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Electrohydrodynamic linear stability analysis of dielectric liquids subjected to unipolar injection in a rectangular enclosure with rigid sidewalls

Published online by Cambridge University Press:  10 October 2014

A. T. Pérez ,
P. A. Vázquez ,
Jian Wu  and
P. Traoré
A. T. Pérez*
Affiliation:
Departamento de Electrónica y Electromagnetismo, Universidad de Sevilla, Facultad de Física, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain
P. A. Vázquez
Affiliation:
Departamento de Física Aplicada III, Universidad de Sevilla, ESI, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain
Jian Wu
Affiliation:
Institut PPRIME, Département Fluide-Thermique-Combustion, Boulevard Pierre et Marie Curie, BP 30179, 86962 Futuroscope-Chasseneuil, France
P. Traoré
Affiliation:
Institut PPRIME, Département Fluide-Thermique-Combustion, Boulevard Pierre et Marie Curie, BP 30179, 86962 Futuroscope-Chasseneuil, France
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the linear stability threshold of a dielectric liquid subjected to unipolar injection in a two-dimensional rectangular enclosure with rigid boundaries. A finite element formulation transforms the set of linear partial differential equations that governs the system into a set of algebraic equations. The resulting system poses an eigenvalue problem. We calculate the linear stability threshold, as well as the velocity field and charge density distribution, as a function of the aspect ratio of the domain. The stability parameter as a function of the aspect ratio describes paths of symmetry-breaking bifurcation. The symmetry properties of the different linear modes determine whether these paths cross each other or not. The resulting structure has important consequences in the nonlinear behaviour of the system after the bifurcation points.

JFM classification

Complex Fluids: Dielectrics Instability: Instability MHD and Electrohydrodynamics: MHD and Electrohydrodynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 758 , 10 November 2014 , pp. 586 - 602
Copyright
© 2014 Cambridge University Press 

