Linear stability analysis of thermocapillary flow in a slowly rotating shallow annular pool using spectral element method

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The stability of thermocapillary flow in a slowly rotating shallow annular pool was investigated by using the Legendre spectral element method. The silicon melt (Pr=0.011), filling in a pool with adiabatic free surface and bottom, was heated at the outer cylindrical wall and cooled at the inner cylindrical wall. The critical stability conditions for different dimensionless rotation rates ω, ranging from 0 to 2000, were determined by linear stability analysis. Moreover, the energy analysis was applied to further illustrate the underlying mechanism of the flow instability. The results indicate that there is one Hopf bifurcation for ω<940 and ω>1185. Thermocapillary flow is the dominant factor of the instability for ω<940, and the pool rotation becomes the key role for the first instability for ω>1185. Three turning points were observed in the interval 940⩽ω⩽1185, corresponding to three transitions between two-dimensional steady flow and three-dimensional oscillatory flow, owing to the competition of two driving forces with increasing Marangoni number at a fixed ω. With pool rotation, the results exhibit that the flow instability is deduced to occur initially in the zone near the cold wall with the evidence of the extreme velocity gradient and also the distribution of the local kinetic energy.