## 5 Results and Discussion
### Heatfunction  
Heat transfer within a cavity occurs through two main methods: conduction and convection (Figure 5). For any Da and at low values of \(\widetilde{\mathrm{Ri}}\) and Gr, conduction dominates. However, at high values of \(\widetilde{\mathrm{Ri}}\) and Gr, the convection becomes the dominant. Recently, heatlines have been considered a main function to illustrate the heat transfer within the cavity [30], and still, most of the CFD-ready packages don't include it. As a fact, the heat transfers from the hot surface towards the end on the cold surface, so the heatlines will follow these rules. Heatlines are perpendicular to isothermal walls and parallel to insulated walls. The maximum value of heatlines directly depends on the maximum value of the isotherms. The positive value of the heat function represents anti-clockwise heat flow, while the negative value represents clockwise heat flow. Thermal boundary conditions have a greater impact on heat transfer than velocity gradient; therefore, dense heatlines are located near heat sources. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\left(\left|\widetilde{\theta}_{\max}\right|_{\widetilde{\Phi},\widetilde{ \mathrm{Ha}}}-\left|\widetilde{\theta}_{\max}\right|_{\widetilde{\Phi}=0, \widetilde{\mathrm{Ha}}}\right)\times 100/\left|\theta_{\max}\right|_{\widetilde{\Phi}=0, \widetilde{\mathrm{Ha}}}\) & \\ \hline \multirow{3}{*}{\(\widetilde{\mathrm{Ri}}=1\)} & \(\widetilde{\mathrm{Ha}}=25\) & \(-6.3891\) & \(-12.227\) & \(-17.634\) \\  & \(\widetilde{\mathrm{Ha}}=75\) & \(-5.5019\) & \(-11.008\) & \(-16.57\) \\  & \(\widetilde{\mathrm{Ha}}=0\) & \(-7.8314\) & \(-15.135\) & \(-22.155\) \\  & \(\widetilde{\mathrm{Ha}}=75\) & \(-9.1976\) & \(-17.813\) & \(-25.79\) \\  & \(\widetilde{\mathrm{Ha}}=0\) & \(-11.144\) & \(-20.828\) & \(-29.298\) \\ \(\widetilde{\mathrm{Ri}}=100\) & \(\widetilde{\mathrm{Ha}}=50\) & \(-11.314\) & \(-21.105\) & \(-29.637\) \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The percentage change in maximum isotherms, case-2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(\left(\left|\theta_{\max}\right|_{\widetilde{\Phi},\widetilde{\mathrm{Ha}}}- \left|\theta_{\max}\right|_{\widetilde{\Phi}=0,\widetilde{\mathrm{Ha}}}\right) \times 100/\left|\theta_{\max}\right|_{\widetilde{\Phi}=0,\widetilde{\mathrm{Ha}}}\) & \\ \hline \multirow{3}{*}{\(\widetilde{\mathrm{Ri}}=10\)} & \(\widetilde{\mathrm{Ha}}=0\) & \(-6.8709\) & \(-13.0376\) & \(-18.6837\) \\  & \(\widetilde{\mathrm{Ha}}=75\) & \(-5.6269\) & \(-11.1926\) & \(-16.7592\) \\ \(\widetilde{\mathrm{Ri}}=25\) & \(-7.2647\) & \(-14.1153\) & \(-20.6392\) \\  & \(\widetilde{\mathrm{Ha}}=75\) & \(-8.7446\) & \(-16.9719\) & \(-24.6290\) \\ \(\widetilde{\mathrm{Ri}}=0\) & \(\widetilde{\mathrm{Ha}}=0\) & \(-10.4928\) & \(-19.6325\) & \(-27.6950\) \\  & \(\widetilde{\mathrm{Ha}}=50\) & \(-10.8242\) & \(-20.2358\) & \(-28.4800\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: The percentage change in maximum isotherms, case-3. _Case-1:_ At low values of \(\widetilde{\mathrm{Ri}}\), where \(\widetilde{\mathrm{Ri}}\) = 1; heat transfer primarily occurs through conduction from the heat sources to the lower portions of the vertical isothermal moving wall, which is more than 90% of the total heat. The circular heatlines in the upper portion of the enclosure have an opposite direction to the heatlines in the lower portion of the same half of the enclosure. For example, when \(\widetilde{\mathrm{Ha}}=0\) and \(\widetilde{\Phi}=0\), the heatlines transfer to 0-25% of the vertical wall length (lower portion), and this percentage increases to 50% at \(\widetilde{\mathrm{Ha}}=75\) and \(\widetilde{\Phi}=0.15\). An increase in \(\widetilde{\mathrm{Ha}}\) and \(\widetilde{\Phi}\) causes a decrease in the circular heatlines in the upper portion. When \(\widetilde{\mathrm{Ri}}=10\), the circular heatlines appear, and the straight heatlines transfer directly to 30% of the vertical walls' length at \(\widetilde{\mathrm{Ha}}=0\) and \(\widetilde{\Phi}=0\).