Skip to main content Accessibility help

An extended Landau–Levich model for the dragging of a thin liquid film with a propagating surface acoustic wave

  • Matvey Morozov (a1) and Ofer Manor (a1)


In this paper we revisit the Landau and Levich analysis of a coating flow in the case where the flow in the thin liquid film is supported by a Rayleigh surface acoustic wave (SAW), propagating in the solid substrate. Our theoretical analysis reveals that the geometry of the film evolves under the action of the propagating SAW in a manner that is similar to the evolution of films that are being deposited using the dip coating technique. We show that in a steady state the thin-film evolution equation reduces to a generalized Landau–Levich equation with the dragging velocity, imposed by the SAW, depending on the local film thickness. We demonstrate that the generalized Landau–Levich equation has a branch of stable steady state solutions and a branch of unstable solutions. The branches meet at a saddle-node bifurcation point corresponding to the threshold value of the SAW intensity. Below the threshold value no steady states were found and our numerical computations suggest a gradual thinning of the liquid film from its initial geometry.


Corresponding author

Email address for correspondence:


Hide All
Afanasiev, K., Münch, A. & Wagner, B. 2007 Landau–Levich problem for non-newtonian liquids. Phys. Rev. E 76, 036307.
Ajaev, V. S. & Homsy, G. M. 2006 Modeling shapes and dynamics of confined bubbles. Annu. Rev. Fluid Mech. 38, 277303.
Altshuler, G. & Manor, O. 2015 Spreading dynamics of a partially wetting water film atop a MHz substrate vibration. Phys. Fluids 27, 102103.
Altshuler, G. & Manor, O. 2016 Free films of a partially wetting liquid under the influence of a propagating MHz surface acoustic wave. Phys. Fluids 28, 072102.
Aussillous, P. & Quéré, D. 2000 Quick deposition of a fluid on the wall of a tube. Phys. Fluids 12, 23672371.
Benilov, E. S., Chapman, S. J., McLeod, J. B., Ockendon, J. R. & Zubkov, V. S. 2010 On liquid films on an inclined plate. J. Fluid Mech. 663, 5369.
Benny, D. 1966 Long waves on liquid films. J. Math. Phys. 45, 150155.
Beyer, R. T. 1974 Nonlinear Acoustics. Navy Sea Systems Command.
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods. Dover.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166188.
Campana, D. M., Ubal, S., Giavedoni, M. D. & Saita, F. A. 2010 Numerical prediction of the film thickening due to surfactants in the Landau–Levich problem. Phys. Fluids 22, 032103.
Chan, D. Y. C., Klaseboer, E. & Manica, R. 2011 Film drainage and coalescence between deformable drops and bubbles. Soft Matt. 7, 22352264.
Dixit, H. N. & Homsy, G. M. 2013 The elastic Landau–Levich problem. J. Fluid Mech. 732, 528.
Eckart, C. 1948 Vortices and streams caused by sound waves. Phys. Rev. 73, 6876.
Friend, J. & Yeo, L. Y. 2011 Microscale acoustofluidics: Microfluidics driven via acoustics and ultrasonics. Rev. Mod. Phys. 83, 647704.
Huerre, A., Miralles, V. & Joullien, M. C. 2014 Bubbles and foams in microfluidics. Soft Matt. 10, 68886902.
Klaseboer, E., Gupta, R. & Manica, R. 2014 An extended Bretherton model for long Taylor bubbles at moderate capillary numbers. Phys. Fluids 26, 032107.
Krantz, W. & Goren, S. 1970 Finite-amplitude, long waves on liquid films flowing down a plane. Ind. Engng Chem. Fundam. 9, 107113.
Krechetnikov, R. & Homsy, G. M. 2006 Surfactant effects in the Landau–Levich problem. J. Fluid Mech. 559, 429450.
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochem. URSS 17, 141153.
Lee, C. P. & Wang, T. G. 1993 Acoustic radiation pressure. J. Acoust. Soc. Am. 94, 10991109.
Manor, O., Rezk, A. R., Friend, J. R. & Yeo, L. Y. 2015 Dynamics of liquid films exposed to high-frequency surface vibration. Phys. Rev. E 91, 053015.
Nyborg, W. L. 1953 Acoustic streaming due to attenuated plane waves. J. Acoust. Soc. Am. 25, 6875.
Park, C. W. 1991 Effects of insoluble surfactants on dip coating. J. Colloid Interface Sci. 146, 382394.
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele Shaw cells: theory. J. Fluid Mech. 139, 291308.
Parker, D. F. 1988 Stratification effects on nonlinear elastic surface waves. Phys. Earth Planet. Inter. 50, 1625.
Quéré, D. 1999 Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31, 347384.
Rayleigh Lord 1884 On the circulation of air observed in Kundt’s tubes, and on some allied acoustical problems. Phil. Trans. R. Soc. Lond. 175, 121.
Rezk, A. R., Manor, O., Friend, J. R. & Yeo, L. Y. 2012 Unique fingering instabilities and soliton-like wave propagation in thin acoustowetting films. Nat. Commun. 3, 1167.
Rezk, A. R., Manor, O., Yeo, L. Y. & Friend, J. R. 2014 Double flow reversal in thin liquid films driven by megahertz-order surface vibration. Proc. R. Soc. Lond. A 470, 20130765.
Ruschak, K. J. 1985 Coating flows. Annu. Rev. Fluid Mech. 17, 6589.
Schwartz, L. W., Princen, H. M. & Kiss, A. D. 1986 On the motion of bubbles in capillary tubes. J. Fluid Mech. 172, 259275.
Shklyaev, S., Alabuzhev, A. A. & Khenner, M. 2009 Influence of a longitudinal and tilted vibration on stability and dewetting of a liquid film. Phys. Rev. E 79, 051603.
Snoeijer, J. H., Ziegler, J., Andreotti, B., Fermigier, M. & Eggers, J. 2008 Thick films of viscous fluid coating a plate withdrawn from a liquid reservoir. Phys. Rev. Lett. 100, 244502.
Spencer, A. J. M. 1970 The static theory of finite elasticity. IMA J. Appl. Maths 6 (2), 164200.
Spiers, R. P., Subbaraman, C. V. & Wilkinson, W. L. 1975 Free coating of non-Newtonian liquids onto a vertical surface. Chem. Engng Sci. 30, 379395.
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 2953.
Wilmanski, B. & Albers, K. 2014 Continuum Thermodynamics: Part II Applications and Examples, Series on Advances in Mathematics for Applied Sciences, vol. 85. World Scientific.
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16, 209221.
Wong, H., Fatt, I. & Radke, C. J. 1996 Deposition and thinning of the human tear film. J. Colloid Interface Sci. 184, 4451.
Wong, H., Radke, C. J. & Morris, S. 1995a The motion of long bubbles in polygonal capillaries. Part 1. Thin films. J. Fluid Mech. 292, 7194.
Wong, H., Radke, C. J. & Morris, S. 1995b The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow. J. Fluid Mech. 292, 95110.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

An extended Landau–Levich model for the dragging of a thin liquid film with a propagating surface acoustic wave

  • Matvey Morozov (a1) and Ofer Manor (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.