Linear Stability Analysis Of Thermocapillary Flow In Annular Pools

When the temperature distribution on the free surface of the fluid is not uniform,the unbalanced surface tension will drive the fluid motion and form thermocapillary flow.The thermocapillary flow is widely encountered in many industrial applications.For example,in the Czochralski crystal growth technique,the instability of thermocapillary flow has an important effect on the quality of crystal growth.In order to improve the quality of single crystal growth,it is necessary to study the critical parameters and instability mechanism for the thermocapillary flow instability.In addition,the thermocapillary flow contains a wealth of nonlinear dynamic phenomena,and the study of its instability is helpful to promote the development of bifurcation theory.For this reason,this thesis systematically studies the effects of the Prandtl number(Pr),the aspect ratio,the volume ratio and the rotation on the instability and instability mechanism of thermocapillary flow in annular pools by the linear stability analysis based on the high-precision spectral element method.The main research contents and results are as follows:In a stationary annular pool heated from inner wall,the effects of Pr,aspect ratio and volume ratio on the thermocapillary flow instability were studied.Five types of instability have been found depending on the Pr.When the Pr is small,the flow instability is caused by the hydrodynamic inertia effect,and when the Pr is large,the instability is caused by the hydrothermal wave instability mechanism arising from the thermocapillary effect.For the intermediate Pr range,the above two mechanisms will coexist and compete with each other,making the instability mechanism more complex,and may cause the multiple instability phenomena.With the increase of the aspect ratio,the critical Marangoni number,oscillation frequency and wave number of the thermocapillary flow decrease,and the instability mode will change from the oscillatory instability mode to the stationary instability mode.For small-Prandtl-number fluid(Pr=0.011),the flow instability is always caused by the inertial effect;for medium-Prandtl-number fluid(Pr=1.4),the instability is mainly caused by the thermocapillary effect,while the inertial effect will increase with the increase of aspect ratio.For the thermocapillary flow of small-Prandtl-number fluid,as the volume ratio increases,the flow stability decreases first,then increases,and finally decreases;while the critical wave number and oscillation frequency of the instability decrease monotonically.The flow instability at different volume ratios is essentially caused by the inertial effect.The effect of rotation on the thermocapillary flow instability of medium-Prandtl-number fluid in an annular pool heated from inner wall was studied.When the rotation rate of the liquid pool is small,the flow stability decreases first and then increases with the rotation rate,while when the rotation rate is large,the flow stability decreases monotonously with the rotation rate.In particular,the basic-state flow is very stable in a certain range of intermediate rotation rate,and no flow instability is found within the parameters range calculated in this paper.Rotation affects the instability mechanism by affecting the flow structure of the basic-state flow,but it does not provide a lot of energy for the flow instability.The effects of the thickness of the liquid layer and the rotation on the buoyancy-thermocapillary flow of medium-Prandtl-number fluid in an annular pool heated from outer wall were studied.The results show that the buoyancy-thermocapillary flow is more stable than the pure thermocapillary flow,and the influence of the buoyancy on the thermocapillary flow increases with the increase of liquid layer thickness.Weak rotation destabilizes the flow,while strong rotation stabilizes the flow.The flow instability is mainly caused by the thermocapillary effect,and the rotation will not provide a lot of energy for the instability.The effect of the crystal or crucible rotation on the thermocapillary flow instability of small-Prandtl-number fluid in a shallow Czochralski configuration was studied.For the crucible rotation,when the rotation rate is small,the flow stability decreases first and then increases with the increase of rotation rate.The flow instability is caused by the radial shear of the basic flow driven by the thermocapillary force.When the rotation rate is large,the flow stability changes in a small range,and the flow instability is caused by the azimuthal shear of the basic flow driven by the pool rotation.In particular,in a certain rotation rate range,the flow will undergo three transitions with the increase of Marangoni number,owing to the competition and interaction between the thermocapillary force and the rotation.For the crystal rotation,when the rotation rate is small,the flow stability first decreases and then increases as the rotation rate increases,and the propagation direction of the oscillating wave is opposite to the direction of crystal rotation;when the rotation rate is large,the flow stability decreases with the increase of rotation rate,and the propagation direction of the oscillating wave is the same as the crystal rotation.Since the influence of the crystal rotation on the basic flow field is small,the instability mechanism is always the radial shear instability caused by the basic-state thermocapillary force.