INSTABILITY IN THE SOLUTAL-THERMOCAPILLARY LIQUID LAYER WITH TWO FREE SURFACE
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摘要: 溶质−热毛细对流是流体界面的浓度和温度分布不均导致的表面张力梯度驱动的流动, 它主要存在于空间微重力环境、小尺度流动等表面张力占主导的情况中, 例如晶体生长、微流控、合金浇筑凝固、有机薄液膜生长等. 对其流动进行稳定性分析具有重要意义. 本文采用线性稳定性理论研究了双自由面溶质−热毛细液层对流的不稳定性, 得到了两种负毛细力比(η)下的临界Marangoni数与Prandtl数(Pr)的函数关系, 并分析了临界模态的流场和能量机制. 研究发现: 溶质−热毛细对流和纯热毛细对流的临界模态有较大的差别, 前者是同向流向波、逆向流向波、展向稳态模态和逆向斜波, 后者是逆向斜波和逆向流向波. 在Pr较大时, Pr增加会降低流动稳定性; 在其他参数下, Pr增加会增强流动稳定性. 在中低Pr, 溶质毛细力使流动更加不稳定; 在大Pr时, 溶质毛细力的出现可能使流动更加稳定; 在其他参数下, 溶质毛细力会减弱流动稳定性. 流动稳定性不随η单调变化. 在多数情况下, 扰动浓度场与扰动温度场都是相似的. 能量分析表明: 扰动动能的主要能量来源是表面张力做功, 但其中溶质毛细力和热毛细力做功的正负性与参数有关.
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关键词:
- 溶质−热毛细 /
- 液层 /
- 不稳定性 /
- Marangoni数 /
- 双自由面
Abstract: Solute-thermocapillary convection is a flow driven by a surface tension gradient caused by uneven concentration and temperature distribution at the fluid interface. It mainly appears in microgravity environment space or small-scale flow where the surface tension dominates, such as crystal growth, microfluidic, alloy pouring and solidification, organic thin liquid film growth, etc. The stability of this flow is of great significance of these applications. In the present work, the convective instability in the solutal-thermocapillary liquid layer with two free surfaces is examined by linear stability analysis. The relation between the critical Marangoni number and the Prandtl numbers (Pr) is obtained at different capillary ratio (η). The critical modes of solute-thermocapillary flow and pure thermocapillary flow are quite different. The former are downstream streamwise wave, upstream streamwise wave, spanwise stationary mode and upstream oblique waves, but the latter are upstream oblique waves and upstream streamwise wave. When Pr is larger, the flow stability will be weaker when Pr increases. At other parameters, the flow stability will be stronger when Pr increases. In the middle or low Pr, solute capillary force makes the flow more unstabler; at high Pr, solute capillary force may make the flow more stable. Flow stability does not change monotonously from η. In most cases, the distributions of perturbation concentration field and temperature field are similar. The energy analysis shows the main energy source of perturbation kinetic energy is the surface capillary force, but the work done by solute capillary force and thermal capillary force may be either positive or negative.-
Key words:
- solutal-thermocapillary /
- liquid layer /
- instability /
- Marangoni number /
- two free surfaces
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图 1 双自由面溶质−热毛细液层对流的示意图
Figure 1. Schematic of the solutal-thermocapillary liquid layer with two free surfaces
图 2 临界参数随
$ Pr $ 的变化曲线: (a)$M{a_{\rm{c}}}$ ; (b)波传播角$ \theta $ ; (c)波数$ k $ ; (d)角频率$ \omega $ . 曲线对应: UOW ($ {a_1} $ ,$ {e_3} $ ), SSM ($ {a_2} $ ,$ {b_2} $ ,$ {d_3} $ ), DSW ($ {b_3} $ ,$ {b_1} $ ,$ {c_2} $ ,$ {c_3} $ ), USW ($ {a_3} $ ,$ {c_3} $ ,$ {d_2} $ ,$ {c_1} $ ,$ {f_3} $ )Figure 2. The variation of critical parameter with
