Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-05T12:06:56.767Z Has data issue: false hasContentIssue false
  • English
  • Français

Longwave Marangoni convection in a surfactant solution between poorly conducting boundaries

Published online by Cambridge University Press:  08 February 2013

Sergey Shklyaev  and
Alexander A. Nepomnyashchy
Sergey Shklyaev*
Affiliation:
Department of Chemical Engineering, University of Puerto Rico – Mayagüez, Mayagüez, PR 00681, USA Institute of Continuous Media Mechanics, Ural Branch of Russian Academy of Sciences, Perm 614013, Russia
Alexander A. Nepomnyashchy
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel Minerva Center for Nonlinear Physics of Complex Systems, Technion – Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider Marangoni convection in a heated layer of a binary liquid. The solute is a surfactant, which is present in both surface and bulk phases; the bulk gradient of the concentration is formed due to the Soret effect. Linear stability analysis demonstrates a well-pronounced stabilization of the layer due to the adsorption kinetics and advection of the surface phase. We derive nonlinear amplitude equations for longwave perturbations in the case of fast sorption kinetics (small Langmuir number) and demonstrate that with increase in the effect of the adsorption, subcritical excitation occurs. In the case of a finite Langmuir number, the weakly nonlinear problem is ill-posed. A physical mechanism of subcritical bifurcation is discussed.

JFM classification

Convection: Marangoni convection Nonlinear Dynamical Systems: Pattern formation Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 718 , 10 March 2013 , pp. 428 - 456
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

