## 2 Governing Equations and Material
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\frac{ \partial^{2}T}{\partial y^{2}}+\frac{\partial}{\partial y}\bigg{(}\frac{16 \alpha\varphi^{*}T_{\infty}^{3}}{3k^{*}}\frac{\partial T}{\partial y}\bigg{)} +\frac{Q_{o}}{\rho C_{p}}(T-T_{\infty}) \tag{3}\] \[+\lambda_{t}^{*}\frac{Q_{o}}{\rho C_{p}}\bigg{(}v\frac{\partial T }{\partial y}+u\frac{\partial T}{\partial x}\bigg{)}\] \[+\lambda_{t}^{*}\tau D_{B}\bigg{(}u\frac{\partial^{2}C}{\partial x \partial y}\frac{\partial T}{\partial y}+u\frac{\partial C}{\partial y}\frac{ \partial^{2}T}{\partial x\partial y}+v\frac{\partial^{2}C}{\partial y^{2}} \frac{\partial T}{\partial y}+v\frac{\partial C}{\partial y}\frac{\partial^{ 2}T}{\partial y^{2}}\bigg{)}\] \[+\lambda_{t}^{*}\tau\frac{D_{T}}{T_{\infty}}\bigg{(}\frac{ \partial T}{\partial y}\bigg{)}^{2}+2\lambda_{t}^{*}\tau\frac{D_{T}}{T_{ \infty}}\bigg{(}u\frac{\partial T}{\partial y}\frac{\partial^{2}T}{\partial y ^{2}}+v\frac{\partial^{2}T}{\partial y^{2}}\frac{\partial T}{\partial y}\bigg{)}\] \[-\lambda_{t}\bigg{[}u^{2}\frac{\partial^{2}T}{\partial x^{2}}+v^{2 }\frac{\partial^{2}T}{\partial y^{2}}+2 uv\frac{\partial^{2}T}{\partial x \partial y}+\frac{\partial T}{\partial x}\bigg{(}u\frac{u}{\partial x}+v\frac{ \partial u}{\partial y}\bigg{)}\] \[+\frac{\partial T}{\partial y}\bigg{(}u\frac{\partial v}{\partial x }+v\frac{\partial v}{\partial y}\bigg{)}\bigg{]}.\]  
\[u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}=D_{B}\frac{ \partial^{2}C}{\partial y^{2}}+\lambda_{c}^{*}\bigg{(}\frac{D_{T}}{T_{\infty}} \bigg{)}\bigg{(}\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial v}{ \partial y}\frac{\partial^{2}T}{\partial y^{2}}+\frac{\partial u}{\partial y }\frac{\partial^{3}T}{\partial x\partial y}\bigg{)} \tag{4}\] \[-Kr^{2}(C-C_{\infty})\bigg{(}\frac{T}{T_{\infty}}\bigg{)}^{n}exp \bigg{(}\frac{-E_{a}}{k\ T}\bigg{)}-\lambda_{c}^{*}k_{c}\bigg{(}v\frac{ \partial C}{\partial y}+u\frac{\partial C}{\partial x}\bigg{)}\] \[-\lambda_{c}^{*}\bigg{[}u^{2}\frac{\partial^{2}C}{\partial x^{2} }+v^{2}\frac{\partial^{2}C}{\partial y^{2}}+2 uv\frac{\partial^{2}C}{\partial x \,\partial y}+\frac{\partial C}{\partial x}\bigg{(}u\frac{\partial u}{ \partial x}+v\frac{\partial u}{\partial y}\bigg{)}\] \[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\frac{ \partial^{2}T}{\partial y^{2}}+\frac{\partial}{\partial y}\bigg{(}\frac{16 \alpha\varphi^{*}T_{\infty}^{3}}{3k^{*}}\frac{\partial T}{\partial y}\bigg{)} +\frac{Q_{o}}{\rho C_{p}}(T-T_{\infty}) \tag{3}\]0  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\frac{ \partial^{2}T}{\partial y^{2}}+\frac{\partial}{\partial y}\bigg{(}\frac{16 \alpha\varphi^{*}T_{\infty}^{3}}{3k^{*}}\frac{\partial T}{\partial y}\bigg{)} +\frac{Q_{o}}{\rho C_{p}}(T-T_{\infty}) \tag{3}\]1  
having boundary conditions  
\[\left. \begin{array}{l}u=U_{w}=cx,\ v=0,-k\frac{\partial T}{\partial y}=h_{f}( T_{w}-T),\\ -D_{B}\frac{\partial C}{\partial y}=h_{g}(C_{w}-C),\qquad-D_{m}\frac{\partial N }{\partial y}=h_{n}(N_{w}-N)\text{ {at} }y=0\\ v\to 0,\ \frac{\partial u}{\partial y}\to 0,\ T\to T_{w},C\to C_{w},\ N\to N_{w}\text{ {as} }y\to\infty.\end{array} \right\} \tag{6}\]  
The above governing equations are solved by following suitable transformations:  
\[\xi=y\sqrt{\frac{c}{v}},\ u=cxf^{\prime},\ v=-\sqrt{cv}\ f,\ \theta=\frac{T \ -\ T_{\infty}}{T_{w}\ -\ T_{\infty}},\phi=\frac{C-C_{\infty}}{C_{w}\ -C_{\infty}},\ \chi=\frac{N-N_{\infty}}{N_{w}-N_{ \infty}}. \tag{7}\]  
where \(f,\theta,\phi,\ \text{and}\ \chi\) are functions of \(\xi.\) The following equations are obtained:  
\[