## 2 Mathematical Model for MHD Flow over Non-Linearly Stretched Sheet  
Consider an unsteady laminar 2D Hiemenz flow of incompressible nanofluid over a porous wedge sheet exposed to solar radiation, which is shown in Figure 1. Transparent porous medium is supposed to have no net flow of heat between fluid and porous wedge. Constant magnetic field M\({}_{0}\) is applied normally to the sheet. \(T_{\omega}\) is the constant temperature at the wedge surface. As \(y{\rightarrow}\infty\), ambient surrounding temperature is achieved. As we move away from the porous sheet, fluid, and surroundings maintain the same and constant temperature \(T_{\infty}\). Due to small value of Reynolds number, the induced magnetic field is supposed negligible in comparison of applied magnetic field. From literature [44; 45], it is observed that if thermal diffusivity of the fluid and porous matrix is of the same order, thermal dispersion effect becomes nominal. \(q_{rad}^{{}^{\prime\prime}}\) is the intensity of radiation flux for non-reflecting transparent wedge sheet. As \(q_{rad}^{{}^{\prime\prime}}\) penetrates the porous sheet, flux is absorbed by the fluid flowing over the sheet according to the absorption coefficient [46]. Due to incident solar radiations on wedge porous sheet, the heat is absorbed by the nanofluid and it is transported from plate to the surroundings. It is assumed that the working fluid have absorption properties. The thermophysical properties are given in Table 1 (see [PERSON] et al. [47]) Water-based nanofluids containing copper and silver are considered. The governing equations for flow and thermal field under above assumptions are modeled as [48],  
Figure 1: Flow model over a porous wedge sheet. \[\frac{\partial\bar{\omega}}{\partial\bar{x}}+\frac{\partial\bar{v}}{\partial\bar{y} }=0 \tag{1}\]  
\[\frac{\partial\bar{\omega}}{\partial\bar{t}}+\bar{\omega}\frac{\partial\bar{v}} {\partial\bar{x}}+\bar{v}\frac{\partial\bar{v}}{\partial\bar{y}}=\frac{1}{ \varrho_{lm}}\Big{[}\Big{(}\frac{\partial W}{\partial t}+W\frac{dW}{dx}\Big{)} \varrho_{lm}\]  
\[+\delta_{lm}\frac{\partial^{2}\bar{\omega}}{\partial\bar{y}^{2}}+(\alpha \alpha)_{lm}f(T-T_{\infty})cos\frac{\chi}{2}-\Big{(}\gamma M_{0}^{2}+\frac{v _{lm}}{\varrho_{lm}}\Big{)}(\bar{\omega}-W)\Big{]} \tag{2}\]  
\[\frac{\partial T}{\partial\bar{t}}+\bar{\omega}\frac{\partial T}{\partial\bar {x}}+\bar{v}\frac{\partial T}{\partial\bar{y}}=\beta_{lm}\frac{\partial^{2}T} {\partial\bar{y}^{2}}-\frac{1}{(\alpha c_{p})_{lm}}\frac{\partial^{{}^{\prime \prime}}_{rad}}{\partial y}-\frac{H_{0}}{(\alpha c_{p})_{lm}}(T-T_{\infty}) \tag{3}\]  
The boundary conditions for these equations are,  
\[\bar{\omega}=0,\bar{v}=-v_{0},T=T_{w}+c_{1}x^{n_{1}}\quad at\quad\bar{y}=0; \tag{4}\]  
\[\bar{\omega}\to W=\frac{v_{h}x^{m}}{\lambda^{m+1}},T\to T_{\infty}\quad as \quad\bar{y}\to\infty\]  
where power indexes \(c_{1}\) and \(n_{1}\) are constant. Suction/injection velocity Y\({}_{0}\) must be greater than zero and suction/injection temperature \(T_{w}\) must be less than zero at the porous plate. With these conditions, the flow velocity can be written as \(U(x,t)=\frac{v_{h}x^{m}}{\lambda^{m+1}}\)[49]. By applying Rossland approximation for solar radiation [50], we can write \(q^{{}^{\prime\prime}}_{rad}=-\frac{4T_{1}}{3f^{\prime}}\frac{\partial T^{4}} {\partial y}\). Here, \(\gamma_{1}\) is the Stefan-Boltzman constant. \(k^{*}\) represents mean absorption coefficient. Surface and ambient conditions are represented by \(w\) and \(\infty\), respectively. Velocity components along the \(x\)-axis and \(y\)-axis are denoted by \(\bar{\omega}\) and \(v\), respectively. \(H_{0}(T_{\infty}-T)\) represents per unit volume heat absorbed/released. Nanofluid thermal diffusivity, local temperature, effective dynamic viscosity, effective density of the nanofluid are denoted by \(\beta_{hm}\), \(T\), \(\delta_{hm}\), and \(\varrho_{hm}\), respectively. \(\bar{f}\) represents the gravitational acceleration, while \(q^{{}^{\prime\prime}}_{rad}\) is the applied radiation absorption heat transfer.