Agrawal, M. L. & Neuman, R. D. 1988 Surface diffusion in monomolecular films. J. Colloid Interface Sci. 121, 366–380.Google Scholar
Andoh, Y. & Yasuoka, K. 2007 Molecular understanding of dynamical properties of the vapor/ethanol-aqueous-solution interface. Mol. Simulation 33, 139–145.Google Scholar
Berg, J. C. & Acrivos, A. 1965 The effect of surface active agents on convection cells induced by surface tension. Chem. Engng Sci. 20, 737–745.Google Scholar
Bernoff, A. J. 1988 Slowly varying fully nonlinear wavetrains in the Ginzburg–Landay equation. Physica D 30, 363–381.Google Scholar
Bernoff, A. J. & Bertozzi, A. L. 1995 Singularities in a modified Kuramoto–Sivashinsky equation describing interface motion for phase transition. Physica D 85, 375–404.Google Scholar
Blackburn, H. M., Sherwin, S. J. & Barkley, D. 2008 Convective instability and transient growth in steady and pulsatile stenotic flows. J. Fluid Mech. 607, 267–277.Google Scholar
Chen, C. F. & Chen, C. C. 1994 Effect of surface tension on the stability of a binary fluid layer under reduced gravity. Phys. Fluids 6, 1482–1490.Google Scholar
Clune, T., Depassier, M. C. & Knobloch, E. 1996 Long wavelength oscillatory instability in binary fluids. In Instabilities and Nonequilibrium Structures (ed. Tirapegui, E. & Zeller, W.), pp. 75–84. Kluwer.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
Edmonstone, B. D., Matar, O. K. & Craster, R. V. 2006 A note on the coating of an inclined plane in the presence of soluble surfactant. J. Colloid Interface Sci. 293, 222–229.Google Scholar
Goldobin, D. S. & Lyubimov, D. V. 2007 Soret-driven convection of binary mixture in a horizontal porous layer in the presence of a heat or concentration source. Sov. Phys. JETP 104, 830–836.Google Scholar
Golovin, A. A., Nepomnyashchy, A. A. & Pismen, L. M. 1995 Pattern formation in large-scale Marangoni convection with deformable interface. Physica D 81, 117–147.Google Scholar
Golubitsky, M., Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh–Bérnard convection. Physica D 10, 249–276.Google Scholar
Hamid, A. S. A., Kechil, S. A. & Aziz, A. S. A. 2011 Marangoni instability in a fluid layer with insoluble surfactant. World Acad. Sci. Engng Technol. 58, 24–28.Google Scholar
Hoyle, R. B. 2006 Pattern Formation: An Introduction to Methods. Cambridge University Press.Google Scholar
Hurle, D. T. J. & Jakeman, E. 1971 Soret-driven thermosolutal convection. J. Fluid Mech. 47, 667–687.Google Scholar
Johnson, D. O. & Stebe, K. J. 1996 Experimental confirmation of the oscillating bubble technique with comparison to the pendant bubble method: the adsorption dynamics of 1-decanol. J. Colloid Interface Sci. 182, 526–538.Google Scholar
Kalitzova-Kurteva, P., Slavtchev, S. & Oron, A. 2009 Long-wavelength monotonic Marangoni instability in a binary mixture layer with nonlinear Soret effect. J. Theor. Appl. Mech. (Sofia) 39 (1), 31–54.Google Scholar
Kostarev, K. G., Zuev, A. L. & Viviani, A. 2009 Experimental considerations of solutocapillary flow initiation on bubble/drop interface in the presence of a soluble surfactant. Microgravity Sci. Technol. 21, 59–65.Google Scholar
Kumar, N., Couzis, A. & Maldarelli, C. 2003 Measurement of the kinetic rate constants for the adsorption of superspreading trisiloxanes to an air/aqueous interface and the relevance of these measurements to the mechanism of superspreading. J. Colloid Interface Sci. 267, 272–285.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics. Pergamon.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena. Cambridge University Press.CrossRefGoogle Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice-Hall.Google Scholar
Li, Z. X., Lu, J. R., Styrkas, D. A. & Thomas, R. K. 1993 The structure of the surface of ethanol/water mixtures. Mol. Phys. 80, 925–939.Google Scholar
Lin, S.-Y., Lu, T.-L. & Hwang, W.-B. 1996 Adsorption kinetics of decanol at the air–water interface. Langmuir 11, 555–562.CrossRefGoogle Scholar
MacLeod, C. A. & Radke, C. J. 1994 Surfactant exchange kinetics at the air–water interface from the dynamic tension of growing liquid drop. J. Colloid Interface Sci. 166, 73–88.Google Scholar
Matar, O. K. & Craster, R. V. 2009 Dynamics of surfactant-assisted spreading. Soft Matt. 5, 3801–3809.Google Scholar
Mikishev, A. B. & Nepomnyashchy, A. A. 2010a Long-wavelength Marangoni convection in a liquid layer with insoluble surfactant: linear theory. Microgravity Sci. Technol. 22, 415–423.Google Scholar
Mikishev, A. B. & Nepomnyashchy, A. A. 2010b Nonlinear large-scale Marangoni convection in a heated liquid layer with insoluble surfactant. Phys. Rev. E 82, 046306.Google Scholar
Nepomnyashchy, A. A., Golovin, A. A., Gubareva, V. & Panfilov, V. 2004 Global feedback control of a long-wave morphological instability. Physica D 199, 61–81.Google Scholar
Nepomnyashchy, A. A. & Simanovskii, I. B. 1986 Thermocapillary convection in two-layer systems in the presence of a surface active agent at the interface. Fluid Dyn. 21, 169–174.Google Scholar
Nepomnyashchy, A. A. & Simanovskii, I. B. 1988 Occurrence of thermocapillary convection in a two-layer system in the presence of a soluble surfactant. Fluid Dyn. 23, 302–306.Google Scholar
Nepomnyashchy, A. A. & Simanovskii, I. B. 1992 Excitation of thermal-capillary convection at a deformable interface in systems with a surface-active agent. J. Appl. Mech. Tech. Phys. 33, 379–385.Google Scholar
