We study the longwave Marangoni convection in two-layer films under the influence of a low frequency vibration. A linear stability analysis is performed by means of the Floquet theory. A competition of subharmonic, synchronous, and quasiperiodic modes is considered. It has been found that the monotonic instability, which exists at constant gravity, is transformed into a synchronous instability, which is critical in a wide range of vibration amplitude. At parameters where oscillatory instability exists, the longwave quasiperiodic mode remains critical until a subharmonic mode becomes critical with the growth of the vibration amplitude.
REFERENCES
1.
M.
Faraday
, “On a peculiar class of acoustic figures; and on certain forms assumed by groups of particles upon vibrating elastic surfaces
,” Philos. Trans. R. Soc. London
121
, 299
(1831
).2.
D. V.
Lyubimov
, T. P.
Lyubimova
, and A. A.
Cherepanov
, Dynamics of Interfaces in Vibration Fields
(Fizmatlit
, Moscow
, 2004
) (in Russian).3.
F. J.
Mancebo
and J. M.
Vega
, “Faraday instability threshold in large-aspect-ratio containers
,” J. Fluid Mech.
467
, 307
(2002
).4.
G. Z.
Gershuni
and D. V.
Lyubimov
, Thermal Vibrational Convection
(Wiley
, New York
, 1998
).5.
P. M.
Gresho
and R. L.
Sani
, “The effect of gravity modulation on the stability of a heated fluid layer
,” J. Fluid Mech.
40
, 783
(1970
).6.
G. Z.
Gershuni
, E. M.
Zhukhovitskii
, and I. S.
Iurkov
, “On the convective stability in the presence of periodically varying parameter
,” J. Appl. Math. Mech.
34
, 442
(1970
).7.
G.
Venezian
, “Effect of modulation on the onset of thermal convection
,” J. Fluid Mech.
35
, 243
(1969
).8.
J. R.
Rogers
, W.
Pesch
, O.
Brausch
, and M. F.
Schatz
, “Complex-ordered patterns in shaken convection
,” Phys. Rev. E
71
, 066214
(2005
).9.
Y.
Shu
, B. Q.
Li
, and B. R.
Ramaprian
, “Convection in modulated thermal gradients and gravity: Experimental measurements and numerical simulations
,” Int. J. Heat Mass Transfer
48
, 145
(2005
).10.
B. L.
Smorodin
, G. Z.
Gershuni
, and M. G.
Velarde
, “On the parametric excitation of thermoelectric instability in a liquid layer open to air
,” Int. J. Heat and Mass Transfer
42
, 3159
(1999
).11.
R. V.
Birikh
, V. A.
Briskman
, A. A.
Cherepanov
, and M. G.
Velarde
, “Faraday ripples, parametric resonance, and the Marangoni effect
,” J. Colloid Interface Sci.
238
, 16
(2001
).12.
J. R. L.
Skarda
, “Instability of a gravity-modulated fluid layer with surface tension variation
,” J. Fluid Mech.
434
, 243
(2001
).13.
B. L.
Smorodin
, A. B.
Mikishev
, A. A.
Nepomnyashchy
, and B. I.
Myznikova
, “Thermocapillary instability of a liquid layer under heat flux modulation
,” Phys. Fluids
21
, 062102
(2009
).14.
V. I.
Yudovich
, S. M.
Zenkovskaya
, V. A.
Novossiadliy
, and A. L.
Shleykel
, “Parametric excitation of waves on a free boundary of a horizontal fluid layer
,” C. R. Mec.
332
, 257
(2004
).15.
S. M.
Zenkovskaya
, V. A.
Novosyadlyi
, and A. L.
Shleikel
, “The effect of vertical vibration on the onset of thermocapillary convection in a horizontal liquid layer
,” J. Appl. Math. Mech.
71
, 247
(2007
).16.
U.
Thiele
, J. M.
Vega
, and E.
Knobloch
, “Long-wave Marangoni instability with vibration
,” J. Fluid Mech.
546
, 61
(2006
).17.
P.
Colinet
, J.-C.
Legros
, and M. G.
