Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T09:08:48.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Effect of surfactant on two-layer channel flow

Published online by Cambridge University Press:  25 October 2013

Arghya Samanta
Arghya Samanta*
Affiliation:
Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of insoluble surfactant on the interfacial waves in connection with a two-layer channel flow is investigated for low to moderate values of the Reynolds number. Previous studies focusing on Stokes flow (Frenkel & Halpern, Phys. Fluids, vol. 14, 2002, p. L45; Halpern & Frenkel, J. Fluid Mech., vol. 485, 2003, pp. 191–220) are extended by including the inertial effect and the study of low-Reynolds-number flow (Blyth & Pozrikidis, J. Fluid Mech., vol. 521, 2004b, pp. 241–250) is enlarged up to moderate Reynolds number. Linear stability analysis based on the Orr–Sommerfeld boundary value problem identifies a surfactant mode together with an interface mode. The presence of surfactant on the interfacial mode is stabilizing at high viscosity ratio and destabilizing at low viscosity ratio. The threshold of instability is determined as a function of the Marangoni number. A long-wave model is developed to predict the families of nonlinear waves in the neighbourhood of the threshold of instability. Far from the threshold, wave dynamics is explored under the framework of a three-equation model in terms of lower layer flow rate ${q}_{2} (x, t)$, lower liquid-layer thickness $h(x, t)$ and surfactant concentration $\Gamma (x, t)$. Primary instability analysis of a three-equation model captures the result of the Orr–Sommerfeld boundary value problem very well for quite large values of wavenumber. In the nonlinear regime, travelling wave solutions demonstrate deceleration of maximum amplitude and acceleration of speed with the Marangoni number at high viscosity ratio $m\gt 1$ and show completely the opposite behaviour at low viscosity ratio $m\lt 1$. However, both maximum amplitude and speed attain a fixed value with increasing Reynolds number and this leads to saturation of instability.

JFM classification

Waves/Free-surface Flows: Channel flow Instability: Nonlinear instability Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 735 , 25 November 2013 , pp. 519 - 552
Copyright
©2013 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

