## 3 Orientation and Formulation of Physical Problem  
It is considered that the 2D Cross nanofluid is flowing, facing a stagnation point in a porous medium. Furthermore, fluid is considered to be incompressible very near to the stagnation point and the surface coincides with the plane of equation \(y=0\). The zero-mass flux and mixed convective temperature conditions can be considered at the boundary surface of the sheet. The active and passive controls of nanoparticles at the surface of the sheet are controlled by zero mass flux nanoparticle conditions. The stagnation point is at the point \(O\). The fluid region is taken as \(y>0\), and the above sheet is porous, which can be seen in Figure 1. Continuity, momentum, temperature, and concentration equations are derived from the law of conservation of momentum, [PERSON]'s equation, the second law of thermodynamics, and [PERSON]'s second law of diffusion respectively. The variation in fluid, temperature and concentration are monitored with the utilization of variable viscosity [32], variable thermal conductivity [33; 34], and variable molecular diffusivity phenomena, respectively. The governing system of equations in the light of the above-mentioned assumptions is given below. \[u_{x}+v_{y}=0 \tag{4}\]  
\[\rho_{f}[uu_{x}+vv_{x}]=\frac{\partial}{\partial y}\Bigg{\{}\frac{\mu_{0}}{ \left[1+\Gamma^{2}\big{(}u_{y}\big{)}^{\mu}\right]}\Bigg{\}}+\frac{\sigma B_{0 }^{2}}{\rho}\sin^{2}(\omega)(u-u_{e})+u_{e}\frac{du_{e}}{dx}+g\beta_{\rm T}(T -T_{\infty}) \tag{5}\]  
\[\big{[}uT_{x}+vT_{y}\big{]}=\frac{1}{\rho c_{p}}\frac{\partial}{\partial y} \big{[}K(T)T_{y}\big{]}+\tau\bigg{[}D_{B}T_{y}C_{y}+\frac{D_{\rm T}}{T_{\infty }}\Big{[}\big{(}T_{y}\big{)}^{2}\Big{]}\bigg{]}+\frac{16\sigma^{*}}{3k^{\mu} c_{p}c_{p}}\bigg{[}\frac{\partial}{\partial y}\Big{(}T_{y}T^{3}\Big{)}\bigg{]} \tag{6}\]  
\[\text{where }\Bigg{[} \begin{array}{l}K(T)=k_{\infty}\Big{\{}1+\epsilon_{1} \frac{(T-T_{\infty})}{\Delta T}\Big{\}},\text{and }D(C)=D_{\infty}\Big{\{}1+\epsilon_{2}\frac{(C-C_{\infty})}{\Delta C}\Big{\}} \\ \frac{1}{\mu}=\frac{1}{\mu_{\infty}}[1+\gamma_{1}(T-T_{\infty})],\text{or }\frac{1}{\mu}=\alpha\{T-T_{\rm T}\}\end{array} .\]  
Following the boundary conditions must be considered for Equations (4)-(7). \[\left. \begin{array}{l}u_{w}=cx,-K(T)T_{y}=(T-T_{W}),v=v_{w},\Big{[}D_{B}C_{ y}+\frac{D_{\rm T}}{T_{\infty}}\big{(}T_{y}\big{)}\Big{]}=0,\text{at }y=0,\\ C\rightarrow\mathcal{C}_{\infty},u\to u_{e}=ax,T\to T_{\infty},as \,y\,\rightarrow\infty\end{array} \right\} \tag{8}\]  
The system of PDE (4)-(8) is passed through these similar transformations  
\[\left. \begin{array}{l}u=xcf^{\prime}(\eta),\quad\eta=y\big{(}\frac{v}{c} \big{)}^{-1/2},v=-\sqrt{cv}f(\eta)\\ \theta(\eta)=\frac{T-T_{\infty}}{T_{w}-T_{0}},\phi(\eta)=\frac{C-C_{\infty}} {C_{w}-C_{0}}\end{array} \right\} \tag{9}\]  
\[f^{\prime\prime}\left(1+(w\epsilon_{1}f^{\prime\prime})^{n}\right)^{-2}\{[1- (n-1)(Wef^{\prime})^{n}]\}+f^{\prime\prime}(\frac{\theta^{\prime}}{\theta_{r -\theta}})-\big{(}\frac{\theta_{r-\theta}}{\theta_{r}}\big{)}[f^{\prime 2}-ff^{ \prime\prime}+G\theta+A^{2}-1-M^{2}\sin^{2}(\omega)f^{\prime}] \tag{10}\]  
\[(1+\epsilon_{1})\theta^{\prime\prime}+\epsilon_{1}\theta^{\prime 2}+\frac{4}{3}N \frac{d}{d\eta}\bigg{[}\Big{\{}\theta\Big{(}\theta_{f}-1\Big{)}+1\Big{\}}^{3 }\theta^{\prime}\bigg{]}+\text{Pr}\Big{[}Nb\phi^{\prime}\theta^{\prime}+N_{l} \theta^{\prime 2}+\big{(}f\theta^{\prime}-f^{\prime}\theta\big{)}\Big{]}=0 \tag{11}\]  
\[(1+\epsilon_{2})\phi^{\prime\prime}+\epsilon_{2}\phi^{\prime}+\frac{N_{l}}{N_ {b}}\theta^{\prime\prime}+Sc\big{(}f\phi^{\prime}-f^{\prime}\phi\big{)}=0 \tag{12}\]  
\[f=s,f^{\prime}=1,\theta^{\prime}=-\gamma(1-\theta),N_{b}\phi^{\prime}+N_{l} \theta^{\prime}=0\,\,at\,\eta=0\\[