Spatio-Temporal Instability of Two-Layer Liquid Film at Small Reynolds Numbers

WANG Zhi-liang; S. P. Lin; ZHOU Zhe-wei

doi:10.3879/j.issn.1000-0887.2010.01.001

Volume 31 Issue 1

Turn off MathJax

Article Contents

Abstract

References

Article Navigation > Applied Mathematics and Mechanics > 2010 > 31(1): 1-11

WANG Zhi-liang, S. P. Lin, ZHOU Zhe-wei. Spatio-Temporal Instability of Two-Layer Liquid Film at Small Reynolds Numbers[J]. Applied Mathematics and Mechanics, 2010, 31(1): 1-11. doi: 10.3879/j.issn.1000-0887.2010.01.001

Citation:

PDF( 520 KB)

Spatio-Temporal Instability of Two-Layer Liquid Film at Small Reynolds Numbers

doi: 10.3879/j.issn.1000-0887.2010.01.001

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China;
Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699, USA

Received Date: 2009-10-20
Rev Recd Date: 2009-11-19
Publish Date: 2010-01-15

Abstract

Abstract

The on set of the instability with respect to spatio-temporally growing disturbances in a viscosity-stratified two-layer liquid fiml flow was analyzed. The known results obtained from the temporal theory of instability showed that the flow was un stable in the lmiit of zero Reynolds numbers. The present theory predicted the neutral stability in the same limit. The discrepancy was explained. And based on the mechanical energy equation, a new mechanism of instability was found. The new mechanism was associated with the convective nature of the disturbance which was not Galilei invariant.
- flow instability,
- fiml coating,
- energy budget,
- low Reynolds number,
- Galilei invariant

FullText(HTML)

References(23)

