## 5 Results and Discussion
### Isotherms  
The isotherms tend to cluster around the heat sources at the bottom wall in all cases. As the nanoparticle volume fraction increases at all values of \(\stackrel{{\sim}}{{\mathrm{Ha}}}\) and \(\stackrel{{\sim}}{{\mathrm{Ri}}}\), the maximum temperature decreases. This is because of increasing the effect of conduction, leading to a decrease in temperature inside the cavity. When \(\stackrel{{\sim}}{{\mathrm{Ha}}}\) increases, the maximum temperature for all values of \(\stackrel{{\sim}}{{\Phi}}\) and \(\stackrel{{\sim}}{{\mathrm{Ri}}}\) increases too. This happens because there is a decrease in heat transfer from the heat sources at the bottom of the cavity to the upper region and an increase in heat transfer in the transverse direction. The maximum temperature also increases as \(\stackrel{{\sim}}{{\mathrm{Ri}}}\) increases. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\left(\left|\stackrel{{\sim}}{{\Phi}}_{\max}\right|_{\stackrel{{ \sim}}{{\Phi}},\stackrel{{\sim}}{{\mathrm{Ha}}}}-\left| \stackrel{{\sim}}{{\Psi}}\right|_{\stackrel{{ \sim}}{{\Phi}}=0,\stackrel{{\sim}}{{\mathrm{Ha}}}}\right)\times 100/ \left|\stackrel{{\sim}}{{\Psi}}_{\max}\right|_{\stackrel{{ \sim}}{{\Phi}}=0,\stackrel{{\sim}}{{\mathrm{Ha}}}}\) & \\ \hline \multicolumn{5}{c}{\(\stackrel{{\sim}}{{\Phi}}=0.05\)} & \(\stackrel{{\sim}}{{\Phi}}=0.1\) & \(\stackrel{{\sim}}{{\Phi}}=0.15\) \\ \hline \multirow{2}{*}{\(\stackrel{{\sim}}{{\mathrm{Ri}}}=1\)} & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=25\) & \(-0.3475\) & \(-0.2911\) & \(0.1095\) \\  & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=75\) & \(-0.9306\) & \(-0.7790\) & \(0.3023\) \\ \multirow{2}{*}{\(\stackrel{{\sim}}{{\mathrm{Ri}}}=10\)} & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=0\) & \(-0.1130\) & \(-0.2203\) & \(-0.3216\) \\  & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=50\) & \(-0.8070\) & \(-0.7646\) & \(1.2492\) \\ \multirow{2}{*}{\(\stackrel{{\sim}}{{\mathrm{Ri}}}=100\)} & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=0\) & \(-1.4046\) & \(-2.4334\) & \(-3.2035\) \\  & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=75\) & \(-1.2811\) & \(-1.3874\) & \(-0.5031\) \\ \hline \end{tabular}
\end{table} 
Table 8: The percentage change in maximum streamfunction, case-2. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(\left(\left|\stackrel{{\sim}}{{\Psi}}_{\max}\right|_{ \stackrel{{\sim}}{{\Phi}},\stackrel{{\sim}}{{ \mathrm{Ha}}}}-\left|\stackrel{{\sim}}{{\Psi}}_{\max}\right|_{ \stackrel{{\sim}}{{\Phi}}=0,\stackrel{{\sim}}{{ \mathrm{Ha}}}}\right)\times 100/\left|\stackrel{{\sim}}{{\Psi}}_{\max}\right|_{ \stackrel{{\sim}}{{\Phi}}=0,\stackrel{{\sim}}{{ \mathrm{Ha}}}}\) & \\ \hline \multicolumn{5}{c}{\(\stackrel{{\sim}}{{\Phi}}=0.05\)} & \(\stackrel{{\sim}}{{\Phi}}=0.1\) & \(\stackrel{{\sim}}{{\Phi}}=0.15\) \\ \hline \multirow{2}{*}{\(\stackrel{{\sim}}{{\mathrm{Ri}}}=1\)} & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=0\) & \(-0.0076\) & \(-0.0105\) & \(-0.0093\) \\  & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=75\) & \(-0.8915\) & \(-0.7368\) & \(0.3146\) \\ \multirow{2}{*}{\(\stackrel{{\sim}}{{\mathrm{Ri}}}=100\)} & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=0\) & \(-0.3534\) & \(-0.6851\) & \(-0.9711\) \\  & \(\stackrel{{\sim}}{{\mathrm{Ha}}}=50\) & \(-0.5563\) & \(-0.2997\) & \(-0.6740\) \\ \hline \end{tabular}
\end{table} 
Table 9: The percentage change in maximum streamfunction, case-3. Case-1:Figure 4 shows the isotherms for different cases. The dense isotherms are located near the heat sources and at the cavity center. The symmetrical isotherms about the vertical line at the cavity center are due to the symmetrical boundary conditions in the vertical direction. The effect of \(\widetilde{\Phi}\) on the isotherms' shape decreases as \(\widetilde{\text{Ha}}\) increases (as shown in Figure 4). However, the impact of \(\widetilde{\text{Ha}}\) on the isotherms is noticeable for all values of \(\widetilde{\text{Ri}}\). An increase in \(\widetilde{\text{Ha}}\) tries to spread the heat (isotherms) in the transverse direction, thereby reducing the concentration of isotherms in the center region.