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The electrostatically forced Faraday instability: theory and experiments

Published online by Cambridge University Press:  14 January 2019

Kevin Ward Open the ORCID record for Kevin Ward [Opens in a new window] ,
Satoshi Matsumoto  and
Ranga Narayanan
Kevin Ward*
Affiliation:
University of Florida, Department of Chemical Engineering, Gainesville, FL 32611, USA University of Lille, IEMN CNRS 8520, 59655, Lille, France
Satoshi Matsumoto
Affiliation:
Human Spaceflight Technology Directorate, Japan Aerospace Exploration Agency, Tsukuba, Ibaraki 305-8505, Japan
Ranga Narayanan*
Affiliation:
University of Florida, Department of Chemical Engineering, Gainesville, FL 32611, USA
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
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Abstract

The onset of interfacial instability in two-fluid systems using a viscous, leaky dielectric model is studied. The instability arises as a result of resonance between the parametric frequency of an imposed electric field and the system’s natural frequency. In addition to a rigorous model that uses Floquet instability analysis, where both viscous and charge effects are considered, this study also provides convincing validating experiments. In other results, it is shown that (a) the imposition of a periodic electrostatic potential acts to counter gravity and this countering effect becomes more effective if a DC voltage is also added, (b) a critical DC voltage exists at which the interface becomes unstable such that no parametric frequency is required to completely destabilize the interface and (c) the leaky dielectric model approaches a model for a perfect dielectric/perfect conductor pair as the conductivity ratio becomes large. It is also shown via experiments that parametric resonant instability using electrostatic forcing may be reliably used to estimate interfacial tension to sufficient accuracy.

JFM classification

Waves/Free-surface Flows: Faraday waves Instability: Parametric instability Nonlinear Dynamical Systems: Pattern formation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 696 - 731
Copyright
© 2019 Cambridge University Press 

