Three-dimensional numerical simulation of Marangoni instabilities in liquid bridges : influence of geometrical aspect ratio

Oscillatory Marangoni convection in silicone oil liquid bridges with different geometrical aspect ratios is investigated by three-dimensional and time-dependent numerical simulations, based on control volume methods in staggered cylindrical non uniform grids. The three-dimensional oscillatory flow regimes are studied and compared with previous experimental and theoretical results. The results show that the critical wave number (m), related to the azimuthal spatio-temporal flow structure, is a monotonically decreasing function of the geometrical aspect ratio of the liquid bridge (defined as ratio of the length to the diameter). For this function a general correlation formula is found, that is in agreement with the previous experimental findings. The critical Marangoni number and the oscillation frequency are decreasing functions of the aspect ratio; however, the critical Marangoni number, based on the axial length of the bridge, does not change much with the aspect ratio. For each aspect ratio investigated, the onset of the instability from the axi-symmetric steady state to the three-dimensional oscillatory one is characterized by the appearance of a standing wave regime that exhibits, after a certain time, a second transition to a travelling wave regime. The standing wave regime is more stable for lower aspect ratios since it lasts for a long time. This behaviour is explained on the basis of the propagation velocity of the disturbances in the liquid phase; for this velocity a general correlation law is found as function of the aspect ratio and of the Marangoni number