References

[1]	Yih C S. Instability due to viscosity stratification[J]. J Fluid Mech, 1967, 27(2): 337-352. doi: 10.1017/S0022112067000357
[2]	Renardy Y. Instability at the interface between two shearing fluids in a channel[J]. Phys Fluids, 1985, 28(12): 3441-3443. doi: 10.1063/1.865346
[3]	Renardy Y. The thin-layer effect and interfacial stability in a two-layer Couette flow with similar liquids[J]. Phys Fluids, 1987, 30(6): 1627-1637. doi: 10.1063/1.866227
[4]	Chen K P. Interfacial instabilities in stratified shear flows involving multiple viscous and viscoelastic fluids[J]. Appl Mech Rev, 1995, 48(11): 763-776. doi: 10.1115/1.3005092
[5]	Tilley B S, Davis S H, Bankoff S G. Linear stability theory of two-layer fluid flow in an inclined channel[J]. Phys Fluids, 1994, 6(12): 3906-3922. doi: 10.1063/1.868382
[6]	Tilley B S, Davis S H, Bankoff S G. Nonlinear long-wave stability of superposed fluids in an inclined channel[J]. J Fluid Mech,1994, 277: 55-83. doi: 10.1017/S0022112094002685
[7]	Hooper A P. Long-wave instability at the interface between two viscous fluids: thin layer effects[J]. Phys Fluids, 1985, 28(6): 1613-1618. doi: 10.1063/1.864952
[8]	Hooper A P, Boyd W G C. Shear-flow instability at the interface between two viscous fluids[J]. J Fluid Mech, 1983, 128: 507-528. doi: 10.1017/S0022112083000580
[9]	Hooper A P, Grimshaw R. Nonlinear instability at the interface between two viscous fluids[J]. Phys Fluids, 1985, 28(1): 37-45. doi: 10.1063/1.865160
[10]	Kao T W. Role of viscosity stratification in the stability of two-layer flow down an incline[J]. J Fluid Mech, 1968, 33: 561-572. doi: 10.1017/S0022112068001515
[11]	Loewenherz D S, Lawrence C J. The effect of viscosity stratification on the stability of a free surface flow at low Reynolds number[J]. Phys Fluids A, 1989, 1(10): 1686-1693. doi: 10.1063/1.857533
[12]	Chen K P. Wave formation in the gravity-driven low-Reynolds number flow of two liquid films down an inclined plane[J]. Phys Fluids A, 1993, 5(12): 3038-3048.
[13]	Wang C K, Seaborg J J, Lin S P. Instability of multi-layered liquid films[J]. Phys Fluids,1978, 21(10): 1669-1673. doi: 10.1063/1.862106
[14]	Jiang W Y, Helenbrook B, Lin S P. Inertialess instability of a two-layer liquid film flow[J]. Phys Fluids,2004, 16(3): 652-663. doi: 10.1063/1.1642657
[15]	Weinstein S J, Kurz M R. Long-wavelength instabilities in three-layer flow down an incline[J]. Phys Fluids A, 1991, 3(11): 2680-2687. doi: 10.1063/1.858158
[16]	Weinstein S J, Chen K P. Large growth rate instabilities in three-layer flow down an incline in the limit of zero Reynolds number[J]. Phys Fluids, 1999, 11(11): 3270-3282. doi: 10.1063/1.870187
[17]	Kliakhandler I L, Sivashinsky G I. Viscous damping and instabilities in stratified liquid film flowing down a slightly inclined plane[J]. Phys Fluids, 1997, 9(1): 23-30. doi: 10.1063/1.869165
[18]	Pozrikidis C. Gravity-driven creeping flow of two adjacent layers through a channel and down a plane wall[J]. J Fluid Mech,1998, 371: 345-376. doi: 10.1017/S0022112098002213
[19]	Kliakhandler I L. Long interfacial waves in multilayer thin films and coupled Kuramoto-Sivashinsky equations[J]. J Fluid Mech,1999, 391: 45-65. doi: 10.1017/S0022112099005297
[20]	Jiang W Y, Helenbrook B, Lin S P. Low-Reynolds-number instabilities in three-layer flow down an inclined wall[J]. J Fluid Mech,2005, 539: 387-416. doi: 10.1017/S0022112005005781
[21]	Brooke Benjamin T. The development of three-dimensional disturbances in an unstable film of liquid flowing down an inclined plane[J]. J Fluid Mech, 1961, 10(3): 401-419. doi: 10.1017/S0022112061001001
[22]	Yih C S. Stability of two-dimensional parallel flows for three-dimensional disturbances[J]. Quart Appl Math,1955, 12: 434-435.
[23]	Squire H B. On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls[J]. Proc R Soc Lond A,1933, 142(847): 621-628. doi: 10.1098/rspa.1933.0193

