Hostname: page-component-54dcc4c588-smtgx Total loading time: 0 Render date: 2025-10-03T20:19:58.150Z Has data issue: false hasContentIssue false
  • English
  • Français

Surface waves in a sheared liquid film on a horizontal plane at moderate Reynolds numbers

Published online by Cambridge University Press:  19 August 2025

Kai-Xin Hu Open the ORCID record for Kai-Xin Hu [Opens in a new window] ,
Kang Du ,
Dan Wu  and
Qi-Sheng Chen
Kai-Xin Hu*
Affiliation:
Zhejiang Provincial Engineering Research Center for the Safety of Pressure Vessel and Pipeline, Ningbo University, Ningbo, Zhejiang 315211, PR China Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education, Ningbo, Zhejiang 315211, PR China
Kang Du
Affiliation:
Zhejiang Provincial Engineering Research Center for the Safety of Pressure Vessel and Pipeline, Ningbo University, Ningbo, Zhejiang 315211, PR China Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education, Ningbo, Zhejiang 315211, PR China
Dan Wu
Affiliation:
Zhejiang Provincial Engineering Research Center for the Safety of Pressure Vessel and Pipeline, Ningbo University, Ningbo, Zhejiang 315211, PR China Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education, Ningbo, Zhejiang 315211, PR China
Qi-Sheng Chen
Affiliation:
School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100190, PR China Key Laboratory of Microgravity, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
*
Corresponding author: Kai-Xin Hu, hjhhkx@126.com
Get access Rights & Permissions [Opens in a new window]

Abstract

When a liquid film on a horizontal plate is driven in motion by a shear stress, surface waves are easily generated. This paper studies such flow at moderate Reynolds numbers, where the surface tension and inertial force are equally important. The governing equations for two-dimensional flows are derived using the long-wave approximation along with the integral boundary-layer theory. For small disturbances, the dispersion relation and neutral curves are determined by the linear stability analysis. For finite-amplitude perturbations, the numerical simulation suggests that the oscillations generated by the perturbation in a certain place continuously spread to the surrounding areas. When the effects of surface tension and gravity reach equilibrium, steady-state solutions will emerge, which include two cases: solitary waves and periodic waves. The former have heteroclinic trajectories between two stationary points, while the latter include five patterns at different parameters. In addition, there are also periodic waves that do not converge after a long period of time. During these evolution processes, strange attractors appear in the phase space. By examining the Poincaré section and the sensitivity to initial values, we demonstrate that these waves can be divided into two types: quasi-periodic and chaotic solutions. The specific type depends on parameters and initial conditions.

JFM classification

Instability: Nonlinear instability Interfacial Flows (free surface): Thin films Nonlinear Dynamical Systems: Chaos

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1017 , 25 August 2025 , A26
Check for updates
Copyright
© The Author(s), 2025. 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.)

