# Magnetohydrodynamics Flow Past a Moving Vertical Thin Needle in a Nanofluid with Stability Analysis
## 3 Results and Discussion
The influence of needle size \(c\) on the variation of the shear stress \(f^{\prime\prime}(c)\), local heat transfer \(-\theta^{\prime}(c)\) and local mass transfer \(-\phi^{\prime}(c)\) are graphically presented in Figure 3a-c for various values of velocity ratio parameter \(\varepsilon\). As we can see, the dual solutions are more pronounced when the needle moves in the opposite direction of the free stream flow (\(\varepsilon<0\)). Obviously, only unique solutions exist when the needle moves in the same direction with the free stream flow (\(\varepsilon>0\)). The range of dual solutions that exist is in between \(\varepsilon_{c}<\varepsilon\leq 0\), and beyond the critical value, no similarity solutions were reported. It is observed that the increase in the needle size caused a decrement in the value of shear stress and local heat transfer. However, the increment of local mass transfer is reported for higher values of \(c\). It is clear from the Figure 3b,c that the graph of the local mass transfer is a reflection of the graph of local heat transfer, where the value of \(Nb=Nt\). These three figures also indicate that the range of the solutions is widely expanded for the thinner surface of the needle (\(c=0.1\)). The effect of the velocity ratio parameter \(\varepsilon\) on the skin friction coefficients for several values of magnetic parameter \(M\) is shown in Figure 4. The magnitude of the skin friction coefficients decreases for \(M=0.0\) to \(M=0.1\) with higher value of \(\varepsilon\). This implies that the larger magnetic parameter leads to decrease the momentum boundary layer thickness as well as the shear stress occurs on the needle surface, and, consequently, decreases the skin friction coefficients. Figure 2: Variation of (**a**) shear stress, (**b**) local heat flux and (**c**) local mass flux with mixed convection parameter \(\lambda\) for different values of magnetic parameter \(M\). Figure 5a,b display the effect of Brownian motion parameter \(Nb\) and Lewis number on the local Sherwood number or mass transfer with various values of thermophoresis parameter \(Nt\). Interestingly, the Lewis number represents the ratio of thermal diffusivity to mass diffusivity in the boundary layer regime. In addition, it helps to explain the characteristics of the fluid flow where there are simultaneous heat and mass transfer by convection. As can be seen in Figure 5a,b, the local Sherwood number is higher for smaller value of \(Le\) (\(Le=5\)) compared to a greater one (\(Le=30\)). This follows the above statement that the reduction of the Lewis number leads to increasing the mass diffusivity as well as a mass transfer coefficient, by which these two functions are inversely proportional to each other. Furthermore, the local Sherwood number is increasing with higher value of \(Nt\). It is clear that the higher value of mass transfer is obtained for \(Nb=0.1\) followed by \(Nb=0.3\) and \(Nb=0.5\). This implies that the stronger rate of Brownian motion reduces the performance of mass transfer  
Figure 4: Variation of skin friction coefficients with velocity ratio parameter \(\varepsilon\) for different values of magnetic parameter \(M\). Figure 3: Variation of (**a**) shear stress, (**b**) local heat flux and (**c**) local mass flux with velocity ratio parameter \(\varepsilon\) for different needle size \(c\). between the needle surface and the free stream flow. Such situation occurs due to random motion of the particles suspended in the fluid caused by the continuous collision between the nanoparticles and base fluid particles. Conspicuously, the Brownian motion parameter does not change the surface temperature gradient as well as the rate of heat transfer (see [21]). Mathematically, when substituting the condition \(\phi^{\prime}(c)+(Nt/Nb)\theta^{\prime}(c)=0\) directly into the energy Equation (3), one may observe that the resulting equation does not contain the parameter \(Nb\), and this is the reason why the graph for the local Nusselt number is not presented here.