We consider thin liquid films on a horizontal, solid, and completely wetting substrate. The substrate is subjected to oscillatory accelerations in the normal direction and/or in the horizontal direction. A linear Floquet analysis shows that the planar film surface loses stability if amplitudes and frequencies of the harmonic oscillations meet certain criteria. Based on the long wave lubrication approximation, we present an integral boundary layer model where the z component is integrated out and the spatial dimension is reduced by one. The linear stability analysis of this model shows good agreement with the exact problem and with the linearized long wave equations. Pattern formation in the nonlinear regime is computed numerically from the long wave model in two and three spatial dimensions. Normal oscillations show the traditional Faraday patterns such as squares and hexagons. Lateral oscillations cause a pattern formation scenario similar to spinodal dewetting, namely coarsening and no rupture. For certain amplitude and frequency ranges, combined lateral and normal oscillations can give rise to one or more traveling drops. Finally, we discuss the control of a drop's motion in the horizontal plane.
REFERENCES
1.
M.
Faraday
, “On the forms and states by fluids in contact with vibrating elastic surfaces
,” Philos. Trans. R. Soc. London
121
, 319
(1831
).2.
T. B.
Benjamin
and F.
Ursell
, “The stability of the plane free surface of a liquid in vertical periodic motion
,” Proc. R. Soc. London, Ser. A
225
, 505
(1954
).3.
A.
Stephenson
, “On a new type of dynamical stability
,” Memoirs and Proceedings of the Manchester Literary and Philosophical Society
52
, 1
(1908
).4.
K.
Kumar
and L. S.
Tuckerman
, “Parametric instability of the interface between two fluids
,” J. Fluid Mech.
279
, 49
(1994
).5.
J.
Beyer
and R.
Friedrich
, “Faraday instability: linear analysis for viscous fluids
,” Phys. Rev. E
51
, 1162
(1995
).6.
J.
Bechhoefer
, V.
Ego
, S.
Manneville
, and B.
Johnson
, “An experimental study of the onset of parametrically pumped surface waves in viscous fluids
,” J. Fluid Mech.
288
, 325
(1995
).7.
C.
Wagner
, H.-W.
Müller
, and K.
Knorr
, “Pattern formation at the bicritical point of the Faraday instability
,” Phys. Rev. E
68
, 066204
(2003
).8.
W. S.
Edwards
and S.
Fauve
, “Patterns and quasi-patterns in the Faraday experiment
,” J. Fluid Mech.
278
, 123
(1994
).9.
B.
Christian
, P.
Alstrom
, and M. T.
Levinsen
, “Ordered capillary-wave states: quasicrystals, hexagons and radial waves
,” Phys. Rev. Lett.
68
, 2157
(1992
).10.
W. S.
Edwards
and S.
Fauve
, “Parametrically excited quasicrystalline surface waves
,” Phys. Rev. E
47
, R788
(1993
).11.
Y.
Ding
and P.
Umbanhowar
, “Enhanced Faraday pattern stability with three-frequency driving
,” Phys. Rev. E
73
, 046305
(2006
).12.
C.-S.
Yih
, “Instability of unsteady flows or configurations
,” J. Fluid Mech.
31
, 737
(1968
).13.
C.-S.
Yih
, “Stability of liquid flow down an inclined plane
,” Phys. Fluids
6
, 321
(1963
).14.
T. B.
Benjamin
, “Wave formation in laminar flow down an inclined plane
,” J. Fluid Mech.
2
, 554
(1957
).15.
A. C.
Or
, “Finite-wavelength instability in a horizontal liquid layer on an oscillating plane
,” J. Fluid Mech.
335
, 213
(1997
).16.
S.
Shklyaev
, A. A.
Alabuzhev
, and M.
Khenner
, “Influence of a longitudinal and tilted vibration on stability and dewetting of a liquid film
,” Phys. Rev. E
79
, 051603
(2009
).17.
E. S.
Benilov
and M.
Chugunova
, “Waves in liquid films on vibrating substrates
,” Phys. Rev. E
81
, 036302
(2010
).18.
J.
Porter
, I.
Tinao
, A.
Laveron-Simavilla
, and C. A.
Lopez
, “Pattern selection in a horizontally vibrated container
,” Fluid Dyn. Res.
44
, 065501
(2012
).19.
D.
Bensimon
, P.
Kolodner
, C. M.
Surko
, H.
Williams
, and V.
Croquette
, “Competing and coexisting dynamical states of travelling-wave convection in an annulus
,” J. Fluid Mech.
217
, 441
(1990
).20.
P.
Chen
and K.-A.
Wu
, “Subcritical bifurcations and nonlinear balloons in Faraday waves
,” Phys. Rev. Lett.
85
, 3813
(2000
).21.
