## 4 Results and Discussion  
A comprehensive parametric analysis is implemented to examine the stimulus of key parameters on axial velocity (\(F\)), radial velocity (\(H\)), temperature (\(\theta\)), and NVF (\(\phi\)). Figure 2a-d illustrates the impact of the magnetic parameter (\(0\leq H\alpha\leq 0.9\)) on axial velocity (\(F\)), radial velocity (\(H\)), temperature (\(\theta\)), and NVF (\(\phi\)). The magnetic field tempts the retarding body force (so-called the Lorentz force) which reduces the flow and causes \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{Ha}\) & \(\mathbf{Ma}\) & \(\mathbf{R}\) & \begin{tabular}{c} **Finite Difference** \\ **Method (FDM)** \\ \end{tabular} & \
\begin{tabular}{c} **MATLAB** \\ **bvp5c Method** \\ \end{tabular} &
\begin{tabular}{c} **Absolute** \\ **Difference** \\ \end{tabular} \\ \hline
0.5 & 0.5 & 0.4 & \(-0.216620\) & \(-0.216620\) & 0.000000 \\ \hline
0.7 & & & \(-0.206954\) & \(-0.206954\) & 0.000000 \\ \hline
0.9 & & & \(-0.198161\) & \(-0.198162\) & 0.000001 \\ \hline
1.1 & & & \(-0.190115\) & \(-0.190116\) & 0.000001 \\ \hline
1.3 & & & \(-0.182713\) & \(-0.182714\) & 0.000001 \\ \hline  & 0.1 & & \(-0.050179\) & \(-0.050180\) & 0.000001 \\ \hline  & 0.2 & & \(-0.095904\) & \(-0.095904\) & 0.000000 \\ \hline  & 0.3 & & \(-0.138483\) & \(-0.138484\) & 0.000001 \\ \hline  & 0.4 & & \(-0.178583\) & \(-0.178583\) & 0.000000 \\ \hline
0.5 & & & \(-0.216620\) & \(-0.216620\) & 0.000000 \\ \hline  & 0.2 & \(-0.216620\) & \(-0.216620\) & 0.000000 \\ \hline  & 0.4 & \(-0.246955\) & \(-0.246955\) & 0.000000 \\ \hline  & 0.6 & \(-0.276190\) & \(-0.276190\) & 0.000000 \\ \hline  & 0.8 & \(-0.304428\) & \(-0.304429\) & 0.000001 \\ \hline  & 1.0 & \(-0.331755\) & \(-0.331756\) & 0.000001 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of \(F^{\prime}(1)\) values obtained by FDM and bvp5c method when \(\beta_{c}=0.5\) and \(\xi_{max}=8\). an upsurge in the thickness of the thermal and solute boundary layers. The electrically conductive fluid, under the Lorentz force perpendicular to the disk, creates the maximum resistance to flow, which implies a decrease in the axial velocity while observing the opposite behavior for the radial velocity. [PERSON] et al. [8] reported similar results for the applied magnetic field. Furthermore, the results of [PERSON] et al. [16], [PERSON] et al. [17], and [PERSON] et al. [26] are in agreement with our results. [PERSON] et al. [26] concluded that the applied magnetic field could be used to control the rheology of working fluids in various industrial applications. Figure 3a-d depicts the stimulus of the Marangoni number \((Ma)\) on the \(F(\xi),\ H(\xi),\ \theta(\xi)\) and \(\phi(\xi)\). It illustrated that an increase in \(Ma\) leads to an abundance of radial flow, thereby increasing the radial velocity and decreasing the azimuth velocity. The thermal surface layer thickness shrinks as \(Ma\) upsurges, this is due to the reduction in thermal diffusion that occurs as \(Ma\) escalates. Similar to the thermal layer, the solu  
increase. The effect of Marangoni ratio parameter on \(F(\xi),\ H(\xi),\ \theta(\xi),\ \theta(\xi)\), and \(\phi(\xi)\) is depicted in Figure 4a-d. The outcomes of the Marangoni ratio parameter (\(R\)) and Marangoni number are qualitatively similar. As can be seen, the Marangoni ratio parameter (\(R\)) appears in the velocity boundary condition \(\left(1+\frac{1}{\beta_{c}}\right)\)\(F^{\prime}(0)=-2M(1+R)\), as a factor of Marangoni number \(Ma\). Thereby, the consequences of the Marangoni ratio parameter on the flow system are similar to those of the Marangoni number \((Ma)\). The results of the Marangoni parameter are in collaboration with those reported by [PERSON] (2017), [PERSON] et al. (2019), and [PERSON] et al. (2016). Figure 3: (a) \(F(\xi)\), (b) \(H(\xi)\), (c) \(\theta(\xi)\), and (d) \(\phi(\xi)\) for different values of \(Ma\). Brownian motion is the random and uncontrolled movement of particles in a fluid when they constantly collide with other molecules [3].