## 2 Mathematical Modeling  
We have assumed the laminar and incompressible thin film flow of second grade fluid containing carbon nanotubes (CNTs) over a linearly extending sheet with Cattaneo-Christov model of heat flux. The stretching sheet is positioned at the starting point of a coordinate system \((x,y)\) as displayed in Figure 1. The stretching sheet moves along x-direction with a velocity \(U_{w}(x,t)=\frac{h\pi}{(1-at)}\) where \(\alpha\) and \(b\) are constants, and temperature is considered as \(T_{w}(x,y)\). The width of the thin film flow is \(h(x,y)\). Along the extending surface, a variable magnetic field of \(B(x,t)=\frac{B_{0}}{(1-at)^{\frac{1}{2}}}\) is applied with an allied angle \(\epsilon\). The stream function \(\xi\) is taken such that \(u=\frac{\partial\xi}{\partial y}\) and \(v=-\frac{\partial\xi}{\partial x}\). The Cauchy stress tensor in a second-grade fluid is [43; 44]  
\[\mathrm{T}=\mu\mathrm{A}_{1}+\alpha_{1}\mathrm{A}_{2}+\alpha_{1}\mathrm{A}_{1} ^{2}-p\mathrm{I}, \tag{1}\]  
\[\mathrm{A}_{1}=(\mathrm{grad}V)+(\mathrm{grad}V)^{\mathrm{T}}, \tag{2}\]  
\[\mathrm{A}_{2}=\frac{d\mathrm{A}_{1}}{dt}+\mathrm{A}_{1}(\mathrm{grad}V)+( \mathrm{grad}V)^{\mathrm{T}}\mathrm{A}_{1}, \tag{3}\]  
where \(\alpha_{1}\) and \(\alpha_{2}\) are material parameters, \(\mu\) is dynamic viscosity, \(p\) is the pressure, \(\mathrm{I}\) is the identity tensor, \(\mathrm{A}_{1}\) and \(\mathrm{A}_{2}\) are Rivlin-Ericksen tensors defined above, \(d/dt\) is the material time derivative, and \(V\) is the velocity of the fluid. The Clausius-Duhem inequality is verified and Helmholtz free energy is minimum in equilibrium for the fluid locally at rest when [43; 44]  
\[\mu\geq 0,\alpha_{1}\geq 0,\alpha_{1}+\alpha_{2}=0. \tag{4}\]  
When \(\alpha_{1}+\alpha_{2}=0\), the second grade fluid equation reduces to viscous fluid. Applying the above assumptions, the governing equations can be specified as [46] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Reference \#**} & **Second Grade** & **Film** & \multirow{2}{*}{**CNTs**} & **Heat Flux Model of** & **Entropy** \\  & **Nanfluid** & **Thickness** & & **Cattaneo-Christov** & **Generation** \\ \hline Ref. [17] & \(\times\) & \(\surd\) & \(\times\) & \(\times\) & \(\times\) \\ Ref. [19] & \(\times\) & \(\surd\) & \(\times\) & \(\times\) & \(\times\) \\ Ref. [23] & \(\times\) & \(\surd\) & SWCNTs & \(\times\) & \(\times\) \\ Ref. [24] & \(\times\) & \(\surd\) & \(\times\) & \(\times\) & \(\times\) \\ Ref. [25] & \(\times\) & \(\surd\) & \(\times\) & \(\times\) & \(\times\) \\ Ref. [26] & \(\times\) & \(\surd\) & \(\times\) & \(\times\) & \(\times\) \\ Current analysis & \(\surd\) & \(\surd\) & SWCNTs/MWCNTs & \(\surd\) & \(\surd\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Studies on the thin film flows (\(\surd\) = Effect is present, \(\times\) = Effect is not present). Figure 1: Gometry of the flow problem. \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{5}\]  
\[v\frac{\partial u}{\partial y}+u\frac{\partial u}{\partial x}+\frac{\partial u}{ \partial t}=v_{nf}\frac{\partial^{2}u}{\partial y^{2}}+\frac{\alpha_{1}}{ \partial v_{nf}}\bigg{[}\frac{\partial^{2}u}{\partial x\partial y^{2}}+\frac{ \delta^{2}v_{nf}^{2}}{\partial y^{2}}\frac{\partial u}{\partial y}+v\frac{ \partial^{2}u}{\partial y^{3}}+u\frac{\partial^{2}u}{\partial x\partial y^{2}} +\frac{\delta^{2}u}{\partial y^{3}}\frac{\partial u}{\partial x}\bigg{]}+\frac{ \alpha_{nf}}{\rho_{nf}}\beta^{2}(t)u\cos^{2}\varepsilon, \tag{6}\]  
\[