## 5 Graphical Analysis  
This section exhibits the behavior of innumerable parameters on velocity \(f^{\prime}(\zeta)\), temperature \(\theta(\zeta)\) and concentration \(g(\zeta)\) profile. Figure 2 illustrates the performance on \(f^{\prime}(\zeta)\) or the velocity ratio parameter \(\lambda\). In our models, the velocity profile shows an increasing behavior near the surface of the needle for \(0\leq\lambda\leq 0.5\); however, for \(\lambda>0.5\), when it's far away from the needle surface, a decreasing nature is observed. Figure 3 displays the result on \(f^{\prime}(\zeta)\) for inertia coefficient \(F_{r}\). This figure illustrates that the inertial forces increase with growth in \(F_{r}\). A downfall of \(f^{\prime}(\zeta)\) appears as diminution of thickness of the boundary layer for different values of \(F_{r}\) that oppose the fluid motion. The influence of \(\lambda\) on \(\theta(\zeta)\) is highlighted in Figure 4. An up rise for increasing values of \(\lambda\) as heat intensely intrudes inside the fluid can be seen for \(\theta(\zeta)\). Figure 5 describes the impact of the Prandtl number Pr on \(\theta(\zeta)\). Pr is the ratio of momentum to thermal diffusivity, so, as values of Pr increase, the momentum diffusion and thermal diffusion drop. The outcome of larger values of Pr results in the diminution of \(\theta(\zeta)\) and the thermal boundary layer. Figure 6a,b illustrates the result of non-uniform source parameter on \(\theta(\zeta)\). As \(C>0,D>0\) corresponds to the internal heat source, an escalation can be analyzed for the strengthening the conduct of \(C,D\) for the thermal boundary layer. Higher values of \(C,D\) respond as heat generators, which are energy, and whose output is in the form of an increasing temperature profile \(\theta(\zeta)\). In Figure 7a,b, \(C<0,D<0\) behave as heat sinks that are the parameters that control the flow and transfer of heat. Additionally, decreasing values act as absorbers of heat for a non-uniform heat sink. This shows a reduction in the thickness of the boundary layer, and \(\theta(\zeta)\) exhibits a downfall. The response of different values of the thermal stratification parameter \(S\) on the temperature profile \(\theta(\zeta)\) can be noticed in Figure 8. It is noticed that the \(\theta(\zeta)\) shows a deteriorating nature for larger values of \(S\) due to the difference between the surface temperature and the ambient temperature. Figures 9-11 provide analyses of homogeneous reaction strength \(K\), heterogeneous reaction strength \(K_{s}\), and the Schmidt number \(Sc\). The outcome of  
Figure 5: Effect of the Prandtl number (Pr) on \(\theta(\zeta)\). Figure 6: Effect of \(C>0\) and \(D>0\) on \(\theta(\zeta)\). Figure 7: Effect of \(C<0\) and \(D<0\) on \(\theta(\zeta)\). Figure 8: Effect of thermal stratification (\(S\)) on \(\theta(\zeta)\). Figure 9: Effect of \(K\) on \(g(\zeta)\). Figure 12 the drag force coefficient \(C_{f}\) is plotted against thermal stratification \(S\) and inertia coefficient \(F_{r}\). An increasing behavior of \(C_{f}\) is seen for higher values of \(S\). For higher values of \(F_{r}\), the drag force coefficient \(C_{f}\) reduces. Figure 13 depicts the effect of \(\mathrm{P_{r}}\) and \(S\) on the heat transfer rate \(Nu_{x}\). It is noted that \(Nu_{x}\) is an increasing function of \(\mathrm{P_{r}}\). For escalating values of \(S\), the heat transfer rate decreases. The graphical result of \(Nu_{x}\) versus \(Sc\) for varying \(K\) is shown in Figure 14. It is observed that heat transfer rate increases for higher values of \(Sc\), whereas \(Nu_{x}\) is a diminishing function of \(K_{s}\). Figure 11: Effect of the Schmidt number (\(Sc\)) on \(g(\zeta)\). Figure 12: Effect of \(S\) and \(F_{r}\) on \(4n^{1/2}f^{\prime}(n)\). Table 1 depicts the numerically calculated values of \(f^{\prime\prime}\left(n\right)\) for numerous estimates of \(n\) done by [PERSON] et al. [38] and [PERSON] et al. [45], but in limiting cases. An outstanding harmony between the results is found.