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本文研究了二维黏性流体薄膜沿非平整不均匀加热基底流动时非线性表面波的演化及其流动稳定性. 利用长波摄动法推导出非平整线性加热基底上非线性表面波的零阶和一阶演化方程，基于所得演化方程，绘制出正弦波纹基底上液膜的表面波形图，并研究液膜流动的线性稳定性，分析了各无量纲参数对液膜线性稳定性的影响. 分析结果表明：在正弦波纹基底上，液膜自由表面随同壁面作相同频率的正弦型波动，且液膜厚度沿流动方向逐渐变小；Marangoni数为稳定影响因素，随Marangoni数的增大，液膜稳定区域增大；Peclet数和倾角均为不稳定影响因素，随Peclet数和倾角的增大，液膜稳定区域减小；在非平整基底的波峰和波谷处，Peclet数、Marangoni数和倾角对稳定性的影响趋势一致，但基底波谷处的液膜稳定区域小于波峰处区域，流动更易失稳.This paper studies mainly the evolution and linear stability of the nonlinear surface waves of a two-dimensional viscous liquid film along an uneven inclined non-uniformly heated wall. A long wave perturbation method is used to derive zero- and first-order evolutions equations of the nonlinear surface wave flowing on an uneven substrate. Based on the obtained evolution equations, the diagram of evolution progress for film surface wave on a sinusoidal corrugated substrate is drawn, the linear stability analysis is also studied, and the effect of various parameters on the flow stability of liquid membrane is analysed. Theoretical results demonstrate that the free surface of the film shows sine wave and has the same frequency as the substrate, and the film thickness will decrease gradually along the flow direction. Marangoni number gives a stabilizing effect, the stable zone increases with the increase of Marangoni number. While, Peclet number and the angle theta are unstable factors, the stable region decreases with the increase of them. Besides, the trends that Marangoni number, Peclet number and angle theta may impact on the stability of the film whice are consistent with one another, so a liquid film is easy to destabilize at the trough of the substrate.
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Keywords:
- uneven substrate /
- heated liquid film /
- nonlinear surface wave /
- linear stability
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[31] Ye X M, Li C X, Yan W P 2011 Acta Mech. Sin. 43 461 (in Chinese) [叶学民, 李春曦, 阎维平 2011 力学学报 43 461]
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[1] Aziz R C, Hashim I 2010 Chin. Phys. Lett. 27 110202
[2] [3] Guo J H, Dai S Q, Dai Q 2010 Acta Phys. Sin. 59 2601 (in Chinese) [郭加宏, 戴世强, 代钦 2010 59 2601]
[4] [5] Zhang X J, Huang Y, Guo Y B, Tian Y, Meng Y G 2013 Chin. Phys. B 22 016202
[6] [7] [8] Wierschem A, Aksel N 2003 Physica D 186 221
[9] Tatiana G R, Yu H Y, Karsten L 2011 Heat Transfer Eng. 32 705
[10] [11] [12] Matar O K, Craster R V, Kumar S 2007 Phys. Rev. E 76 056301
[13] [14] Sisoev G M, Matar O K, Craster R V 2010 Chem. Eng. Sci. 65 950
[15] Li C X, Pei J J, Ye X M 2013 Acta Phys. Sin. 21 214704 (in Chinese) [李春曦, 裴建军, 叶学民 2013 21 214704]
[16] [17] Laohalertdecha S, Wongwises S 2010 Int. J Heat Mass Tran. 53 2924
[18] [19] Li Z, Hu G H, Zhou J J, Zhou Z W 2011 Acta Mech. Sinica. 43 699 (in Chinese) [李振, 胡国辉, 周继杰, 周哲玮 2011 力学学报 43 699]
[20] [21] [22] Zhang F, Zhao X G, Geng J 2006 J. Nanjin Univ. Tech. 28 93 (in Chinese) [张锋, 赵贤广, 耿皎 2006 南京工业大学学报 28 93]
[23] Shi J S, Zhang Q Z 2010 Chin. J. Appl. Mech. 27 166 (in Chinese) [师晋生, 张巧珍 2010 应用力学学报 27 166]
[24] [25] Samanta A 2008 Physica D 237 2587
[26] [27] Sadiq M R, Usha R, Joo S W 2010 Chem. Eng. Sci. 65 4443
[28] [29] [30] Mukhopadhyay A 2011 Acta Mech. 216 225
[31] Ye X M, Li C X, Yan W P 2011 Acta Mech. Sin. 43 461 (in Chinese) [叶学民, 李春曦, 阎维平 2011 力学学报 43 461]
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