# Stability Analysis of Mixed Convection Flow towards a Moving Thin Needle in Nanofluid
## 3 Results and Discussion  
To ensure the accuracy of the numerical method used in this study, we made comparison results of \(f^{\prime\prime}(c)\) with those of [PERSON] et al. [19] and [PERSON] et al. [24] for different values of needle size, \(c\), when \(\lambda=0\) and \(Pr=1\) (see Table 1). Our computations are in excellent agreement with the work of [PERSON] et al. [19] and [PERSON] et al. [24] for all considered values of \(c\). Thus, to test the accuracy of our code, we compute the numerical results for this problem. The system of Equations (8) and (9) with the conditions in (10) is solved using an implemented bvp4c package in MATLAB software. The bvp4c package is a method used to solve the boundary value problems for ordinary differential equations. This package applies the finite difference method, where the solution can be obtained using an initial guess supplied at an initial mesh point and changes step size to get the specified accuracy. However, to solve these boundary value problems, it is necessary to first reduce it to a system of first order ordinary differential equations. Depending on the values of the parameters used, we take a suitable finite value of \(\eta\rightarrow\infty\)as \(\eta=10\)-\(140\). In addition, we assume the range of \(\phi\) is between zero and \(0.2\) with \(\phi=0\), representing the regular fluid, and the Prandtl number is set to \(Pr=6.2\). The thermophysical properties of the copper, titania, alumina and the base fluid can be referred in the paper of [PERSON] and [PERSON] [30]. Figures 2 and 3 display the variation of shear stress \(f^{\prime\prime}(c)\) and local heat flux \(-\theta^{\prime}(c)\) with mixed convection parameter \(\lambda\) for various values of nanoparticles volume fraction \(\phi\) for Cu nanoparticles. Interestingly, in Figure 2, the shear stress at the wall increases with the increase of the parameter \(\phi\). As a result, more particles are suspended in the base fluid, and the thermal conductivity of nanoparticles becomes stronger. Besides, it is seen from Figure 3 that upon increasing the size of the nanoparticles, the local heat flux on the surface reduces. It is also good to know that an increment in the size of the nanoparticles leads to a decrease in the ratio of thermal conductivity. This occurs due to the fact that the existence of nanoparticles in the base fluid plays a significant role to increase the heat transfer. However, Figure 3 implies that the increase in \(\phi\) is to reduce the rate of heat transfer on the needle surface. It is clear from the graphs that the critical values of \(\lambda\), say \(\lambda_{c}\), in which the upper and the lower branches are connected is getting higher, when the \(\phi\) value increases. It is also found that the dual solutions are likely to exist when the flow is opposing (\(\lambda<0\)). In summation, the dual solutions are in the range of \(\lambda_{c}<\lambda\leq 0.3\). When \(\lambda<\lambda_{c}\), no similarity solutions exist for Equations (8) and (9). The effect of velocity ratio parameter \(\varepsilon\) on the variation of shear stress and local heat flux for different nanoparticles are illustrated in Figures 4 and 5 for the opposing flow (\(\lambda=-0.2\)). It follows \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\varepsilon\) & [PERSON] et al. [19] & [PERSON] et al. [24] & Present Results \\ \hline
0.01 & 8.4924 & 8.491454 & 8.491455 \\
0.1 & 1.2888 & 1.288778 & 1.288778 \\
0.2 & - & 0.751665 & 0.751665 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison results of \(f^{\prime\prime}(c)\) with those of [PERSON] et al. [19] and [PERSON] et al. [24] when \(\lambda=0\) and \(Pr=1\). Figure 2: Effect of mixed convection parameter \(\lambda\) on the variation of shear stress for various values of nanoparticle volume fraction \(\phi\). that the dual similarity solutions exist in the range of \(\varepsilon_{c}<\varepsilon\leq 0\).