Oron, A. & Nepomnyashchy, A. A. 2004 Long-wavelength thermocapillary instability with the Soret effect. Phys. Rev. E 69, 016313.Google Scholar
Palmer, H. J. & Berg, J. C. 1972 Hydrodynamic stability of surfactant solutions heated from below. J. Fluid Mech. 51, 385–402.Google Scholar
Pan, R., Green, J. & Maldarelli, C. 1998 Theory and experiment on the measurement of kinetic rate constants for surfactant exchange at an air/water interface. J. Colloid Interface Sci. 205, 213–230.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.Google Scholar
Penfold, J., Richardson, R. M., Zarbakhsh, A., Webster, J. R. P., Bucknall, D. G., Rennie, A. R., Jones, R. A. L., Cosgrove, T., Thomas, R. K., Higgins, J. S., Fletcher, P. D. I., Dickinson, E., Roser, S. J., McLure, I. A., Hillman, A. R., Richards, R. W., Staples, E. J., Burgess, A. N., Simistero, E. A. & Whitep, J. W. 1997 Recent advances in the study of chemical surfaces and interfaces by specular neutron reaction. J. Chem. Soc. Faraday Trans. 93, 3899–3917.Google Scholar
Pismen, L. 1988 Selection of long-scale oscillatory convective patterns. Phys. Rev. A 38, 2564–2572.Google Scholar
Podolny, A., Oron, A. & Nepomnyashchy, A. A. 2005 Long-wave Marangoni instability in a binary-liquid layer with deformable interface in the presence of Soret effect: linear theory. Phys. Fluids 17, 104104.Google Scholar
Podolny, A., Oron, A. & Nepomnyashchy, A. A. 2006 Linear and nonlinear theory of longwave Marangoni instability with the Soret effect at finite Biot numbers. Phys. Fluids 18, 054104.CrossRefGoogle Scholar
Raina, G., Kulkarni, G. U. & Rao, C. N. R. 2001 Mass spectrometric determination of the surface compositions of ethanolwater mixtures. Int. J. Mass Spectrometry 212, 267–271.Google Scholar
Roberts, M., Swift, J. W. & Wagner, D. H. 1986 The Hopf bifurcation on a hexagonal lattice. In Multiparameter Bifurcation Theory (ed. Golubitsky, M. & Guckenheimer, J.), Contemp. Math. , vol. 56, pp. 283–318. American Mathematical Society.Google Scholar
Ryabitskii, E. A. 1993 Oscillatory thermocapillary instability of a liquid layer in the presence of a surfactant. Fluid Dyn. 28, 3–6.Google Scholar
Ryabitskii, E. A. 1996 Thermocapillary instability of a liquid layer in the presence of a soluble surfactant. Fluid Dyn. 31, 1–5.Google Scholar
Ryabitskii, E. A. 2000 Thermocapillary instability of a liquid layer with allowance for the Soret effect. Fluid Dyn. 35, 315–320.Google Scholar
Scriven, L. E. & Sternling, C. V. 1964 On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity. J. Fluid Mech. 19, 321–340.Google Scholar
Shin, J. Y. & Abbott, N. L. 2001 Combining molecular dynamics simulations and transition state theory to evaluate the sorption rate constants for decanol at the surface of water. Langmuir 17, 8434–8443.Google Scholar
Shklyaev, S. & Nepomnyashchy, A. A. 2011 Marangoni instability of a heated liquid layer in the presence of a soluble surfactant. Oscillatory mode for small Lewis number. Eur. Phys. J. ST 192, 155–161.Google Scholar
Shklyaev, S., Nepomnyashchy, A. A. & Oron, A. 2007 Three-dimensional oscillatory long-wave Marangoni convection in a binary liquid layer with the Soret effect: bifurcation analysis. Phys. Fluids 19, 072105.Google Scholar
Shklyaev, S., Nepomnyashchy, A. A. & Oron, A. 2009a Long-wave oscillatory convection in a binary liquid: hexagonal patterns. Europhys. Lett. 86, 14005.Google Scholar
Shklyaev, S., Nepomnyashchy, A. A. & Oron, A. 2009b Marangoni convection in a binary liquid layer with Soret effect at small Lewis number: linear stability analysis. Phys. Fluids 21, 054101.CrossRefGoogle Scholar
Shklyaev, S., Nepomnyashchy, A. A. & Oron, A. 2011 Oscillatory long-wave Marangoni convection in a layer of a binary liquid: hexagonal patterns. Phys. Rev. E 84, 056327.Google Scholar
Silber, M. & Knobloch, E. 1991 Hopf bifurcation on a square lattice. Nonlinearity 4, 1063–1106.CrossRefGoogle Scholar
Sivashinsky, G. I. 1983 On cellular instability in the solidification of a dilute binary alloy. Physica D 8, 243–248.Google Scholar
Slavtchev, S., Kalitzova-Kurteva, P. & Oron, A. 2009 Long-wavelength oscillatory Marangoni instability in a binary mixture layer with nonlinear Soret effect. J. Theor. Appl. Mech. (Sofia) 39 (3), 63–78.Google Scholar
Smith, K. A. 1966 On convective instability induced by surface-tension gradients. J. Fluid Mech. 24, 401–414.Google Scholar
Tarek, M., Bandyopadhyay, S. & Klein, M. L. 1998 Molecular dynamics studies of aqueous surfactants systems. J. Molecular Liquids 78, 1–6.Google Scholar
Valkovska, D. S. & Danov, K. D. 2000 Determination of bulk and surface diffusion coefficients from experimental data for thin liquid film drainage. J. Colloid Interface Sci. 223, 314–316.Google Scholar
VanHook, S. J., Schatz, M., Swift, J. B., McCormick, W. D. & Swinney, H. L. 1997 Long-wavelength surface-tension-driven Bénard convection: experiment and theory. J. Fluid Mech. 345, 45–78.Google Scholar
Vidal, A. & Acrivos, A. 1966 Nature of the neutral state in surface-tension driven convection. Phys. Fluids 9, 615–616.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 2010 A mechanism of Marangoni instability in evaporating thin liquid films due to soluble surfactant. Phys. Fluids 22, 022102.Google Scholar