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Absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow: a linear analysis

Published online by Cambridge University Press:  16 January 2019

Fang Li ,
Bo-Fu Wang ,
Zhen-Hua Wan Open the ORCID record for Zhen-Hua Wan [Opens in a new window] ,
Jian Wu Open the ORCID record for Jian Wu [Opens in a new window]  and
Mengqi Zhang Open the ORCID record for Mengqi Zhang [Opens in a new window]
Fang Li
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Anhui 230027, PR China
Bo-Fu Wang
Affiliation:
Shanghai Institute of Applied Mathematics and Mechanics, and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, PR China
Zhen-Hua Wan
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Anhui 230027, PR China
Jian Wu
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
Mengqi Zhang*
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, 117575Singapore
*
†Email address for correspondence: mpezmq@nus.edu.sg
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Abstract

We present a study of absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow (EHD-Poiseuille). The electric field is imposed on two infinite flat plates filled with a non-conducting dielectric fluid with unipolar ion injection. Mathematically, the dispersion relation of the linearised problem is studied based on the asymptotic response of an impulse disturbance imposed on the base EHD-Poiseuille flow. Transverse, longitudinal and oblique rolls are investigated to identify the saddle point satisfying the pinching condition in the corresponding complex wavenumber space. It is found that when the ratio of Coulomb force to viscous force increases, the transverse rolls can transit from convective instability to absolute instability. The ratio of hydrodynamic mobility to electric mobility, which exerts negligible effect on the linear stability criterion when the cross-flow is small, has significant influence on the convective–absolute instability transition, especially when the ratio is small. As we change the value of the mobility ratio, a saddle point shift phenomenon occurs in the case of transverse rolls. The unstable longitudinal rolls are convectively unstable as long as there is a cross-flow, a result which is deduced from a one-mode Galerkin approximation. Longitudinal rolls have a larger growth rate than transverse rolls except for a small cross-flow. Finally, regarding the oblique rolls, a numerical search for the saddle point simultaneously in the complex streamwise and transverse wavenumber spaces always yields an absolute transverse wavenumber of zero, implying that oblique rolls give way to transverse rolls when the flow is unstable.

JFM classification

Instability: Absolute/convective instability MHD and Electrohydrodynamics: MHD and Electrohydrodynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 816 - 844
Copyright
© 2019 Cambridge University Press 

