## 2 Materials and Methods  
The incompressible Newtonian CNT flow due to a linearly stretched semi-infinite plate embedded in MHD with mass transpiration is investigated, with [PERSON]'s slip taken into account. The flow is in the x-direction, with the y-axis perpendicular to it (see Figure 1), and a homogeneous magnetic field of intensity is supplied along it. The equations of conservation of mass and momentum are described as follows. \[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=v_{nf}\frac{ \partial^{2}u}{\partial y^{2}}-\frac{\sigma_{nf}B_{0}^{2}\sin^{2}(\tau)u}{ \rho_{nf}}, \tag{2}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{\kappa_{nf }}{(\rho\mathcal{C}_{p})_{nf}}\frac{\partial^{2}T}{\partial y^{2}}-\frac{1}{ (\rho\mathcal{C}_{p})_{nf}}\frac{\partial q_{f}}{\partial y}, \tag{3}\]The boundary conditions are [PERSON] et al. (2022). \[ \begin{array}{c}u(x,0)={\it d}ax+A\frac{\partial u}{\partial y},\
u(x,0)=V_{c},u( x,\infty)\to 0,\\ \theta(0)=0,\qquad\theta(0)=1+\tilde{\xi}\theta^{\prime}(0)\end{array} \Bigg{\}}. \tag{4}\]  
where \(A\) is the first order slip parameter. We now introduce the similarity variable. \[ \begin{array}{c}\psi=\sqrt{\alpha v_{f}}xf(\eta),\eta=\sqrt{\frac{a}{v_{f}}y },\theta(\eta)=\frac{T-T_{\infty}}{T_{\omega}-T_{\infty}}\\ u=\frac{\partial\psi}{\partial y}=axf^{\prime}(\eta),\upsilon=-\frac{ \partial\psi}{\partial x}=-af^{\prime}(\eta).\end{array} \Bigg{\}}. \tag{5}\]  
where \(\upsilon_{nf}\) is the effective kinematic viscosity, \(\kappa_{nf}\) is the thermal conductivity, \(T\) is temperature, and \(\left(\rho_{cp}\right)_{nf}\) is the heat capacity, \(\eta\) is the similarity variables, and \(\psi\) is the stream function defines as follows ([PERSON], 2014). \[ \begin{array}{c}\rho_{nf}=\ (1-\phi)\rho_{f}+\phi\rho_{CNT},\qquad\upsilon_{nf} =\ \frac{\mu_{nf}}{\rho_{nf}},\\ \left(\rho_{cp}\right)_{nf}=\ (1-\phi)\left(\rho_{cp}\right)_{f}+\phi\left( \rho_{cp}\right)_{CNT}\ \alpha_{nf}\ =\ \frac{\kappa_{nf}}{\left(\rho_{cp}\right)_{nf}},\\ \frac{\kappa_{nf}}{k_{f}}\ =\ \frac{1-\phi+2\phi\frac{\kappa_{CNT}}{ kT_{\omega}-k_{f}}\log\frac{\kappa_{CNT}+\kappa_{f}}{2\kappa_{f}}}{1-\phi+2 \phi\frac{\kappa_{f}}{kT_{\omega}-k_{f}}\log\frac{\kappa_{CNT}+\kappa_{f}}{2 \kappa_{f}}},\\ \frac{\sigma_{nf}}{\sigma_{f}}\ =\ 1+\frac{3\phi\left(\frac{\kappa_{CNT}}{ \sigma_{f}}-1\right)}{\frac{\kappa_{CNT}}{\sigma_{f}}-2-\left(\frac{\sigma_{CNT }}{\sigma_{f}}-1\right)\phi}.\end{array} \Bigg{\}} \tag{6}\]  
where \(\rho_{nf}\) the density of the Nano fluid is, \(\phi\) is the solid volume fraction of the Nano fluid, and \(\sigma_{nf}\) is the electrical conductivity of the Nano fluid. [PERSON] (2015) approximation and Radiative heat flux \(q_{r}\) is simplified as follows using [PERSON] et al. (2016, 2017, 2018):  
\[q_{r}=-\frac{4\sigma^{*}}{3k^{*}}\frac{\partial T^{4}}{\partial y}, \tag{7}\]  
Figure 1: Physical model of the stretching/shrinking surface. where \(\sigma^{*}\) is the Stefan-Boltzmann constant and absorption coefficient is \(k^{*}\), \(q_{r}\) is the radioactive flux. Expanding \(T^{4}\) in a Taylor expression we obtain. \[T^{4}\cong 4T_{\infty}^{3}T-3T_{\infty}^{4}. \tag{8}\]  
By using Equations (7) and (8) in (3), reduce to  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{\kappa_{ nf}}{\left(\rho_{cp}\right)_{nf}}\Bigg{(}1+\frac{16\sigma^{*}T_{\infty}^{3}}{3k^{*} \kappa_{nf}}\Bigg{)}\frac{\partial^{2}T}{\partial y^{2}}, \tag{9}\]  
With the introduction of the stream function the continuity Equation (1) is satisfied, and the momentum Equations (2) and (3) transform into  
\[f^{\prime\prime}+\alpha\Big{[}ff^{\prime\prime}-{f^{\prime}}^{2}\Big{]}-\gamma \sin^{2}(\tau)f^{\prime}=0, \tag{10}\]  
\[\tau(1+N_{r})\theta^{\prime\prime}+f\theta^{\prime}=0. \tag{11}\]  
where \(\alpha=(1-\phi)^{2.5}\Big{(}(1-\phi)+\phi\Big{(}\frac{\rho_{s}}{\rho_{f}} \Big{)}\Big{)}\), and \(\gamma=M(1-\phi)^{2.5}\).