## 2 Mathematical Modelling
### Magnetic Dipole  
The scalar potential of the magnetic dipole is stated as,  
\[\Omega=\frac{xl}{2\pi(x^{2}+\left(r+c\right)^{2})}, \tag{13}\]  
In Equation (13), \(l\) specifies the magnetic strength. The horizontal and vertical part of the magnetic field in \(x-\) and \(r-\) directions are defined as,  
\[\frac{\partial H}{\partial x}=-\frac{\partial\Omega}{\partial x}=\frac{x^{2}- \left(r+c\right)^{2}}{2\pi(x^{2}+\left(r+c\right)^{2})}, \tag{14}\]  
\[\frac{\partial H}{\partial r}=-\frac{\partial\Omega}{\partial r}=\frac{2x(r+ c)}{2\pi(x^{2}+\left(r+c\right)^{2})^{2}}. \tag{15}\]  
The absolute magnetic field and magnetic body force is stated as,  
\[H=\sqrt{\left(\frac{\partial H}{\partial x}\right)^{2}+\left(\frac{\partial H }{\partial r}\right)^{2}}. \tag{16}\]  
where the components in the square root are defined as,  
\[\frac{\partial H}{\partial x}=\frac{-2x}{2\pi(r+c)^{4}}, \tag{17}\]  
\[\frac{\partial H}{\partial r}=\frac{1}{2\pi}\Bigg{(}\frac{-2}{\left(r+b\right) ^{3}}+\frac{4x}{\left(r+c\right)^{5}}\Bigg{)}. \tag{18}\]  
The intensity of the magnetic field became stronger and formed a linear relationship between temperature and magnetic variation:  
\[M=K^{*}(T-T_{\infty}). \tag{19}\]  
For the discussion of the ferrodynamic interaction, we may suppose that the magnetic field is applied in a non-homogenous way, and fluid temperature must be less than the Curie temperature.  
obtained with the help of the MATLAB built-in function Bvp4c. The comparison table of the current problem with data published by [PERSON] et al. [1] and [PERSON] [2] is presented in Table 2. It is shown that a stronger value of the Prandtl number improves the temperature gradient. Table 3 shows the variation in the skin friction coefficient, and the rates of heat transfer, mass transfer, and microorganism transfer against \(M\), \(\alpha\), and \(\phi_{1}\). It is quite interesting to see that the skin friction coefficient improves for the improved values of \(M\), while heat, mass, and microorganism transfer rates decline for the stronger values of \(M\). Further, \(\alpha\) and \(\phi_{1}\) show increasing behavior against the skin friction, and the heat, mass, and microorganism transfer rates. The effect of \(L_{b}\), \(P_{e}\), \(\alpha\), and \(\delta\) on the microorganism transfer rate by the influence of Ag and MgO is found in Table 4. It is noted that a larger estimation of \(L_{b}\), \(P_{e}\), \(\alpha\), and \(\delta\) improves the microorganism transfer rate for both of the nanoparticles.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(L_{b}\) & \(P_{e}\) & \(\alpha\) & \(\delta\) & \multicolumn{2}{c}{\(-\mathbf{h^{{}^{\prime}}}\)**(0)**} \\ \cline{3-6}  & & & & **Ag** & **MgO** \\ \hline
0.5 & 0.5 & 0.1 & 0.2 & 1.7591 & 1.7606 \\
0.6 & 0.5 & 0.1 & 0.2 & 1.8422 & 1.8439 \\
0.7 & 0.5 & 0.1 & 0.2 & 1.9210 & 1.9227 \\
0.5 & 0.1 & 0.1 & 0.2 & 1.4693 & 1.4707 \\
0.5 & 0.2 & 0.1 & 0.2 & 1.6329 & 1.6345 \\
0.5 & 0.3 & 0.1 & 0.2 & 1.7987 & 1.8003 \\
0.5 & 0.5 & 0.2 & 0.2 & 2.1662 & 2.1679 \\
0.5 & 0.5 & 0.3 & 0.2 & 2.1962 & 2.1978 \\
0.5 & 0.5 & 0.4 & 0.2 & 2.2262 & 2.2278 \\
0.5 & 0.5 & 0.1 & 0.3 & 2.1996 & 2.2014 \\
0.5 & 0.5 & 0.1 & 0.4 & 2.2631 & 2.2649 \\
0.5 & 0.5 & 0.1 & 0.5 & 2.3267 & 2.3285 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Variation of different parameters against the microorganism transfer rate.