# Stability Analysis of Mixed Convection Flow towards a Moving Thin Needle in Nanofluid
## 3 Results and Discussion
This is due to the complications of the proposed model, in which we consider the thin needle. The axial velocity and temperature profiles for different needle sizes, \(c\), are plotted in Figures 10 and 11 for \(\varepsilon=-2\), \(\phi=0.1\) and \(\lambda=-0.1\) (opposing flow) for Cu nanoparticles. From Figure 10, it is found that the velocity increases as the needle size decreases for the upper branch, while it decreases for the lower branch. The physical reason is due to the thinner surface of the needle, which leads to the decrease in the drag force that occurs  
Figure 6: Effect of mixed convection parameter \(\lambda\) on the variation of shear stress for different needle sizes \(c\). Figure 7: Effect of mixed convection parameter \(\lambda\) on the variation of local heat flux for different needle sizes \(c\). fluid particles in the flow. Note that the velocity boundary layer thickness for the upper branch decreases with the reduction in the needle size. Figure 11 shows that when the size of the needle decreases, the temperature increases, as well, while the thermal boundary layer thickness decreases in the upper branch. Figure 8: Variation of skin friction coefficients with nanoparticle volume fraction \(\phi\) for different nanoparticles when \(c=0.1\) and \(c=0.2\). Figure 9: Variation of local Nusselt numbers with nanoparticle volume fraction \(\phi\) for different nanoparticles when \(c=0.1\) and \(c=0.2\). Figures 12 and 13 illustrate the velocity and the temperature profiles for varying mixed convection parameter \(\lambda\) for \(\varepsilon=-2\), \(\phi=0.1\) and \(c=0.1\) for Cu nanoparticles. It shows that the velocity and temperature boundary layers are highly influenced by this parameter. In Figure 12, an increment in the values of \(\lambda\) results in an increase in velocity profiles and consequently decreases the momentum boundary layer thickness for the upper branch. A reverse trend is observed for the lower branch. Meanwhile, in Figure 13, as the \(\lambda\) value increases for the upper branch, the temperature increases and the thermal boundary layer thickness decreases respectively. However, the results are inverted for the lower branch. Figure 2:0 Effect of needle size \(c\) on the variation of temperature profiles. Figure 2:1 Effect of needle size \(c\) on the variation of velocity profiles. the lower branch. Based on Figures 10-13, we noticed that all the profiles obtained have fulfilled the requirement of the far field boundary condition (10) asymptotically, thus supporting the numerical results, as well as the dual nature of the solutions obtained in the current study. From this study, the stability of the dual solutions is determined by using the bvp4c package through MATLAB software. This analysis is performed to identify which of the upper or lower branch is linearly stable and physically realizable. The stability of the solutions depends on the smallest eigenvalues obtained. The unknown eigenvalue \(\gamma\) is presented in Equation (18), and to compute the  
Figure 2:2 Effect of mixed convection parameter \(\lambda\) on the variation of velocity profiles. Figure 2:3 Effect of mixed convection parameter \(\lambda\) on the variation of temperature profiles. values of \(\gamma\), we solve the linear eigenvalue problems given in Equations (22) and (23) together with the conditions (24). Tables 2 and 3 show the smallest eigenvalue \(\gamma\) with respect to several values of the involved parameters of interest. The tables indicate that the negative value of \(\gamma\) refers to an initial growth of disturbance, and the flow is in unstable mode. Meanwhile, the positive value of \(\gamma\) denotes an initial decay of disturbance, and the flow is said to be in a stable mode. It is worth knowing that the stable solution always gives a good physical meaning that can be realized. In Table 2, as the values of \(\lambda\) are approaching \(\lambda_{c}\), the smallest eigenvalue \(\gamma\) tends to zero either from the positive side or negative side.