# Magnetohydrodynamics Flow Past a Moving Vertical Thin Needle in a Nanofluid with Stability Analysis
## 3 Results and Discussion
The effects of needle size \(c\) on the skin friction coefficient, the local Nusselt number and local Sherwood number are illustrated in Figure 6a-c, respectively, for different values of thermophoresis parameter \(Nt\). It is observed that reducing the needle size leads to increasing the skin friction coefficients, heat transfer as well as the mass transfer rate of the needle. Physically, the surface of the needle in contact with the fluid particles decreases when the size of the needle is getting smaller. This causes the friction force that occurs on the needle surface and the fluid flow will be reduced and this phenomenon leads to increasing the skin friction coefficients. Furthermore, the rate of heat transfer between the nanofluid flow and the needle surface is higher for \(c=0.1\) compared to \(c=0.2\). From the physical point of view, this occurs due to the thinner surface of the needle that allows the heat to diffuse rapidly. Similar observations were obtained for the mass transfer coefficients when the needle thickness decreases. In addition, the absolute value of the skin friction coefficients and the local Sherwood number increase for higher values of \(Nt\). Note that the presence of thermophoresis in the flow is to influence the nanoparticle migration in the opposite direction of the temperature gradient (warmer to colder region). Thus, the increase in the values of \(Nt\) helps the nanoparticles to penetrate deeper in fluid and causes the reduction in the momentum, thermal and concentration boundary layer thickness. This situation increases the coefficient of the skin friction, heat and also mass transfer on the needle surface. Figures 7 and 8 display the variation of the velocity, temperature and concentration profiles for the influence of the magnetic field \(M\) and mixed convection parameter \(\lambda\). As we can see, all of these profiles approach the endpoint boundary conditions Equation (10) asymptotically with various shapes, thus supporting the validity of the presents results obtained in this study. It is important to note that the existence of the dual velocity, temperature as well as concentration profiles in these figures has supported the dual nature of the solutions obtained in Figures 2 and 3. As expected, the thickness of the hydrodynamic boundary layer for the lower branch solution is always thicker than the upper branch solution. Figure 5: Variation of local Sherwood number with thermophoresis parameter \(Nt\) when (**a**) \(Le=5\) and (**b**) \(Le=30\) for different values of Brownian motion \(Nb\). Figure 6: Variation of (**a**) skin friction coefficients; (**b**) local Nusselt number and (**c**) local Sherwood number with thermophoresis parameter \(Nt\) for different values of needle size \(c\). Figure 7: Effects of the magnetic parameter \(M\) on the (**a**) velocity profiles; (**b**) temperature profiles; and (**c**) concentration profiles. In this study, the determination of the stability of the solutions obtained is done using a stability analysis. The fact is, to perform this analysis, we must have more than one solution. The main function of this analysis is to identify which of the upper or lower branch solutions is linearly stable and physically relevant by using the bvp4c package in MATLAB software. The stability of the solutions depends on the sign of the smallest eigenvalues \(\gamma\) obtained. The unknown eigenvalue \(\gamma\) is given in Equation (24) and, to obtain the value, we must solve the linear eigenvalue Equations (25)-(27) with the respective boundary conditions Equation (28). Table 2 shows the smallest eigenvalues \(\gamma\) for different values of magnetic field and mixed convection parameter. It is clear from Table 2 that the positive value of \(\gamma\) represents an initial decay of disturbance in the boundary layer separation and the flow is said to be stable. Meanwhile, the negative value of \(\gamma\) represents an initial growth of disturbance that will interrupt the boundary layer separation, and, consequently, the flow will be unstable.