## 1 Introduction  
Marangoni convection is very common in many industrial applications, such as crystal growth [1; 2; 3], welding [4; 5; 6; 7], and evaporation [8; 9]. During crystal growth, the buoyancy convection is weakened under microgravity, and Marangoni convection becomes the dominant convection. The prepared crystals are not only large in size but also uniformly distributed [10]. The floating zone method is a crucial method for producing high-qualitysemiconductor materials. During the floating zone's crystal growth, the semiconductor materials in a molten state are frequently doped with impurities. The non-uniform distribution of impurities on the free surface of the melt can generate the solute capillary convection due to the concentration gradient. The convection is coupled with thermocapillary convection driven by the temperature gradient, and they form Marangoni convection on the free surface eventually [11; 12]. A liquid bridge model can be utilized to explore this phenomenon. The liquid bridge model is a model in which a fluid medium is located between two coaxial solid disks, and the upper and lower disks experience various temperature and concentration differences, respectively, thereby forming the thermo-solutocapillary convection on the surface of the liquid [13; 14]. After the upper disk is heated and a certain amount of solute introduced, the surface tension based on temperature and concentration begins to change and forms Marangoni stress at the free interface, resulting in the fluid movement at the free interface. In situations where the disparity in temperature and concentration between the solid disks is small, a steady axisymmetric vortex structure will appear inside the liquid bridge. Simultaneously, the time-based periodic oscillation occurs internally when the temperature and concentration differences reach critical values. [PERSON] et al. [15] observed this phenomenon through experiments for the first time. During the floating zone technique for crystal growth, the primary factor responsible for generating micron-level impurity fringes within the crystal is oscillation [16]. Many scholars applied magnetic field [17; 18], transverse vibration [19; 20; 21; 22], and geometric size adjustment [23; 24; 25] to investigate the internal flow and oscillatory behavior of the thermo-solutocapillary convection aiming to suppress the oscillation. [PERSON] et al. [26] first studied the transition state of the liquid bridge from the static to oscillation at a low Prandtl number. The non-contact detection method was used in the experiment to premix tin oxide particles in the fluid. The results exhibit that the temperature begins to oscillate periodically and the corresponding oscillation frequency and amplitude are obtained when the Marangoni number increases to 43.3 in a high vacuum environment. Simultaneously, the frequency and amplitude of oscillation increase when the Marangoni number reaches 91. [PERSON] et al. [27] investigated the flow structure of buoyant thermocapillary convection and the critical condition of oscillation under diverse height-diameter and volume ratios in ground experiments. [PERSON] and [PERSON] [28] carried out a numerical research of oscillating thermocapillary flow in a 5 cSt silicone oil-filled liquid enclosure. Under microgravity conditions, the finite volume method was applied to calculate in the liquid bridge with various volume ratios to determine the dynamic evolution of temperature disturbance during the beginning of oscillation instability. [PERSON] et al. [29; 30] conducted a numerical study on the thermocapillary convection of nanofluids with medium and large Prandtl numbers in a rectangular cavity. The critical temperature difference of the unsteady flow is obtained via varying the thermal gradient, and the influence of the volume fraction of nanoparticles on the critical temperature gradient is analyzed. Finally, the corresponding oscillation laws are obtained.