Relative Articles

[1]	ZHAI Guoqing, CHEN Qiaoyu, TONG Dongbing, ZHOU Wuneng. Fixed-Time Asymptotic Stability and Energy Consumption Estimation of Nonlinear Systems[J]. Applied Mathematics and Mechanics, 2023, 44(10): 1180-1186. doi: 10.21656/1000-0887.440041
[2]	KONG Wei, LI Jia. Intermittent Turbulence Characteristics in the Stokes Layer for a Transitional Reynolds Number[J]. Applied Mathematics and Mechanics, 2020, 41(10): 1171-1182. doi: 10.21656/1000-0887.400382
[3]	HUANG Wei-li, CAI Jian-le. Conformal Invariance for the Nonholonomic System of Chetaev’s Type With Variable Mass[J]. Applied Mathematics and Mechanics, 2012, 33(11): 1294-1303. doi: 10.3879/j.issn.1000-0887.2012.11.005
[4]	ZHAO Lu-hai-bo, HU Guo-hui, ZHOU Zhe-wei. Linear Instability of Ultra-Thin Liquid Films Flowing Down a Cylindrical Fibre[J]. Applied Mathematics and Mechanics, 2011, 32(2): 127-134. doi: 10.3879/j.issn.1000-0887.2011.02.001
[5]	WANG Zhi-liang, S. P. Lin, ZHOU Zhe-wei. Formation of Radially Expanding Liquid Sheet by Impinging Two Round Jets[J]. Applied Mathematics and Mechanics, 2010, 31(8): 891-900. doi: 10.3879/j.issn.1000-0887.2010.08.001
[6]	TANG Qiong, CHEN Chuan-miao, LIU Luo-hua. Space-Time Finite Element Method for the Schrodinger Equation and Its Conservation[J]. Applied Mathematics and Mechanics, 2006, 27(3): 300-304.
[7]	WANG Yan-xia, HU Guo-hui, ZHOU Zhe-wei. Floquet Stability Analysis of Two-Layer Flows With Periodic Fluctuation in a Vertical Pipe[J]. Applied Mathematics and Mechanics, 2006, 27(8): 883-890.
[8]	FANG Jian-hui, CHEN Pei-sheng, ZHANG Jun. Form Invariance and Lie Symmetry of Variable Mass Nonholonomic Mechanical System[J]. Applied Mathematics and Mechanics, 2005, 26(2): 187-192.
[9]	YOU Zhen-jiang, LIN Jian-zhong. Effects of Tensor Closure Models and 3-D Orientation on the Stability of Fiber Suspensions in a Channel Flow[J]. Applied Mathematics and Mechanics, 2005, 26(3): 281-286.
[10]	LUO Shao-kai. Form Invariance and Noether Symmetrical Conserved Quantity of Relativistic Birkhoffian Systems[J]. Applied Mathematics and Mechanics, 2003, 24(4): 414-422.
[11]	LIN Jian-zhong, YOU Zhen-jiang. Stability Analysis in Spatial Mode for Channel Flow of Fiber Suspensions[J]. Applied Mathematics and Mechanics, 2003, 24(8): 771-778.
[12]	CHEN Xiang-wei, LUO Shao-kai, MEI Feng-xiang. A Form Invariance of Constrained Birkhoffian System[J]. Applied Mathematics and Mechanics, 2002, 23(1): 47-51.
[13]	Qiang Shizhong, Wang Xiaoguo, Tang Maolin, Liu Min. On the Arbitrary Difference Precise Integration Method and Its Numerical Stability[J]. Applied Mathematics and Mechanics, 1999, 20(3): 256-262.
[14]	Liu Jie. nvariant Manifolds and Their Stability in a Three-Dimensional Measure-preserving Mapping System[J]. Applied Mathematics and Mechanics, 1995, 16(10): 879-888.
[15]	Tian Hong-jiong, Kuang Jiao-xun. Numerical Stahility Analysis of Numerical Nethods for Volterra integral Equations with Delay Argument[J]. Applied Mathematics and Mechanics, 1995, 16(5): 451-457.
[16]	Liang Tian-lin, Zhou Ling-yun. Principle of Equal Effects of General Relativity and Invariance of Classical Mechanics Equations[J]. Applied Mathematics and Mechanics, 1993, 14(4): 343-347.
[17]	Zhou Heng. The Re-examination of the Weakly Nonlinear Theory of Hydrodynamic Stability[J]. Applied Mathematics and Mechanics, 1991, 12(3): 203-208.
[18]	Wang Fa-min, Ng. Tiak Wan. Asymptotic Analysis of Stability Problem of Plane Poiseuille Flow for High Reynolds Number[J]. Applied Mathematics and Mechanics, 1988, 9(4): 317-326.
[19]	Xu Ye-yi. First Integral and Stability of Motion in the Critical Cases[J]. Applied Mathematics and Mechanics, 1985, 6(5): 447-454.
[20]	Wu Wang-yi, Richard Skalak. The Stokes Flow from Half-Space to Semi-Infinite Circular Cylinder[J]. Applied Mathematics and Mechanics, 1985, 6(1): 9-23.