Hostname: page-component-7d684dbfc8-7nm9g Total loading time: 0 Render date: 2023-09-22T11:22:08.125Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Nonlinear evolution, travelling waves, and secondary instability of sheared-film flows with insoluble surfactants

Published online by Cambridge University Press:  14 December 2007

Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA


The nonlinear development of the interfacial-surfactant instability is studied for the semi-infinite plane Couette film flow. Disturbances whose spatial period is close to the marginal wavelength of the long-wave instability are considered first. Appropriate weakly nonlinear partial differential equations (PDEs) which couple the disturbances of the film thickness and the surfactant concentration are obtained from the strongly nonlinear lubrication-approximation PDEs. In a rescaled form each of the two systems of PDEs is controlled by a single parameter C, the ‘shear-Marangoni number’. From the weakly nonlinear PDEs, a single Stuart–Landau ordinary differential equation (ODE) for an amplitude describing the unstable fundamental mode is derived. By comparing the solutions of the Stuart–Landau equation with numerical simulations of the underlying weakly and strongly nonlinear PDEs, it is verified that the Stuart–Landau equation closely approximates the small-amplitude saturation to travelling waves, and that the error of the approximation converges to zero at the marginal stability curve. In contrast to all previous stability work on flows that combine interfacial shear and surfactant, some analytical nonlinear results are obtained. The Hopf bifurcation to travelling waves is supercritical for C < Cs and subcritical for C > Cs, where Cs is approximately 0.29. This is confirmed with a numerical continuation and bifurcation technique for ODEs. For the subcritical cases, there are two values of equilibrium amplitude for a range of C near Cs, but the travelling wave with the smaller amplitude is unstable as a periodic orbit of the associated dynamical system (whose independent variable is the spatial coordinate). By using the Bloch (‘Floquet’) disturbance modes in the linearized PDEs, it transpires that all the small-amplitude travelling-wave equilibria are unstable to sufficiently long-wave disturbances. This theoretical result is confirmed by numerical simulations which invariably show the large-amplitude saturation of the disturbances. In view of this secondary instability, the existence of small-amplitude periodic solutions (on the real line) bifurcating from the uniform flow at the marginal values of the shear-Marangoni number does not contradict the earlier conclusions that the interfacial-surfactant instability has a strongly nonlinear character, in the sense that there are no small-amplitude attractors such that the entire evolution towards them is captured by weakly-nonlinear equations. This suggests that, in general, for flowing-film instabilities that have zero wavenumber at criticality, the saturated disturbance amplitudes do not always have to decrease to zero as the control parameter approaches its value at criticality.

Copyright © Cambridge University Press 2008

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.)



