## 2 Problem Formulation  
Steady, three-dimensional, mixed convection, MHD laminar boundary layer flow and heat transfer with electrically conducting fluid over a nonlinear, vertical, permeable, and stretching/shrinking sheet has been considered for investigation. Besides, this study has employed water-based nanofluid with varied types of metallic solid nanoparticles, namely copper (Cu) and silver (Ag). The thermophysical properties of these base fluids and nanoparticles are presented in Table 1[40; 41; 42]. Moreover, the sheet is extended nonlinearly in two lateral \(x\)- and \(y\)- directions, as illustrated in Figure 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Physical Properties** & **Water** & **Cu** & **Ag** \\ \hline \(C_{p}\) (\(\mathrm{J}\) kg\({}^{-1}\) K\({}^{-1}\)) & 4179 & 385 & 235 \\ \(\rho\) (\(\mathrm{k}\) gm\({}^{-1}\)) & 997.1 & 8933 & 10,500 \\ \(k\) (Wm\({}^{-1}\) K\({}^{-1}\)) & 0.613 & 400 & 429 \\ \(\beta\times 10^{-5}\) (K\({}^{-1}\)) & 21 & 1.67 & 1.89 \\ \(\sigma\) (Sm\({}^{-1}\)) & \(5.5\times 10^{-6}\) & \(59.6\times 10^{6}\) & \(6.3\times 10^{7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Thermophysical properties of base fluid and nanoparticles. It further portrays that coordinate \(z\)- is measured in the direction of normal to the sheet, while coordinates \(x\) and \(y\) are in the plane of the sheet with \(x\)-axis measured in vertical direction. Besides, it has been assumed that the flow subjected to the variable of transverse magnetic field \(B\), is applied parallel to \(z\)-axis. The induced magnetic field is also assumed to be small compared to the applied magnetic field, hence it is neglected. Other than that, the vertical sheet is assumed to move continuously in both \(x\)- and \(y\)-directions with velocities, \(u_{w}\) and \(v_{w}\), respectively, while the mass flux velocity is \(w_{w}\), the temperature of the sheet is \(T_{w}\) and the constant temperature of the ambient fluid is \(T_{\infty}\), which are determined later. Under these conditions, the governing boundary layer equations for both flow and heat transfer are given in the following:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w} {\partial z}=0, \tag{1}\]  
\[u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u }{\partial z}=v_{nf}\frac{\partial^{2}u}{\partial z^{2}}-\frac{\sigma_{nf}B^{ 2}u}{\rho_{nf}}+\frac{g(\rho\beta)_{nf}}{\rho_{nf}}(T-T_{\infty}), \tag{2}\]  
\[u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v }{\partial z}=v_{nf}\frac{\partial^{2}v}{\partial z^{2}}-\frac{\sigma_{nf}B^{ 2}v}{\rho_{nf}}, \tag{3}\]  
\[u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T }{\partial z}=\alpha_{nf}\frac{\partial^{2}T}{\partial z^{2}}, \tag{4}\]  
along with the following boundary conditions:  
\[ \begin{array}{ll}u=cu_{w}+N_{1}\frac{\partial u}{\partial z},&v=cv_{w}+N_{1} \frac{\partial v}{\partial z},&w=w_{w},&T=T_{w}+K_{1}\frac{\partial T}{ \partial z}\text{ at }z=0,\\ u\to 0,&v\to 0,&T\to T_{\infty}\text{ as }z\rightarrow\infty,\end{array}  \tag{5}\]  
where \(u\), \(v\) and \(w\) are the velocity components along the \(x\)-, \(y\)- and \(z\)- axes, respectively, \(c\) is the stretching/shrinking parameter with \(c<0\) for a shrinking sheet and \(c>0\) for a stretching sheet, \(g\) refers to the acceleration due to gravity, \(B\) is the constant magnetic field, \(\
u_{nf}\) denotes the kinematic viscosity of the nanofluid, \(\left(\rho\beta\right)_{nf}\) denotes the thermal expansion coefficient of the nanofluid, \(\rho_{nf}\) refers to the density of the nanofluid, \(\sigma_{nf}\) is the electrical conductivity of the nanofluid and \(\alpha_{nf}\) is the thermal  
Figure 1: Physical model and coordinate system. diffusivity of the nanofluid. Furthermore, the effective properties of the nanofluid are described in the following (see [40; 41; 43; 44; 45]):  
\[