due to the increase in the conduction effect. The values of the streamfunction depend strongly on the values of \(\widetilde{\text{Ha}}\) and \(\widetilde{\Phi}\), as the effect of adding nanoparticles significantly depends on the value of \(\widetilde{\text{Ha}}\) and \(\widetilde{\text{Ri}}\). At a Rayleigh number (\(\widetilde{\text{Ri}}\)) of 1, adding nanoparticles (\(\widetilde{\Phi}\)) between 0.05 and 0.1 will cause the maximum streamfunction to decrease. This is due to an increase in the buoyancy force, which in turn causes a decrease in fluid movement. Additionally, the maximum value of the streamfunction will decline as the Hartmann number (\(\widetilde{\text{Ha}}\)) decreases. This is because there is an increase in the transverse movement, accompanied by a decrease in fluid movement along the Y-axis (buoyancy force). At a certain \(\widetilde{\text{Ha}}\) value (non-zero), increasing \(\widetilde{\Phi}\) from 0.1 to \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\widetilde{\text{Re}}\) & \(\widetilde{\text{Gr}}\) & **Ref.** & **Present** \\ \hline
50 & \(10^{4}\) & 0.110 & 0.115 \\
100 & \(10^{3}\) & 0.097 & 0.102 \\
250 & \(10^{4}\) & 0.108 & 0.113 \\
500 & \(10^{5}\) & 0.135 & 0.139 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison between the present study and [30] for average maximum streamfunction (\(\widetilde{\text{bu}}_{\text{max}}\)) at \(\widetilde{\text{Da}}\) = 10\({}^{-2}\) and \(\widetilde{\text{Ha}}\) = 0. 0.15 will drop the maximum streamfunction. This is due to an increase in the movement of fluid in the transverse direction, coupled with a low fluid velocity (\(\widetilde{\text{Ri}}\) = 1). As \(\widetilde{\text{Ha}}\) increases, this increase in the streamfunction becomes more noticeable (Table 6). At \(\widetilde{\text{Ri}}\) values of 10 and 100, the maximum streamfunction decreases with an increase in \(\widetilde{\text{Ha}}\) for all \(\widetilde{\Phi}\) values due to high convection effects. The impact of \(\widetilde{\text{Ha}}\) is more significant at a low value \(\widetilde{\Phi}\) and high \(\widetilde{\text{Ri}}\) values. The strength of these circulations increases as the \(\widetilde{\text{Ri}}\) number increases due to an increase in the buoyancy force inside the cavity. The percentage reduction in the maximum streamfunction due to an increase in \(\widetilde{\Phi}\) at a specific \(\widetilde{\text{Ha}}\) value depends mainly on the \(\widetilde{\text{Ri}}\) value. It increases as \(\widetilde{\Phi}\) increases at high \(\widetilde{\text{Ri}}\) values and decreases at low \(\widetilde{\text{Ri}}\) values (\(\widetilde{\text{Ri}}\) = 1). Table 7 presents the percentage change in the maximum streamfunction for some values of \(\widetilde{\Phi}\) and \(\widetilde{\text{Ha}}\) in certain cases. Figure 3: _Cont._  
_Case-2:_ when the vertical walls move in opposite directions, two elliptical shapes of streamfunction are created. These shapes are bounded by lines to form a \(\infty\) shape.