Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T12:07:13.189Z Has data issue: false hasContentIssue false
  • English
  • Français

Falling film with insoluble surfactants: effects of surface elasticity and surface viscosities

Published online by Cambridge University Press:  21 February 2020

Tao Hu ,
Qingfei Fu  and
Lijun Yang Open the ORCID record for Lijun Yang [Opens in a new window]
Tao Hu
Affiliation:
School of Astronautics, Beihang University, 100083Beijing, China
Qingfei Fu
Affiliation:
School of Astronautics, Beihang University, 100083Beijing, China Beijing Advanced Innovation Center for Big Data-Based Precision Medicine, Beihang University, 100083Beijing, China
Lijun Yang*
Affiliation:
School of Astronautics, Beihang University, 100083Beijing, China Beijing Advanced Innovation Center for Big Data-Based Precision Medicine, Beihang University, 100083Beijing, China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of a thin falling film with both surface elasticity and surface viscosities induced by insoluble surfactants on its free surface is studied. Based on the full Navier–Stokes equations and surfactant concentration equation with corresponding boundary conditions, a weighted residual model (WRM) is derived to investigate the long-wave instability of the thin film incorporating the influence of surfactants. The Chebyshev spectral collocation method is employed to solve the linear stability of the film. The results show good agreement between the WRM and full equations. It is found that surface elasticity decreases the temporal growth rate and increases the critical Reynolds number, showing a stabilizing impact on the film. And the surface viscosity effect slightly reduces the growth rate and cutoff wavenumber while it does not alter the critical Reynolds number. Nonlinear travelling wave solutions are obtained using the WRM equations. As the surface elasticity is enhanced, the speed of travelling waves gradually approaches the corresponding linear neutral value, implying that the dispersion effect is damped; and the amplitudes of both fast waves and slow waves are suppressed by surface elasticity. Moreover, the bifurcation diagram of travelling waves is influenced by the surface viscosity, which basically promotes the speed of travelling waves with relatively large wavelengths. As the surface viscosity effect becomes stronger, for fast waves the amplitude of the humps slightly increases while that of the troughs becomes smaller for slow waves.

JFM classification

Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 889 , 25 April 2020 , A16
Copyright
© Beijing University of Aeronautics and Astronautics, 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

