## 1 Introduction
[PERSON] and [PERSON] [31] verified through experiments that the stability of the liquid bridge is significantly affected by the heat loss of the bridge. [PERSON] et al. [32] examined the impact of free surface heat loss on the oscillatory regime of thermocapillary-driven convection in a high Prandtl number liquid bridge via experimental and numerical analyses. They concluded that the heat loss effect is caused by the interaction between the dynamic surface morphology of the liquid bridge and the heat loss from the hot wall. [PERSON] et al. [33] conducted experimental investigations on the oscillatory flow of thermocapillary convection in a liquid bridge and numerically analyzed the impact of heat loss on the initial stage of surface oscillation. Thermo-solutocapillary convection is more complicated than pure thermocapillary convection. Since [PERSON] [34] first reported the thermo-solutocapillary convection, the study of thermo-solutocapillary convection has attracted more and more attention. [PERSON] and [PERSON] [35] paid attention to the dynamic free surface of thermo-solutocapillary convection. The numerical results exhibit that the thermo-solutocapillary convection with various capillary ratios displays three distinct modes of free surface shape at low Marangoni values(small temperature difference). [PERSON] et al. [36] discussed the impact of the thermal and solute Marangoni numbers on Czochralski growth under microgravity conditions. [PERSON] and [PERSON] [37] investigated how the concentration distribution in SiGe alloy changes during growth, as a function of the growth rate. The concentration gradient no longer maintains a uniform distribution on the free interface when the growth rate (\(0.1<V_{g}<4\)) is considered, and the initiation of flow is primarily attributed to the concentration gradient present at the melting interface which creates a diffusive boundary layer. In this case, the instability of the growth zone comes from the flow inertia and the interface concentration gradient. The instability is not limited to strong concentration regions, but spreads to the whole melt. [PERSON] et al. [38] studied the coupling of thermo-solutocapillary convection in the SixGeI-x system of a liquid bridge of a half floating zone by numerical simulation. The impact of the aspect ratio on the stability of capillary convection is also studied. [PERSON] and [PERSON] [39] studied the coupling of solute and thermocapillary convection under buoyancy via numerical methods. The law of heat and mass transfer rate variation is obtained within a certain range of thermal Marangoni numbers (a certain temperature). [PERSON] and [PERSON] [40] studied Marangoni instability in binary mixtures heated from the free surface, taking into account the impact of thermocapillary convection, solutocapillary convection, and buoyancy. They analyzed the stability characteristics of the polymer located at the deformable interface and the rigid substrate with poor conductivity. Meanwhile, they concluded that the system has two different kinds of oscillatory instability besides the monotone perturbation. [PERSON] and [PERSON] [41] considered the Soret effect and studied the influence of the thermal Marangoni number on thermo-solutocapillary convection in a liquid bridge containing a solution of n-decane and n-hexane under microgravity by a numerical method. The results show that the thermo-solutocapillary convection wave occurs when the \(Ma_{T}\) number is large (large temperature difference), and the \(Ma_{T}\) number has an impact on the flow field structure of convection oscillation in the liquid bridge. Moreover, the fluctuation of thermo-solutocapillary convection increases with the increase in \(Ma_{T}\) number. [PERSON] and [PERSON] [42] numerically investigated the feasibility of axial vibration on the evolution of convection in a liquid bridge under the neglect of gravity. It was observed that the convection varied with temperature when the solute Marangoni number remained constant.