## 4 Results and Discussion  
This segment is prepared to investigate the impacts of involved parameters \(M,\lambda,Q,\Pr,N_{t},N_{b},\gamma,M_{1},k_{0}\) and \(Sc\) on dimensionless velocity, temperature and concentration, Nusselt number, and skin friction coefficient. The influence of the velocity ratio parameter \((\lambda)\) on velocity distribution is drawn in Figure 2. It is observed that when \(0<\lambda<0.5\), the axial velocity expands just close to the surface of the needle and diminishes far from it. However, a contrary pattern is noticed for the values of ratio parameter greater than zero. Figure 3 portrays the impact of different estimations of the velocity ratio parameter \((\lambda)\) on the temperature profile. It is witnessed that for escalating values of ratio \(\lambda\), the temperature profile is enhanced. In Figure 4, the output of \(\lambda\) on the nanoparticle concentration profile is demonstrated. It is examined that for mounting values of \(\lambda\), the concentration profile \(\varphi(\eta)\). decreases. Figure 5 reveals the needle velocity for various estimations of the melting parameter \((M_{e})\). It is examined that for the mounting valuation of \(M_{e}\), the velocity profile increases. For a stronger \(M_{e}\), more molecular motion is observed, thus increasing the fluid velocity. In Figure 6, a variation of the magnetic parameter \((M)\) on the velocity field is portrayed. It is found that an increase in \((M)\) causes a decreased velocity profile. Sturdier Lorentz force hinders the movement of the fluid motion and thus decreases the velocity of the fluid. Figure 7 is plotted to show the trend of the porosity parameter \((k_{0})\) on the velocity profile. It is reported that the fluid velocity dwindles for the mounting estimation of \((k_{0})\). Physically, the motion of the fluid is hindered due to the presence of porous media, which is why a reduced velocity profile is witnessed. In Figure 8, a variation of heat absorption/generation coefficient \((Q_{0})\) on the temperature distribution is plotted. For rising estimates of \(Q_{0}\), a higher temperature profile is observed. Figure 9 depicts the attributes of \(M_{e}\) on the temperature. It is examined that with an increase in values of \(M_{e}\), the temperature profile reduces due to enhancement in thermal boundary layer thickness. Moreover, a sheet on normal temperature when dipped in the hot water. This causes the temperature to decrease while melting heat values may increase. The impact of Prandtl number (Pr) on the temperature profile is depicted in Figure 10. The increasing estimations of Pr results in a reduction of temperature field. Higher estimates of Prandtl number lower the thermal diffusivity, thus declining the temperature of the fluid. Figure 11 is illustrated to observe the behavior of non-linear chemical reaction \((n)\) on the concentration profile. It is witnessed that the thickness of the species distribution increases as \(n\) increases. Figure 12 indicates the concentration profile for varying thermophoresis parameter \((N_{t})\). The temperature gradient is directly proportional to the larger values of \(N_{t}\) that, accordingly, upsurges the concentration profile and its related concentration boundary layer thickness. Figure 13 is designed to highlight the Schmidt number \((S_{c})\) behavior on the nanoparticle concentration \(\phi(\eta)\). \(S_{c}\) is the ratio of momentum to the mass diffusion. For a growing estimation \(S_{c}\), a reduction in the mass diffusion coefficient causes a thinner concentration boundary layer. The response of chemical reaction parameter \((\gamma)\) on the concentration profile \(\varphi(\eta)\) is observed in Figure 14. It is clearly examined that concentration \(\varphi(\eta)\) is decreased via \(\gamma\). In Figure 15, a retarding effect of \(\lambda\) against magnetic parameter \(M\) can be seen for skin friction. It is reported that the thinner boundary layer is accompanying larger values of \(\lambda\), which results in a higher velocity gradient close to the wall.