Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-19T02:25:29.978Z Has data issue: false hasContentIssue false
  • English
  • Français

Marangoni instabilities associated with heated surfactant-laden falling films

Published online by Cambridge University Press:  28 January 2020

S. J. D. D’Alessio Open the ORCID record for S. J. D. D’Alessio [Opens in a new window] ,
J. P. Pascal ,
E. Ellaban  and
C. Ruyer-Quil
S. J. D. D’Alessio*
Affiliation:
Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
J. P. Pascal
Affiliation:
Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada
E. Ellaban
Affiliation:
Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada
C. Ruyer-Quil
Affiliation:
Université de Savoie Mont Blanc, CNRS, LOCIE 73000 Chambéry, France
*
†Email address for correspondence: sdalessio@uwaterloo.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

Investigated in this paper is the stability of the gravity-driven flow of a liquid layer laden with soluble surfactant down a heated incline. A mathematical model incorporating variations in surface tension with surfactant concentration and temperature has been formulated. A linear stability analysis is carried out both asymptotically for small wavenumbers and numerically for arbitrary wavenumbers. An expression for the critical Reynolds number has been derived which accounts for thermocapillary and solutocapillary effects, and reduces to known documented results for special cases. Also, a nonlinear reduced model has been derived using weighted residuals, and solved numerically to simulate the instability of the equilibrium flow and the development of permanent surface waves that arise. The nonlinear simulations were found to be in good agreement with the linear stability analysis.

JFM classification

Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 887 , 25 March 2020 , A20
Copyright
© The Author(s), 2020. 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