Adebayo, I., Xie, Z., Che, Z. & Matar, O. K. 2017 Doubly excited pulse waves on thin liquid films flowing down an inclined plane: an experimental and numerical study. Phys. Rev. E 96 (1), 013118.Google ScholarPubMed
Alekseenko, S. V., Antipin, V. A., Guzanov, V. V., Kharlamov, S. M. & Markovich, D. M. 2005 Three-dimensional solitary waves on falling liquid film at low Reynolds numbers. Phys. Fluids 17 (12), 121704.CrossRefGoogle Scholar
Alekseenko, S. V., Nakoryakov, V. E. & Pokusaev, B. G. 1994 Wave Flow of Liquid Films. Begell House.Google Scholar
Alekseenko, S. V., Nakoryakov, V. Y. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31 (9), 1446–1460.CrossRefGoogle Scholar
Amaouche, M., Djema, A. & Ait Abderrahmane, H. 2012 Film flow for power-law fluids: modeling and linear stability. Eur. J. Mech. (B/Fluids) 34, 70–84.CrossRefGoogle Scholar
Bender, A., Stephan, P. & Gambaryan-Roisman, T. 2017 Thin liquid films with time-dependent chemical reactions sheared by an ambient gas flow. Phys. Rev. Fluids 2 (8), 084002.CrossRefGoogle Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–573.CrossRefGoogle Scholar
Benjamin, T. B. 1964 Effects of surface contamination on wave formation in falling liquid films. Arch. Mech. Stosow. 16 (3), 615–626.Google Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45 (1), 150–155.CrossRefGoogle Scholar
Bhat, F. A. & Samanta, A. 2018 Linear stability of a contaminated fluid flow down a slippery inclined plane. Phys. Rev. E 98 (3), 033108.Google Scholar
Bhat, F. A. & Samanta, A. 2019 Linear stability analysis of a surfactant-laden shear-imposed falling film. Phys. Fluids 31 (5), 054103.CrossRefGoogle 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
Chang, H. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103–136.CrossRefGoogle Scholar
Charogiannis, A. & Markides, C. N. 2019 Spatiotemporally resolved heat transfer measurements in falling liquid-films by simultaneous application of planar laser-induced fluorescence (PLIF), particle tracking velocimetry (PTV) and infrared (IR) thermography. Exp. Therm. Fluid Sci. 107, 169–191.CrossRefGoogle Scholar
Charogiannis, A., Denner, F., van Wachem, B. G. M., Kalliadasis, S. & Markides, C. N. 2017 Detailed hydrodynamic characterization of harmonically excited falling-film flows: a combined experimental and computational study. Phys. Rev. Fluids 2 (1), 014002.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.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 (8), 083604.Google Scholar
Dávalos-Orozco, L. A. 2012 The effect of the thermal conductivity and thickness of the wall on the nonlinear instability of a thin film flowing down an incline. Intl J. Non-Linear Mech. 47 (4), 1–7.CrossRefGoogle Scholar
Ding, Z. & Wong, T. N. 2015 Falling liquid films on a slippery substrate with Marangoni effects. Intl J. Heat Mass Transfer 90, 689–701.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 (9), 092105.CrossRefGoogle Scholar
Emmert, R. E. 1954 A study of gas absorption in falling liquid films. Chem. Engng Prog. 50 (2), 87–93.Google Scholar
Fu, Q. F., Hu, T. & Yang, L. J. 2018 Instability of a weakly viscoelastic film flowing down a heated inclined plane. Phys. Fluids 30 (8), 084102.CrossRefGoogle Scholar
Georgantaki, A., Vlachogiannis, M. & Bontozoglou, V. 2012 The effect of soluble surfactants on liquid film flow. J. Phys. Conf. Ser. 395 (1), 012165.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
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
Joo, S. W., Davis, S. H. & Bankoff, S. G. 1991 Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers. J. Fluid Mech. 230, 117–146.CrossRefGoogle Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin viscous liquid films. Zh. Eksp. Teor. Fiz. 19, 105–120.Google Scholar
Karapetsas, G. & Bontozoglou, V. 2014 The role of surfactants on the mechanism of the long-wave instability in liquid film flows. J. Fluid Mech. 741, 139–155.CrossRefGoogle Scholar
Kharlamov, S. M., Guzanov, V. V., Bobylev, A. V., Alekseenko, S. V. & Markovich, D. M. 2015 The transition from two-dimensional to three-dimensional waves in falling liquid films: wave patterns and transverse redistribution of local flow rates. Phys. Fluids 27 (11), 114106.CrossRefGoogle Scholar
Khorrami, M. R., Malik, M. R. & Ash, R. L. 1989 Application of spectral collocation techniques to the stability of swirling flows. J. Comput. Phys. 81 (1), 206–229.CrossRefGoogle Scholar
