## 2 Mathematical Formulation
### Governing Equations  
The governing surface layer equations are (see [8; 9; 10]):  
_Conservation of mass_  
\[\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{\partial w}{\partial z}=0, \tag{1}\]  
_Conservation of momentum_  
\[u\frac{\partial u}{\partial r}+w\frac{\partial u}{\partial z}=\frac{\mu}{\rho} \bigg{(}1+\frac{1}{\beta_{c}}\bigg{)}\frac{\partial^{2}u}{\partial z^{2}}- \frac{\delta B_{0}^{2}}{\rho}u, \tag{2}\]  
_Conservation of thermal energy_  
\[ \begin{array}{l}u\frac{\partial T}{\partial r}+w\frac{\partial T}{\partial z }+\lambda_{T}\Big{[}u^{2}\frac{\partial^{2}T}{\partial r^{2}}+w^{2}\frac{ \partial^{2}T}{\partial z^{2}}+2 uw\frac{\partial^{2}T}{\partial r\partial z}+ \frac{\partial T}{\partial t}\Big{(}u\frac{\partial u}{\partial r}+w\frac{ \partial u}{\partial z}\Big{)}+\frac{\partial T}{\partial z}\Big{(}u\frac{ \partial u}{\partial r}+w\frac{\partial u}{\partial z}\Big{)}\Big{]}\\ =\frac{k}{\rho\
u}\frac{\partial^{2}T}{\partial z^{2}}+\tau\bigg{\{}D_{B} \frac{\partial T}{\partial z}\frac{\partial C}{\partial z}+\frac{D_{T}}{T_{ \infty}}\bigg{(}\frac{\partial T}{\partial z}\Big{)}^{2}\bigg{\}}+\frac{Q_{c} \left(T_{\infty}-T_{\infty}\right)}{\rho\
u_{p}}\exp\left\{-n\sqrt{\Omega/\
u} \,z\right\}+\frac{Q_{t}}{\rho\
u_{p}}(T-T_{\infty}),\end{array}  \tag{3}\]  
_Conservation of nanoparticles volume fraction_  
\[ \begin{array}{l}u\frac{\partial C}{\partial r}+w\frac{\partial C}{\partial z }=D_{B}\frac{\partial^{2}C}{\partial z^{2}}+\frac{D_{T}}{T_{\infty}}\frac{ \partial^{2}T}{\partial z^{2}}\\ -\lambda_{c}\Big{[}u^{2}\frac{\partial^{2}C}{\partial r^{2}}+w^{2}\frac{ \partial^{2}C}{\partial z^{2}}+2 uw\frac{\partial^{2}C}{\partial r\partial z}+ \frac{\partial C}{\partial r}\Big{(}u\frac{\partial u}{\partial r}+w\frac{ \partial u}{\partial z}\Big{)}+\frac{\partial C}{\partial z}\Big{(}u\frac{ \partial u}{\partial r}+w\frac{\partial w}{\partial z}\Big{)}\Big{]},\end{array}  \tag{4}\]  
_Boundary conditions_  
\[ \begin{array}{l}\mu\left(1+\frac{1}{\beta_{c}}\right)\frac{\partial u}{ \partial z}=\frac{\partial w}{\partial r}=\frac{\partial w}{\partial T}\frac{ \partial T}{\partial r}+\frac{\partial w}{\partial C}\frac{\partial C}{ \partial r},\end{array}  \tag{5}\]  
\(T=T_{\infty}+A\left(\frac{r}{R}\right)^{2}\) and \(C=C_{\varpi}=C_{\varpi}+B\left(\frac{r}{R}\right)^{2}\) at \(z=0\)  
\(u\to 0\), \(T\to T_{\infty}\), \(C\to C_{\infty}\) at \(z\to\infty\)  
The second term on the right side of Equation (2) represents the transverse magnetic field, and the second and third terms on the right-hand side of Equation (3) represent the Brownian motion and Thermophoresis phenomenon. The thermophoresis parameter is quantified by nanoparticles particle's thermos-diffusion coefficient \(D_{T}\) and it signifies the capability of nanoparticles to move as a result of the temperature gradient. \(D_{T}>0\) signifies the movement of nanoparticles toward cooler regions while \(D_{T}<0\) designates the movement of nanoparticles toward warmer regions. The last two terms of Equation (3) represent the heat source effects. Figure 1: Physical model. Marangoni convection, induced by a surface tension gradient at the interface, is an important physical phenomenon in conditions of microgravity. In the mid-1860s, [PERSON] discovered the phenomenon whereby the gravity-dominated natural convection of the liquid gradually disappeared in an environment of microgravity, while at the interface of the liquid, surface tension plays an important role and causes a gradient.