Skip to main content Accessibility help

Nonlinear dynamics of electrostatic Faraday instability in thin films

  • Dipin S. Pillai (a1) and R. Narayanan (a1)


The nonlinear evolution of an interface between a perfect conducting liquid and a perfect dielectric gas subject to periodic electrostatic forcing is studied under the long-wave approximation. It is shown that inertial thin films become unstable to finite-wavelength Faraday modes at the onset, prior to the long-wave pillaring instability reported in the lubrication limit. It is further shown that the pillaring-mode instability is subcritical in nature, with the interface approaching either the top or the bottom wall, depending on the liquid–gas holdup. On the other hand, the Faraday modes exhibit subharmonic or harmonic oscillations that nonlinearly saturate to standing waves at low forcing amplitudes. Unlike the pillaring mode, wherein the interface approaches the wall, Faraday modes may exhibit saturated standing waves when the instability is subcritical. At higher forcing amplitudes, the interface may approach either wall, again depending on the liquid–gas holdup. It is also shown that a gravitationally unstable configuration of such thin films, under the long-wave approximation, cannot be stabilized by periodic electrostatic forcing, unlike mechanical Faraday forcing. In this case, it is observed that the interface exhibits oscillatory sliding behaviour, approaching the wall in an ‘earthworm-like’ motion.


Corresponding author

Email address for correspondence:


Hide All
Aylward, G. H. & Findlay, T. J. V. 1994 S.I. Chemical Data, 3rd edn. Wiley.
Bandopadhyay, A. & Hardt, S. 2017 Stability of horizontal viscous fluid layers in a vertical arbitrary time periodic electric field. Phys. Fluids 29, 124101.
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 121, 299340.
Bestehorn, M. 2013 Laterally extended thin liquid films with inertia under external vibrations. Phys. Fluids 25, 114106.
Cimpeanu, R., Papageorgiou, D. T. & Petropoulos, P. G. 2014 On the control and suppression of the Rayleigh–Taylor instability using electric fields. Phys. Fluids 26, 022105.
Craster, R. V. & Matar, O. 2005 Electrically induced pattern formation in thin leaky dielectric films. Phys. Fluids. 17, 032104.
Cross, M. & Greenside, H. 2009 Pattern Formation and Dynamics in Nonequilibrium Systems. Cambridge University Press.
Dietze, G. & Ruyer-Quil, C. 2015 Film in narrow tubes. J. Fluid Mech. 762, 68109.
Dietze, G. F. & Ruyer-Quil, C. 2013 Wavy liquid films in interaction with a confined laminar gas flow. J. Fluid Mech. 722, 348393.
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123148.
Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. A 121, 299340.
Gambhire, P. & Thaokar, R. M. 2010 Electrohydrodynamic instabilities at interfaces subjected to alternating electric field. Phys. Fluids 22, 064103.
Glasner, K. B. 2007 The dynamics of pendant droplets on a one-dimensional surface. Phys. Fluids. 19, 102104.
Israelachvili, J. N. 2011 Intermolecular and Surface Forces. Academic Press.
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452, 11131126.
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4968.
Kumar, S. & Matar, O. K. 2002 Instability of long-wavelength disturbances on gravity-modulated surfactant-covered thin liquid layers. J. Fluid Mech. 466, 249258.
Labrosse, G. 2011 Méthodes numèriques: Méthodes spectrale: Méthodes locales, méthodes globales, problèmes d’Helmotz et de Stokes, équations de Navier–Stokes. Ellipses Marketing.
Lister, J. R., Morrison, N. F. & Rallison, J. M. 2006 Sedimentation of a two-dimensional drop towards a rigid horizontal plate. J. Fluid Mech. 552, 345351.
Lister, J. R., Rallison, J. M. & Rees, S. J. 2010 The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling. J. Fluid Mech. 647, 239264.
Matar, O. K., Kumar, S. & Craster, R. V. 2004 Nonlinear parametric excited surface waves in surfactant-covered thin liquid films. J. Fluid Mech. 520, 243265.
Müller, H. W. 1993 Periodic triangular patterns in the Faraday experiment. Phys. Rev. Lett. 20, 32873290.
Nayfeh, A. H. 1981 Introduction to Perturbation Techniques. Wiley.
Rayleigh, L. 1833 On the crispations of fluid resting upon a vibrating support. Phil. Mag. 16, 5058.
Roberts, S. & Kumar, S. 2009 AC electrohydrodynamic instabilities in thin liquid films. J. Fluid Mech. 631, 255279.
Rojas, N. O., Argentina, M., Erda, C. & Tirapegui, E. 2010 Inertial lubrication theory. Phys. Rev. Lett. 104, 187801.
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, 170183.
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 2764.
Sterman-Cohen, E., Bestehorn, M. & Oron, A. 2017 Rayleigh–Taylor instability in thin liquid films subjected to harmonic vibration. Phys. Fluids. 29, 052105.
Suman, B. & Kumar, S. 2008 Surfactant- and elasticity-induced inertialess instabilities in vertically vibrated liquids. J. Fluid Mech. 610, 407423.
Thaokar, R. M. & Kumaran, V. 2005 Electrohydrodynamic instability of the interface between two fluids confined in a channel. Phys. Fluids. 17 (8), 084104.
Tsai, S. C. & Tsai, C. S. 2013 Linear theory on temporal instability of Megahertz Faraday waves for monodisperse microdroplet ejection. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 60, 17461755.
Ward, K., Matsumoto, S. & Narayanan, R. 2018 The electrostatically forced Faraday instability – theory and experiments. J. Fluid Mech. (submitted).
Ward, K. L.2018 Faraday instability in mechanically and electrostatically forced systems. PhD thesis, University of Florida.
Wu, L. & Chou, S. Y. 2003 Dynamic modeling and scaling of nanostructure formation in the lithographically induced self-assembly and self-construction. Appl. Phys. Lett. 82 (19), 32003202.
Yih, C. S. 1968 Stability of horizontal fluid interface in a periodic vertical electric field. Phys. Fluids 11, 14471449.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Nonlinear dynamics of electrostatic Faraday instability in thin films

  • Dipin S. Pillai (a1) and R. Narayanan (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