## 3 Results and Discussions

Table 4: Calculation conditions (corresponding to Section 3.2). Figure 3: (**a–d**) Velocity vector at different temperature differences (\(R\) = 5 mm, \(Ar\) = 1.0, \(Pr\) = 5.54, \(Le\) = 25.8, \(\Delta C_{C}\) = \(-0.021\), \(\rho_{l}/\rho_{g}\) = 597.9). Figure 4a exhibits the schematic diagrammatic sketch of the working area of the liquid bridge. The free surface on the right side of the liquid bridge is selected as the research object. Figure 4b exhibits the temperature change of the right free surface, and Figure 4c shows the concentration change of the right free surface. It can be observed from Figure 4b that the surface temperature is distributed as a convex function under different temperature differences. Additionally, a notable disparity in the temperature gradient is noticed between the upper and lower ends, with the former exhibiting a significantly larger variation. As the temperature difference between the upper and lower ends increases, the gradient difference between these two ends diminishes. This suggests that the temperature distribution at the interface becomes more uniform as the temperature difference increases. Concerning the concentration change of the free interface, it can be found from Figure 4c that the hot end (z = 0.85 to 1.0) and cold end (z = 0 to 0.15) of the liquid bridge present a large concentration gradient. The concentration increased sharply from 0.3 to 1 near the hot end and decreased sharply from 0.2 to 0 near the cold end. Because the hot and cold ends near the interface are the conversion areas of volume reflow and surface reflow, a large concentration gradient is formed in this area. Simultaneously, it can be found that in the range of height z = 0.15 to 0.85 the concentration shows a decreasing trend as the temperature difference increases. The temperature change presents more uniform than the concentration change. Figure 5 exhibits the transverse and longitudinal velocities of the free surface on the right side, as affected by various temperature differences. It can be found from Figure 5a that the transverse velocity generally presents a transverse S-type distribution. The peak of transverse velocity under different temperature difference conditions is close to the hot end (z = 0.7 to 1.0) and the cold end (z = 0 to 0.3) of the system, and the velocity varies sharply. The velocity at the intermediate position (z = 0.3 to 0.7) does not vary significantly with the temperature difference. The peak values of transverse velocity appear near z = 0.15 and z = 0.85 near the cold and hot ends, respectively. The peak value rises  
Figure 4: (**a**) Schematic diagram of working area of liquid bridge; (**b**) variation of temperature on right free surface; (**c**) variation of concentration on right free surface (\(R\) = 5 mm, \(Ar\) = 1.0, \(Pr\) = 5.54, \(Le\) = 25.8, \(Ca_{C}\) = \(-\)0.021, \(\rho_{l}/\rho_{g}\) = 597.9). with an elevation in the temperature gradient, while the velocity value at the middle height is relatively stable. The velocity value fluctuates greatly as the temperature difference increases. The conversion process between surface flow and volume reflux of the system is accelerated. The acceleration of surface flow will also lead to a big pressure gradient around the interface. The pressure gradient near the interface increases the transverse velocity fluctuation. As listed in Figure 5b, the longitudinal velocity presents a parabolic shape, and there is no complex fluctuation phenomenon. The longitudinal velocities are all positive, indicating a unidirectional fluid flow from the lower to the upper plate. Due to the high concentration of the hot end and the absorption of the interface solute, the interface has a higher concentration gradient. At this time, the solutocapillary force exceeds the thermocapillary force and dominates the orientation of the interfacial fluid motion. Figure 6 exhibits the deformation trend of the free surface on the right side, as affected by various temperature differences. On the whole, the interface deformation is S-type.