Lin, S. P. 1974 Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429.CrossRefGoogle Scholar
Liu, J., Gollub, J. P. & Aarts, D. G. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6 (5), 1702–1712.CrossRefGoogle Scholar
Liu, J., Paul, J. D. & Gollub, J. P. 1993 Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101.CrossRefGoogle Scholar
Liu, J., Schneider, J. B. & Gollub, J. P. 1995 Three-dimensional instabilities of film flows. Phys. Fluids 7 (1), 55–67.CrossRefGoogle Scholar
Martínez-Calvo, A. & Sevilla, A. 2018 Temporal stability of free liquid threads with surface viscoelasticity. J. Fluid Mech. 846, 877–901.CrossRefGoogle Scholar
Ogden, K., Pascal, J. & D’Alessio, S. 2011 Gravity-driven flow over heated, porous, wavy surfaces. Phys. Fluids 23 (12), 122102.CrossRefGoogle Scholar
Oron, A. & Edwards, D. A. 1993 Stability of a falling liquid film in the presence of interfacial viscous stress. Phys. Fluids A 5 (2), 506–508.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.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 (5), 054004.CrossRefGoogle Scholar
Pereira, A. & Kalliadasis, S. 2008 Dynamics of a falling film with solutal Marangoni effect. Phys. Rev. E 78 (3), 036312.Google ScholarPubMed
Ponce-Torres, A., Montanero, J. M., Herrada, M. A., Vega, E. J. & Vega, J. M. 2017 Influence of the surface viscosity on the breakup of a surfactant-laden drop. Phys. Rev. Lett. 118 (2), 024501.CrossRefGoogle ScholarPubMed
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15 (2), 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 (1), 170–183.CrossRefGoogle Scholar
Ruyer-Quil, C., Chakraborty, S. & Dandapat, B. S. 2012 Wavy regime of a power-law film flow. J. Fluid Mech. 692, 220–256.CrossRefGoogle Scholar
Ruyer-Quil, C., Kofman, N., Chasseur, D. & Mergui, S. 2014 Dynamics of falling liquid films. Eur. Phys. J. E 37 (4), 30.CrossRefGoogle ScholarPubMed
Sadiq, I. M. R. & Usha, R. 2008 Thin Newtonian film flow down a porous inclined plane: stability analysis. Phys. Fluids 20 (2), 022105.CrossRefGoogle Scholar
Salvagnini, W. M. & Taqueda, M. E. S. 2004 A falling-film evaporator with film promoters. Ind. Engng Chem. Res. 43 (21), 6832–6835.CrossRefGoogle Scholar
Samanta, A. 2008 Stability of liquid film falling down a vertical non-uniformly heated wall. Physica D 237 (20), 2587–2598.Google Scholar
Samanta, A., Ruyer-Quil, C. & Goyeau, B. 2011 A falling film down a slippery inclined plane. J. Fluid Mech. 684, 353–383.CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C. & Manneville, P. 2006 Wave patterns in film flows: modelling and three-dimensional waves. J. Fluid Mech. 562, 183–222.CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C., Thiele, U., Kabov, O. A., Legros, J. C. & Colinet, P. 2005 Validity domain of the Benney equation including the Marangoni effect for closed and open flows. J. Fluid Mech. 527, 303–335.CrossRefGoogle Scholar
Scriven, L. E. & Sternling, C. V. 1960 The Marangoni effects. Nature 187 (4733), 186–188.CrossRefGoogle Scholar
Shine, S. R. & Nidhi, S. S. 2018 Review on film cooling of liquid rocket engines. Propul. Power Res. 7 (1), 1–18.CrossRefGoogle 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 2 (1), 43–51.Google Scholar
Smith, M. K. 1990 The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485.CrossRefGoogle Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77 (3), 977–1026.CrossRefGoogle Scholar
Stirba, C. & Hurt, D. M. 1955 Turbulence in falling liquid films. AIChE J. 1 (2), 178–184.CrossRefGoogle Scholar
Tailby, S. R. & Portalski, S. 1961 The optimum concentration of surface active agents for the suppression of ripples. Trans. Inst. Chem. 39, 328–336.Google Scholar
Trevelyan, P., Scheid, B., Ruyer-Quil, C. & Kalliadasis, S. 2007 Heated falling films. J. Fluid Mech. 592, 295–334.CrossRefGoogle Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Whitaker, S. 1964 Effect of surface active agents on the stability of falling liquid films. Ind. Engng Chem. Fundam. 3 (2), 132–142.CrossRefGoogle Scholar
Ye, H. Y., Yang, L. J. & Fu, Q. F. 2016 Spatial instability of viscous double-layer liquid sheets. Phys. Fluids 28 (10), 102101.CrossRefGoogle Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6 (3), 321–334.CrossRefGoogle Scholar