Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T22:34:32.177Z Has data issue: false hasContentIssue false
  • English
  • Français

Effect of surfactants on the long-wave stability of two-layer oscillatory film flow

Published online by Cambridge University Press:  05 October 2021

Cheng-Cheng Wang ,
Haibo Huang Open the ORCID record for Haibo Huang [Opens in a new window] ,
Peng Gao Open the ORCID record for Peng Gao [Opens in a new window]  and
Xi-Yun Lu Open the ORCID record for Xi-Yun Lu [Opens in a new window]
Cheng-Cheng Wang
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Haibo Huang*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Peng Gao
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Xi-Yun Lu
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
*
†Email address for correspondence: huanghb@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of the two-layer film flow driven by an oscillatory plate under long-wave disturbances is studied. The influence of key factors, such as thickness ratio ($n$), viscosity ratio ($m$), density ratio ($r$), oscillatory frequency ($\beta$) and insoluble surfactants on the stability behaviours is studied systematically. Four special Floquet patterns are identified, and the corresponding growth rates are obtained by solving the eigenvalue problem of the fourth-order matrix. A small viscosity ratio ($m\le 1$) may stabilize the flow but it depends on the thickness ratio. If the viscosity ratio is not very small ($m>0.1$), in the $(\beta ,n)$-plane, stable and unstable curved stripes appear alternately. In other words, under the circumstances, if the two-layer film flow is unstable, slightly adjusting the thickness of the upper film may make it stable. In particular, if the upper film is thin enough, even under high-frequency oscillation, the flow is always stable. The influence of density ratio is similar, i.e. there are curved stable and unstable stripes in the $(\beta ,r)$-planes. Surface surfactants generally stabilize the flow of the two-layer oscillatory membrane, while interfacial surfactants may stabilize or destabilize the flow but the effect is mild. It is also found that gravity can generally stabilize the flow because it narrows the bandwidth of unstable frequencies.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Instability: Instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 928 , 10 December 2021 , A19
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Blyth, M.G. & Pozrikidis, C. 2004 a Effect of surfactant on the stability of film flow down an inclined plane. J. Fluid Mech. 521, 241–250.CrossRefGoogle Scholar
Blyth, M.G. & Pozrikidis, C. 2004 b Effect of surfactants on the stability of two-layer channel flow. J. Fluid Mech. 505, 59–86.CrossRefGoogle Scholar
Frenkel, A.L. & Halpern, D. 2002 Stokes-flow instability due to interfacial surfactant. Phys. Fluids 14 (7), L45–L48.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2006 Effect of surfactants on the long-wave stability of oscillatory film flow. J. Fluid Mech. 562, 345–354.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2007 Effect of surfactants on the inertialess instability of a two-layer film flow. J. Fluid Mech. 591, 495–507.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2008 a Mechanism of the long-wave inertialess instability of a two-layer film flow. J. Fluid Mech. 608, 379–391.CrossRefGoogle Scholar
Gao, P. & Lu, X.-Y. 2008 b Instability of an oscillatory fluid layer with insoluble surfactants. J. Fluid Mech. 595, 461–490.CrossRefGoogle Scholar
Halpern, D. & Frenkel, A.L. 2003 Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers. J. Fluid Mech. 485, 191–220.CrossRefGoogle Scholar
Kumar, K. & Tuckerman, L.S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.CrossRefGoogle Scholar
Lin, S.P. 1970 Stabilizing effects of surface-active agents on a film flow. AIChE J. 16 (3), 375–379.CrossRefGoogle Scholar
Mohammadi, A. & Smits, A.J. 2017 Linear stability of two-layer Couette flows. J. Fluid Mech. 826, 128–157.CrossRefGoogle Scholar
Or, A.C. 1997 Finite-wavelength instability in a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 335, 213–232.CrossRefGoogle Scholar
Pozrikidis, C. 2003 Effect of surfactants on film flow down a periodic wall. J. Fluid Mech. 496, 105–127.CrossRefGoogle Scholar
Samanta, A. 2013 Effect of surfactant on two-layer channel flow. J. Fluid Mech. 735, 519–552.CrossRefGoogle Scholar
Samanta, A. 2017 Linear stability of a viscoelastic liquid flow on an oscillating plane. J. Fluid Mech. 822, 170–185.CrossRefGoogle Scholar
Stone, H.A. 1990 A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2 (1), 111–112.CrossRefGoogle Scholar
Wei, H.-H. 2005 a Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane. Phys. Fluids 17 (1), 012103.CrossRefGoogle Scholar
Wei, H.-H. 2005 b On the flow-induced Marangoni instability due to the presence of surfactant. J. Fluid Mech. 544, 173–200.CrossRefGoogle Scholar
Wei, H.-H., Halpern, D. & Grotberg, J.B. 2005 Linear stability of a surfactant-laden annular film in a time-periodic pressure-driven flow through a capillary. J. Colloid Interface Sci. 285 (2), 769–780.CrossRefGoogle Scholar
Whitaker, S. & Jones, L.O. 1966 Stability of falling liquid films. Effect of interface and interfacial mass transport. AIChE J. 12 (3), 421–431.CrossRefGoogle Scholar
Yih, C.-S. 1968 Instability of unsteady flows or configurations. Part 1. Instability of a horizontal liquid layer on an oscillating plane. J. Fluid Mech. 31 (4), 737–751.CrossRefGoogle Scholar