## 1 Introduction
References [18; 19; 20; 21; 22; 23; 24] provide a comprehensive of recent advancements concerning power law non-Newtonian fluids, encompassing diverse physical aspects and various flow-generating domains. Several investigations have concentrated on the heat transfer performance of Newtonian fluids in trapezoidal cavities, where mixed convection and natural have been investigated for magnetohydrodynamic (MHD) flows. [PERSON] and [PERSON] [25] examined the impact of a magnetic field on the heat transfer of laminar natural convective in a titled cavity and maximum temperature at the lower side. They found that a higher Hartmann number reduces convective heat transfer for a given inclination. In recent years, various research has examined natural convection under inclined magnetic fields [26; 27; 28]. [PERSON] et al. [29] thoroughly investigated the MHD heat convection in a wavy enclosure with a porous medium, and the lower wall has a maximum temperature. Numerous researchers have examined the impact of incorporating porous media within an enclosure for various purposes, as porous materials can enhance or reduce heat transfer [30; 31; 32]. The flow within a vertical cavity with a wavy shape that was permeated by a porous medium was examined by [PERSON] et al. [33]. They obtained a numerical solution using the finite element method. [PERSON] et al. [34] established a complete structure for the heat transfer effects of free convection in a porous corrugated cavity under MHD flow and a uniform magnetic field, utilizing a properly non-dimensional system that was solved through FEM with high resolution near the surface of corrugation, and presented visualizations of flow patterns and temperature distribution for various parameters. This includes the process of discretizing the governing partial differential equations through a weak formulation. The weak form permits the utilization of lower-degree polynomials for approximating field variables. The weak form for non-Newtonian fluids was introduced by [35; 36]. Subsequently, [PERSON] et al. [37] discussed the optimal control problem associated with a mathematical model that describes the steady flow of a nonlinear-viscous, incompressible fluid within a bounded two-dimensional domain. Weak solutions are constructed to minimize a specified cost functional, subject to a given bounded set of permissible controls. This article investigates the heat transfer effects in a porous cross-shaped cavity under MHD flow and a uniform magnetic field. The research will be conducted through a comprehensive literature review on the subject. Subsequently, a mathematical model has been developed to capture and describe the occurrence of free convection within the cavity. This article explores the non-Newtonian model characterizing shear-thinning and shear-thickening fluid behaviors within a plus-shaped enclosure. The study seeks to elucidate the influence of various factors, encompassing the Reynolds number, magnetic field inclination, the Hartmann number, and power law index. Numerical simulations were then performed to solve the governing partial differential equations using the finite element method, which provides detailed insights into the flow patterns and thermal distribution within the cavity. Graphical results such as streamlines and isotherms were presented to aid in visualizing the simulation outcomes. Finally, based on the results obtained, a conclusion was drawn, which summarized the findings and implications of the study.