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References

Abramson, H. N., Dodge, F. T. & Kana, D. 1965 Liquid surface oscillations in longitudinally excited rigid cylindrical containers. AIAA J. 3 (4), 685–695.Google Scholar
Adamson, A. W. & Gast, A. P. 1997 Physical Chemistry of Surfaces, 6th edn. Wiley.Google Scholar
Amiroudine, S., Zoueshtiagh, F. & Narayanan, R. 2012 Mixing generated by Faraday instability between miscible liquids. Phys. Rev. E 85, 016326.Google Scholar
Bandopadhyay, A. & Hardt, S. 2017 Stability of horizontal viscous fluid layers in a vertical arbitrary time periodic electric field. Phys. Fluids 29, 124101.10.1063/1.4999429Google Scholar
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013a Dual role of gravity on the Faraday threshold for immiscible viscous layers. Phys. Rev. E 88, 063002.Google Scholar
Batson, W., Zoueshtiagh, F. & Narayanan, R. 2013b The Faraday threshold in small cylinders and the sidewall non-ideality. J. Fluid Mech. 729, 496–523.10.1017/jfm.2013.324Google Scholar
Bechhoefer, J., Ego, V., Manneville, S. & Johnson, B. 1995 An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech. 288 (325–350).10.1017/S0022112095001169Google Scholar
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 225 (1163), 505–515.Google Scholar
Boukra, M., Cartellier, A., Ducasse, É., Gajan, P., Lalo, M., Noel, T. & Strzelecki, A. 2009 Use of Faraday instabilities to enhance fuel pulverisation in air-blast atomisers. C. R. Méc. 337 (6), 492–503.10.1016/j.crme.2009.06.027Google Scholar
Briskman, V. A. & Shaidurov, G. F. 1968 Parametric instability of a fluid surface in an alternating electric field. Sov. Phys. Dokl. 13, 540–542.Google Scholar
Cerda, E. A. & Tirapegui, E. L. 1998 Faraday’s instability in viscous fluid. J. Fluid Mech. 368, 195–228.10.1017/S0022112098001578Google Scholar
Ciliberto, S. & Gollub, J. P. 1985 Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. 158, 381–398.10.1017/S0022112085002701Google Scholar
Craster, R. V. & Matar, O. K. 2005 Electrically induced pattern formation in thin leaky dielectric films. Phys. Fluids 17, 032104.10.1063/1.1852459Google Scholar
Devitt, E. B. & Melcher, J. R. 1965 Surface electrohydrodynamics with high-frequency fields. Phys. Fluids 8 (6), 1193–1195.10.1063/1.1761377Google Scholar
Dong, J., de Almeida, V. F. & Tsouris, C. 2001 Formation of liquid columns on liquid–liquid interfaces under applied electric fields. J. Colloid Interface Sci. 242, 327–336.10.1006/jcis.2001.7845Google Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383–409.10.1017/S0022112090003603Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123–148.10.1017/S0022112094003642Google Scholar
Espin, L., Corbett, A. & Kumar, S. 2013 Electrohydrodynamic instabilities in thin viscoelastic films: AC and DC fields. J. Non-Newtonian Fluid Mech. 196, 102–111.10.1016/j.jnnfm.2012.12.013Google Scholar
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. 121, 299–340.Google Scholar
Fragomeni, J. M. & Nunes, A. C. Jr 2003 A study of the effects of welding parameters on electron beam welding in the space environment. Aerosp. Sci. Technol. 7, 373–384.10.1016/S1270-9638(03)00031-2Google Scholar
Gambhire, P. & Thaokar, R. M. 2010 Electrohydrodynamic instabilities at interfaces subjected to alternating electric field. Phys. Fluids 22, 064103.10.1063/1.3431043Google Scholar
Gambhire, P. & Thaokar, R. M. 2012 Role of conductivity in the electrohydrodynamic patterning of air–liquid interfaces. Phys. Rev. E 86, 036301.Google Scholar
Henderson, D. M. & Miles, J. W. 1990 Single-mode Faraday waves in small cylinders. J. Fluid Mech. 213, 95–109.10.1017/S0022112090002233Google Scholar
Hyers, R. & Rogers, J. 2008 A review of electrostatic levitation for materials research. High Temp. Mater. Process. 27 (6), 461–474.10.1515/HTMP.2008.27.6.461Google Scholar
Iino, M., Suzuki, M. & Ikushima, A. 1985 Surface-wave resonance method for measuring surface tension with a very high precision. J. Phys. Colloq. 46 (C10), 813–816.10.1051/jphyscol:198510178Google Scholar
Jacqmin, D. & Duval, W. M. B. 1988 Instabilities caused by oscillating accelerations normal to a viscous fluid–fluid interface. J. Fluid Mech. 196, 495–511.10.1017/S0022112088002794Google Scholar
Johns, L. & Narayanan, R. 2002 Interfacial Instability. Springer.Google Scholar
Khorshidi, B., Jalaal, M. & Esmaeilzadeh, E. 2011 Electrohydrodynamic instability at the interface between two leaky dielectric fluid layers. Colloids Surf. A 380 (1), 207–212.10.1016/j.colsurfa.2011.02.033Google Scholar
Kudrolli, A. & Gollub, J. P. 1996 Patterns and spatiotemporal chaos in parametrically forced surface waves: a systematic survey at large aspect ratio. Physica D 97 (1), 133–154.Google Scholar
Kumar, J. & Subramanian, C. 2015 Liquid encapsulated Czochralski growth of large size gallium arsenide and indium phosphide single crystals and their characterisation: a review. IETE J. Res. 43 (2–3), 125–130.10.1080/03772063.1997.11415971Google Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.Google Scholar
Kumar, S. 1999 Parametrically driven surface waves in viscoelastic liquids. Phys. Fluids 11, 1970–1981.10.1063/1.870061Google Scholar
Kumar, S. 2002 Vibration-induced interfacial instabilities in viscoelastic fluids. Phys. Rev. E 65, 026305.Google Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1 (1), 111–146.10.1146/annurev.fl.01.010169.000551Google Scholar
