## 2 Materials and Methods  
Consider a two-dimensional flow of Williamson fluid that has constant density, variable viscosity, and a temperature gradient over an unsteady stretched surface, in which heat and mass are transmitted instantaneously. The flow coordinates are selected in such a manner that the \(x\)-axis is parallel to the plate and the \(y\)-axis is vertical to it. The stretching velocity of the sheet is in the direction of the \(x\)-axis with magnitude \(U\left(x,t\right)=\frac{bx}{1-at}\), in which \(b>0\) is the stretching velocity constraint and defined in [37; 38; 39]. If \(b<0\) then it will become a shrinking velocity constraint. The temperature field is defined as \(T_{s}(x,t)=T_{0}-T_{ref}\left[\frac{bx^{2}}{2v}\right]\left(1-at\right)^{- \frac{3}{2}}\), and the magnitude is inversely proportional to the distance from the surface. Similarly, the concentration field for the given flow is defined as \(C_{s}(x,t)=C_{0}-C_{ref}\left[\frac{bx^{2}}{2v}\right]\left(1-at\right)^{- \frac{3}{2}}\), where \(T_{0}\) represents the temperature at the surface, \(T_{ref}\) indicates the reference temperature, and \(C_{ref}\) indicates the reference concentration, respectively, as shown in [27; 28; 29; 30], such that \(0\leq T_{ref}\leq T_{0}\) and \(0\leq C_{ref}\leq C_{0}\). The local Reynolds is defined as \(\frac{bx^{2}}{v\left(1-at\right)}\). Firstly, the sheet is fixed to the origin; after that some outer force is applied to stretch the surface in the direction of the x-coordinate axis at a velocity \(U\left(x,t\right)=\frac{b}{1-at}\) in time \(0\leq a<1\). Taking the above suppositions into consideration, the governing equations of continuity, velocity, temperature, and concentration can be expressed as:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0, \tag{1}\]  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial y}=\frac{1}{\rho}\frac{\partial}{\partial y}\left(\mu(T)\frac{ \partial u}{\partial y}\right)+\frac{\sqrt{2}\Gamma}{\rho}\frac{\partial}{ \partial y}\left[\mu(T)\frac{\partial u}{\partial y}\right]\frac{\partial u }{\partial y}, \tag{2}\]\[ \begin{array}{l}\rho c_{P}\left[\frac{\partial T}{\partial t}+u\frac{\partial T}{ \partial x}+v\frac{\partial T}{\partial y}\right]=\frac{\partial}{\partial y} \left[k(T)\frac{\partial T}{\partial y}\right]-\frac{\partial u_{T}}{\partial y }+\mu(T)\left[\left(\frac{\partial u}{\partial y}\right)^{2}+\sqrt{2}\Gamma \left(\frac{\partial u}{\partial y}\right)^{3}\right]\\ +\frac{\rho D_{mk}T_{T}}{c_{s}}\frac{\partial^{2}C}{\partial y^{2}}\end{array} , \tag{3}\]  
\[\left[\frac{\partial C}{\partial t}+u\frac{\partial C}{\partial x}+v\frac{ \partial C}{\partial y}\right]=D_{m}\frac{\partial^{2}C}{\partial y^{2}}+ \frac{D_{m}k_{T}}{T_{m}}\frac{\partial^{2}T}{\partial y^{2}}. \tag{4}\]  
The boundary conditions are:  
\[u=U,\ v=0,\ T=T_{s},\ C=C_{s}, \tag{5}\]  
\[\frac{\partial u}{\partial y}=\frac{\partial T}{\partial y}=\frac{\partial C }{\partial y}=0\quad v=\frac{dh}{dt}=0, \tag{6}\]  
where \(\mu(T)=\frac{\mu_{0}}{(1-\gamma)\frac{T-T_{0}}{T_{ref}(\frac{\mu_{0}^{2}}{2 \gamma})}}\) indicates the variable viscosity in which \(\mu_{0}\) is the fluid viscosity at reference temperature \(T_{0}\) and the coefficient \(\gamma\) expresses the strength of the dependency between \(\mu\) and \(T\). \(K(T)=K_{1}\left(1-\varepsilon\left(\frac{T-T_{0}}{T_{ref}(\frac{\mu_{0}^{2}}{2 \gamma})}\right)\right)\) represents the temperature-dependent thermal conductivity, in which \(\varepsilon\) is the variable thermal conductivity parameter.