## 2 Models and Methods
### Level Set Method
\tag{20}\]  
A uniform staggered grid is employed to discretize the liquid bridge and the surrounding air region, using a central difference scheme for all derivatives except the convection term. The QUICK method is used for convection terms in the Navier-Stokes equation to achieve third-order accuracy, while the central difference scheme is adopted for other terms with first-order accuracy. The level set function equation is evaluated using the QUICK method for third-order accuracy. The Poisson equation for pressure is solved using the successive over relaxation (SOR) method, while the continuum surface force (CSF) model is utilized to handle surface tension at the interface. Additionally, the ENO scheme is implemented to solve the convection term of the level set function equation. Overall, the methods employed in this simulation exhibit high accuracy, and the adopted grid discretization and difference scheme are capable of accurately capturing the characteristics of the liquid bridge and surrounding air. The liquid bridge working medium is a toluene/n-hexane mixed solution, and physical properties [49; 50] are exhibited in Table 1.  
This study applies a variety of different grids for detection calculations to ensure the grid independence. The report of grid refinement is exhibited in Table 2. To validate the code and refine the mesh, we selected the velocity, temperature, and concentration distributions at the intermediate height interface of the liquid bridge as the objects of analysis. Based on the error calculation method utilized, the values for error in longitudinal velocity comparison are 3.6%, 0.7%, and 0.6% for the second group and other groups, respectively. Correspondingly, the error values for the transverse velocity comparison are 5.4%, 1.6%, and 1.6%. Moreover, in the temperature comparisons, the error values are 1.1%, 0.3%, and 0.2%, while in the concentration comparisons, the respective values are 1.2%, 0.4%, and 0.3%. Following the analysis of the results, we concluded that a square grid measuring 81 \(\times\) 41 is sufficient to capture the interface, temperature, concentration, and flow fields with a high degree of accuracy. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Symbol** & **Value** \\ \hline Density & \(\rho_{l}\) & 699 kg/m\({}^{3}\) \\ Diffusion coefficient & \(\alpha_{t}\) & \(1.88\times 10^{-7}\) m\({}^{2}\)/s \\ Dynamic viscosity & \(\mu_{t}\) & \(3.37\times 10^{-4}\) kg/(m-s) \\ Surface tension & \(\sigma_{0}\) & \(2.1\times 10^{-2}\) N/m \\ Temperature surface tension coefficient & \(\sigma_{T}\) & \(9.43\times 10^{-5}\) N/(m-K) \\ Concentration surface tension coefficient & \(\sigma_{C}\) & \(-8.62\times 10^{-3}\) N/m \\ Prandtl number & \(Pr\) & 5.54 \\ Lewis number & \(Le\) & 25.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical properties of toluene/n-hexane mixture at 25 \({}^{\circ}\)C (0.24/0.76). In order to verify the present model and code, we compared present results with [PERSON]'s results [35]. Other calculation parameters are \(R_{\omega}=-10\), \(Le=100\), \(Ar=1.2\), and \(Pr=1\). Figure 2 shows the surface longitudinal velocity and surface deformation in different \(Ma\) numbers, respectively. The present results are consistent with [PERSON]'s results, and these comparisons illustrate the correctness of the present model and code.