## 2 Mathematical Modeling  
Consider a two-dimensional (2-D) incompressible boundary layer flow of Eyring-Powell fluid over an oscillatory stretching sheet concurring with plane y. In this Cartesian coordinate system, \(\overline{x}\) is parallel to the stretching sheet, and \(\overline{y}\) is perpendicular to the stretching sheet. The stretching sheet is kept porous, and the flow is supposed in an unsteady state. The magnetic field is applied in the y direction. It is assumed that \(T_{w}>T_{\infty}\), where \(T_{w}\) is the surface temperature and \(T_{\infty}\) is the temperature as the distance from the surface tends to infinity. The physical model of the problem is shown in Figure 1.  
The above-stated problem satisfies all the conditions [29]. \[\frac{\partial u}{\partial\overline{x}}+\frac{\partial v}{\partial\overline{ y}}=0, \tag{1}\]  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial\overline{x}}+v \frac{\partial u}{\partial\overline{y}}=\left(v+\frac{1}{\rho\alpha C}\right) \frac{\partial^{2}u}{\partial\overline{y}^{2}}-\frac{1}{2\rho\alpha C^{3}} \left(\frac{\partial u}{\partial\overline{y}}\right)^{2}\!\!\frac{\partial^ {2}u}{\partial\overline{y}^{2}}-\left(\frac{v}{k}+\sigma B_{0}^{2}\right)u( t), \tag{2}\]  
\[\rho C_{p}\!\left(\frac{\partial\mathrm{T}}{\partial t}+u\frac{\partial \mathrm{T}}{\partial\overline{x}}+v\frac{\partial\mathrm{T}}{\partial \overline{y}}\right)=k\!\left(\frac{\partial^{2}\mathrm{T}}{\partial\overline {y}^{2}}\right)-\left(\frac{\partial q_{r}}{\partial\overline{y}}\right)+ \frac{Q_{r}}{\rho C_{p}}(\mathrm{T}-\mathrm{T}_{\infty}). \tag{3}\]  
In Equations (1)-(3), \(u\) and \(v\) are the velocity components in the direction of \(\overline{x}\) and \(\overline{y}\), respectively. Also, \(v\) is the kinematic viscosity, \(\sigma\) is the electrical conductivity, \(\alpha\) and \(C\) are the fluid materials, \(\rho\) is the density, \(C_{p}\) is the specific heat, \(k\) is the thermal conductivity, \(Q_{r}\) is the heat source, and \(q_{r}\) is the radiative heat flux. \(q_{r}\) is defined as  
\[q_{r}=-\frac{4\sigma*}{3k*}\frac{\partial\mathrm{T}^{4}}{\partial\overline{y}}, \tag{4}\]  
where \(\sigma*\) (Stefan-Boltzmann constant) and \(k*\) (absorption coefficient). Expending \(\mathrm{T}^{4}\) by Taylor series expansion, we obtained [28]  
\[\mathrm{T}^{4}=\mathrm{T_{\infty}}^{4}+4\mathrm{T_{\infty}}^{3}(\mathrm{T}- \mathrm{T_{\infty}})+6\mathrm{T_{\infty}}^{2}(\mathrm{T}-\mathrm{T_{\infty}}) ^{2}+\ldots. \tag{5}\]  
Figure 1: Geometrical representation of the problem. Neglecting higher terms from Equation (5) and substituting in Equation (3), we have  
\[\rho C_{p}\left(\frac{\partial\mathrm{T}}{\partial t}+u\frac{\partial\mathrm{T}}{ \partial\overline{x}}+v\frac{\partial\mathrm{T}}{\partial\overline{y}}\right)= \left(k+\frac{16\sigma*\mathrm{T_{\infty}}^{3}}{3k*}\right)\frac{\partial^{2} \mathrm{T}}{\partial\overline{y}^{2}}+\frac{Q_{r}}{\rho C_{p}}(\mathrm{T}- \mathrm{T_{\infty}}). \tag{6}\]  
With the following boundary conditions  
\[ \begin{array}{l}u=U_{w}=c\overline{x}\sin\tilde{\varsigma}t,\ v=0,\ \mathrm{T}=\mathrm{T_{w}}\ \mathrm{at}\ \overline{y}=0,\ t>0,\\ u\to 0,\ \mathrm{T}\rightarrow\mathrm{T_{\infty}}\ \mathrm{at}\ y\rightarrow\infty, \end{array}  \tag{7}\]  
we introduce the following dimensionless variables for the non-dimensionalization of the flow problem. \[y=\left(\frac{c}{\
u}\right)^{\frac{1}{2}}\overline{y},\ \tau=t\xi,\ u=c \overline{x}f_{y}(y,\tau),\ v=-(vc)^{\frac{1}{2}}f(y,\tau),\ g(y,\tau)=\frac{ \mathrm{T}-\mathrm{T_{\infty}}}{\mathrm{T_{w}}-\mathrm{T_{\infty}}}. \tag{8}\]  
An influential tool in fluid mechanics is the thought of dimensional analysis and scaling laws; by looking at the physical properties present in a system, we may estimate their size and hence which, for example, might be neglected. In some cases, the system may not have a fixed natural length scale (timescale) while the solution depends on space (time).