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Role of soluble surfactant in linear stability of a liquid film flowing down a compliant substrate

Published online by Cambridge University Press:  13 May 2025

Arghya Samanta Open the ORCID record for Arghya Samanta [Opens in a new window]
Arghya Samanta*
Affiliation:
Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India
*
Corresponding author: Arghya Samanta, arghya@am.iitd.ac.in
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Abstract

A linear stability analysis of a soluble surfactant-laden liquid film flowing down a compliant substrate is performed. Our purpose is to expand the prior studies (Carpenter and Garrad 1985 J. Fluid Mech. 155, 465–510; Alexander et al., 2020 J. Fluid Mech. 900, A40) by incorporating a soluble surfactant into the flow configuration. As a result, we formulate the Orr–Sommerfeld-type boundary value problem and solve it analytically by using the long-wave series expansion as well as numerically by using the Chebyshev spectral collocation method in an arbitrary wavenumber regime for infinitesimal disturbances. The long-wave result reveals that surface instability is stabilized in the presence of a surfactant, whereas it is destabilized in the presence of a compliant substrate. These opposing impacts suggest an analytical relationship between parameters associated with the soluble surfactant and compliant wall, ensuring the same critical Reynolds number for the emergence of surface instability corresponding to both surfactant-laden film flow over a compliant wall and surfactant-free film flow over a non-compliant wall. In the arbitrary wavenumber regime, along with the surface mode, we identify two additional modes based on their distinct phase speeds. Specifically, the wall mode emerges in the finite wavenumber regime, while the shear mode emerges only when the Reynolds number is large. As the surfactant Marangoni number increases, the wall mode destabilizes, resulting in a different outcome from the surface mode. Moreover, increasing the value of the ratio of adsorption and desorption rate constants stabilizes surface instability but destabilizes wall mode instability. As a result, we perceive that the soluble surfactant-laden film flow is linearly more unstable than the insoluble one due to surface instability but linearly more stable than the insoluble one due to wall mode instability. Additionally, we see a peculiar behaviour of base surface surfactant concentration on the primary instability. In fact, it has a specific value depending on adsorption and desorption rate constants below which surface instability stabilizes but wall mode instability destabilizes, whereas above which an opposite phenomenon occurs. Finally, in the high-Reynolds-number regime, we can suppress shear mode instability by raising the surfactant Marangoni number and the ratio of adsorption and desorption rate constants when the angle of inclination is sufficiently small. Unlike surface instability, the base surface surfactant concentration exhibits both stabilizing and destabilizing influences on shear mode instability.

