## 2 Mathematical Model  
Consider a 2D steady-state free convective power law fluid flow with constant density in a plus-shaped cavity. The enclosure is enclosed by vertical walls maintained at constant temperatures, with the right-side lower wall uniformly heated while the left-side upper wall has low temperature. The enclosure is assumed to be impermeable, with no movement of the walls. Therefore, this paper introduces originality in investigating the non-Newtonian model that characterizes the behaviors of shear-thinning and shear-thickening fluids. The Boussinesq assumption has been implemented, and the influence of Rayleigh dissipation is considered negligible. The illustration in Figure 1 depicts the physical layout of the problem. Considering the above assumptions, we utilize the following governing equations to represent the flow phenomenon [38; 39; 40; 41]  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho} \frac{\partial p}{\partial x}+\frac{1}{\rho}\bigg{(}\frac{\partial\tau_{xx}} {\partial x}+\frac{\partial\tau_{xy}}{\partial y}\bigg{)}+\xi_{x} \tag{2}\]  
Figure 1: Schematic diagram representing the problem. \[u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho}\frac{ \partial p}{\partial y}+\frac{1}{\rho}\left(\frac{\partial\tau_{xy}}{\partial x }+\frac{\partial\tau_{yy}}{\partial y}\right)+\xi_{y} \tag{3}\]  
where \(\left(\rho,\ \mu,\ \xi_{x},\ \xi_{y}\right)\) are fluid density, kinematic viscosity, and force terms (magnetic field and Boussinesq approximation). The force includes thermal diffusivity \(\varrho\) and solutal expansions \(\left(\beta_{T},\ \beta_{c}\right)\). \[\left(\xi_{x},\xi_{y}\right)=\left(\varrho B_{0}^{2}\left(v.sin\gamma cos \gamma-u.sin^{2}\gamma\right)-\frac{\mu}{\kappa}u,\ \varrho B_{0}^{2}\left(u.cos\gamma sin\gamma-v.cos^{2}\gamma\right)\right.\]  
\[\left.-\frac{\mu}{\kappa}v+\rho g[\beta_{c}(c-c_{l})+\beta_{T}(T-T_{l})]\right)\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\xi\left(\frac{ \partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}\right) \tag{4}\]  
\[u\frac{\partial c}{\partial x}+v\frac{\partial c}{\partial y}=D\left(\frac{ \partial^{2}c}{\partial x^{2}}+\frac{\partial^{2}c}{\partial y^{2}}\right) \tag{5}\]  
\[\tau_{ij}=2\mu_{a}D_{ij}=\mu_{a}\left(\frac{\partial u_{i}}{\partial x_{j}}+ \frac{\partial u_{j}}{\partial x_{i}}\right)\]  
\[\mu_{a}=K\left\{2\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left( \frac{\partial u}{\partial y}\right)^{2}\right]+\left(\frac{\partial u}{ \partial x}+\frac{\partial u}{\partial y}\right)^{2}\right\}^{\frac{n-1}{2}}\]  
D is magnetic diffusivity and \(\xi\) is the diffusion coefficient. The non-dimensional forms of Equations (1)-(5) are obtained by using dimensionless parameters as  
\[ \begin{array}{c}X=\frac{\tau}{\xi},Y=\frac{\
u}{\xi},U=\frac{uL}{\alpha},V= \frac{vL}{\alpha},(T_{h}-T_{l})\theta=(T-T_{l}),P.\rho\alpha^{2}=p.L^{2},Da= \frac{K}{l^{2}}\\ P_{r}=\frac{K\ L^{2-2n}}{\rho\ A^{2-n}},(c_{h}-c_{l}).C=(c-c_{l}),\ Ra=\frac{\rho\ B_{r}\xi L^{2n+1}\Delta T}{K\ A^{n}},\ Le=\frac{\xi}{D},\ Ha=BH\sqrt{\frac{\varrho}{\mu}}\end{array}  \tag{6}\]  
The boundary conditions are created by inserting Equation (6) into (1)-(5), as follows.