Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-29T06:06:45.320Z Has data issue: false hasContentIssue false
  • English
  • Français

From electrodiffusion theory to the electrohydrodynamics of leaky dielectrics through the weak electrolyte limit

Published online by Cambridge University Press:  14 September 2018

Yoichiro Mori Open the ORCID record for Yoichiro Mori [Opens in a new window]  and
Y.-N. Young
Yoichiro Mori*
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
Y.-N. Young
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
*
†Email address for correspondence: ymori@umn.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The Taylor–Melcher (TM) model is the standard model for describing the dynamics of poorly conducting leaky dielectric fluids under an electric field. The TM model treats the fluids as ohmic conductors, without modelling the underlying ion dynamics. On the other hand, electrodiffusion models, which have been successful in describing electrokinetic phenomena, incorporate ionic concentration dynamics. Mathematical reconciliation of the electrodiffusion picture and the TM model has been a major issue for electrohydrodynamic theory. Here, we derive the TM model from an electrodiffusion model in which we explicitly model the electrochemistry of ion dissociation. We introduce salt dissociation reaction terms in the bulk electrodiffusion equations and take the limit in which the salt dissociation is weak; the assumption of weak dissociation corresponds to the fact that the TM model describes poor conductors. Together with the assumption that the Debye length is small, we derive the TM model with or without the surface charge convection term depending upon the scaling of relevant dimensionless parameters. An important quantity that emerges is the Galvani potential (GP), the jump in voltage across the liquid–liquid interface between the two leaky dielectric media; the GP arises as a natural consequence of the interfacial boundary conditions for the ionic concentrations, and is absent under certain parametric conditions. When the GP is absent, we recover the TM model. Our analysis also reveals the structure of the Debye layer at the liquid–liquid interface, which suggests how interfacial singularities may arise under strong imposed electric fields. In the presence of a non-zero GP, our model predicts that the liquid droplet will drift under an imposed electric field, the velocity of which is computed explicitly to leading order.