JFM classification

Interfacial Flows (free surface): Thin films Instability: Shear-flow instability Aerodynamics: Flow-structure interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1011 , 25 May 2025 , A6
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Alekseenko, S.V., Nakoryakov, V.E. & Pokusaev, B.G. 1994 Wave Flow in Liquid Films. 3rd edn. Begell House (New York).CrossRefGoogle Scholar
Alexander, J.P., Kirk, T.L. & Papageorgiou, D.T. 2020 Stability of falling liquid films on flexible substrates. J. Fluid Mech. 900, A40.CrossRefGoogle Scholar
Benjamin, T.B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2 (06), 554–573.CrossRefGoogle Scholar
Benjamin, T.B. 1960 Effects of a flexible boundary on hydrodynamic stability. J. Fluid Mech. 9 (4), 513–532.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.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
Bruin, G.D. 1974 Stability of a layer of liquid flowing down an inclined plane. J. Engng Math. 8 (3), 259–270.CrossRefGoogle Scholar
Carpenter, P.W. & Gajjar, J.S.B. 1990 A general theory for two- and three-dimensional wall-mode instabilities in boundary layer over isotropic and anisotropic compliant walls. Theor. Comput. Fluid Dyn. 1 (6), 349–378.CrossRefGoogle Scholar
Carpenter, P.W. & Garrad, A.D. 1985 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien–Schlichting instabilities. J. Fluid Mech. 155, 465–510.CrossRefGoogle Scholar
Carpenter, P.W. & Garrad, A.D. 1986 The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 2. Flow-induced surface instabilities. J. Fluid Mech. 170, 199–232.CrossRefGoogle Scholar
Chang, H.C. & Demekhin, E.A. 2002 Complex Wave Dynamics On Thin Films. Elsevier.Google Scholar
Chin, R., Abernath, F. & Bertschy, J. 1986 Gravity and shear wave stability of free surface flows. Part 1. Numerical calculations. J. Fluid Mech. 168 (-1), 501–513.CrossRefGoogle Scholar
Craster, R.V. & Matar, O.K. 2009 Dynamics and the stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.CrossRefGoogle Scholar
D’Alessio, S.J.D., Pascal, J.P., Ellaban, E. & Ruyer-Quil, C. 2020 Marangoni instabilities associated with heated surfactant-laden falling films. J. Fluid Mech. 887, A20.CrossRefGoogle Scholar
Davies, C. & Carpenter, P.W. 1997 Instabilities in a plane channel flow between compliant walls. J. Fluid Mech. 352, 205–243.CrossRefGoogle Scholar
Edmonstone, B.D., Craster, R.V. & Matar, O.K. 2006 Surfactant-induced fingering phenomena beyond the critical micelle concentration. J. Fluid Mech. 564, 105–138.CrossRefGoogle Scholar
Ehrenstein, U. & Rossi, M. 1993 Nonlinear Tollmien-Schlichting waves for plane Poiseuille flow with compliant walls. Eur. J. Mech. B/Fluids 12, 789–810.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 (4), 983–989.CrossRefGoogle Scholar
Gajjar, J.S.B. & Sibanda, P. 1996 The hydrodynamic stability of channel flow with compliant boundaries. Theor. Comput. Fluid Dyn. 8 (2), 105–129.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
Grotberg, J.B. & Jensen, O.E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36 (1), 121–147.CrossRefGoogle Scholar
Gyorqyfalvy, D. 1967 Possibilities of drag reduction by the use of flexible skin. J. Aircraft 4 (3), 186–192.CrossRefGoogle Scholar
Hains, F.D. & Price, J.F. 1962 Effect of a flexible wall on the stability of Poiseuille flow. Phys. Fluids 5 (3), 365–365.CrossRefGoogle Scholar
Halpern, D. & Grotberg, J.B. 1993 Surfactant effects on fluid-elastic instabilities of liquid- lined flexible tubes-model of airway-closure. J. Biomech. Engng Trans. ASME 120 (3), 271–277.CrossRefGoogle Scholar
Hoepffner, J., Bottaro, A. & Favier, J. 2010 Mechanisms of non-modal energy amplification in channel flow between compliant walls. J. Fluid Mech. 642, 489–507.CrossRefGoogle Scholar
Huang, L. 1998 Instabilities in a plane channel flow between compliant walls. J. Fluids Struct. 12 (2), 131–151.CrossRefGoogle Scholar
Jensen, O. & Grotberg, J. 1993 The spreading of heat or soluble surfactant along a thin liquid film. Phys. Fluids A 5 (1), 58–68.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., Ruyer-Quil, C., Scheid, B. & Velarde, M. 2012 Falling Liquid Films. 1st edn. Springer.CrossRefGoogle Scholar
Kapitza, P.L. 1948 Wave flow of a thin viscous fluid layer. I. Free flow. J. Exp. Theor. Phys. 18, 3–20.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
