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

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