# Stability Analysis of Mixed Convection Flow towards a Moving Thin Needle in Nanofluid
## 3 Results and Discussion
Note that Equations (8) and (9) subject to the conditions in (10) possess the dual solutions only when the needle and the free stream moves in the opposite direction (\(\varepsilon<0\)). Furthermore, from these figures, only unique solutions exist for \(\varepsilon>0\) where the needle and the free stream move in the same direction. Moreover, the critical values of \(\varepsilon\) in which the upper and the lower branches intersect increase for \(\mathrm{Al_{2}O_{3}}\) followed by \(\mathrm{TiO_{2}}\) and Cu. Figure 4: Effect of the velocity ratio parameter \(\varepsilon\) on the variation of shear stress for different nanoparticles. Figure 3: Effect of mixed convection parameter \(\lambda\) on the variation of local heat flux for various values of nanoparticle volume fraction \(\phi\). Figures 6 and 7 portray the influence of the mixed convection parameter on the variation of the shear stress and local heat flux for different sizes of needle \(c\) for Cu nanoparticles. It is observed from the figures that the magnitude of the shear stress and local heat flux on the needle surface is higher for \(c=0.1\) compared to that of \(c=0.2\). From the physical point of view, when the size of the needle is getting smaller, the surface of the needle in contact with the fluid particle decreases. This leads to the reduction of the drag force that occurs on the needle surface and the fluid flow. Furthermore, decreasing the needle size causes the heat transfer to accelerate since the heat is easily diffused through a thin surface compared to a thick surface. Besides, the range of \(\lambda\) increases as the size of the needle decreases. In particular, the dual similarity solutions exist when \(\lambda_{c}<\lambda\leq 0.2\) for both values of \(c=0.1\) and \(c=0.2\). The variation of the skin friction coefficient \((Re_{x})^{1/2}C_{f}\) and the local Nusselt number \((Re_{x})^{-1/2}Nu_{x}\) on different nanoparticles are shown in Figures 8 and 9 for various values of \(c\) and \(\phi\). Observation from these figures yields that the skin friction coefficient and the heat transfer rate are higher for \(c=0.1\) compared to \(c=0.2\). This trend implies that the reduction in the needle size causes the thickness of the velocity boundary layer to decrease. Consequently, an increase in the shear stress, as well as the skin friction coefficients occurs on the surface, and this behavior can be seen through Figure 8. Similar observations were obtained on the local Nusselt number in Figure 9 as the value of \(c\) decreases. Physically, the thin surface of the needle allows the heat transfer between the needle and the fluid flow to become faster. Also from Figure 8, the stronger rate of nanoparticle volume fraction leads to a greater magnitude of skin friction coefficient. This is due to the collisions between the suspended nanoparticles and the base fluid particles that enhance the friction occurring on the needle surface. From Figure 9, the rate of heat transfer increases for \(c=0.1\), while it decreases for \(c=0.2\) when the size of the nanoparticle \(\phi\) becomes higher. In contrast with the micrometer scale, nanoparticles possess higher surface area to volume ratio due to a huge number of atoms on the boundaries. This causes them to be highly stable in suspensions. It follows that these suspensions show high thermal conductivity possibly due to enhanced convection between the solid particle and fluid surfaces. This behavior is more suitable for the thinner surface of the needle (\(c=0.1\)). Moreover, the graphs presented in Figures 8 and 9 depict that the Cu nanoparticles has the lowest values of skin friction coefficient and heat transfer rate compared to Al\({}_{2}\)O\({}_{3}\) and TiO\({}_{2}\). Sometimes, there is no  
Figure 5: Effect of the velocity ratio parameter \(\varepsilon\) on the variation of local heat flux for different nanoparticles. clear evidence to completely predict the effects of different nanoparticles on the local Nusselt number.