STABILITY ANALYSIS OF THERMOCAPILLARY LIQUID LAYERS WITH TWO FREE SURFACES FOR A BINGHAM FLUID
-
摘要: 热毛细对流是流体界面温度分布不均导致的表面张力梯度驱动的流动. 它主要存在于空间等微重力环境或小尺度流动等表面张力占主导的情况中. 在很多工业领域, 如晶体生长、聚合物加工、喷墨打印、微流控, 产品质量都与热毛细对流密切相关. 空间3D打印是太空制造的重要技术, 可以支持空间站的在轨长期有人照料的运行和维护, 实现按需制造. 本文以聚合物流体的空间3D打印为应用背景, 采用线性稳定性理论研究了Bingham流体双自由面热毛细液层的稳定性, 得到了在不同Bingham数(B)下的临界Marangoni数(Mac)与Prandtl数(Pr)的函数关系,分析了临界模态的流场和能量机制. 研究发现: 该流动的临界模态包括流向波和斜波模态, 与B, Bi和两界面垂直方向上的温差(Q)相关. B和Bi的增加会增强热毛细对流的稳定性. 当Q = 0时, 扰动温度分布分成对称和反对称两种情况. 当Q > 0时, Pr的增加会减弱流动稳定性. 在小Pr情况下, 扰动温度分布在整个流场, 在大Pr情况下, 扰动温度在栓塞区为零. 能量分析表明: 扰动动能的主要能量来源是表面张力做功, 但小Pr数下基本流也有一定贡献.
-
关键词:
- 热毛细对流 /
- Bingham流体 /
- 稳定性 /
- Marangoni数
Abstract: Thermocapillary convection refers to the fluid motion driven by the temperature-induced surface tension gradient. It mainly exists in the microgravity environment such as space or small-scale flow dominated by surface tension. In many industrial fields, such as crystal growth, polymer processing, inkjet printing, and microfluidic, product quality is closely related to thermocapillary convection. 3D printing is an important technology in space manufacturing, which can support the long-term manned operation and maintenance of the space station in orbit and realize on-demand manufacturing. This paper takes the spatial 3D printing of polymer fluids as the application background, the stability of thermocapillary liquid layers with two free surfaces for a Bingham fluid is studied by using the linear stability analysis. The function relation between the critical Marangoni number (Mac) and Prandtl number (Pr) at different Bingham number (B) is obtained. The flow field and energy mechanism of the critical mode are analyzed. It is found that the critical modes include the streamwise wave and the oblique wave, which are related to B, Bi and the vertical temperature difference (Q) between two interfaces. The increase of B and Bi will enhance the stability. When Q = 0, there are two kinds temperature distribution, which are symmetric and antisymmetric. When Q > 0, the increase of Pr will destabilize the flow. The perturbation temperature is distributed in the whole flow field at small Pr, and the perturbation temperature is zero in the plug region at large Pr. The energy analysis shows that the main energy source of perturbation energy is the work done by surface tension,but for small Pr, the basic flow also makes some contributions.-
Key words:
- thermocapillary /
- Bingham fluid /
- stability /
- Marangoni number
-
图 1 Bingham流体双自由面热毛细液层示意图
Figure 1. The thermocapillary liquid layer with two free surfaces for a Bingham fluid
图 2 基本流的(a)速度分布和(b)垂直温度分布(Ma = 100, Q = 0)
Figure 2. (a) The velocity distribution and (b) vertical temperature distribution at Ma = 100 for the basic flow
图 3 Q = 0, Bi = 0时, 不同B下, (a) Mac 随Pr的变化曲线以及(a)所对应的(b)波数和(c)波传播角. 曲线对应:
$a,c,d,e,g,h$ 为斜波,$b,f,i,j$ 为流向波Figure 3. (a) The variation of Mac with Pr under different B at whenQ = 0 and Bi = 0, (b) wave number and (c) wave propagation angle. The curves correspond to:
$a,c,d,e,g,h$ oblique wave and$b,f,i,j$ streamwise wave
图 4 B = 0.2, Ma = 100时垂直方向上的温度分布
Figure 4. Vertical temperature distribution at B = 0.2 and Ma = 100
图 5 Q = 0.05时, (a) Mac 随Pr的变化曲线以及(a)所对应的(b)波数和(c)波传播角. 曲线对应:
$a,d,g$ 为斜波,$b,e$ 为流向波,$c,h,f$ 为展向稳态模态Figure 5. (a) The variation of Mac with Pr under Q = 0.05, (b) wave number and (c) wave propagation angle. The curves correspond to:
$a,d,g$ oblique wave,$b,e$ streamwise wave,$c,h,f$ spanwise stationary mode
图 6 Q = 0, Bi = 0时不同临界模态所对应的扰动流场: 逆向斜波(B = 0.2, Pr = 0.01)的(a)上屈服面扰动, (b)温度和流线; 逆向斜波(B = 0.4, Pr = 20)的(c)上屈服面扰动, (d)温度和流线
Figure 6. The perturbation flow field of the different preferred modes at Q = 0 and Bi = 0: (a) upper yield surface perturbation, (b) temperature and streamlines of the upstream oblique wave (B = 0.2, Pr = 0.01); (c) upper yield surface perturbation, (d) temperature and streamlines of the upstream oblique wave (B = 0.4, Pr = 20)
图 7 Q = 0, Bi = 5时不同临界模态所对应的扰动流场: 逆向斜波(B = 0.4, Pr = 0.01)的(a)上屈服面扰动, (b)温度和流线; 逆向斜波(B = 0.6, Pr = 0.01)的(c)上屈服面扰动, (d) 温度和流线
Figure 7. The perturbation flow field of the different preferred modes at Q = 0 and Bi = 5: (a) upper yield surface perturbation, (b) temperature and streamlines of the upstream oblique wave (B = 0.4, Pr = 0.01); (c) upper yield surface perturbation, (d) temperature and streamlines of the upstream oblique wave (B = 0.6, Pr = 0.01)
图 8 Q = 0.05, Bi = 0时不同临界模态所对应的扰动流场: 同向流向波(B = 0, Pr = 10)的(a)温度和流线; 同向斜波(B = 0.2, Pr = 0.1)的(b)上屈服面扰动, (c)温度和流线
Figure 8. The perturbation flow field of the different preferred modes at Q = 0.05 and Bi = 0: (a) temperature and streamlines of the downstream streamwise wave (B = 0, Pr = 10); (b) upper yield surface perturbation; (c) temperature and streamlines of the downstream oblique wave (B = 0.2, Pr = 0.1)
表 1 不同参数下各扰动能量变化项的值
Table 1. Values of perturbation energy variation terms at different parameters
Q Pr B Bi $ - N $ $ M $ $ I $ 0 0.01 0 0 −0.037253 0.037222 0.000033 0.2 0 −0.020173 0.017116 0.002953 0.4 0 −0.017003 0.014711 0.001957 0.6 0 −0.011005 0.009559 0.001536 1 0.2 5 −0.215173 0.169342 0.045979 0.4 5 −0.259566 0.22143 0.037879 100 0.2 5 −12.368487 12.372851 0.002954 0.4 5 −12.608616 12.611561 0.001634 0.05 0.01 0 0 −0.037481 0.037471 0.000011 0.1 0 0 −0.112611 0.112588 0.000023 0.1 0.2 0 −0.060235 0.046419 0.010708 
下载: 导出CSV
-
[1] Braescu L, Epure S, Duffar T. Crystal Growth Processes Based on Capillarity : Czochralski, Floating Zone, Shaping and Crucible Techniques. Wiley, 2010: 465-524 [2] Smith MK, Davis SH. Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. Journal of Fluid Mechanics, 1983, 132: 119-144 [3] Patne R, Agnon Y, Oron A. Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient. Journal of Fluid Mechanics, 2020, 906(412): A12 [4] Chen EH, Xu F. Transient Marangoni convection induced by an iso-thermal sidewall of a rectangular liquid pool. Journal of Fluid Mechanics, 2021, 928(406): A6 [5] Pettit D. Saturday morning science videos. URLhttp://mix. msfc. nasa. gov/IMAGES/QTVR/0601211. mov, 2003 [6] Watanabe T, Kowata Y, Ueno I. Flow transition and hydrothermal wave instability of thermocapillary-driven flow in a free rectangular liquid film. International Journal of Heat and Mass Transfer, 2018, 116: 635-641 doi: 10.1016/j.ijheatmasstransfer.2017.09.059 [7] Ueno I, Torii T. Thermocapillary-driven flow in a thin liquid film sustained in a rectangular hole with temperature gradient. Acta Astronautica, 2010, 66(7-8): 1017-1021 doi: 10.1016/j.actaastro.2009.09.027 [8] Messmer B, Lemee T, Ikebukuro K, et al. Confined thermo-capillary flows in a double free-surface film with small Marangoni numbers. International Journal of Heat and Mass Transfer, 2014, 78: 1060-1067 [9] Toshiki W, Yosuke K, Ichiro U. Flow transition and hydrothermal wave instability of thermocapillary-driven flow in a free rectangular liquid film. International Journal of Heat and Mass Transfer, 2018, 116: 635-641 [10] Hu KX, Zhao CZ, Zhang SN, et al. Instabilities of thermocapillary liquid layers with two free surfaces. International Journal of Heat and Mass Transfer, 2021, 173(30): 121217 [11] 赵诚卓, 胡开鑫. 双自由面溶质-热毛细液层的不稳定性. 力学学报, 2022, 54(2): 291-300 (Zhao Chengzhuo, Hu Kaixin. Instability in the solutal-thermocapillary liquid layer with two free surface. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 291-300 (in Chinese) doi: 10.6052/0459-1879-21-148 [12] Hu WR, Imaishi N. Thermocapillary flow in an annular liquid layer painted on a moving fiber. International. Journal of Heat & Mass Transfer, 2000, 43(24): 4457-4466 [13] Toussaint G, Bodiguel H, Doumenc F, et al. Experimental characterization of buoyancy- and surface tension-driven convection during the drying of a polymer solution. International Journal of Heat & Mass Transfer, 2008, 51(17-18): 4228-4237 [14] Jonathan P, Thomas EL, Kooi SE. Focused laser-induced Marangoni dewetting for patterning polymer thin films. Journal of Polymer Science, Part B. Polymer Physics, 2017, 55(6): 542-542 doi: 10.1002/polb.24313 [15] Darhuber AA, Davis JM, Troian SM, et al. Thermocapillary actuation of liquid flow on chemically patterned surfaces. Physics of Fluids, 2003, 15(5): 1295-1304 doi: 10.1063/1.1562628 [16] Basaran OA, Gao HJ, Bhat PP, et al. Nonstandard inkjets. Annual Review of Fluid Mechanics, 2013, 45(1): 85-113 [17] Pojman JA, Downey JP. Polymer research in microgravity: polym-erization and processing. American Chemical Society, 2001, 124(14): 3799 [18] Tong MX, Yang LJ, Fu QF. Thermocapillar instability of a two- dimensional viscoelastic planar liquid sheet in surrounding gas. Physics of Fluids, 2014, 26(3): 50 [19] Hernández IJH, Dávalos-Orozco LA. Competition between stationary and oscillatory viscoelastic thermocapillary convection of a film coating a thick wall. International Journal of Thermal Sciences, 2015, 89: 164-173 doi: 10.1016/j.ijthermalsci.2014.11.003 [20] Hu KX, He M, Chen QS. Instability of