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References

Atten, P. & Lacroix, J. C. 1979 Non-linear hydrodynamic stability of liquids subjected to unipolar injection. J. Méc. 18 (3), 469–510.Google Scholar
Atten, P., Lacroix, J. C. & Malraison, B. 1980 Chaotic motion in a Coulomb force driven instability: large aspect ratio experiments. Phys. Lett. A 79 (4), 255–258.Google Scholar
Atten, P. & Moreau, R. 1972 Stabilité électrohydrodynamique des liquides isolants soumis à une injection unipolaire. J. Méc. 11, 471–520.Google Scholar
Boffi, D. 2010 Finite element approximation of eigenvalue problems. Acta Numerica 19, 1–120.Google Scholar
Castellanos, A. 1991 Coulomb-driven convection in electrohydrodynamics. IEEE Trans. Electr. Insul. 26 (6), 1201–1215.Google Scholar
Castellanos, A.(Ed.) 1998 Electrohydrodynamics, Springer.Google Scholar
Castellanos, A., Atten, P. & Pérez, A. T. 1987 Finite amplitude electroconvection in liquids in the case of weak unipolar injections. Physico-Chem. Hydrodyn. 9 (3–4), 443–452.Google Scholar
Castellanos, A., Ramos, A., González, A., Green, N. G. & Morgan, H. 2003 Electrohydrodynamics and dielectrophoresis in microsystems: scaling laws. J. Phys. D: Appl. Phys. 36 (20), 2584–2597.Google Scholar
Chicón, R., Castellanos, A. & Martín, E. 1997 Numerical modelling of Coulomb-driven convection in insulating liquids. J. Fluid Mech. 344, 43–66.Google Scholar
Cliffe, K. A., Garratt, T. J. & Spence, A. 1994 Eigenvalues of block matrices arising from problems in fluid mechanics. SIAM J. Matrix Anal. Applics. 15 (4), 1310–1318.Google Scholar
Cliffe, K. A. & Winters, K. H. 1986 The use of symmetry in bifurcation calculations and its application to the Bénard problem. J. Comput. Phys. 67, 310–326.Google Scholar
Crowley, J. M. 1986 Fundamentals of Applied Electrostatics. Wiley.Google Scholar
Darabi, J., Ohadi, M. & DeVoe, D. L. 2001 An electrohydrodynamic polarization micropump for electronic cooling applications. J. Microelectromech. Syst. 10, 98–106.Google Scholar
Felici, N. 1969 Phénomenes hydro et aérodynamiques dans la conduction des diélectriques fluides. Rev. Gén. Electrostat. 78, 717–734.Google Scholar
Hernandez, V., Roman, J. E. & Vidal, V. 2005 SLEPc: a scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Softw. 31 (3), 351–362.CrossRefGoogle Scholar
Jalaal, M., Khorshidi, B. & Esmaeilzadeh, E. 2013 Electrohydrodynamic (EHD) mixing of two miscible dielectric liquids. Chem. Engng J. 219, 118–123.Google Scholar
Jones, T. B. 1978 Electrohydrodynamically enhanced heat transfer in liquids – a review. In Advances in Heat Transfer, vol. 14, pp. 107–148. Academic Press.Google Scholar
Lacroix, J. C., Atten, P. & Hopfinger, E. J. 1975 Electroconvection in a dielectric liquid layer subjected to unipolar injection. J. Fluid Mech. 69, 539–563.Google Scholar
Logg, A., Mardal, K. A. & Wells, G. N. 2010 Automated Solution of Differential Equations by the Finite Element Method. Springer.Google Scholar
Logg, A. & Wells, G. N. 2010 DOLFIN: automated finite element computing. ACM Trans. Math. Softw. 37 (2), 20.Google Scholar
Malraison, B. & Atten, P. 1982 Chaotic behavior of instability due to unipolar ion injection in a dielectric liquid. Phys. Rev. Lett. 49 (10), 723–726.Google Scholar
McCluskey, F. M. J. & Atten, P. 1988 Modifications to the wake of a wire across Poiseuille flow due to a unipolar space charge. J. Fluid Mech. 197, 81–104.Google Scholar
Melcher, J. R.(Ed.) 1981 Continuum Electromechanics, MIT Press.Google Scholar
Pearson, M. & Seyed-Yagoobi, J. 2009 Advances in electrohydrodynamic conduction pumping. IEEE Trans. Dielec. Elec. Insul. 16 (2), 424–434.Google Scholar
Pérez, A. T. & Castellanos, A. 1989 Role of charge diffusion in finite-amplitude electroconvection. Phys. Rev. A 40 (10), 5844–5855.CrossRefGoogle ScholarPubMed
Ryu, J. C., Park, H. J., Park, J. K. & Kang, K. H. 2010 New electrohydrodynamic flow caused by the Onsager effect. Phys. Rev. Lett. 104, 104502.Google Scholar
Schneider, J. M. & Watson, P. K. 1970 Electrohydrodyamic stability of space-charge-limited currents in dielectric liquids I: theoretical study. Phys. Fluids 13 (8), 1948–1954.Google Scholar
Seyed-Yagoobi, J. 2005 Electrohydrodynamic pumping of dielectric liquids. J. Electrostat. 63 (6), 861–869.Google Scholar
Seyed-Yagoobi, J. & Bryan, J. E. 1999 Enhancement of heat transfer and mass transport in single-phase and two-phase flows with electrohydrodynamics. In Advances in Heat Transfer, vol. 33, pp. 95–186. Elsevier.Google Scholar
Traoré, P. & Louste, C.2011 Numerical analysis of the effect of an EHD actuator on the flow past a square cylinder. In Proceedings of the 2011 IEEE International Conference on Dielectric Liquids (ICDL), pp. 1–4.Google Scholar
Traoré, Ph. & Pérez, A. T. 2012 Two-dimensional numerical analysis of electroconvection in dielectric liquid subjected to strong unipolar injection. Phys. Fluids 24, 037102.Google Scholar
Vázquez, P. A. & Castellanos, A. 2013 Numerical simulation of {EHD} flows using discontinuous Galerkin finite element methods. Comput. Fluids 84, 270–278.Google Scholar
Vázquez, P. A., Georghiou, G. E. & Castellanos, A. 2006 Characterization of injection instabilities in electrohydrodynamics by numerical modelling: comparison of particle in cell and flux corrected transport methods for electroconvection between two plates. J. Phys. D: Appl. Phys. 39, 2754–2763.Google Scholar
Vázquez, P. A., Georghiou, G. E. & Castellanos, A. 2008 Numerical analysis of the stability of the electrohydrodynamic (EHD) electroconvection between two plates. J. Phys. D: Appl. Phys. 41 (17), 175303.Google Scholar
van de Vooren, A. I. & Dukstra, H. A. 1989 A finite element stability analysis for the Marangoni problem in a rectangular container with rigid sidewalls. Comput. Fluids 17 (3), 467–485.Google Scholar
Winters, K. H., Plesser, Th. & Cliffe, K. A. 1988 The onset of convection in a finite container due to surface tension and buoyancy. Physica D 29 (3), 387–401.CrossRefGoogle Scholar
Wong, P. K., Wang, T. H., Deval, J. H. & Ho, M. C. 2004 Electrokinetics in micro devices for biotechnology applications. IEEE/ASME Trans. Mechatronics 9 (2), 366–376.Google Scholar
Wu, J., Traoré, P. & Louste, C. 2013a An efficient finite volume method for electric field–space charge coupled problems. J. Electrostat. 71 (3), 319–325.Google Scholar
Wu, J., Traoré, P., Vázquez, P. A. & Pérez, A. T. 2013b Onset of convection in a finite two-dimensional container due to unipolar injection of ions. Phys. Rev. E 88, 053018.Google Scholar