This work is devoted to the study of steady thermocapillary-buoyant convection in a system of two horizontal superimposed immiscible liquid layers filling a lateral heated thin annular pool.The governing equations are...This work is devoted to the study of steady thermocapillary-buoyant convection in a system of two horizontal superimposed immiscible liquid layers filling a lateral heated thin annular pool.The governing equations are solved using an asymptotic theory for the aspect ratios ε→ 0.Asymptotic solutions of the velocity and temperature fields are obtained in the core region away from the cylinder walls.In order to validate the asymptotic solutions,numerical simulations are also carried out and the results are compared to each other.It is found that the present asymptotic solutions are valid in most of the core region.And the applicability of the obtained asymptotic solutions decreases with the increase of the aspect ratio and the thickness ratio of the two layers.For a system of gallium arsenide （lower layer） and boron oxide （upper layer）,the buoyancy slightly weakens the thermocapillary convection in the upper layer and strengthens it in the lower layer.展开更多
This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow...This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios,which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt,the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio,the applicability of the present asymptotic solutions decreases gradually.展开更多
Thermocapillary flow of silicon melt(Pr=0.011)in shallow annular pool heated from inner wall was simulated at the dimensionless rotation ratewranging from 0 to 7000.The effect of pool rotation on the stability of the ...Thermocapillary flow of silicon melt(Pr=0.011)in shallow annular pool heated from inner wall was simulated at the dimensionless rotation ratewranging from 0 to 7000.The effect of pool rotation on the stability of the thermocapillary flow was investigated.The steady axisymmetric basic state was solved by using the spectral element method;the critical stability parameters were determined by linear stability analysis;the mechanism of the flow instability was explored by the analysis of energy balance.A stability diagram,exhibiting the variation of the critical Marangoni number versus the dimensionless rotation ratewwas presented.The results reveal that only one Hopf bifurcation point appeared in the intervals ofω<3020 andω>3965,and the corresponding instability was caused by the shear energy,which was provided by the thermocapillary force and pool rotation,respectively.In addition,the competition between thermocapillary force and pool rotation leads to three Hopf bifurcation points in the range of 3020<ω<3965 with the increase of Marangoni number.展开更多
基金supported by the National Natural Science Foundation of China (50776102)the Fundamental Research Funds for the Central Universities (CDJXS1041148)
文摘This work is devoted to the study of steady thermocapillary-buoyant convection in a system of two horizontal superimposed immiscible liquid layers filling a lateral heated thin annular pool.The governing equations are solved using an asymptotic theory for the aspect ratios ε→ 0.Asymptotic solutions of the velocity and temperature fields are obtained in the core region away from the cylinder walls.In order to validate the asymptotic solutions,numerical simulations are also carried out and the results are compared to each other.It is found that the present asymptotic solutions are valid in most of the core region.And the applicability of the obtained asymptotic solutions decreases with the increase of the aspect ratio and the thickness ratio of the two layers.For a system of gallium arsenide （lower layer） and boron oxide （upper layer）,the buoyancy slightly weakens the thermocapillary convection in the upper layer and strengthens it in the lower layer.
基金supported by the National Natural Science Foundation of China (50776102)the Fundamental Research Funds for the Central Universities (CDJXS10142248)
文摘This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios,which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt,the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio,the applicability of the present asymptotic solutions decreases gradually.
基金supported by the National Natural Science Foundation of P. R. China (No. 11572062)Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_17R112)
文摘Thermocapillary flow of silicon melt(Pr=0.011)in shallow annular pool heated from inner wall was simulated at the dimensionless rotation ratewranging from 0 to 7000.The effect of pool rotation on the stability of the thermocapillary flow was investigated.The steady axisymmetric basic state was solved by using the spectral element method;the critical stability parameters were determined by linear stability analysis;the mechanism of the flow instability was explored by the analysis of energy balance.A stability diagram,exhibiting the variation of the critical Marangoni number versus the dimensionless rotation ratewwas presented.The results reveal that only one Hopf bifurcation point appeared in the intervals ofω<3020 andω>3965,and the corresponding instability was caused by the shear energy,which was provided by the thermocapillary force and pool rotation,respectively.In addition,the competition between thermocapillary force and pool rotation leads to three Hopf bifurcation points in the range of 3020<ω<3965 with the increase of Marangoni number.
