Stability of buoyancy-thermocapillary convection in molten silicon liquid bridge between two disks with different radii under gravity

By employing a linear stability analysis based on the spectral element method, we investigated the impact of radius ratio (Gamma(r)) on the stability of buoyancy-thermocapillary convection in a molten silicon liquid bridge (Pr = 0.011). This liquid bridge was located between two coaxial disks with different radii under the influence of gravity. The aspect ratio of the liquid bridge was maintained at Gamma = 1, with a volume ratio Gamma(v) = 1 and a fixed height. To explore the physical mechanisms behind convection instability, a perturbation energy analysis was adopted. The free surface shape was determined using the Young-Laplace equation, and two distinct heating strategies were employed. In a top-heated liquid bridge, the convection stability under gravity is always stronger than under zero-gravity. However, in a bottom-heated liquid bridge, the convection stability under gravity is not consistently stronger than under zero-gravity; specifically, when 0.522 < Gamma(r) < 0.673, the convection stability under gravity is weaker than under zero-gravity. Despite the small height of the liquid bridge (approximately 2 mm), the maximum relative difference of the critical Marangoni number (Ma(c)) between gravity and zero-gravity conditions reaches as high as 227.8%. In a bottom-heated liquid bridge, oscillatory instability occurs at larger radius ratios (Gamma(r) = 0.8) compared to the zero-gravity condition. Furthermore, all instabilities for various radius ratios and heating strategies were found to be of hydrodynamic in nature.