Velarde
, Nonlinear Dynamics of Surface-Tension-Driven Instabilities
(Wiley-VCH
, Berlin
, 2001
).18.
A. A.
Nepomnyashchy
, M. G.
Velarde
, and P.
Colinet
, Interfacial Phenomena and Convection
(Chapman and Hall/CRC Press
, London
, 2001
).19.
R. V.
Birikh
, V. A.
Briskman
, M. G.
Velarde
, and J.-C.
Legros
, Liquid Interfacial Systems. Oscillations and Instability
(Marcel Dekker
, New York, Basel
, 2003
).20.
S. M.
Zenkovskaya
and V. A.
Novosyadlyi
, “Averaging method and long-wave asymptotics in vibrational convection in layers with an interface
,” J. Eng. Math.
69
, 277
(2010
).21.
A. A.
Nepomnyashchy
and I. B.
Simanovskii
, “Marangoni instability in ultrathin two layer films
,” Phys. Fluids
19
, 122103
(2007
).22.
A. A.
Nepomnyashchy
and I. B.
Simanovskii
, “Nonlinear Marangoni waves in a two-layer film in the presence of gravity
,” Phys. Fluids
24
, 032101
(2012
).23.
A. A.
Nepomnyashchy
and I. B.
Simanovskii
, “The influence of vibration on Marangoni waves in two-layer films
,” J. Fluid Mech.
726
, 476
–496
(2013
).24.
J. W.
Scanlon
and L. A.
Segel
, “Finite amplitude cellular convection induced by surface tension
,” J. Fluid Mech.
30
(1
), 149
(1967
).25.
S. J.
Van Hook
, M. F.
Schatz
, J. B.
Swift
, W. D.
McCormic
, and H. L.
Swinney
, “Long-wavelength surface-tension-driven Bénard convection: Experiment and theory
,” J. Fluid Mech.
345
, 45
(1997
).26.
A.
Oron
, S. H.
Davis
, and S. G.
Bankoff
, “Long-scale evolution of thin liquid films
,” Rev. Mod. Phys.
69
, 931
(1997
).27.
S. H.
Davis
, “Thermocapillary instabilities
,” Annu. Rev. Fluid Mech.
19
, 403
(1987
).28.
I.
Nehati
, M.
Dietzel
, and S.
Hardt
, “Conjugated liquid layers driven by the short-wavelength Bénard-Marangoni instability: Experiment and numerical simulation
,” J. Fluid Mech.
783
, 46
(2015
).29.
Ph.
Géoris
, M.
Hennenberg
, G.
Lebon
, and J. C.
Legros
, “Investigation of thermocapillary convection in a three-liquid-layer system
,” J. Fluid Mech.
389
, 209
(1999
).30.
A.
Prakash
and J. N.
Koster
, “Convection in multiple layers of immiscible liquids in a shallow cavity I. Steady natural convection
,” Int. J. Multiphase Flow
20
, 383
(1994
).31.
C. A.
Burkersroda
, A.
Prakash
, and J. N.
Koster
, “Interfacial tension between fluorinert liquids and silicone oil
,” Microgravity Q.
4
, 93
(1994
).32.
M. M.
Degen
, P. W.
Colovas
, and C. D.
Andereck
, “Time-dependent patterns in the two-layer Rayleigh-Bénard systems
,” Phys. Rev. E
57
, 6647
(1998
).33.
B.
Zhou
, Q.
Liu
, and Z.
Tang
, “Rayleigh-Marangoni-Bénard instability in two-layer fluid system
,” Acta Mech. Sin.
20
, 366
(2004
).34.
35.
B. L.
Smorodin
and M. G.
Velarde
, “Electrothermoconvective instability of an ohmic liquid layer in an unsteady electric field
,” J. Electrost.
48
(3-4
), 261
–277
(2000
).36.
B. L.
Smorodin
, B. I.
Myznikova
, and J. C.
Legros
, “Evolution of convective patterns in a binary-mixture layer subjected to a periodical change of the gravity field
,” Phys. Fluids
20
, 094102
(2008
).© 2018 Author(s).
2018
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