## 2 Formulation of the Problem
 \tag{8}\]  
where \(\overrightarrow{v}\) and \(\overrightarrow{u}\) are components of velocity along \(\overrightarrow{y}\) and \(\overrightarrow{x}\) directions, respectively, \(t\) is the time, \(\overrightarrow{U}_{w}(x,t)\) is the stretching sheet velocity, \(T_{w}(x,t)\) is the stretching surface temperature, \(C_{w}(x,t)\) represents the concentration of the stretching surface, \(T_{\infty}\) represents the temperature, which is far from the extending surface with \(T_{w}>T_{\infty}\) and \(C_{\infty}\) is the representation of concentration which is far away from the  
Figure 1: The schematic diagram of the Maxwell nanofluid flow. extending surface with \(C_{w}>C_{\infty}\). The term \(\overrightarrow{V}_{w}=-\sqrt{\frac{vdL_{w}}{2x}}f(0)\) shows the mass transmission at the surface with \(\overrightarrow{V}_{w}<0\) for suction and \(\overrightarrow{V}_{w}>0\) for injection. The source/sink is non-uniform term \(q^{\prime\prime\prime}\), defined as:  
\[q^{\prime\prime\prime}=\frac{\kappa L_{w}(x,t)}{xv}\big{(}A(T_{w}-T_{\infty}) f^{\prime}+(T-T_{\infty})B\big{)}, \tag{9}\]  
where \(A\) and \(B\) represent temperature- and space-dependent coefficients. When \(A\) and \(B\) greater than zero, they link to inner heat source, and, when \(A\) and \(B\) are less than zero, they correspond to inner heat sink. The sheet is stretched, thus flow of nanofluid is caused, and it moves with the surface velocity, temperature and concentration in the form  
\[\overrightarrow{U}_{w}(x,t)=\frac{ax}{1-ct},T_{w}(x,t)=T_{\infty}+\frac{ax}{1- ct},C_{w}(x,t)=C_{\infty}+\frac{ax}{1-ct}, \tag{10}\]  
where \(c\) and \(a\) represents stretching parameter respectively, which are positive constants with \(ct<1,c\geq 0\). It is perceived that the extending rate \(\frac{a}{1-ct}\) is increases with time t since \(a>0\). The similarity variables are presented as  
\[\eta=\sqrt{\frac{a}{v(1-ct)}y},\overrightarrow{u}=\frac{ax}{1-ct}f^{\prime}( \eta),\ \overrightarrow{v}=-\sqrt{\frac{va}{1-ct}}f(\eta),\theta(\eta)=\frac{T-T_{ \infty}}{T_{w}-T_{\infty}},\ \ \phi(\eta)=\frac{C-C_{\infty}}{C_{w}-C_{\infty}}. \tag{11}\]  
In Equation (6), applying the Rosseland approximation for thermal radiation, the heat flux \(q_{r}\) is demarcated as \(q_{r}=-\frac{4\sigma^{*}}{3K^{*}}\frac{\partial T^{4}}{\partial y}\), where \(\sigma^{*}\) is called the of Stefan-Boltzmann constant and \(K^{*}\) is known as the mean absorption coefficient. The temperature differences are such that the term \(T^{4}\) within the flow is a function of the temperature. Using Taylor series, \(T^{4}\) takes the form of free watercourse temperature \(T_{\infty}\):  
\[T^{4}=T_{\infty}^{4}+4T_{\infty}^{3}(T-T_{\infty})+6T_{\infty}^{2}(T-T_{ \infty})^{2}+..., \tag{12}\]  
Ignoring those terms exhibiting higher-order and beyond \((T-T_{\infty})\), the obtain result is  
\[T^{4}\cong 4T_{\infty}^{3}T-3T_{\infty}^{4}, \tag{13}\]  
Thus, adding Equation (13), \(q_{r}\) reduce to the form of  
\[q_{r}=-\frac{16T_{\infty}^{3}\sigma^{*}}{3K^{*}}\frac{\partial^{2}T}{\partial y ^{2}}, \tag{14}\]  
The thermophoresis influence is discussed with the help of average velocity attained by minute particles as compared to gas particles, where temperature variations is brought. In the flow, the temperature variations in \(x\)-direction is less than in \(y\)-direction, which is why only the \(V_{T}\) along \(y\)-axis is considered. Thus, the velocity of thermophoretic \(V_{T}\), as seems in Equation (10), is articulated as:  
\[\overrightarrow{V}_{T}=-\frac{k_{1}v}{T_{R}}\frac{\partial T}{\partial y}, \tag{15}\]  
in which \(T_{R}\) is the reference temperature coefficient.