thermocapillary liquid layers for Oldroyd-B fluid. Physics of Fluids, 2016, 28(3): 93-127 [21] Hu KX, He M, Chen QS, et al. Linear stability of thermocapillary liquid layers of a shear-thinning fluid. Physics of Fluids, 2017, 29(7): 85-112 [22] Ghorbel E. A viscoplastic constitutive model for polymeric materials. International Journal of Plasticity, 2008, 24(11): 2032-2058 doi: 10.1016/j.ijplas.2008.01.003 [23] Balmforth NJ, Burbidge AS, Craster RV, et al. Visco-plastic models of isothermal lava domes. Fluid Mech., 2000, 403: 37-65 [24] 杨飞飞. 基于宾汉流体支撑的凝胶3D打印工艺研究. [博士论文]. 杭州: 浙江大学, 2017: 6Yang Feifei. The study on the process of three-dimensional gel printing based on Bingham fluid support. [PhD Thesis]. Hangzhou: Zhejiang University, 2017: 6 (in Chinese) [25] Nouar C, Bottaro A. Stability of the flow of a Bingham fluid in a channel: eigenvalue sensitivity, minimal defects and scaling laws of transition. Journal of Fluid Mechanics, 2010, 642: 349-372 doi: 10.1017/S0022112009991832 [26] Hu KX, He M, Chen QS, et al. On the stability of thermocapillary convection of a Bingham fluid in an infinite liquid layer. International Journal of Heat and Mass Transfer, 2018, 122: 993-1002 doi: 10.1016/j.ijheatmasstransfer.2018.02.048 [27] 王世芳, 夏坤. Bingham流体在低渗透多孔介质中球向渗流的分形模型. 华中师范大学学报(自然科学报), 2021, 55(4): 554-558 (Wang Shifang, Xia Kun. The fractal model for spherical seepage of Bingham fluid in the porous media with low permeability. Journal of Huazhong Normal University (Natural Sciences) , 2021, 55(4): 554-558 (in Chinese) [28] 韩建国, 毕耀, 阎培渝等. 使用同轴双圆柱流变仪获取Bingham流体流变参数. 硅酸盐学报, 2021, 49(2): 323-330Han Jianguo, Bi Yao, Yan Peiyu, et al. The measurement of Bingham fluid rheological parameters by Couette rheometer. J. Chin. Cheram. Soc., 2021, 49(2): 323-330(in Chinese) ) [29] Baioumy B, Chebbi R, Jabbar NA. Bingham fluid flow in the entrance region of a pipe. Journal of Fluids Engineering, 2021, 143(2): 024503 doi: 10.1115/1.4048610 [30] Fusi L. A finite difference scheme for the unsteady planar motion of a Bingham fluid. Journal of Non-Newtonian Fluid Mechanics, 2022, 299: 104702 doi: 10.1016/j.jnnfm.2021.104702 [31] Maurya A, Tiwari N, Chhabra RP. Onset of flow reversal in a v-ertical t-channel: bingham fluids. Chemical Engineering & Technology, 2022, 45(3): 425-431 [32] Esmaeili E, Grassia P, Torres UC. Squeeze film flow of viscoplastic Bingham fluid between non-parallel plates. Journal of Non-Newtonian Fluid Mechanics, 2022, 305: 104817 doi: 10.1016/j.jnnfm.2022.104817 [33] Zhang QT, Liu WC, Dahi TA. Numerical study on non-Newtonian Bingham fluid flow in development of heavy oil reservoirs using radiofrequency heating method. Energy, 2022, 239: 122385 [34] Dou R, Tang WZ, Hu KX, et al. Ceramic paste for space stereolithography 3D printing technology in microgravity environment. Journal of the European Ceramic Society, 2022, 42(9): 3968-3975 doi: 10.1016/j.jeurceramsoc.2022.03.030 -
下载:
点击查看大图
计量
- 文章访问数: 370
- HTML全文浏览量: 113
- PDF下载量: 52
- 被引次数: 0