$ Pr $ . (a)$M{a_{\rm{c}}}$ , (b) wave propagation angle$ \theta $ , (c) wave number$ k $ and (d) angular frequency$ \omega $ . The curves correspond to: UOW ($ {a_1} $ ,$ {e_3} $ ), SSM ($ {a_2} $ ,$ {b_2} $ ,$ {d_3} $ ), DSW ($ {b_3} $ ,$ {b_1} $ ,$ {c_2} $ ,$ {c_3} $ ), USW ($ {a_3} $ ,$ {c_3} $ ,$ {d_2} $ ,$ {c_1} $ ,$ {f_3} $ )
图 3
$ \eta = - 0.5 $ 时临界模态的扰动流场. (a) SSM ($ Pr = 0.01 $ ), (b) SSM ($ Pr = 50 $ ), (c) DSW ($ Pr = 70 $ ), (d) USW ($ Pr = 100 $ )Figure 3. The perturbation flow field of the different preferred modes at
$ \eta = - 0.5 $ . (a) SSM ($ Pr = 30 $ ), (b) SSM ($ Pr = 0.01 $ ), (c) DSW ($ Pr = $ 70), (d) USW ($ Pr = 100 $ )
图 4
$ \eta = - 2 $ 时不同临界模态所对应的扰动流场. (a) USW ($ Pr = 1 $ ), (b) DSW ($ Pr = 50 $ ), (c) USW ($ Pr = 100 $ )Figure 4. The perturbation flow field of the different preferred modes at
$ \eta = - 2 $ . (a) DSW ($ Pr = 1 $ ), (b) USW ($ Pr = 50 $ ), (c) DSW ($ Pr = 100 $ )表 1 298.15 K时甲苯/正己烷混合溶液(C0 = 26.17%)的物性参数
Table 1. Physical properties of toluene/n-henxane mixture at T0 = 298.15 K and C0 = 26.17%
Physical properties Value Density $\rho /({\rm{kg}} \cdot {\rm{m}}^{ - 3} )$ $ 699.35 $ Thermal diffusivity $ \xi /({{\rm{m}}}^{2}\cdot{{\rm{s}}}^{-1} )$ $ 8.70 \times {10^{ - 8}} $ Solute diffusion coefficient $ D/({{\rm{m}}}^{2}\cdot{{\rm{s}}}^{-1} )$ $ 3.37 \times {10^{ - 9}} $ Soret coefficient $ \zeta /{{\rm{K}}^{ - 1}} $ $ 4.02 \times {10^{ - 3}} $ Dynamic viscosity $ \mu /({\rm{kg}}\cdot{{\rm{m}}}^{-1}\cdot{{\rm{s}}}^{-1} )$ $ 3.37 \times {10^{ - 4}} $ Dynamic viscosity $ Sc $ $ 142.99 $ Temperature coefficient of surface tension $ {\gamma }_{{\rm{T}}}/({\rm{N}}\cdot{{\rm{m}}}^{-1}\cdot{{\rm{K}}}^{-1} )$ $ 9.34 \times {10^{ - 5}} $ Concentration coefficient of surface tension $ {\gamma }_{{\rm{C}}}/({\rm{N}}\cdot{{\rm{m}}}^{-1} )$ $ - 8.62 \times {10^{ - 3}} $ Prandtl number$ Pr $ $ 5.54 $ 
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表 2 不同参数下各扰动能量变化项的值
Table 2. Values of perturbation energy variation terms at different parameters.
Pr η −N M I MT MC θ 0.01 −0.5 −362.156 0 362.155 6 0.000 2 −0.075 1 362.230 7 90 30 −0.5 −207.597 3 207.594 3 0.000 4 −386.273 2 593.867 5 90 50 −0.5 −7.590 7 7.588 7 0.002 0 −11.631 3 19.220 0 90 70 −0.5 −0.968 0 0.971 9 −0.003 8 2.025 7 −1.053 8 0 100 −0.5 −3.649 1 3.645 6 0.003 5 6.603 7 −2.958 0 180 200 −0.5 −10.465 3 10.463 8 0.001 6 7.276 4 3.187 4 180 0.1 −2 −5.313 0 5.317 9 −0.004 9 −0.004 4 5.322 3 180 10 −2 −4.706 1 4.709 8 −0.003 7 −1.750 6 6.460 5 180 40 −2 −9.238 8 9.232 2 0.006 6 2.135 2 7.097 0 0 70 −2 −7.777 6 7.770 8 0.006 9 −3.676 0 11.446 7 0 100 −2 −2.349 7 2.356 6 −0.006 9 −2.403 4 4.760 0 180 150 −2 −20.198 4 20.195 4 0.003 1 75.575 8 −55.380 5 90 0.01 0 −0.037 3 0.000 0 0.037 2 —— —— 107 1 0 −0.241 7 0.000 3 0.241 4 —— —— 114.1 10 0 −0.790 0 −0.011 1 0.801 1 —— —— 0 100 0 −2.446 4 −0.006 7 2.453 1 —— —— 0
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