Aksel, N. & Schörner, M. 2018 Films over topography: from creeping flow to linear stability, theory, and experiments, a review. Acta Mech. 229, 1453–1482.CrossRefGoogle Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
Benjamin, T. B. 1964 Effects of surface contamination on wave formation in falling liquid films. Arch. Mech. Stos. 16, 615–626.Google Scholar
Blyth, M. G. & Pozrikidis, C. 2004 Effect of surfactant on the stability of film flow down an inclined plane. J. Fluid Mech. 521, 241–250.CrossRefGoogle Scholar
D’Alessio, S. J. D., Pascal, J. P., Jasmine, H. A. & Ogden, K. A. 2010 Film flow over heated wavy inclined surfaces. J. Fluid Mech. 665, 418–456.CrossRefGoogle Scholar
D’Alessio, S. J. D. & Pascal, J. P. 2016 Thermosolutal Marangoni effects on the inclined flow of a binary liquid with variable density. II. Nonlinear analysis and simulations. Phys. Rev. Fluids 1, 083604.Google Scholar
D’Alessio, S. J. D., Pascal, J. P. & Jasmine, H. A. 2009 Instability in gravity-driven flow over uneven surfaces. Phys. Fluids 21, 062105.CrossRefGoogle Scholar
Ellaban, E., Pascal, J. P. & D’Alessio, S. J. D. 2017 Instability of a binary liquid film flowing down a slippery heated plate. Phys. Fluids 29, 092105.CrossRefGoogle Scholar
Emmert, R. E. & Pigford, R. 1954 A study of gas absorption in falling liquid films. Chem. Engng Prog. 50, 87–93.Google Scholar
Floryan, J. M., Davis, S. H. & Kelly, R. E. 1987 Instabilities of a liquid film flowing down a slightly inclined plane. Phys. Fluids 30, 983–989.CrossRefGoogle Scholar
Georgantaki, A., Vlachogiannis, M. & Bontozoglou, V. 2016 Measurements of the stabilisation of liquid film flow by the soluble surfactant sodium dodecyl sulfate (SDS). Intl J. Multiphase Flow 86, 28–34.CrossRefGoogle Scholar
Goussis, D. A. & Kelly, R. E. 1991 Surface wave and thermocapillary instabilities in a liquid film flow. J. Fluid Mech. 223, 25–45.CrossRefGoogle Scholar
Ji, W. & Setterwall, F. 1995 Effect of heat transfer additives on the instabilities of an absorbing falling film. Chem. Engng Sci. 50, 3077–3097.CrossRefGoogle Scholar
Ji, W. & Setterwall, F. 1994 On the instabilities of vertical falling liquid films in the presence of surface-active solute. J. Fluid Mech. 278, 297–323.CrossRefGoogle Scholar
Kalliadasis, S., Demekhin, E. A., Ruyer-Quil, C. & Velarde, M. G. 2003a Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J. Fluid Mech. 492, 303–338.CrossRefGoogle Scholar
Kalliadasis, S., Kiyashko, A. & Demekhin, E. A. 2003b Marangoni instability of a thin liquid film heated from below by a local heat source. J. Fluid Mech. 475, 377–408.CrossRefGoogle Scholar
Kalliadasis, S., Demekhin, E. A., Ruyer-Quil, C. & Velarde, M. G. 2012 Falling Liquid Films. Springer.CrossRefGoogle Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin layers of a viscous fluid. Part III. Experimental study of undulatory flow conditions. Soc. Phys. J. Exp. Theor. Phys. 19, 105–120.Google Scholar
Karapetsas, G. & Bontozoglou, V. 2013 The primary instability of falling films in the presence of soluble surfactants. J. Fluid Mech. 729, 123–150.CrossRefGoogle Scholar
Karapetsas, G. & Bontozoglou, V. 2014 The role of surfactants on the mechanism of long-wave instability in liquid film flows. J. Fluid Mech. 741, 139–155.CrossRefGoogle Scholar
Leveque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.CrossRefGoogle Scholar
LeVeque, R. J. & Yee, H. C. 1990 A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86, 187–210.CrossRefGoogle Scholar
Nepomnyashchy, A. A., Velarde, M. G. & Colinet, P. 2002 Interfacial Phenomena and Convection. CRC.Google Scholar
Ogden, K. A., D’Alessio, S. J. D. & Pascal, J. P. 2011 Gravity-driven flow over heated, porous, wavy surfaces. Phys. Fluids 23, 122102.CrossRefGoogle Scholar
Pascal, J. P., D’Alessio, S. J. D. & Ellaban, E. 2019 Stability of inclined flow of a liquid film with soluble surfactants and variable mass density. Phys. Rev. Fluids 4, 054004.CrossRefGoogle Scholar
Pascal, J. P. & D’Alessio, S. J. D. 2016 Thermosolutal Marangoni effects on the inclined flow of a binary liquid with variable density. I. Linear stability analysis. Phys. Rev. Fluids 1, 083603.CrossRefGoogle Scholar
Pascal, J. P. & D’Alessio, S. J. D. 2010 Instability in gravity-driven flow over uneven permeable surfaces. Intl J. Multiphase Flow 36, 449–459.CrossRefGoogle Scholar
Pascal, J. P., D’Alessio, S. J. D. & Hasan, M. 2018 Instability of gravity-driven flow of a heated power-law fluid with temperature dependent consistency. AIP Adv. 8, 105215.CrossRefGoogle Scholar
Pearson, J. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.CrossRefGoogle Scholar
Pereira, A., Trevelyan, P. M. J., Thiele, U. & Kalliadasis, S. 2007 Dynamics of a horizontal thin liquid film in the presence of reactive surfactants. Phys. Fluids 19, 112102.CrossRefGoogle Scholar
Pereira, A. & Kalliadasis, S. 2008 Dynamics of a falling film with solutal Marangoni effect. Phys. Rev. E 78, 036312.Google ScholarPubMed
Pereira, A. & Kalliadasis, S. 2011 On the transport equation for an interfacial quantity. Eur. Phys. J. Appl. Phys. 44 (2), 211–214.CrossRefGoogle Scholar
Roberts, A. J. 1995 Low-dimensional models of thin film fluid dynamics. Phys. Lett. A 212, 63–71.CrossRefGoogle Scholar
Ruyer-Quil, C., Scheid, B., Kalliadasis, S., Velarde, M. G. & Zeytounian, R. Kh. 2005 Thermocapillary long waves in a liquid film flow. Part 1. Low-dimensional formulation. J. Fluid Mech. 538, 199–222.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14, 170–183.CrossRefGoogle Scholar
Ruyer-Quil, C., Kofman, N., Chasseur, D. & Mergui, S. 2014 Dynamics of falling films. Eur. Phys. J. E 37 (4), 1–17.Google Scholar
Scheid, B., Ruyer-Quil, C., Kalliadasis, S., Velarde, M. G. & Zeytounian, R. Kh. 2005 Thermocapillary long waves in a liquid film flow. Part 2. Linear stability and nonlinear waves. J. Fluid Mech. 538, 223–244.CrossRefGoogle Scholar
Sheludko, A. 1967 Thin liquid films. Adv. Colloid Interface Sci. 1, 391–464.CrossRefGoogle Scholar
Spurk, J. H. & Aksel, N. 2008 Fluid Mechanics, 2nd edn. Springer.Google Scholar
Srivastava, A. & Tiwari, N. 2018 Effect of an insoluble surfactant on the dynamics of a thin liquid film flowing over a non-uniformly heated substrate. Eur. Phys. J. E 41, 1–12.Google Scholar
Sternling, C. V. & Scriven, L. E. 1959 Interfacial turbulence: hydrodynamic stability and the Marangoni effect. AIChE J. 5, 514–523.CrossRefGoogle Scholar
Stirba, C. & Hurt, D. 1955 Turbulence in falling liquid films. AIChE J. 1, 178–184.CrossRefGoogle Scholar
Tailby, S. & Portalski, S. 1961 The optimum concentration of surface active agents for the suppression of ripples. Trans. Inst. Chem. Engrs 39, 328–336.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in Matlab. SIAM.CrossRefGoogle Scholar
Trevelyan, P. M. J., Scheid, B., Ruyer-Quil, C. & Kalliadasis, S. 2007 Heated falling films. J. Fluid Mech. 592, 295–334.CrossRefGoogle Scholar
Veremieiev, S. & Wacks, D. H. 2019 Modelling gravity-driven film flow on inclined corrugated substrate using a high fidelity weighted residual integral boundary-layer method. Phys. Fluids 31, 022101.CrossRefGoogle Scholar
Wei, H. 2005 Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane. Phys. Fluids 17, 012103.CrossRefGoogle Scholar
Whitaker, S. 1964 Effect of surface active agents on the stability of falling liquid films. Ind. Engng Chem. Fundam. 3, 132–142.CrossRefGoogle Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar