## 2 Mathematical Modeling
\begin{array}{l}\Big{(}\rho c_{p}\Big{)}_{nf}\bigg{(}\frac{\partial T}{ \partial y}+u\frac{\partial T}{\partial x}+\frac{\partial T}{\partial t}\Big{)} +\lambda_{2}\bigg{[}\frac{\partial^{2}T}{\partial t^{2}}+\frac{\partial u}{ \partial t}\frac{\partial T}{\partial x}+2u\frac{\partial^{2}T}{\partial t \partial x}+2v\frac{\partial^{2}T}{\partial t\partial y}+\frac{\partial v}{ \partial t}\frac{\partial T}{\partial y}\\ +v\frac{\partial T}{\partial y}\frac{\partial v}{\partial y}+u^{2}\frac{ \partial T}{\partial x^{2}}+v^{2}\frac{\partial^{2}T}{\partial v^{2}}+u\frac{ \partial T}{\partial x}\frac{\partial u}{\partial x}+2 uv\frac{\partial^{2}T}{ \partial x\partial y}+u\frac{\partial T}{\partial y}\frac{\partial v}{\partial x }+v\frac{\partial T}{\partial x}\frac{\partial u}{\partial y}\bigg{]}=\\ \Big{(}k_{nf}+\frac{16}{3}\frac{T_{0}^{3,\sigma^{*}}}{K}\Big{)}\frac{\partial^ {2}T}{\partial y^{2}}+\alpha_{1}\bigg{[}\frac{\partial u}{\partial y}\frac{ \partial^{2}u}{\partial t\partial y}+u\frac{\partial u}{\partial y}\frac{ \partial^{2}u}{\partial x\partial y}+v\frac{\partial u}{\partial y}\frac{ \partial^{2}u}{\partial y^{3}}\bigg{]}+q^{\prime\prime\prime},\end{array}  \tag{7}\]  
The boundary conditions are defined as:  
\[ \begin{array}{l}-\xi_{x}=0,\xi_{y}=U_{w},T=T_{s},\text{at}\,y=0,\\ \xi_{yy}=0,-\xi_{x}=h_{t},T=0,\text{as}\,y=h(t).\end{array}  \tag{8}\]  
In the above equations, \(u\) and \(v\) are velocity components along \(x-\) and \(y-\) directions respectively, \(v_{nf}\) is the kinematic viscosity of the nanofluid, \(\alpha_{1}\) is the non- dimensionless second grade fluid parameter, \(\rho_{nf}\) is the density of the nanofluid, \(\sigma_{nf}\) is the electrical conductivity of the nanofluid, \(c_{p}\) is the specific heat on the nanofluid, \(\lambda_{2}\) is the Cattaneo-Christov parameter, \(k_{nf}\) is the thermal conductivity of the nanofluid, \(\sigma^{*}\) is the Stephan-Boltzmann constant, \(k^{*}\) is the mean absorption coefficient, and \(q^{\prime\prime\prime}\) is the non-uniform heat source/sink term. \(q^{\prime\prime\prime}\) is defined as  
\[q^{\prime\prime\prime}=\frac{k_{f}U_{w}(T_{s}-T_{0})}{xv_{f}}\bigg{(}B^{*} \frac{(T-T_{0})}{(T_{s}-T_{0})}+A^{*}f^{\prime}\bigg{)}. \tag{9}\]  
According to [PERSON] model [47]  
\[ \begin{array}{l}\mu_{nf}=\frac{\mu_{f}}{(1-\phi)^{5/2}},\rho_{nf}=\phi\rho_{ CNT}+(1-\phi)\rho_{f},\alpha_{nf}=\frac{k_{nf}}{\rho_{nf}(c_{p})_{nf}},v_{nf}= \frac{\mu_{nf}}{\rho_{nf}},\\ \frac{\alpha_{nf}}{\sigma_{f}}=1+\frac{3\phi\phi-3\phi}{\phi+2-\phi\phi+\phi}, \sigma=\frac{\sigma_{CNT}}{\sigma_{f}},\frac{k_{nf}}{k_{f}}=\frac{2\phi\frac{k_ {CNT}}{\sqrt{k_{f}}}\ln\Big{(}\frac{k_{CNT}+k_{f}}{2k_{f}}\Big{)}+(1-\phi)}{2 \phi\frac{k_{f}}{\sqrt{k_{f}}}\ln\Big{(}\frac{k_{CNT}+k_{f}}{2k_{f}}\Big{)}+(1 -\phi)}.\end{array}  \tag{10}\]  
The similarity variables are defined as  
\[ \begin{array}{l}\xi=\beta\sqrt{\frac{bv_{f}}{(1-at)}}xf(\eta),\theta=\frac{T- T_{0}}{T_{s}-T_{0}},T_{s}=T_{0}-\frac{T_{r}}{(1-at)^{1.5}}\bigg{(}\frac{bx^{2}}{2 v_{f}}\bigg{)}\theta(\eta),\eta=\sqrt{\frac{b}{v_{f}(1-at)}}\frac{y}{\beta}.\end{array}  \tag{11}\]  
Using Equation (11), the continuity equation is obvious, and Equations (6) and (7) yield  
\[ \begin{array}{l}f^{\prime\prime\prime}+(1-\phi)^{2.5}\bigg{(}1-\phi+\phi \frac{\rho_{CNT}}{\rho_{f}}\bigg{)}\lambda\Big{[}ff^{\prime\prime}-{f^{\prime 2}}-S\Big{(}f^{ \prime}+\frac{1}{2}\eta f^{\prime\prime}\Big{)}\Big{]}+\alpha(1-\phi)^{2.5}\\ \Big{(}2 ff^{\prime\prime\prime}+S\Big{(}2f^{\prime\prime\prime}+\frac{1}{2}\eta f ^{(iv)}\Big{)}-ff^{(iv)}-f^{\prime\prime 2}\Big{)}-(1-\phi)^{2.5}\frac{\alpha_{nf}}{\sigma_{f}}Mf^{ \prime}\cos^{2}\varepsilon=0,\end{array}  \tag{12}\]  
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