Article purchase

Temporarily unavailable

References

Bae, B., Ahn, T., Jeong, J., Kim, K. & Yun, B. 2017 Characteristics of an interfacial wave in a horizontal air–water stratified flow. Intl J. Multiphase Flow 97, 197–205.CrossRefGoogle Scholar
Chang, H.C. & Demekhin, E.A. 2002 Complex Wave Dynamics on Thin Films. Springer.Google Scholar
Craster, R.V. & Matar, O.K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 8l (3), 1131–1198.CrossRefGoogle Scholar
Denner, F., Charogiannis, A., Pradas, M., Markides, C.N., van Wachem, B.G. & Kalliadasis, S. 2018 Solitary waves on falling liquid films in the inertia-dominated regime. J. Fluid Mech. 837, 491–519.CrossRefGoogle Scholar
Frank, A.M. 2006 Shear driven solitary waves on a liquid film. Phys. Rev. E 74 (6), 065301.CrossRefGoogle ScholarPubMed
Hu, K.-X., Du, K. & Chen, Q.-S. 2024 Nonlinear waves in a sheared liquid film on a horizontal plane at small Reynolds numbers. J. Fluid Mech. 999, A14.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (ed.Godreche, C. & Manneville, P.), pp. 81–294. Cambridge University Press.CrossRefGoogle Scholar
Jurman, L.A. & McCready, M.J. 1989 Study of waves on thin liquid films sheared by turbulent gas flows. Phys. Fluids 1 (3), 522–536.CrossRefGoogle Scholar
Kadam, Y., Patibandla, R. & Roy, A. 2023 Wind-generated waves on a water layer of finite depth. J. Fluid Mech. 967, A12.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M.G. 2011 Falling Liquid Films. Springer Science & Business Media.Google Scholar
Kuznetsov, V.V. & Fominykh, E.Y. 2020 Evaporation of a liquid film in a microchannel under the action of a co-current dry gas flow. Microgravity Sci. Technol. 32 (2), 245–258.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
Náraigh, L.Ó, Spelt, P.D.M. & Shaw, S.J. 2013 Absolute linear instability in laminar and turbulent gas–liquid two-layer channel flow. J. Fluid Mech. 714, 58–94.CrossRefGoogle Scholar
Nosoko, T. & Miyara, A. 2004 The evolution and subsequent dynamics of waves on a vertically falling liquid film. Phys. Fluids 16 (4), 1118–1126.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
Oron, A., Gottlieb, O. & Novbari, E. 2009 Numerical analysis of a weighted-residual integral boundary-layer model for nonlinear dynamics of falling liquid films. Eur. J. Mech. B/Fluids 28 (1), 1–36.CrossRefGoogle Scholar
Özgen, S., Degrez, G. & Sarma, G.S.R. 1998 Two-fluid boundary layer stability. Phys. Fluids 10 (11), 2746–2757.CrossRefGoogle Scholar
Park, C.D. & Nosoko, T. 2003 Three-dimensional wave dynamics on a falling film and associated mass transfer. AIChE J. 49 (11), 2715–2727.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.CrossRefGoogle Scholar
Shinan, C., Weidong, Y., Mengjie, S., Mengyao, L. & Qiyu, S. 2019 Investigation on wavy characteristics of shear-driven water film using the planar laser induced fluorescence method. Intl J. Multiphase Flow 118, 242–253.CrossRefGoogle Scholar
Smith, M.K. & Davis, S.H. 1982 The instability of sheared liquid layers. J. Fluid Mech. 121, 187–206.CrossRefGoogle Scholar
Strogatz, S. 2015 Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering. Taylor & Francis Group.Google Scholar
Tomlin, R.J., Papageorgiou, D.T. & Pavliotis, G.A. 2017 Three-dimensional wave evolution on electrified falling films. J. Fluid Mech. 822, 54–79.CrossRefGoogle Scholar
Tseluiko, D. & Kalliadasis, S. 2011 Nonlinear waves in counter-current gas–liquid film flow. J. Fluid Mech. 673, 19–59.CrossRefGoogle Scholar
Tsvelodub, O. & Arkhipov, D. 2013 Nonlinear wave simulation on a surface of liquid film entrained by turbulent gas flow at weightlessness. Microgravity Sci. Technol. 25, 179–186.CrossRefGoogle Scholar
Tsvelodub, O.Y. & Bocharov, A.A. 2018 Simulating solitary waves on a flowing down liquid film entrained by a gas stream. Eur. J. Mech. B/Fluids 72, 449–455.CrossRefGoogle Scholar
Tsvelodub, O.Y. & Vozhakov, I.S. 2022 Waves on the surface of a thin liquid film entrained by a turbulent gas flow in a narrow channel. Eur. J. Mech. B/Fluids 96, 78–89.CrossRefGoogle Scholar
Valluri, P., Matar, O.K., Hewitt, G.F. & Mendes, M.A. 2005 Thin film flow over structured packings at moderate Reynolds numbers. Chem. Engng Sci. 60, 1965–1975.CrossRefGoogle Scholar
Vellingiri, R., Tseluiko, D. & Kalliadasis, S. 2015 Absolute and convective instabilities in counter-current gas–liquid film flows. J. Fluid Mech. 763, 166–201.CrossRefGoogle Scholar
Vozhakov, I.S., Arkhipov, D.G. & Tsvelodub, O.Y. 2015 Simulating nonlinear waves on the surface of thin liquid film entrained by turbulent gas flow. Thermophys. Aeromech. 22 (2), 191–202.CrossRefGoogle Scholar
Vozhakov, I.S., Cherdantsev, A.V. & Arkhipov, D.G. 2016 Modelling secondary instability of co-current a thin gas-sheared film. Fluid Dyn. Res. 48 (6), 061420.CrossRefGoogle Scholar
Wray, A.W., Matar, O.K. & Papageorgiou, D.T. 2017 Accurate low-order modeling of electrified falling films at moderate Reynolds number. Phys. Rev. Fluids 2 (6), 063701.CrossRefGoogle Scholar
Yuan, C., Li, J., Jiang, Z., Chen, X. & Zhang, Y. 2019 Experimental investigation of liquid film cooling in hypersonic flow. Phys. Fluids 31 (4), 041701.CrossRefGoogle Scholar
Supplementary material: File

Hu et al. supplementary movie

Movie 1
Download Hu et al. supplementary movie(File)
File 38.5 MB