## 2 Mathematical Formulation
The values of permeability \(K\) and inertia coefficient \(C_{i}\) for the non-Darcian effects on solid-fluid combinations are used by employing [PERSON]'s model [33; 34]:  
\[K=\frac{d^{2}\varepsilon^{3}}{150(1-\varepsilon)^{2}} \tag{10}\]  
\[C_{i}=\frac{1.75(1-\varepsilon)}{\varepsilon^{3}d} \tag{11}\]  
The governing equations can be transformed using the dimensionless stream function, \(f(\xi,\eta)\), dimensionless temperature, \(\theta(\xi,\eta)\), and dimensionless concentration, \(\phi(\xi,\eta)\), with the following parameters:  
\[ \begin{array}{ccccc}\eta(x,y)=\frac{y}{x}Ra_{x}{}^{1/5},&\xi(x)=\frac{x^{2} }{K}Ra_{x}{}^{-2/5},&f(\xi,\eta)=\frac{\Psi(x,y)}{\alpha_{e}Ra_{x}{}^{1/5}},\\ \theta(\xi,\eta)=\frac{T-T_{\infty}}{T^{\kappa}}Ra_{x}{}^{1/5},&\phi=\frac{C- C_{\infty}}{C_{\infty}-C_{\infty}}=\frac{C-C_{\infty}}{\Delta C},&Nt=\frac{T_{ \infty}}{T^{\kappa}}Ra_{x}{}^{1/5},&T^{\kappa}=\frac{x_{\theta}(x)}{k}\end{array}  \tag{12}\]  
where \(\psi(x,y)\) is the stream function satisfying Equation (3) with \(u=\partial\psi/\partial y\) and \(v=-\partial\psi/\partial x\). Substituting Equation (12) into Equations (4)-(7), we obtain the following system:  
\[\frac{1}{\varepsilon}f^{\prime\prime\prime}+\theta+N\phi-\frac{1}{5\varepsilon ^{2}\text{Pr}}\Big{[}(2n+3)(f^{\prime})^{2}-(n+4)ff^{\prime}\Big{]}-\xi f^{ \prime}-\xi^{5/4}\frac{\gamma^{1/2}}{\text{Pr}}(f^{\prime})^{2}=\frac{2(1-n)} {5\varepsilon^{2}\text{Pr}}\xi\bigg{(}f^{\prime}\frac{\partial f^{\prime}}{ \partial\xi}-f^{\prime\prime}\frac{\partial f}{\partial\xi}\bigg{)} \tag{13}\]  
\[\theta^{\prime\prime}+\frac{n+4}{5}f\theta^{\prime}+\frac{-1-4n}{5}f^{\prime} \theta=\frac{2(1-n)}{5}\xi\bigg{(}f^{\prime}\frac{\partial\theta}{\partial \xi}-\theta^{\prime}\frac{\partial f}{\partial\xi}\bigg{)} \tag{14}\]  
\[\frac{1}{Sc}\phi^{\prime\prime}+\bigg{(}\frac{n+4}{5\text{Pr}}f+\kappa\frac{1}{ Nt+\theta}\theta^{\prime\prime}-\kappa\bigg{(}\frac{1}{Nt+\theta}\bigg{)}^{2}( \theta^{\prime})^{2}\bigg{]}\phi=\frac{2(1-n)}{5\text{Pr}}\xi\bigg{(}f^{ \prime}\frac{\partial\phi}{\partial\xi}-\phi^{\prime}\frac{\partial f}{ \partial\xi}\bigg{)} \tag{15}\]  
\[\eta=0,\quad f(\xi,0)=f_{w}\quad f^{\prime}(\xi,0)=0\quad\theta^{\prime}(\xi,0)= -1\quad\phi(\xi,0)=1\]\[\eta\rightarrow\infty,\quad f^{\prime}(\xi,\infty)=0\quad\theta(\xi,\infty)=0 \quad\phi(\xi,\infty)=0 \tag{16}\]  
where the primes denotes partial differentiation with respect to \(\eta\), \(Ra_{x}=(g\beta_{T}T^{*}x^{3})/\alpha_{e}v\) is the Rayleigh number, and \(T^{*}\) is the characteristic temperature. \(\Pr=v/\alpha_{e}\) is the Prandtl number (0.7 used for air). \(Sc=v/D_{e}\) is the Schmidt number (0.1 to 2.5 is selected for gas species diffusing into the air [35]). \(N=(\beta_{C}\Delta C/\beta_{T}T^{*})Ra_{x}{}^{1/5}\) is the buoyancy ratio defined as the driving force intensity of the concentration source to the wall heat flux under natural convection. The parameter \(N>0\) coincides with the aiding buoyancy, whereas \(N<0\) represents the opposing buoyancy. Additionally, \(\gamma=(K^{5/2}\bar{C}_{i}^{2}g\beta_{T}T^{*})/\alpha_{e}v\) is the dimensionless inertia parameter, and \(Nt\) is the thermophoretic parameter from the driving force by the temperature gradient acting on the diffused species. \(f_{w}\) is the dimensionless wall suction/blowing velocity; \(f_{w}<0\) for surface blowing, and \(f_{w}>0\) for surface suction. The important physical quantities of our interests are the local Nusselt number, \(Nu_{x}\), for heat transfer, and the Sherwood number, \(Sh_{x}\), for mass transfer, and mass transfer conductance, \(g_{mx}\).