## 2 Models and Methods
### Governing Equations and Boundary Conditions  
This study mainly investigates the thermo-solutocapillary convection under microgravity conditions. To ensure the precision of the final outcomes, the surface deformation is considered. This study uses the level set method to capture the interface displacement in the application of a gas-liquid two-phase system. Figure 1: Schematic of liquid bridge model and layout of monitoring points. To facilitate computational efficiency, it is imperative that the governing equations be transformed into a dimensionless form. The characteristic density and viscosity are \(\rho_{l}\) and \(\mu_{l}\). The characteristic length is \(L=H=0.5R\), and the characteristic velocity is \(U=(\sigma_{T}\cdot\Delta T)/\mu_{l}\). The subscripts \(g\) and \(l\) represent gas and liquid, respectively. The ensuing expression encapsulates the dimensionless parameters designated:  
\[\overline{x}=\frac{x}{L},\ \overline{z}=\frac{z}{L},\ \overline{u}=\frac{u}{U},\ \overline{v}=\frac{v}{U},\ \overline{p}=\frac{\rho}{\rho_{l}},\ \overline{\mu}=\frac{\mu}{\mu_{l}},\ \overline{p}=\frac{p}{\rho_{l}U^{2}},\ \overline{t}=\frac{t}{L/U} \tag{1}\]  
where the physical quantity with the \"-\" symbol as the superscript represents a dimensionless quantity. \(\overline{x}\) and \(\overline{z}\) are utilized to represent dimensionless distances in the horizontal and vertical directions, respectively. \(\overline{u}\) and \(\overline{v}\) refer to dimensionless transverse and longitudinal velocities, respectively. \(\overline{p}\), \(\overline{\mu}\), \(\overline{p}\), and \(\overline{t}\) refer to dimensionless density, viscosity, pressure, and time, respectively. The governing equations consist of the following dimensionless mass conservation, Navier-Stokes, energy conservation and concentration diffusion equations, and interface capture function [35], as listed in Formulas (2)-(6). Among them, the energy conservation and concentration diffusion equations are derived based on the continuum assumption and the conservation of mass, momentum, and energy [44], using the equation of state and [PERSON]'s law of diffusion, respectively. \[\
abla\cdot\mathbf{V}=0 \tag{2}\]  
\[ \begin{array}{ll}\overline{\rho}\Big{[}\frac{\partial\mathbf{V}}{\partial \overline{t}}+\
abla(\mathbf{V}\cdot\mathbf{V})\Big{]}&=-\
abla\overline{P}+ \
abla\Big{(}\frac{2\overline{P}\ \mathrm{D}^{\prime}}{Re}\Big{)}+\frac{(Ca_{T}\
abla_{ \phi}\phi+Ca_{c}\
abla_{S}C^{\prime})}{We}\\ &+\frac{[(1-Ca_{T}\rho-Ca_{c}C)\cdot\kappa(\varphi)(\varphi)(\varphi)\
abla \varphi]}{We\overline{p}}\\ \end{array}  \tag{3}\]  
\[\frac{\partial\Theta}{\partial\overline{t}}+\mathbf{V}\cdot\
abla\Theta=\frac {\
abla^{2}\Theta}{Ma} \tag{4}\]  
\[\frac{\partial C^{\prime}}{\partial\overline{t}}+\mathbf{V}\cdot\
abla C^{ \prime}=\frac{\
abla^{2}C^{\prime}}{(Ma\cdot Le)} \tag{5}\]  
\[\frac{\partial\varphi}{\partial\overline{t}}+\mathbf{V}\cdot\
abla\varphi=0 \tag{6}\]  
where \(\mathbf{V}=(\overline{u},\overline{v}\ )\) represents the dimensionless fluid velocity, \(D^{\prime}\) represents the viscous stress tensor, \(\kappa\) represents the interface curvature, \(\delta\) represents the Dirac delta function, \(\varphi\) represents the level set function, and \(\
abla_{s}=(I-m)\cdot\
abla\) represents the free interface gradient operator. \(\Theta\) and \(C^{\prime}\) refer to the dimensionless temperature and concentration, respectively, taking \(\Theta=(T-T_{0})/(T_{1}-T_{0})\) and \(C^{\prime}=(C-C_{0})/(C_{1}-C_{0})\). Moreover, the dimensionless quantity is expressed as the [PERSON] number \(Re=(\rho_{l}UH)/\mu_{l}\), [PERSON] number \(We=(\rho_{l}U^{2}H)/\sigma\), [PERSON] number \(Pr=\mu y(\rho_{l}\alpha)\). [PERSON] number \(Ma=Re\cdot Pr\), thermal capillary number \(Ca_{T}=(\sigma_{T}\cdot\Delta T)/\sigma_{0}\), solutal capillary number \(Ca_{C}=(\sigma_{C}\cdot\Delta C)/\sigma_{0}\), and [PERSON] number \(Le=\alpha_{l}/D_{l}\), where \(\alpha_{l}\) and \(D_{l}\) refer to the thermal and solutal diffusivity, respectively.