# Magnetohydrodynamics Flow Past a Moving Vertical Thin Needle in a Nanofluid with Stability Analysis
## 3 Results and Discussion  
To provide a clear understanding into the present problem, the numerical computations are carried out for several values of the involved parameters which include magnetic, mixed convection, needle size, Brownian motion, thermophoresis and velocity ratio parameter. Validation of the present code has been done by computing the numerical results of \(f^{\prime\prime}(c)\) for \(M=\lambda=Nr=\varepsilon=0\) and \(Pr=1\) for some values of needle size \(c\). Table 1 shows the data produced by the present code and those of [PERSON] et al. [21]. As can be seen in Table 1, both pieces of data are found to be in excellent agreement, hence, the present code can be used confidently. The system of highly nonlinear ordinary differential Equations (7)-(9) together with the boundary conditions Equation (10) is solved numerically using the bvp4c package in MATLAB software. The bvp4c package is one of the popular techniques for solving the boundary value problems for ordinary differential equations. Furthermore, this package considers the finite difference method, where the solution can be obtained using an initial guess supplied at an initial mesh point and change step size to get the specified accuracy. It is necessary to reduce the boundary value problems to a system of first order ordinary differential equations before being used. The suitable finite value of \(\eta\) is considered from 10 to \(\eta_{\infty}=60\), where \(\eta_{\infty}\) refers to \(\eta\to\infty\) which lies very well outside the momentum, thermal and concentration boundary layers. However, it should be mentioned that the present solution is only non-similar because the parameters \(M\) and \(Nr\) are the function of the independent variable \(x\). Figure 2a-c present the variation of the shear stress \(f^{\prime\prime}(c)\), local heat flux \(-\theta^{\prime}(c)\) and local mass flux \(-\phi^{\prime}(c)\) with mixed convection parameter \(\lambda\) for different values of magnetic parameter \(M\). It can easily be observed that the values of \(f^{\prime\prime}(c)\) and \(-\theta^{\prime}(c)\) are noticed to decrease when the imposed magnetic field is larger. Nevertheless, the reverse trend is noted for \(-\phi^{\prime}(c)\) as the parameter \(M\) increases. The rising values of magnetic parameter offer large resistances on the fluid particles and consequently, cause the heat to be generated in the fluid. Thus, a drag force known as a Lorentz force is formed due to the imposition of vertical magnetic field. The presence of these forces may resist and decelerate the motion of the nanofluid flow over the needle surface, which in turn reduces the wall friction and temperature, respectively. Other than that, the stronger magnetic field faster the boundary layer separation by which the upper and lower branch solutions intersect. The critical values of \(\lambda\), which given by \(\lambda_{c}\) are found to be increased as the values of magnetic parameter increases. Moreover, it is possible for the Equations (7)-(9) to get dual solutions for the assisting flow (\(\lambda>0\)) compared to the opposing flow (\(\lambda<0\)). For the opposing flow, the buoyancy forces oppose the fluid motion, retarding the fluid in the boundary layer and also act as an adverse pressure gradient. However, there are dual solutions only in the range \(\lambda_{c}<\lambda\leq 0\), a unique solution when \(\lambda>0\) (assisting flow) and no solution when \(\lambda<\lambda_{c}\). \begin{table}
\begin{tabular}{c c c} \hline \hline \(c\) & **[PERSON] et al. [21]** & **Present Work** \\ \hline
0.1 & 1.2888171 & 1.2888299 \\
0.01 & 8.4924360 & 8.4924452 \\
0.001 & 62.163672 & 62.163606 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Numerical values of \(f^{\prime\prime}(c)\) when \(M=\lambda=Nr=\varepsilon=0\) and \(Pr=1\) for some values of needle size \(c\).