## 2 Mathematical Formulation  
Figure 1 shows the physical model for a two-dimensional steady natural convection flow over a vertical permeable surface with a power-law wall heat flux \(q_{w}(x)\) and a constant concentration \(C_{w}\). The ambient temperature and concentration are \(T_{\infty}\) and \(C_{\infty}\), respectively. The coordinates are \(x\), measured along the surface from its leading edge, and the \(y\) normal to the system. The following assumptions are made for the system:  
1. The boundary layer approximations can be applied to the flow. 2. The convective fluid and porous structure are everywhere in local thermodynamic equilibrium. There is no phase change of the fluid in the medium, and thermal dispersion effects are neglected. 4. The Boussinesq approximation is employed; that is, \[\rho=\rho_{\infty}[1-\beta_{T}(T-T_{\infty})-\beta_{C}(C-C_{\infty})]\] (1) where the subscript \"\(\infty\)\" refers to the ambient condition. The volumetric expansion coefficients for thermal and concentration are \[\beta_{T}=-\frac{1}{\rho}\bigg{(}\frac{\partial\rho}{\partial T}\bigg{)}_{P,C},\beta_{C}=-\frac{1}{\rho}\bigg{(}\frac{\partial\rho}{\partial C}\bigg{)}_{T,P}\] (2)  
The value of \(C_{\infty}\) is set to zero in the problem as the concentration of species far from the wall; \(C_{\infty}\) is sufficiently small [31]. The thermophysical properties are maintained constant. The conservation equations for fluid flow through an isotropic and homogeneous saturated porous medium can be written as follows:  
1. Continuity equation \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\] (3)
2. Momentum equation \[\frac{\rho}{\varepsilon^{2}}(u\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial y})=\rho_{\infty}g[\beta_{T}(T-T_{\infty})+\beta_{C}(C-C_{\infty} )]+\frac{\mu}{\varepsilon}\frac{\partial^{2}u}{\partial y^{2}}-\frac{\mu}{K}u -\rho C_{i}u^{2}\] (4)
3. Energy equation \[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha_{e} \frac{\partial^{2}T}{\partial y^{2}}\] (5)
4. Concentration equation \[u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}+\frac{\partial }{\partial y}(CV_{T})=D_{e}\frac{\partial^{2}C}{\partial y^{2}}\] (6)  
Figure 1: Schematic of natural convection flow over a vertical permeable surface physical model and coordinate system. The boundary conditions for Equations (3) to (6) are  
\[ \begin{array}{ccccc}x=0,&y>0,&T=T_{\infty}&C=C_{\infty}&u=0\\ x>0,&y=0,&q_{w}(x)=Ax^{n}&C=C_{w}&u=0&v=v_{w}\\ &y\rightarrow\infty,&T=T_{\infty}&C=C_{\infty}&u=0\end{array}  \tag{7}\]  
where \(u\) and \(v\) are flow velocities in the \(x\) and \(y\) direction, respectively. \(T\) is the temperature, \(\mu\) is the dynamic viscosity, and \(k\) is the thermal conductivity. \(A\) is the constant strength of wall heat source, and \(n\) is a power-law exponent of heat flux. \(\rho\) is the fluid density. \(\varepsilon\) and \(K\) are the porosity and permeability of the porous medium, respectively. \(v_{w}\) represents the suction/blowing velocity of the permeable surface, and \(g\) is the acceleration due to gravity. \(C_{i}\) is the transport property related to the inertia effect. \(\alpha_{e}\) and \(D_{e}\) are the effective thermal and mass diffusivity of the porous medium, respectively. \(V_{T}\) is the thermophoretic velocity given by [PERSON] et al. [7], as follows:  
\[V_{T}=-\kappa\frac{\upsilon}{T}\frac{\partial T}{\partial y} \tag{8}\]  
where \(\kappa\) is the thermophoretic coefficient defined using [32]  
\[\kappa=\frac{2C_{s}(\lambda_{g}/\lambda_{p}+C_{t}Kn)C_{c}}{(1+3C_{m}Kn)(1+2 \lambda_{g}/\lambda_{p}+2C_{t}Kn)} \tag{9}\]  
where \(C_{s}\), \(C_{t}\), and \(C_{m}\) are constants. \(\lambda_{g}\) and \(\lambda_{p}\) are the thermal conductivities of the fluid and diffused particles, respectively. \(C_{c}=1+Kn(C_{1}+C_{2}e^{-C_{3}/Kn})\) is the Cunningham correction factor, and \(Kn\) is the Knudsen number.