Alba, K., Khayat, R. E. & Pandher, S. 2008 Steady two-layer gravity-driven thin film flow. Phys. Rev. E 77, 056304.Google Scholar
Alba, K., Laure, P. & Khayat, R. E. 2011 Transient two-layer thin film flow inside a channel. Phys. Rev. E 84, 026320.Google Scholar
Amaouche, M., Mehidi, N. & Amatousse, N. 2007 Linear stability of a two-layer film flow down an inclined channel: a second-order weighted residual approach. Phys. Fluids 19, 084106.Google Scholar
Bassom, A. P., Blyth, M. G. & Papageorgiou, D. T. 2010 Nonlinear development of two-layer Coutte–Poiseuille flow in the presence of surfactant. Phys. Fluids 22, 102102.Google Scholar
Blyth, M. G. & Pozrikidis, C. 2004a Effect of inertia on the Marangoni instability of two-layer channel flow, part II: normal-mode analysis. J. Engng Maths 50, 329–341.Google Scholar
Blyth, M. G. & Pozrikidis, C. 2004b Effect of surfactant on the stability of film flow down an inclined plane. J. Fluid Mech. 521, 241–250.Google Scholar
Boomkamp, P. A. M. & Miesen, R. H. M. 1996 Classification of instabilities in parallel two-phase flow. Intl J. Multiphase Flow 22, 67–88.Google Scholar
Cao, Q., Sarkar, K. & Prasad, A. K. 2004 Direct numerical simulations of two-layer viscosity-stratified flow. Intl J. Multiphase Flow 30, 1485–1508.CrossRefGoogle Scholar
Charru, F. & Fabre, J. 1994 Long waves at the interface between two viscous fluids. Phys. Fluids 6, 1223–1235.Google Scholar
Colistra, J. 2006 On the interfacial dynamics of multi-liquid channel flow–flow regime characterization and the effect of gravity. Master’s thesis, Lehigh University.Google Scholar
Dietze, G. F. & Ruyer-Quil, C. 2013 Wavy liquid films in interaction with a confined laminar gas flow. J. Fluid Mech. 722, 348–393.Google Scholar
Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B. & Wang, X.-J. 2007 Auto07: continuation and bifurcation software for ordinary differential equations. Tech. Rep.. Department of Computer Science, Concordia University, Montreal, Canada (available by FTP from ftp.cs.concordia.ca in directory pub/doedel/auto).Google Scholar
Frank, A. M. 2006 Shear driven solitary waves on a liquid film. Phys. Rev. E 74, 065301.Google Scholar
Frank, A. M. 2008 Numerical simulation of gas driven waves in a liquid film. Phys. Fluids 20, 122102.Google Scholar
Frenkel, A. L. & Halpern, D. 2002 Stokes-flow instability due to interfacial surfactant. Phys. Fluids 14, L45.Google Scholar
Frenkel, A. L. & Halpern, D. 2005 Effect of inertia on the insoluble-surfactant instability of shear flow. Phys. Rev. E 71, 016302.Google Scholar
Frenkel, A. L. & Halpern, D. 2006 Strongly nonlinear nature of interfacial surfactant instability of Couette flow. Intl J. Pure Appl. Maths 29, 205–223.Google Scholar
Gao, P. & Lu, X.-Y. 2007 Effect of surfactats on the inertialess instability of a two-layer film flow. J. Fluid Mech. 591, 495–507.Google Scholar
Gaster, M. 1962 A note on the relation between temporally increasing and spatially increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14, 222–224.Google 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.Google Scholar
Halpern, D. & Frenkel, A. L. 2008 Nonlinear evolution, travelling waves, and secondary instability of shear-film flows with insoluble surfactants. J. Fluid Mech. 594, 125–156.Google Scholar
Halpern, D. & Grotberg, J. B. 1993 Surfactant effects on fluid elastic instabilities of an oscillating Couette film flow. Trans. ASME: J. Biomech. Engng 115, 271–277.Google Scholar
Hooper, A. P. & Grimshaw, R. 1985 Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28, 37–45.Google Scholar
Joseph, D. D. & Renardy, Y. 1993 Fundamentals of Two-Fluid Dynamics. Mathematical theory and Applications, vol. I, Springer.Google Scholar
Kabov, O. A., Zaitsev, D. V., Cheverda, V. V. & Bar-Cohen, A. 2011 Evaporation and flow dynamics of thin shear-driven liquid films in microgap channels. Exp. Therm. Fluid Sci. 35, 825–831.Google Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films Applied Mathematical Science, 176th edn. Springer.Google Scholar
Kao, T. W. & Park, C. 1972 Experimental investigations of the stability of chanel flows. J. Fluid Mech. 52, 401–423.Google Scholar
Kelley, R. E., Goussis, D. A., Lin, S. P. & Hsu, F. K. 1989 The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A 1, 819–828.Google Scholar
Khomami, B. & Su, K. C. 2000 An experimental/theoretical investigation of interfacial instabilities in superposed pressure-driven channel flow of Newtonian and well characterized viscoelastic fluids part I: linear stability and encapsulation effects. J. Non-Newtonian Fluid Mech. 91, 59–84.Google Scholar
Li, J. & Renardy, Y. 2000 Numerical study of flows of two immiscible liquids at low Reynolds number. SIAM Rev. 42, 417–439.Google Scholar
Liu, J. & Gollub, J. P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.Google Scholar
Malamataris, N. A., Vlachogiannis, M. & Bontozoglou, V. 2002 Solitary waves on inclined films: flow structure and binary interactions. Phys. Fluids 14, 1082–1094.Google Scholar
Mavromoustaki, A., Matar, O. K. & Craster, R. V. 2010 Shock-wave solutions in two-layer channel flow. I. One-dimensional flows. Phys. Fluids 22, 112102.Google Scholar
Mavromoustaki, A., Matar, O. K. & Craster, R. V. 2011 Shock-wave solutions in two-layer channel flow. II. Linear and nonlinear stability. Phys. Fluids 23, 112101.Google Scholar
Pinarbasi, A. & Liakopoulos, A. 1995 The effect of variable viscosity on the interfacial stability of two-layer Poiseuille flow. Phys. Fluids 7, 1318–1324.Google Scholar
Pozrikidis, C. 2004 Instability of multi-layer channel and film flows. Adv. Appl. Maths 40, 179–239.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modelling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.Google Scholar
Samanta, A. 2009 Effect of electric field on the stability of an oscillatory contaminated film flow. Phys. Fluids 21, 114101.Google Scholar
Samanta, A., Goyeau, B. & Ruyer-Quil, C. R. 2013 A falling film on a porous medium. J. Fluid Mech. 716, 414–444.Google Scholar
Samanta, A., Ruyer-Quil, C. R. & Goyeau, B. 2011 Falling film down a slippery inclined plane. J. Fluid Mech. 684, 353–383.Google Scholar
Segin, T. M., Tilley, B. S. & Kondic, L. 2005 On undercompressive shocks and flooding in countercurrent two-layer flows. J. Fluid Mech. 532, 217–242.Google Scholar
Severtson, Y. C. & Aidun, C. K. 1996 Stability of two-layer stratified flow in inclined channels: applications to air entrainment in coating systems. J. Fluid Mech. 312, 173–200.Google Scholar
Shkadov, V. Y. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Izv. Akad. Nauk SSSR, Mekh. Zhidk Gaza 1, 43–51 (translation in Fluid Dyn. 2, 29–34, 1970 Faraday Press).Google Scholar
Sisoev, G. M. & Bennett, C. J. 2013 Linear growth in two-fluid plane Poiseuille flow. Fluid Dyn. Res. 45, 015503.Google Scholar
Sisoev, G. M., Matar, O. K., Sileri, D. & Lawrence, C. J. 2009 Wave regimes in two-layer microchannel flow. Chem. Engng Sci. 64, 3094–3102.Google Scholar
Slattery, J. C. 1974 Interfacial effects in the entrapment and displacement of residual oil. AIChE J. 20, 1145–1154.Google Scholar
Smith, M. K. 1990 The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485.Google Scholar
South, M. J. & Hooper, A. P. 1999 Linear growth in two-fluid plane Poiseuille flow. J. Fluid Mech. 381, 121–139.Google Scholar
Tilley, B. S., Davis, S. H. & Bankoff, S. G. 1994a Linear stability theory of two-layer fluid flow in an inclined channel. Phys. Fluids 6, 3906–3922.Google Scholar
Tilley, B. S., Davis, S. H. & Bankoff, S. G. 1994b Nonlinear long-wave stability of superposed fluids in an inclined channel. J. Fluid Mech. 277, 55–83.Google Scholar
Vempati, B., Oztekin, A. & Neti, S. 2010 Stability of two-layered fluid flows in an inclined channel. Acta Mech. 209, 187–199.Google Scholar
Wei, H. H. 2005 Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane. Phys. Fluids 17, 012103.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 1987 Analysis of superposed fluids by the finite element method: linear stability and flow development. Intl J. Numer. Meth. Fluids 7, 247–261.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 1988 Linear stability of plane Poiseuille flow of two superposed fluids. Phys. Fluids 31, 3225–3238.Google Scholar
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.Google Scholar
Zhang, J., Miksis, M. J., Bankoff, S. G. & Tryggvason, G. 2002 Nonlinear dynamics of an interface in an inclined channel. Phys. Fluids 14, 1877–1885.Google Scholar