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References

Allen, P. H. G. & Karayiannis, T. G. 1995 Electrohydrodynamic enhancement of heat transfer and fluid flows. Heat Recovery Systems CHP 15 (5), 389–423.Google Scholar
Atten, P., Caputo, J. G., Malraison, B. & Gagne, Y. 1984 Détermination de dimension d’attracteurs pour differents écoulements. J. Méc. Theor. Appl. 133–156 (numéro spécial).Google Scholar
Atten, P. & Honda, T. 1982 The electroviscous effect and its explanation I-The electrohydrodynamic origin; study under unipolar D.C. injection. J. Electrostat. 11 (3), 225–245.Google Scholar
Atten, P. & Lacroix, J. C. 1974 Stabilité hydrodynamique non-linéaire d’un liquide isolant soumis à une injection unipolaire forte. C. R. Acad. Sci. Paris T.278, B 385–387.Google Scholar
Atten, P. & Lacroix, J. C. 1979 Non-linear hydrodynamic stability of liquids subjected to unipolar injection. J. Méc. 18, 469–510.Google Scholar
Atten, P., McCluskey, F. M. J. & Lahjomri, A. C. 1987 The electrohydrodynamic origin of turbulence in electrostatic precipitators. IEEE Trans. Ind. Applics. IA‐23 (4), 705–711.Google Scholar
Atten, P. & Moreau, R. 1972 Stabilité electrohydrodynamique des liquides isolants soumis à une injection unipolaire. J. Méc. 11, 471–520.Google Scholar
Bers, A. 1983 Space-time evolution of plasma instabilities-absolute and convective. In Handbook of Plasma Physics (ed. Rosenbluth, M. N. & Sagdeev, R. Z.), vol. 1, pp. 451–517. North-Holland.Google Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. Massachusetts.Google Scholar
Carrière, P. & Monkewitz, P. A. 1999 Convective versus absolute instability in mixed Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 384, 243–262.Google Scholar
Carrière, P., Monkewitz, P. A. & Martinand, D. 2004 Envelope equations for the Rayleigh–Bénard–Poiseuille system. Part 1. Spatially homogeneous case. J. Fluid Mech. 502, 153–174.Google Scholar
Castellanos, A. 1991 Coulomb-driven convection in electrohydrodynamics. IEEE Trans. Elec. Insul. 26 (6), 1201–1215.Google Scholar
Castellanos, A. 1998 Electrohydrodynamics. Springer.Google Scholar
Castellanos, A. & Agrait, N. 1992 Unipolar injection induced instabilities in plane parallel flows. IEEE Trans. Ind. Applics. 28 (3), 513–519.Google Scholar
Chakraborty, S., Liao, I.-C., Adler, A. & Leong, K. W. 2009 Electrohydrodynamics: a facile technique to fabricate drug delivery systems. Adv. Drug Deliv. Rev. 61 (12), 1043–1054.Google Scholar
Chen, C.-H., Lin, H., Lele, S. K. & Santiago, J. G. 2005 Convective and absolute electrokinetic instability with conductivity gradients. J. Fluid Mech. 524, 263–303.Google Scholar
Chomaz, J. M. 1992 Absolute and convective instabilities in nonlinear systems. Phys. Rev. Lett. 69, 1931–1934.Google Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Cross, M. C. 1980 Derivation of the amplitude equation at the Rayleigh–Bénard instability. Phys. Fluids 23 (9), 1727–1731.Google Scholar
Darabi, J., Rada, M., Ohadi, M. & Lawler, J. 2002 Design, fabrication, and testing of an electrohydrodynamic ion-drag micropump. J. Microelectromech. Syst. 11 (6), 684–690.Google Scholar
Deissler, R. J. 1987 The convective nature of instability in plane Poiseuille flow. Phys. Fluids 30 (8), 2303–2305.Google Scholar
Delbende, I., Chomaz, J.-M. & Huerre, P. 1998 Absolute/convective instabilities in the Batchelor vortex: a numerical study of the linear impulse response. J. Fluid Mech. 355, 229–254.Google Scholar
Dubois, M. 1982 Experimental aspects of the transition to turbulence in Rayleigh–Bénard convection. In Stability of Thermodynamics Systems (ed. Casas-Vázquez, J. & Lebon, G.), pp. 177–191. Springer.Google Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447–468.Google Scholar
Félici, N. 1971 DC conduction in liquid dielectrics. Part II: electrohydrodynamic phenomena. Direct Current Power Electron. 2, 147–165.Google Scholar
Frisch, U. & Morf, R. 1981 Intermittency in nonlinear dynamics and singularities at complex times. Phys. Rev. A 23, 2673–2705.Google Scholar
Fujimura, K. & Kelly, R. E. 1988 Stability of unstably stratified shear flow between parallel plates. Fluid Dyn. Res. 2 (4), 281–292.Google Scholar
Fujimura, K. & Kelly, R. E. 1995 Interaction between longitudinal convection rolls and transverse waves in unstably stratified plane Poiseuille flow. Phys. Fluids 7 (1), 68–79.Google Scholar
Gage, K. S. & Reid, W. H. 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33 (01), 21–32.Google Scholar
Gaster, M. 1965 On the generation of spatially growing waves in a boundary layer. J. Fluid Mech. 22 (03), 433–441.Google Scholar
Gaster, M. 1968 Growth of disturbances in both space and time. Phys. Fluids 11 (4), 723–727.Google Scholar
