## 2 Mathematical Model  
We have consider a mathematical model for MHD steady flow of change in viscosity on Eyring-Powell nano-fluid flow over a nonlinear stretching sheet along with double stratification. The flow is produced by a nonlinear stretching sheet. Moreover, the highlights of thermal conductivity and zero mass flux condition are taken into account. The sheet is stretched with velocity \(u_{w}(x,y)=U_{0}(x+c)^{m-1}\), assume the sheet is designated with profile \(y=A(x+c)^{\frac{m-1}{2}}\) here \(U_{0}\) stretching rate, \(c\) dimensionless constant, \(A\) represents constant, \(m\) represents stretching index. Moreover, it is supposed that the model needs to be satisfied individual for \(m\
eq 1\), because for \(m=1\), it decreases to flat surface. A magnetic field \(B_{o}\) is applied normal to the sheet, vertical to the surface (see in Figure 1). The assumption of rate process is used to obtained the Eyring Powell model (1994) in order to explain the shear of non-Newtonian flow. [PERSON] et al. [26] evaluated the shear stress of in Eyring Powell model as  
\[\tau_{i,j}=\mu\frac{\partial\dot{a}_{i}}{\partial\dot{a}_{j}}+\frac{1}{\beta_ {1}}sinh^{-1}\left(\frac{1\partial\dot{a}_{i}}{c\partial\dot{a}_{j}}\right) \tag{1}\]  
here \(\mu\) represents the dynamic viscosity, \(c\) and \(\beta_{1}\) are represents parameters of the fluid. We take the second order approximation as  
\[sinh^{-1}\left(\frac{1\partial\dot{a}_{i}}{c\partial\dot{a}_{j}}\right)\cong \left(\frac{1\partial\dot{a}_{i}}{c\partial\dot{a}_{j}}\right)-\frac{1}{6}( \frac{1\partial\dot{a}_{i}}{c\partial\dot{a}_{j}})^{3} \tag{2}\]  
Under the above assumptions and boundary layer approximation leads to the governing expressions [37; 38; 26; 39] (Eyring Powell nano-fluid model, energy and concentration formulation) are stated from operation (3)-(6) as follows:  
\[\frac{\partial\dot{a}}{\partial x}+\frac{\partial\phi}{\partial y} =0, \tag{3}\] \[\dot{a}\frac{\partial\dot{a}}{\partial x}+\phi\frac{\partial\dot {a}}{\partial y} =\frac{\partial}{\partial y}(\mu(T)\frac{\partial\dot{a}}{\partial y })+\frac{1}{\rho\partial c}\frac{\partial^{2}\dot{a}}{\partial^{2}y}-\frac{1} {2\rho\delta c^{3}}(\frac{\partial\dot{a}}{\partial y})^{2}(\frac{\partial^ {2}\dot{a}}{\partial y^{2}})-\sigma\frac{B_{0}^{2}\dot{a}}{\rho},\] (4) \[\dot{a}\frac{\partial\dot{T}}{\partial x}+\phi\frac{\partial\dot {T}}{\partial y} =\frac{\partial}{\partial\dot{g}}\left(k(\hat{T})\frac{\partial\hat{T}}{ \partial\dot{g}}\right)+\tau[(\frac{D_{\hat{T}}}{\hat{T}_{\infty}})(\frac{ \partial\hat{T}}{\partial y})^{2}+D_{B}(\frac{\partial C}{\partial y}\frac{ \partial\hat{T}}{\partial y})]\] (5) \[\dot{a}\frac{\partial C}{\partial x}+\phi\frac{\partial C}{ \partial y} =D_{B}(\frac{\partial^{2}C}{\partial^{2}y})+\tau(\frac{D_{\hat{T}}}{\hat{T} _{\infty}})(\frac{\partial^{2}\hat{T}}{\partial y^{2}})-k_{r}(\frac{\hat{T}}{ \hat{T}_{\infty}})^{n}\exp(\frac{-E}{\varsigma\hat{T}})(C-C_{\infty}). \tag{6}\]  
Subjected to boundary condition  
\[\dot{a}(x,y)=\dot{a}_{w}(x)=U_{0}(x+c)^{m},\quad v(x,y)=0 \tag{7}\] \[\hat{T}(x,y)=\hat{T}_{w}(x,y)+N\frac{\partial\hat{T}}{\partial y},\] \[\left(D_{B}\frac{\partial C}{\partial y}\right)+\left(\frac{D_{ \hat{T}}}{\hat{T}_{\infty}}\right)\left(\frac{\partial\hat{T}}{\partial y} \right)=0,\quad\text{at}\quad y=A(x+c)^{\frac{1-m}{2}},\] (8) \[\dot{a}(x,y)\to 0,\quad\hat{T}\to\hat{T}_{\infty}\quad C \to C_{\infty}\quad\text{as}\quad y\to\infty. \tag{9}\]In the above expression the velocity components \(\hat{u}\) and \(\hat{v}\) are in the \(x\) and \(y\) - directions, respectively, \(\mu(T)\) denotes viscosity, \(\rho\) the density of the nanoparticle, \(\hat{T}\) denotes the temperature of the nanofluid, \(\hat{C}\) represent the concentration of the nanofluid, \(C_{w}\) and \(\hat{T}_{w}\) the concentration and temperature and along the stretching sheet. \(\hat{C}_{\infty}\) and \(\hat{T}_{\infty}\) the ambient concentration and temperature, \((\rho c)_{f}\) the heat capacitance of the base fluid, \(D_{\hat{T}}\) the thermostereis coefficient.