Linear Stability Analysis Of Thermocapillary Flow In Annular Pool And Liquid Bridge

Thermalcapillary flow is widely exists in the industrial process:floating-zone method and Czochralski method,for example.The flow instability is of great importance on the research of microgravity fluid physics and Space materials science during the last twenty years.The half-floating-zone model and annular-pool model,simplified from the floating method and Czochralski method separately,are adopted extensively in the study of Thermalcapillary flow.This paper investigates the instability of thermalcapillary flow in the floating-zone model and annular-pool model by using linear stability analysis based on spectral element method and introduces the simplified infinite-length model and finite-length model,the ?₁-?₂ model as well as multi-pole pair non-uniform rotating magnetic field to the linear stability analysis method.(1)The instability of thermalcapilly flow in the shallow annular pool heated from inner wall was investigated under microgravity environment.The results show that the flow instability with small Prandtl number is influenced by the pool rotation.The instability is caused by the hydrodynamic mechanism.The flow stability decreases and then increases with the rotating velocity inceases.Two unstable flow are found when rotating velocity rises to a certain range,corresponding to three Hopf bifurcation points.The curve of critical Marangoni number with respect to rotating velocity of the thermalcapillary flow heated from inner wall of shallow annular pool is similar to that of the pool heated from outer wall.The value of the Marangoni number is higher which indicates that it helps improving the flow stability by heating from the inner wall.Meanwhile,energy analysis shows that the instability of the thermalcapillary flow in the shallow annular pool is caused by the grad of velocity field triggered by rotating?thermalcapillary force or both of them.The flow instability is caused by hydrodynamic mechanism and no direct contribution comes from the thermalcapillary mechanism.The flow stability in the half-zone liquid bridge is studied and the critical parameters within 0.001-5 range of Pr number were obtained.Results show that the flow instability of low Prandtl number(0.001-0.178)is caused by hydrodynamic mechanism,rather than the thermalcapillary mechnism.For higher Prandtl numbers(Prandtl>0.7),the instability is mainly caused by the thermalcapillary mechnism,as the work done by the thermalcapillary force along the free surface is much greater than the energy transfer between the basic state and the disturbance velocity.For the Prandtl number between 0.178 and 0.7,both the thermalcapillary and hydrodynamic mechanism contribute to the flow instability.The critical curve of Reynolds number indicates that the flow converts from two dimensional axisymmetric flow to three dimensional oscillating flow.The critical Reynolds number increases with the increase of Prandtl number,then decreases after reaches to the max at 0.07.With the increasing Prandtl number,the initial position of flow instability is gradually approaching to the symmetry axis from the initial free surface.The infinite-length model,the finite-length model and the ?₁-?₂ model were introduced to the linear stability analysis based on spectral element method for investigating the stability of thermocapillary flow.The difference between the three models in solving the thermalcapillary flow problem under the rotating magnetic field indicates shows that the basic state and the critical parameters obtained from the finite-length model and ?₁-?₂ model agree with each other,but vary from the infinite-length model with a large margin of error.By using ?₁-?₂ model,the thermalcapillary flow in half-zone under the action of external rotating magnetic field was simulated and analyzed.Results indicate that.The external magnetic field improves the flow stability of the half-zone liquid bridge,with a much higher critical Marangoni number.The hydrodynamic mechanism brings out the flow instability and the thermalcapillary force or the Lorenz force caused by external magnetic field makes no contribution to the flow instability.The critical Marangoni number increases firstly,declines later and increases finally with the increasing Taylor number of the rotating magnetic field?The external magnetic field gives no direct effect on the flow instability.The Lorenz force affects the flow instability by changing the energy transformation between the basic solution and the disturbance field.The disturbance velocity direction is opposite to the thermalcapillary force,which indicates the thermalcapillary suppress the flow instability.For the first time this paper introduce the multi-pole pair external non-uniform rotating magnetic field model to linear stability analysis based on spectral element method.The flow stability of half-floating zone liquid bridge under rotating magnetic is investigated.Results show effect of different polar pairs of non-uniform magnetic field is basically the same as in 1-polar pair situation on the basic state of the thermalcapillary flows in the half-zone liquid bridge.The influences on the flow stability are qualitatively similar,but different quantitatively.The critical curve changes rapider with the Taylor number under 1-polar pair situation and slower under 3-polar pair situation.