Effects of Pr and pool curvature on thermocapillary flow instabilities in annular pool

Abstract

We linearly analyzed the effect of the Prandtl number (Pr) on the stability of thermocapillary flow in shallow annular pools with an aspect ratio Γ = (Ro - Ri)/d = 20 and two radius ratios ΓR = Ri/Ro= 0.50 and 0.98039, where d, Ro, and Ri are the liquid depth and the radii of the heated outer wall and the cooled inner wall, respectively. The results for Pr ∈ [0, 102] show that the steady axisymmetric thermocapillary flows in these annular pools become unstable against oscillatory instability modes, OSC1, OSC2 and hydrothermal wave (HTW). Two co-dimension-two bifurcations occur at Pr*1 and Pr*2. The critical mode for Pr ≤ Pr*1 is OSC2 with almost constant critical Reynolds number Rec, large wave number and high frequency. OSC1 with smaller wave number and lower frequency is the critical mode for Pr*1 ≤ Pr ≤ Pr*2 and Rec slightly decreases with increasing Pr. HTW is the critical mode for Pr ≥ Pr*2 and Rec decreases with increasing Pr. The values of (Pr*1, Pr*2) are (0.00953, 0.03054) for ΓR = 0.50, and (0.01919, 0.01946) for ΓR = 0.98039. Energy-budget analyses reveal that the instabilities in low-Pr range are caused by instabilities in the steady toroidal vortex (or vortices) near the cold wall.