Instabilities of thermocapillary-buoyancy flow in a rotating annular pool for medium-Prandtl-number fluid

The instabilities of the steady axisymmetric thermocapillary-buoyancy flow in a rotating annular pool were investigated by linear stability analysis. The critical instability parameters for the thermocapillary-buoyancy flow (normal gravity) and the pure thermocapillary flow (microgravity) were compared under different pool depths and rotation rates. The results show that the thermocapillary-buoyancy flow is more stable than the pure thermocapillary flow due to the stabilizing effect of the gravity (buoyancy) force. Two types of oscillatory instabilities were observed depending on the different rotation rates, and the propagation direction of the hydrothermal wave is also affected by the rotation rate.

Figure 1

Geometry of the annular pool.

Figure 2

Dependence of ${\mathrm{Ma}}_c$ and relative difference on the aspect ratio Г ($\varepsilon =0.5$, $\mathrm{Pr}=1.4$, $\omega =0$).

Figure 3

Dependence of (a) ${\mathrm{Ma}}_c$, (b) critical frequency $f_c$, and (c) ${\omega }_{\mathrm{HTW}}/\omega$ on the rotation rate ω ($\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$).

Figure 4

Surface patterns of the perturbation temperature under the critical mode at (a) $\omega =6$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=7531$ and (b) $\omega =25$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=8104$ for pure thermocapillary flow. (c) $\omega =6$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=11102$; and (d) $\omega =25$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=11734$ for thermocapillary-buoyancy flow.

Figure 5

Disturbance energy balance under the critical mode at (a) $\omega =6$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=7531$ and (b) $\omega =25$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=8104$ for pure thermocapillary flow. (c) $\omega =6$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=11102$ and (d) $\omega =25$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=11734$ for thermocapillary-buoyancy flow.

Figure 6

Basic state and critical mode on a $r\text{−}z$ cut plane where the perturbation temperature takes the maximum value for $\omega =6$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=7531$ under zero gravity. (a) Quasi-two-dimensional streamlines of the basic flow and distribution of the basic-state temperature $T_0$ (isolines). (b) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T1}$ (isolines). (c) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T2}$ (isolines).

Figure 7

Basic state and critical mode on a $r\text{−}z$ cut plane where the perturbation temperature takes the maximum value for $\omega =6$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=11102$ under normal gravity. (a) Quasi-tw- dimensional streamlines of the basic flow and distribution of the basic-state temperature $T_0$ (isolines). (b) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T1}$ (isolines). (c) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T2}$ (isolines).

Figure 8

Surface patterns of the perturbation temperature under the critical mode at (a) $\omega =150$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=12839$ for pure thermocapillary flow and (b) $\omega =150$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=17045$ for thermocapillary-buoyancy flow.

Figure 9

Disturbance energy balance under the critical mode at (a) $\omega =150$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=12839$ for pure thermocapillary flow and (b) $\omega =150$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=17045$ for thermocapillary-buoyancy flow.

Figure 10

Basic state and critical mode on a $r\text{−}z$ cut plane where the perturbation temperature takes the maximum value for $\omega =150$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=12839$ under zero gravity. (a) Quasi-two-dimensional streamlines of the basic flow and distribution of the basic-state temperature $T_0$ (isolines). (b) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T1}$ (isolines). (c) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T2}$ (isolines).

Figure 11

Basic state and critical mode on a $r\text{−}z$ cut plane where the perturbation temperature takes the maximum value for $\omega =150$, $\mathrm{\Gamma }=0.25$, $\varepsilon =0.5$, $\mathrm{Pr}=1.4$, ${\mathrm{Ma}}_c=17045$ under normal gravity. (a) Quasi-two-dimensional streamlines of the basic flow and distribution of the basic-state temperature $T_0$ (isolines). (b) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T1}$ (isolines). (c) Disturbance velocity vectors (arrows) and perturbation temperature combined with the local thermal energy $i_{T2}$ (isolines).