## 4 Results and Discussions  
The study utilized the finite element method to investigate the heat transfer properties of a power law fluid contained within a plus-shaped enclosure. The analysis meticulously examined variations in velocity and temperature distribution in relation to distinct magnitudes of crucial parameters, such as the Prandtl number \(Pr\), the power law index \(n\), and the Rayleigh number \(Ra\). This method facilitated a comprehensive understanding of the complex interplay between different factors and their impact on the fluid's behavior. Moreover, the study also calculated local and mean heat transfer rates and K.E. measurements related to various variables. The fluctuations in Nu concerning the power law fluid \(n\) are presented in Table 2 for a comprehensive understanding of their relationship. The analysis demonstrates that when the power law index \(n\) increases, a simultaneous decline is observed in heat transfer. This is attributed to the fluid's behavior transitioning from pseudoplastic to dilatant as the power law index increases. Hence, the increase in fluid viscosity as an effect contributes to reducing the motion and heat transfer processes within the flow domain. In Figure 3 with \(Ha=0\), it has been found that the enclosure is primarily occupied by a single convective cell, which is aligned diagonally within the enclosure. The fluid moves near the walls, while the temperature moves closer to the central point. The diagonal walls are considered to be adiabatic. This leads to the formation of a counterclockwise flow pattern within the cavity. When Darcy's number rises \(10^{-4}\leq Da\leq 10^{-2}\), the main convective cell expands horizontally, and secondary cells emerge in opposite corners, reflecting a stronger convective process within the enclosure. Moreover, an increase in the Da leads to higher stream function values, signifying a notable impact on fluid movement within the cavity. The results show that as Da increases, there is a decrease in fluid resistance inside the cavity, which, in turn, leads to a considerable enhancement in convective fluid intensity. As a result, the fluid flow properties are positively influenced by the \(Da\) number, improving the overall behavior of the fluid within the cavity. Figure 4 illustrates the changes in temperature distribution as the Darcy number increases, which is associated with the consistent heating applied to the side walls. When \(Ha=0\) and all evaluated power law indices and isotherm lines parallel to vertical walls are noticeable at a low Darcy number \(Da=10^{-4}\), this indicates that conduction is the dominant mode of heat transfer in this case. As the Darcy number increases, heat transfer diagonally from right to left walls occurs more rapidly, owing to the enhanced convective flow within the enclosure. The thermal gradients surrounding the vertical side walls are considerably steeper than those observed at the lower Darcy number. As the \(Ha\) number increases, its impact on isotherms indicates a substantial reduction in the thermal convection effect. At low \(Ha\) numbers, conduction is the main mode of heat transfer, causing the temperature distribution to be less influenced by the applied magnetic field. Moreover, when \(Da\) (ranging from low to high) and the \(Ha\) number increase, the isotherms are adversely affected, resulting in the restraint of convective flow within the enclosure. Figure 5 demonstrates the impact of different Rayleigh numbers \(10^{4}\leq Ra\leq 10^{6}\) and Hartmann numbers \(Ha=0,40\) on the streamlines within the cavity for two different rheological behaviors of the fluid, \(n=0.7\) (shear-thinning behavior) and \(n=1.2\) (shear-thickening behavior). When \(Ha=0\), the discussion primarily centered around the influence of Ra and fluid rheology on the streamlines inside the cavity. The results showed that for both the pseudoplastic behavior \(n=0.7\) and dilatant behavior \(n=1.2\) fluids, as the Ra increased from \(10^{4}\) to \(10^{6}\) the fluid motion became more ordered and stable.