Katsiavria, A. & Bontozoglou, V. 2020 Stability of liquid film flow laden with the soluble surfactant sodium dodecyl sulphate: predictions versus experimental data. J. Fluid Mech. 894, A18.CrossRefGoogle Scholar
Kramer, M.O. 1957 Boundary-layer stabilization by distributed damping. J. Aero. Sci. 24, 459.Google Scholar
Kumar, S. & Matar, O.K. 2004 Dewetting of thin liquid films near soft elastomeric layers. J. Colloid Interface Sci. 273 (2), 581–588.CrossRefGoogle ScholarPubMed
Landhal, M.T. 1962 On the stability of a laminar incompressible boundary layer over a flexible surface. J. Fluid Mech. 13 (4), 609–632.CrossRefGoogle Scholar
Lin, S.P. 1967 Instability of a liquid film flowing down an inclined plane. Phys. Fluids 10 (2), 308–313.CrossRefGoogle Scholar
Malik, M., Skote, M. & Bouffanais, R. 2018 Growth mechanisms of perturbations in boundary layers over a compliant wall. Phys. Rev. Fluids 3 (1), 013903.CrossRefGoogle Scholar
Matar, O.K., Craster, R.V. & Kumar, S. 2007 Falling films on flexible inclines. Phys. Rev. E 76 (5), 056301.CrossRefGoogle ScholarPubMed
Matar, O.K. & Kumar, S. 2004 Rupture of a surfactant-covered thin liquid film on a flexible wall. SIAM J. Appl. Maths 6 (6), 2144–2166.CrossRefGoogle Scholar
Matar, O.K. & Kumar, S. 2007 Dynamics and stability of flow down a flexible incline. J. Engng Maths 57 (2), 145–158.CrossRefGoogle Scholar
Nepomnyashchy, A., Velarde, M.G. & Colinet, P. 2001 Interfacial Phenomena and Convection. Chapman and Hall/CRC.CrossRefGoogle Scholar
Oron, A., Davis, S.H. & Bankoff, S.G. 1997 Long scale evolution of thin films. Rev. Mod. Phys. 69 (3), 931–980.CrossRefGoogle Scholar
Peng, J., Jiang, L.Y., Zhuge, W.L. & Zhang, Y.J. 2016 Falling film on a flexible wall in the presence of insoluble surfactant. J. Engng Maths 97 (1), 33–48.CrossRefGoogle Scholar
Pereira, A. & Kalliadasis, S. 2008 On the transport equation for an interfacial quantity. Eur. Phys. J. Appl. Phys. 44 (2), 211–214.CrossRefGoogle Scholar
Riley, J.J., el Hak, M.G. & Metcalfe, R.W. 1988 Compliant coatings. Annu. Rev. Fluid Mech. 20 (1), 393–420.CrossRefGoogle Scholar
Rotenberry, J.M. 1992 Finite-amplitude shear waves in channel compliant boundaries. Phys. Fluids 4 (2), 270–276.CrossRefGoogle Scholar
Samanta, A. 2013 a Effect of surfactant on two-layer channel flow. J. Fluid Mech. 735, 519–552.CrossRefGoogle Scholar
Samanta, A. 2013 b Shear wave instability for electrified falling films. Phys. Rev. E 88 (5), 053002.CrossRefGoogle ScholarPubMed
Samanta, A. 2014 Effect of surfactants on the instability of a two-layer film flow down an inclined plane. Phys. Fluids 26 (9), 094105.CrossRefGoogle Scholar
Samanta, A. 2019 Effect of electric field on an oscillatory film flow. Phys. Fluids 31 (3), 034109.CrossRefGoogle Scholar
Samanta, A. 2020 Linear stability of a plane Couette–Poiseuille flow overlying a porous layer. Intl J. Multiphase Flow 123, 103160.CrossRefGoogle Scholar
Samanta, A. 2021 a Effect of surfactant on the linear stability of a shear-imposed fluid flowing down a compliant substrate. J. Fluid Mech. 920, A23.CrossRefGoogle Scholar
Samanta, A. 2021 b Modal analysis of a viscous fluid falling over a compliant wall. Proc. R. Soc. Lond. A: Math. Phys. Engng Sci. 477 (2253), 20210487.Google Scholar
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sen, P.K. & Arora, D.S. 1988 On the stability of laminar boundary-layer flow over a flat plate with a compliant surface. J. Fluid Mech. 197, 201–240.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 1, 43–51.Google Scholar
Sisoev, G.M., Matar, O.K., Craster, R.V. & Kumar, S. 2010 Coherent wave structures on falling fluid films flowing down a flexible wall. Chem. Engng Sci. 65 (2), 950–961.CrossRefGoogle Scholar
Smith, M.K. 1990 The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485.CrossRefGoogle Scholar
Tsvelodub, O.Y. 1977 Stability of plane Poiseuille flow in an elastic channel. Zh. Prikl. Mekh. Tekh. Fiz. 5, 75–80.Google Scholar
Wei, H.H. 2005 Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane. Phy. Fluids 17 (1), 012103.CrossRefGoogle Scholar
Yang, H., Jiang, L.Y., Hu, K.X. & Peng, J. 2018 Numerical study of the surfactant-covered falling film flowing down a flexible wall. Eur. J. Mech. B Fluids 72, 422–431.CrossRefGoogle Scholar
Yih, C.S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6 (3), 321–334.CrossRefGoogle Scholar