## 5 Discussion of Graphical Results  
The problem posed by nonlinear system of Equations (29)-(34) under boundary conditions (35) has been solved in MATLAB symbolic software using the Nachtsheim-Swigert [26] iteration technique along with a sixth-order Runge-Kutta iterative process. This technique is extremely efficient and stable and has been deployed in numerous previous investigations by the authors- see [33; 34; 35; 36]. In order to consider the results, carry out calculations for various parameter values, for instance, Prandtl number \(Pr\), Prandtl number with reciprocal magnetization \(\gamma\), radiation variable \(N\), magnetic force parameter \(\beta\), velocity exponent parameter \(p\), temperature exponent parameter \(\lambda\), buoyancy force parameter \(\zeta\). Results are graphically plotted in Figures 2-18. In Figures 2-4, it is shown how a magnetic parameter affects the induced magnetic fields' speed, temperature, and direction. It is evident that the velocity profile decreases with escalating magnetic force parameter \(\beta\) and increases with the increasing buoyancy force parameter \(\zeta\) as shown in Figure 2. The momentum boundary layer thickness is, therefore, increased with greater magnetic Lorentz retarding force (flow retardation) whereas it is reduced with the thermal buoyancy effect. By increasing the magnetic force parameter \(\beta\) the temperature profile increases whereas it is reduced by increasing the upthrust force parameter \(\zeta\) as shown in Figure 3. The additional energy that the magnetic polymer expends \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{P_{r}}\) &
\begin{tabular}{c} **Present** \\ **(Local Nusselt Number)** \\ \end{tabular} & **[PERSON] et al. [16]** \\ \hline
0.07 & 0.3393 & 0.3381 \\ \hline
0.5 & 0.8285 & 0.8274 \\ \hline
2.0 & 1.5204 & 1.5214 \\ \hline
6.8 & 2.5951 & 2.5978 \\ \hline
10.0 & 3.0758 & 3.0790 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The Local Nusselt number for various \(P_{r}\) when p = 1.0, \(\lambda\) = N = 0.0, \(\gamma\) = 1.0, \(\delta\) = 3.0, \(\xi\) = 0.0, \(\beta\) = 1. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\delta\) & **Present** & **[PERSON] and [PERSON] [48]** & **[PERSON] et al. [16]** \\ \hline
0.1 & \(-\)0.9698 & \(-\)0.9694 & \(-\)0.9694 \\ \hline
0.2 & \(-\)0.9187 & \(-\)0.9181 & \(-\)0.9181 \\ \hline
0.5 & \(-\)0.6675 & \(-\)0.6673 & \(-\)0.6673 \\ \hline
2.0 & 2.0174 & 2.0175 & 2.0175 \\ \hline
5.0 to drag against the imposed magnetic field is undoubtedly lost as thermal energy. This warms the environment and thickens the thermal-boundary layer. Increasing thermal buoyancy, however, only assists momentum development and counteracts thermal diffusion in the regime by suppressing convection currents and reducing thermal boundary layer thickness. It is also seen that the induced magnetic field profile decreases with increasing magnetic force parameter \(\beta\) whereas it increases with increasing buoyancy force parameter \(\xi\) as shown in Figure 4. Magnetic induction is, therefore, inhibited with Lorentz body force whereas it is encouraged with thermal buoyancy effect; this manifests in a respective thinning and thickening of magnetic boundary layer thickness, as well as numerous additional investigations, including those by [PERSON] [11] and [PERSON] [12] among others. It is also important to note that Figure 4 specifies 0.5 for the magnetic Prandtl number's inverse \(=\frac{u_{\xi}}{v}\). Physically as elaborated by [PERSON] and [PERSON] [49]. When this parameter is unity the momentum and magnetic boundary layer thicknesses are approximately equal. Magnetic diffusion rate, however, outpaces momentum diffusion rate for \(\gamma=0.5\). This encourages the influence of Lorentzian body force which counteracts magnetic induction and decreases the magnetic boundary layer thickness. Asymptotically smooth distributions are computed in the free stream indicating that a sufficiently large infinity boundary condition is prescribed in the MATLAB Runge-Kutta code.