P.
Chen
, “Nonlinear wave dynamics in Faraday instabilities
,” Phys. Rev. E
65
, 036308
(2002
).22.
S.
Ubal
, M. D.
Giavedoni
, and F. A.
Saita
, “A numerical analysis of the influence of the liquid depth on two-dimensional Faraday waves
,” Phys. Fluids
15
, 3099
(2003
).23.
N.
Perinet
, D.
Juric
, and L. S.
Tuckerman
, “Numerical simulation of Faraday waves
,” J. Fluid Mech.
635
, 1
(2009
).24.
A.
Oron
, S. H.
Davis
, and S. G.
Bankoff
, “Long-scale evolution of thin liquid films
,” Rev. Mod. Phys.
69
, 931
(1997
).25.
N. O.
Rojas
, M.
Argentina
, E.
Cerda
, and E.
Tirapegui
, “Inertial lubrication theory
,” Phys. Rev. Lett.
104
, 187801
(2010
).26.
P. G.
de Gennes
, “Wetting: statics and dynamics
,” Rev. Mod. Phys.
57
, 827
(1985
).27.
If we speak about drops, we always mean a separated local elevation of the surface with small contact angle. Within lubrication theory, the surface remains a unique function of the horizontal coordinate(s).
28.
P.
Brunet
, J.
Eggers
, and R. D.
Deegan
, “Vibration-induced climbing of drops
,” Phys. Rev. Lett.
99
, 144501
(2007
).29.
E. S.
Benilov
, “Drops climbing uphill on a slowly oscillating substrate
,” Phys. Rev. E
82
, 026320
(2010
).30.
E. S.
Benilov
and C. P.
Cummins
, “Thick drops on a slowly oscillating substrate
,” Phys. Rev. E
88
, 023013
(2013
).31.
W. H.
Press
, B. P.
Flannery
, S. A.
Teukolsky
, and W. T.
Vetterling
, Numerical Recipes
, 3rd ed. (Cambridge University Press
, New York, USA
, 2007
).32.
E.
Anderson
, Z.
Bai
, C.
Bischof
, S.
Blackford
, J.
Demmel
, J.
Dongarra
, J.
Du Croz
, A.
Greenbaum
, S.
Hammarling
, A.
McKenney
, and D.
Sorensen
, LAPACK Users’ Guide
, 3rd ed. (Society for Industrial and Applied Mathematics
, Philadelphia, USA
, 1999
).33.
G.
Pucci
, E.
Fort
, M.
Ben Amar
, and Y.
Couder
, “Mutual adaptation of a Faraday instability pattern with its flexible boundaries in floating fluid drops
,” Phys. Rev. Lett.
106
, 024503
(2011
).34.
R. V.
Craster
and O. K.
Matar
, “Dynamics and stability of thin liquid films
,” Rev. Mod. Phys.
81
, 1131
(2009
).35.
C.
Ruyer-Quil
and P.
Manneville
, “Improved modeling of flows down inclined planes
,” Eur. Phys, J. B
15
, 357
(2000
).36.
W.
Guo
, G.
Labrosse
, and R.
Narayanan
, The Application of the Chebyshev-Spectral Method in Transport Phenomena
(Springer Press
, Heidelberg
, 2012
).37.
A.
Oron
, O.
Gottlieb
, and E.
Novbari
, “Weighted-residual integral boundary-layer model of temporally excited falling liquid films
,” Eur. J. Mech. B/Fluids
28
, 37
(2009
).38.
N. O.
Rojas
, M.
Argentina
, E.
Cerda
, and E.
Tirapegui
, “Faraday patterns in lubricated thin films
,” Eur. Phys. J. D
62
, 25
(2011
).39.
M.
Bestehorn
, A.
Pototsky
, and U.
Thiele
, “3D large scale Marangoni convection in liquid films
,” Eur. Phys. J. B
33
, 457
(2003
).40.
A.
Oron
, “Nonlinear dynamics of three-dimensional long-wave Marangoni instability in thin liquid films
,” Phys. Fluids
12
, 1633
(2000
).41.
M.
Bestehorn
and K.
Neuffer
, “Surface patterns of laterally extended thin liquid films in three dimensions
,” Phys. Rev. Lett.
87
, 046101
(2001
).42.
K.
John
and U.
Thiele
, “Self-ratcheting Stokes drops by oblique vibrations
,” Phys. Rev. Lett.
104
, 107801
(2010
).43.
M. P. K.
Kundu
, I. M.
Cohen
, and D. R.
Dowling
, Fluid Mechanics
, 5th ed. (Academic Press
, Waltham, USA
, 2012
).© 2013 AIP Publishing LLC.
2013
AIP Publishing LLC
You do not currently have access to this content.