Gaster, M. 1975 A theoretical model of a wave packet in the boundary layer on a flat plate. Proc. R. Soc. Lond. A 347 (1649), 271–289.Google Scholar
Gaster, M. 1981 Propagation of linear wave packets in laminar boundary layers. AIAA J. 19 (4), 419–423.Google Scholar
Grandjean, E. & Monkewitz, P. A. 2009 Experimental investigation into localized instabilities of mixed Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 640, 401–419.Google Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Kelly, R. E. 1994 The onset and development of thermal convection in fully developed shear flows. Adv. Appl. Mech. 31, 35–112.Google Scholar
Kikuchi, H. 2001 Electrohydrodynamics in Dusty and Dirty Plasmas. Kluwer Academic Publishers.Google Scholar
Kourmatzis, A. & Shrimpton, J. S. 2014 Electrohydrodynamic inter-electrode flow and liquid jet characteristics in charge injection atomizers. Exp. Fluids 55 (3), 1688.Google Scholar
Kourmatzis, A. & Shrimpton, J. S. 2016 Characteristics of electrohydrodynamic roll structures in laminar planar Couette flow. J. Phys. D: Appl. Phys. 49 (4), 045503.Google Scholar
Kupfer, K., Bers, A. & Ram, A. K. 1987 The cusp map in the complex-frequency plane for absolute instabilities. Phys. Fluids 30 (10), 3075–3082.Google Scholar
Lacroix, J. C., Atten, P. & Hopfinger, E. J. 1975 Electro-convection in a dielectric liquid layer subjected to unipolar injection. J. Fluid Mech. 69 (03), 539–563.Google Scholar
Lara, J. L., Castellanos, A. & Pontiga, F. 1997 Destabilization of plane Poiseuille flow of insulating liquids by unipolar charge injection. Phys. Fluids 9 (2), 399–406.Google Scholar
Leonard, G. L., Mitchner, M. & Self, S. A. 1983 An experimental study of the electrohydrodynamic flow in electrostatic precipitators. J. Fluid Mech. 127, 123–140.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
Martinand, D., Carrière, P. & Monkewitz, P. A. 2006 Three-dimensional global instability modes associated with a localized hot spot in Rayleigh–Bénard–Poiseuille convection. J. Fluid Mech. 551, 275–301.Google Scholar
McCluskey, F. M. J. & Atten, P. 1984 Entrainment of an injected unipolar space charge by a forced flow in a rectangular channel. J. Electrostat. 15 (3), 329–342.Google Scholar
Melcher, J. R. & Firebaugh, M. S. 1967 Traveling-wave bulk electroconvection induced across a temperature gradient. Phys. Fluids 10 (6), 1178–1185.Google Scholar
Müller, H. W., Lücke, M. & Kamps, M. 1992 Transversal convection patterns in horizontal shear flow. Phys. Rev. A 45 (6), 3714–3726.Google Scholar
Müller, H. W., Tveitereid, M. & Trainoff, S. 1993 Rayleigh–Bénard problem with imposed weak through-flow: two coupled Ginzburg–Landau equations. Phys. Rev. E 48, 263–272.Google Scholar
Pardon, G., Ladhani, L., Sandström, N., Ettori, M., Lobov, G. & van der Wijngaart, W. 2015 Aerosol sampling using an electrostatic precipitator integrated with a microfluidic interface. Sensors Actuators B 212, 344–352.Google Scholar
Pearlstein, A. J. 1985 On the two-dimensionality of the critical disturbances for stratified viscous plane parallel shear flows. Phys. Fluids 28 (2), 751–753.Google Scholar
Pérez, A. T. & Castellanos, A. 1989 Role of charge diffusion in finite-amplitude electroconvection. Phys. Rev. A 40 (10), 5844–5855.Google Scholar
Richardson, A. T. & Deo, B. J. S. 1986 Nonlinear stability bounds for plane Poiseuille flow subjected to unipolar charge injection. Q. J. Mech. Appl. Maths 39 (1), 25–40.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schoppa, W. & Hussain, F. 1998 A large-scale control strategy for drag reduction in turbulent boundary layers. Phys. Fluids 10 (5), 1049–1051.Google Scholar
Shrimpton, J. S. 2009 Charge Injection Systems: Physical Principles, Experimental and Theoretical Work. Springer.Google Scholar
Stuetzer, O. M. 1960 Ion drag pumps. J. Appl. Phys. 31 (1), 136–146.Google Scholar
Suslov, S. A. 2006 Numerical aspects of searching convective/absolute instability transition. J. Comput. Phys. 212 (1), 188–217.Google Scholar
Tveitereid, M. & Müller, H. W. 1994 Pattern selection at the onset of Rayleigh–Bénard convection in a horizontal shear flow. Phys. Rev. E 50, 1219–1226.Google Scholar
White, H. J. 1965 Industrial Electrostatic Precipitation. Addison-Wesley.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley.Google Scholar
Yin, X. & Sun, D. 2003 Vortex Stability. National Defense Industry Press.Google Scholar
Zhang, M. 2016 Weakly nonlinear stability analysis of subcritical electrohydrodynamic flow subject to strong unipolar injection. J. Fluid Mech. 792, 328–363.Google Scholar
Zhang, M., Martinelli, F., Wu, J., Schmid, P. J. & Quadrio, M. 2015 Modal and nonmodal stability analysis of electrohydrodynamic flow with and without cross-flow. J. Fluid Mech. 770, 319–349.Google Scholar