## 2 Formulation of the Problem  
Consider the two-dimensional time dependent and thermally and electrically conducting flow of Maxwell nanofluid moving on the vertical continuous stretching sheet which passes through a thin slot. The flow is along \(y\)-direction which is shown schematically in Figure 1. The uniform magnetic field \(B_{0}\) and electric field \(E_{0}\) are applied to the thermal extrusion forming stretching sheet along the \(y\)-direction. The concentration \(C_{w}\) and temperature \(T_{w}\) of the nanoparticles are kept uniform where these values are supposed to be greater than ambient concentration and temperature. The fluid flow is considered to be limited in a region where \(y>0\). The internal heat is taken to be non-uniform in absorption/generation to know the concentration and temperature variances between the fluid and the surfaces. The velocity is expected to be directly proportional to the space taken from the slit. The governing equation of energy, mass and momentum in existence of magnetic field and heat source is expressed as Equations (1)-(3). \[div\overrightarrow{V}=0, \tag{1}\]  
\[\rho\frac{d\overrightarrow{V}}{dt}=divT+\overrightarrow{J}\times\overrightarrow {B}, \tag{2}\]  
\[(\rho c)_{p}\frac{DT}{Dt}=T.\frac{\delta\overrightarrow{V}}{\delta x}+K\frac{ \partial^{2}T}{\partial t^{2}}+Q(T-T_{\infty}). \tag{3}\]  
where \(\overrightarrow{B}\) denotes the magnetic field in the direction of \(y\)-axis, \(\overrightarrow{J}\) is the current density and defined as \(\overrightarrow{J}=\sigma(\overrightarrow{V}\times\overrightarrow{B})\), \(k\) is thermal conductivity and represents specific heat, \(\sigma\) is electrical conductivity, and \(\overrightarrow{V}\) is fluid velocity where \(\overrightarrow{V}=\{u(x,y),v(x,y),0\}\). Here, \(\overrightarrow{J}\times\overrightarrow{B}\) is the Lorentz force and it is reduced to the form of \(\overrightarrow{J}\times\overrightarrow{B}=-\sigma\overrightarrow{B}_{0} 2\overrightarrow{V}\). The flow of the nanofluid and heat transfer is assumed in unsteady state which is incompressible, laminar and stable. Keeping in view the above deliberation, the basic equations of continuity, velocity, energy and concentration are reduced as:  
\[\frac{\partial\overrightarrow{u}}{\partial x}+\frac{\partial \overrightarrow{v}}{\partial y}=0, \tag{4}\]  
\[\frac{\partial\overrightarrow{u}}{\partial t}+\overrightarrow{u}\frac{ \partial\overrightarrow{u}}{\partial x}+\overrightarrow{v}\frac{\partial \overrightarrow{u}}{\partial y}+\lambda_{1}\left(\overrightarrow{u} \frac{2}{u}\frac{\partial^{2}\overrightarrow{u}}{\partial x^{2}}+ \overrightarrow{v}^{2}\frac{\partial^{2}\overrightarrow{u}}{\partial x^{2} }+2\overrightarrow{u}\overrightarrow{v}\frac{\partial^{2}\overrightarrow{u} }{\partial x\partial y}\right)=\
u\frac{\partial^{2}\overrightarrow{u}}{ \partial y^{2}}-\frac{\sigma\overrightarrow{B}_{0}}{\rho_{f}}\overrightarrow {u}-\frac{\sigma\overrightarrow{B}_{0}\overrightarrow{v}}{\rho_{f}}, \tag{5}\]  
\[\frac{\partial T}{\partial t}+\overrightarrow{u}\frac{\partial T}{\partial x}+ \overrightarrow{v}\frac{\partial T}{\partial y}=\frac{k}{\rho c_{p}}\frac{ \partial^{2}T}{\partial y^{2}}-\frac{1}{\rho c_{p}}\frac{\partial q_{p}}{ \partial y}+\frac{1}{\rho c_{p}}(q^{\prime\prime\prime}), \tag{6}\]  
\[\frac{\partial C}{\partial t}+\overrightarrow{u}\frac{\partial C}{\partial x}+ \overrightarrow{v}\frac{\partial C}{\partial y}=D_{m}\frac{\partial^{2}C}{ \partial y^{2}}-\frac{\partial}{\partial y}\left(\overrightarrow{V}_{T}C \right), \tag{7}\]  
The relevant boundary conditions are:  
\[ \begin{array}{l}\overrightarrow{u}=\overrightarrow{U}_{w}(x,t),\ \ \overrightarrow{v}= \overrightarrow{V}_{w},\ \ T=T_{w}(x,t),\ \ C=C_{w}(x,t),\ \ \ at\ y=0,\\ \overrightarrow{u}\to 0,\ \ T\to T_{\infty},\ \ C\to C_{\infty},\ \ \text{at}\ \ y\to\infty.\end{array}