Müller, H. W. 1993 Periodic triangular patterns in the Faraday experiment. Phys. Rev. Lett. 71 (20), 3287–3290.Google Scholar
Ozen, O., Aubry, N., Papageorgiou, D. T. & Petropoulos, P. G. 2006 Electrohydrodynamic linear stability of two immiscible fluids in channel flow. Electrochim. Acta 51, 5316–5323.Google Scholar
Pease, L. F. & Russel, W. B. 2002 Linear stability analysis of thin leaky dielectric films subjected to electric fields. J. Non-Newtonian Fluid Mech. 102, 233–250.10.1016/S0377-0257(01)00180-XGoogle Scholar
Pontiga, F. & Castellanos, A. 1994 Physical mechanisms of instability in a liquid layer subjected to an electric field and a thermal gradient. Phys. Fluids 6 (5), 1684–1701.10.1063/1.868231Google Scholar
Rayleigh, Lord 1883 VII. On the crispations of fluid resting upon a vibrating support. Phil. Mag. 16 (97), 50–58.10.1080/14786448308627392Google Scholar
Raynal, F., Kumar, S. & Fauve, S. 1999 Faraday instability with a polymer solution. Eur. Phys. J. B 9, 175–178.10.1007/s100510050753Google Scholar
Reddy, M. N. & Esmaeeli, A. 2009 The EHD-driven fluid flow and deformation of a liquid jet by a transverse electric field. Intl J. Multiphase Flow 35 (11), 1051–1065.10.1016/j.ijmultiphaseflow.2009.06.008Google Scholar
Roberts, S. A. & Kumar, S. 2009 AC electrohydrodynamic instabilities in thin liquid films. J. Fluid Mech. 631, 255–279.10.1017/S0022112009006843Google Scholar
Roberts, S. A. & Kumar, S. 2010 Electrohydrodynamic instabilities in thin liquid trilayer films. Phys. Fluids 22 (12), 122102.10.1063/1.3520134Google Scholar
Rosen, M. J., Wang, H., Shen, P. & Zhu, Y. 2005 Ultralow interfacial tension for enhanced oil recovery at very low surfactant concentrations. Langmuir 21 (9), 3749–3756.10.1021/la0400959Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.10.1146/annurev.fluid.29.1.27Google Scholar
Schwabe, D. 1988 Surface-tension-driven flow in crystal growth melts. In Superhard Materials, Convection, and Optical Devices, pp. 75–112. Springer.10.1007/978-3-642-73205-8_2Google Scholar
Shankar, V. & Sharma, A. 2004 Instability of the interface between thin fluid films subjected to electric fields. J. Colloid Interface Sci. 274, 294–308.10.1016/j.jcis.2003.12.024Google Scholar
Smorodin, B. L. & Taraut, A. V. 2010 Parametric convection of a low-conducting liquid in an alternating electric field. Fluid Dyn. 45 (1), 1–9.10.1134/S0015462810010011Google Scholar
Sterman-Cohen, E., Bestehorn, M. & Oron, A. 2017 Rayleigh–Taylor instability in thin liquid films subjected to harmonic vibration. Phys. Fluids 29, 052105.Google Scholar
Suman, B. & Kumar, S. 2008 Surfactant- and elasticity-induced inertialess instabilities in vertically vibrated liquids. J. Fluid Mech. 610, 407–423.10.1017/S0022112008002772Google Scholar
Taylor, G. I. 1966 Studies in electrohydrodynamics. Part I. The circulation produced in a drop by electrical field. Proc. R. Soc. Lond. A 291 (1425), 159–166.Google Scholar
Taylor, G. I. & McEwan, A. D. 1965 The stability of a horizontal fluid interface in a vertical electric field. J. Fluid Mech. 22 (1), 1–15.10.1017/S0022112065000538Google Scholar
Tipton, C. R. & Mullin, T. 2004 An experimental study of Faraday waves formed on the interface between two immiscible liquids. Phys. Fluids 16 (7), 2336–2341.10.1063/1.1718354Google Scholar
Troyon, F. & Gruber, R. 1971 Theory of the dynamic stabilization of the Rayleigh–Taylor instability. Phys. Fluids 14, 2069–2073.10.1063/1.1693294Google Scholar
Tsai, C., Mao, R., Tsai, S., Shahverdi, K., Zhu, Y., Lin, S., Hsu, Y., Boss, G., Brenner, M., Mahon, S. & Smaldone, G. 2017 Faraday waves-based integrated ultrasonic micro-droplet generator and applications. Micromachines 851, 56–69.10.3390/mi8020056Google Scholar
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 (8), 1746–1755.10.1109/TUFFC.2013.2755Google Scholar
Tseluiko, D. & Papageorgiou, D. T. 2007 Nonlinear dynamics of electrified thin liquid films. J. Appl. Maths 67 (5), 1310–1329.Google Scholar
Tsukahara, S., Tsuruta, T. & Fujiwara, T. 2013 Surface tension determination through measurements of resonance oscillation of a small surface using dielectric force by a localized alternating current electric field. Analyst 138, 2110–2117.10.1039/c3an36260dGoogle Scholar
Wolf, G. H. 1970 Dynamic stabilization of the interchange instability of a liquid–gas interface. Phys. Rev. Lett. 24 (9), 444–446.10.1103/PhysRevLett.24.444Google Scholar
Workman, G. L. & Kaukler, W. F.1989 Laser welding in space. Tech. Rep. 19910018227. National Aeronautics and Space Administration.Google Scholar
Yang, Q., Li, B. Q. & Xu, F. 2017 Electrohydrodynamic Rayleigh–Taylor instability in leaky dielectric fluids. Intl J. Heat Mass Transfer 109, 690–704.10.1016/j.ijheatmasstransfer.2017.02.049Google Scholar
Yih, C.S. 1968 Stability of a horizontal fluid interface in a periodic vertical electric field. Phys. Fluids 11 (7), 1447–1449.10.1063/1.1692127Google Scholar
Zhou, L., Das, S. & Ellis, B. R. 2016 Effect of surfactant adsorption on the wettability alteration of gas-bearing shales. Environ. Engng Sci. 33 (10), 766–777.Google Scholar

Ward et al. supplementary movie

Example video of an electrostatically forced Faraday instability within an infinite geometry. The system is that described by Test A in Table 2, forced at a frequency of 4.75 Hz and an amplitude of 5 kV.

Download Ward et al. supplementary movie(Video)
Video 5.7 MB