Babchin, A. J., Frenkel, A. L., Levich, B. G. & Sivashinsky, G. I. 1983 Flow-induced nonlinear effects in thin liquid film stability. Ann. NY Acad. Sci. 404, 426428.CrossRefGoogle Scholar
Barthelet, P., Charru, F. & Fabre, J. 1995 Experimental study of interfacial long waves in a 2-layer shear-flow. J. Fluid Mech. 303, 2353.CrossRefGoogle Scholar
Blennerhassett, P. J. 1980 On the generation of waves by wind. Proc. R. Soc. Lond. A 298, 451494.Google Scholar
Blyth, M. G. & Pozrikidis, C. 2004 a Effect of inertia on the Marangoni instability of two-layer channel flow, Part II: normal-mode analysis. J. Engng Maths 50, 329341.CrossRefGoogle Scholar
Blyth, M. G. & Pozrikidis, C. 2004 b Effect of surfactants on the stability of two-layer channel flow. J. Fluid Mech. 505, 5986.CrossRefGoogle Scholar
Chang, H. C., Demekhin, E. A. & Kopelevich, D. I. 1993 Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433480.CrossRefGoogle Scholar
Chen, K. & Joseph, D. D. 1991 Lubricated pipelining: stability of core-annular flow. Part 4. Ginzburg–Landau equations. J. Fluid Mech. 227, 587615.CrossRefGoogle Scholar
Cheng, M. & Chang, H. C. 1990 A generalized side-band stability theory via center manifold projection. Phys. Fluids A 2, 13641379.CrossRefGoogle Scholar
Doedel, E. J., Champneys, A. R., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B. & Wang, X. 2006 Auto-07P: Continuation and Bifurcation Software for Ordinary Differential Equations.Google Scholar
Dong, L. & Johnson, D. 2005 Experimental and theoretical study of the interfacial instability between two shear fluids in a channel Couette flow. Int. J. Heat Fluid Flow 26, 133140.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Frenkel, A. L. 1991 Stability of an oscillating Kolmogorov flow. Phys. Fluids A 3, 17181729.CrossRefGoogle Scholar
Frenkel, A. L., Babchin, A. J., Levich, B. G., Shlang, T. & Sivashinsky, G. I. 1987 Annular flow can keep unstable flow from breakup: nonlinear saturation of capillary instability. J. Colloid Interface Sci. 115, 225233.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2002 Stokes-flow instability due to interfacial surfactant. Phys. Fluids 14, L45L48.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2005 Effect of inertia on the insoluble-surfactant instability of a shear flow. Phys. Rev. E 71, 016302.Google ScholarPubMed
Frenkel, A. L. & Halpern, D. 2006 Strongly nonlinear nature of interfacial-surfactant instability of Couette flow. Intl J. Pure Appl. Maths 29, 205224. [arXiv:nlin/0601025]Google Scholar
Frenkel, A. L. & Indireshkumar, K. 1996 Derivation and simulations of evolution equations of wavy film flows. In Math Modeling and Simulation in Hydrodynamic Stability (ed. Riahi, D. N.), pp. 3581. World Scientific.CrossRefGoogle Scholar
Gao, G. P. & Lu, X. Y. 2006 Effect of surfactants on the long-wave stability of oscillatory film flow. J. Fluid Mech. 562, 345354.CrossRefGoogle Scholar
Gjevik, B. 1970 Occurrrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 19181925.CrossRefGoogle Scholar
Glendinning, P. 1994 Stability, Instability, and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press.CrossRefGoogle Scholar
Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and Groups in Bifurcation Theory: Volume 2. Springer.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, 191220.CrossRefGoogle Scholar
Halpern, D., Naire, S., Jensen, O. E. & Gaver, D. P. 2005 Unsteady bubble propagation in a flexible channel: predictions of a viscous stick-slip instability. J. Fluid Mech. 528, 5386.CrossRefGoogle Scholar
Herbert, T. 1988 Secondary instability of boundary-layers. Annu. Rev. Fluid Mech. 20, 487526.CrossRefGoogle Scholar
Hooper, A. P. 1985 Long-wave instability at the interface between two viscous fluids: Thin layer effects. Phys. Fluids 28, 16131618.CrossRefGoogle Scholar
Hooper, A. P. & Grimshaw, R. 1985 Nonlinear instability at the interface between two viscous fluids. Phys. Fluids 28, 3745.CrossRefGoogle Scholar
Indireshkumar, K. & Frenkel, A. L. 1996 Spatiotemporal patterns in a 3-D film flow. In Advances in Multi-Fluid Flows (Proc. AMS–IMS–SIAM Summer Research Conference, Seattle, 1995) (ed. Renardy, Y.), pp. 288–309. Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar
Jensen, O. E. & Grotberg, J. B. 1992 Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture. J. Fluid Mech. 240, 259288.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. 1993 Fundamentals of Two-Fluid Dynamics, vol I: Mathematical Theory and Applications. Springer.Google Scholar
Kelly, R. E. 1967 On stability of an inviscid shear layer which is periodic in space and time. J. Fluid Mech. 27, 657689.CrossRefGoogle Scholar
Leshansky, A. M. & Rubinstein, B. Y. 2005 Nonlinear rupture of thin liquid films on solid surfaces. Phys. Rev. E 71, 040601.Google ScholarPubMed
Levy, R. & Shearer, M. 2006 The motion of a thin liquid film driven by surfactant and gravity. SIAM J. Appl. Maths 66, 15881609.CrossRefGoogle Scholar
Lin, S. P. 1974 Finite-amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417429.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long–scale evolution of thin liquid films. Rev. Mode. Phys. 69, 931980.CrossRefGoogle Scholar
Pugh, J. D. & Saffman, P. G. 1998 Two-dimensional superharmonic stability of finite-amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295307.CrossRefGoogle Scholar
Renardy, Y. 1989 Weakly nonlinear behavior of periodic disturbances in two-layer Couette–Poiseuille flow. Phys. Fluids A 1, 16661676.CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C., Thiele, U., Kabov, O. A., Legros, J. C. & Colinet, P. 2005 Validity of the Benney equation including the Marangoni effect for closed and open flows. J. Fluid Mech. 527, 303335.CrossRefGoogle Scholar
Schwartz, L. W., Weidner, D. E. & Eley, R. R. 1995 An analysis of the effect of surfactant on the leveling behavior of a thin liquid coating layer. Langmuir 11, 36903693.CrossRefGoogle Scholar
Shlang, T., Sivashinsky, G.I., Babchin, A. J. & Frenkel, A. L. 1985 Irregular wavy flow due to viscous stratification. J. Phys. (Paris) 46, 863866.CrossRefGoogle Scholar
Sivashinsky, G. I. & Michelson, D. M. 1980 On irregular wavy flow of a liquid film down a vertical plane. Prog. Theor. Phys. 63, 21122114.CrossRefGoogle Scholar
Stuart, J. T. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 1. The basic behaviour in plane Poiseuille flow. J. Fluid Mech. 9, 353370.CrossRefGoogle Scholar
Watson, J. 1960 On the nonlinear mechanics of wave disturbances in stable and unstable parallel flows. Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. J. Fluid Mech. 9, 371389.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, 012103.CrossRefGoogle Scholar
Wei, H. H. 2005 b Marangoni destabilization on a core-annular film flow due to the presence of surfactant. Phys. Fluids 17, 027101.CrossRefGoogle Scholar
Wei, H. H. 2005 c On the flow-induced Marangoni instability due to the presence of surfactant. J. Fluid Mech. 544, 173200.CrossRefGoogle Scholar
Wei, H. H. 2007 Role of base flows on surfactant-driven interfacial instabilities. Phys. Rev. E 75, 036306.Google ScholarPubMed
Wei, H. H. & Rumschitzki, D. S. 2005 The effects of insoluble surfactants on the linear stability of a core-annular flow. J. Fluid Mech. 541, 115142.CrossRefGoogle Scholar
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337352.CrossRefGoogle Scholar