## 4 Results and Discussions
The streamlines became more twisted on the vertical side, and the convective cells in the bottom right and upper left corners became more noticeable. For the pseudoplastic behavior of the fluid, the fluid velocity increased significantly, and the fluid moved in a more circular motion, while for the shear-thickening fluid, the velocity decreased slightly, and the fluid moved in a more linear motion. The behavior of the fluid is significantly influenced by the presence of a magnetic field, particularly for the maximum value of the Hartmann number. The fluid's flow became more regular and steadier in the presence of the magnetic field, resulting in weaker and less intense convective cells within the enclosure. In the case of the shear-thinning fluid, the impact of the Rayleigh number \(Ra\) and the rheological attributes on the streamlines was insignificant. A rise in Rayleigh number for shear-thickening fluid resulted in more vertically elongated streamlines. The convective cells in the lower and upper corners became more pronounced and distinct. Figure 3: Variation of streamlines with \(Ha\), \(Da\), and \(n\) values at specified parameters of \(Ra=10^{4}\), \(Pr=0.7\), and \(\gamma=0^{\circ}\). The impact of kinetic energy at power law indices \(n\) and various Rayleigh numbers \(Ra\) is presented in Table 3. The kinetic energy is calculated based on the FEM simulations conducted in the study. The results show that the kinetic energy escalates with an increase in the Ra. Moreover, \(n=0.7\) (shear-thinning fluid) exhibits higher kinetic energy values as compared to the shear-thickening fluid \(n=1.2\) and Newtonian fluid \(n=1\) for all evaluated \(Ra\) values, indicating better fluid motion and convective heat transfer. Figure 6 represents the influence of Ra on the isotherms within the cavity for two different rheological behaviors of the fluid, shear-thickening and shear-thinning while keeping the Hartmann number \(Ha=0\). For the shear-thinning fluid \(n=0.7\), at low Rayleigh number \(Ra=10^{4}\), the temperature distribution is dominated by conduction, and the isotherms are along the vertical walls. With an increase in the \(Ra\), the isotherms become more prolonged, indicating the onset of convection. At higher Rayleigh numbers \(Ra=10^{6}\), the isotherms become more irregular, indicating strong convection within the enclosure. For the shear-thickening fluid \(n=1.2\), the impact of the Rayleigh number on  
Figure 4: Variation of isotherms with \(Ha,\ Da\), and \(n\) values at specified parameters of \(Ra=10^{4}\), \(Pr=0.7\), and \(\gamma=0^{\circ}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rayleigh Number** & **Shear-Thinning** & **Newtonian** & **Shear-Thickening** \\ \(Ra\) & \(n=0.7\) & \(n=1\) & \(n=1.2\) \\ \hline
1000 & 0.000604989 & 0.000506759 & 0.000457145 \\
10,000 & 0.088720673 & 0.054054614 & 0.035492132 \\
100,000 & 20.27295603 & 8.693457243 & 3.339311009 \\
1,000,000 & 763.1102155 & 485.2226004 & 155.4152072 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of kinetic Energy for different n and Ra. the isotherms is similar, but the overall temperature distribution is less affected compared to the shear-thinning fluid. For a fixed Hartmann number of 40 and a varying Rayleigh number ranging from \(10^{4}\) to \(10^{6}\), significant changes in the cavity's isotherms are observed for pseudoplastic behavior \(n=0.7\) and dilatant behavior \(n=1.2\) fluids. With an increase in the Rayleigh number, the isotherms for \(n=0.7\) exhibit slight distortion, resulting in a more pronounced temperature gradient along the side walls of the enclosure. When \(n=1.2\), the Rayleigh number increases because of thermal distribution, resulting in an overall decrease in the temperature gradient. When \(Ra\) increases, several changes occur in the isotherms within the cavity. Specifically, the thermal gradient is more prominent in the case of the pseudoplastic behavior as compared to the dilatant behavior fluid.