JFM classification

Drops and Bubbles: Drops and Bubbles Drops and Bubbles: Electrohydrodynamic effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 67 - 130
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aris, R. 1990 Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover.Google Scholar
Baygents, J. C. & Saville, D. A. 1990 The circulation produced in a drop by an electric field: a high field strength electrokinetic model. In AIP Conference Proceedings, vol. 197, pp. 7–17. AIP.Google Scholar
Baygents, J. C. & Saville, D. A. 1991 Electrophoresis of drops and bubbles. J. Chem. Soc. Faraday Trans. 87 (12), 1883–1898.Google Scholar
Bazant, M. Z. 2015 Electrokinetics meets electrohydrodynamics. J. Fluid Mech. 782, 1–4.Google Scholar
Bazant, M. Z., Kilic, M. S., Storey, B. D. & Ajdari, A. 2009 Towards an understanding of induced-charge electrokinetics at large applied voltages in concentrated solutions. Adv. Colloid Interface Sci. 152 (1), 48–88.Google Scholar
Berry, J. D., Davidson, M. R. & Harvie, D. J. E. 2013 A multiphase electrokinetic flow model for electrolytes with liquid/liquid interfaces. J. Comput. Phys. 251, 209–222.Google Scholar
Booth, F. 1951 The cataphoresis of spherical fluid droplets in electrolytes. J. Chem. Phys. 19 (11), 1331–1336.Google Scholar
Brosseau, Q. & Vlahovska, P. M. 2017 Streaming from the equator of a drop in an external electric field. Phys. Rev. Lett. 119 (3), 034501.Google Scholar
Bruus, H. 2007 Theoretical Microfluidics. Oxford University Press.Google Scholar
Chang, H.-C. & Yeo, L. Y. 2010 Electrokinetically Driven Microfluidics and Nanofluidics. Cambridge University Press.Google Scholar
Chen, C.-H. 2011 Electrohydrodynamic stability. In Electrokinetics and Electrohydrodynamics in Microsystems (ed. Ramos, A.), pp. 177–220. Springer-Verlag.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
Das, D. & Saintillan, D. 2017a Electrohydrodynamics of viscous drops in strong electric fields: numerical simulations. J. Fluid Mech. 829, 127–152.Google Scholar
Das, D. & Saintillan, D. 2017b A nonlinear small-deformation theory for transient droplet electrohydrodynamics. J. Fluid Mech. 810, 225–253.Google Scholar
Delgado, Á. V. 2001 Interfacial Electrokinetics and Electrophoresis, vol. 106. CRC Press.Google Scholar
Feng, J. Q. & Scott, T. C. 1996 A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J. Fluid Mech. 311, 289–326.Google Scholar
Girault, H. H. J. & Schiffrin, D. J. 1989 Electrochemistry of liquid-liquid interfaces. Electroanalyt. Chem. 15, 1–141.Google Scholar
He, H., Salipante, P. F. & Vlahovska, P. M. 2013 Electrorotation of a viscous droplet in a uniform direct current electric field. Phys. Fluids 25 (3), 032106.Google Scholar
Hu, W.-F., Lai, M.-C. & Young, Y.-N. 2015 A hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations. J. Comput. Phys. 282, 47–61.Google Scholar
Hung, L. Q. 1980 Electrochemical properties of the interface between two immiscible electrolyte solutions. Part i. Equilibrium situation and galvani potential difference. J. Electroanalyt. Chem. Interfacial Electrochemistry 115 (2), 159–174.Google Scholar
Lanauze, J. A., Walker, L. M. & Khair, A. S. 2015 Nonlinear electrohydrodynamics of slightly deformed oblate drops. J. Fluid Mech. 774, 245–266.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.Google Scholar
López-Herrera, J. M., Popinet, S. & Herrada, M. A. 2011 A charge-conservative approach for simulating electrohydrodynamic two-phase flows using volume-of-fluid. J. Comput. Phys. 230 (5), 1939–1955.Google Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–147.Google Scholar
de la Mora, J. F. 2007 The fluid dynamics of taylor cones. Annu. Rev. Fluid Mech. 39, 217–243.Google Scholar
Mori, Y., Liu, C. & Eisenberg, R. S. 2011 A model of electrodiffusion and osmotic water flow and its energetic structures. Physica D 240, 1835–1852.Google Scholar
Olver, F. W. J. 2010 NIST Handbook of Mathematical Functions Hardback and CD-ROM. Cambridge University Press.Google Scholar
Pascall, A. J. & Squires, T. M. 2011 Electrokinetics at liquid/liquid interfaces. J. Fluid Mech. 684, 163–191.Google Scholar
Reymond, F., Fermın, D., Lee, H. J. & Girault, H. H. 2000 Electrochemistry at liquid/liquid interfaces: methodology and potential applications. Electrochim. Acta 45 (15), 2647–2662.Google Scholar
Roberts, S. A. & Kumar, S. 2009 Ac electrohydrodynamic instabilities in thin liquid films. J. Fluid Mech. 631, 255–279.Google Scholar
Roberts, S. A. & Kumar, S. 2010 Electrohydrodynamic instabilities in thin liquid trilayer films. Phys. Fluids 22 (12), 122102.Google Scholar
Rubinstein, I. 1990 Electro-Diffusion of Ions. SIAM.Google Scholar
Salipante, P. F. & Vlahovska, P. M. 2010 Electrohydrodynamics of drops in strong uniform dc electric fields. Phys. Fluids 22 (11), 112110.Google Scholar
Salipante, P. F. & Vlahovska, P. M. 2013 Electrohydrodynamic rotations of a viscous droplet. Phys. Rev. E 88 (4), 043003.Google Scholar
Saville, D. A. 1977 Electrokinetic effects with small particles. Annu. Rev. Fluid Mech. 9 (1), 321–337.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64.Google Scholar
Schnitzer, O. & Yariv, E. 2015 The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description. J. Fluid Mech. 773, 1–33.Google Scholar
Sengupta, R., Walker, L. M. & Khair, A. S. 2017 The role of surface charge convection in the electrohydrodynamics and breakup of prolate drops. J. Fluid Mech. 833, 29–53.Google Scholar
Squires, T. M. & Bazant, M. Z. 2004 Induced-charge electro-osmosis. J. Fluid Mech. 509, 217–252.Google Scholar
Steel, B. J., Stokes, J. M. & Stokes, R. H. 1958 Individual ion mobilities in mixtures of non-electrolytes and water. J. Phys. Chem. 62 (12), 1514–1516.Google Scholar
Taylor, G. 1966 Studies in electrohydrodynamics. i. The circulation produced in a drop by electrical field. Proc. R. Soc. Lond. A 291 (1425), 159–166.Google Scholar
Tomar, G., Gerlach, D., Biswas, G., Alleborn, N., Sharma, A., Durst, F., Welch, S. W. J. & Delgado, A. 2007 Two-phase electrohydrodynamic simulations using a volume-of-fluid approach. J. Comput. Phys. 227 (2), 1267–1285.Google Scholar
Vizika, O. & Saville, D. A. 1992 The electrohydrodynamic deformation of drops suspended in liquids in steady and oscillatory electric fields. J. Fluid Mech. 239, 1–21.Google Scholar
Vlahovska, P. M. 2016 Electrohydrodynamic instabilities of viscous drops. Phys. Rev. Fluids 1 (6), 060504.Google Scholar
Xu, X. & Homsy, G. M. 2006 The settling velocity and shape distortion of drops in a uniform electric field. J. Fluid Mech. 564, 395–414.Google Scholar
Zholkovskij, E. K., Masliyah, J. H. & Czarnecki, J. 2002 An electrokinetic model of drop deformation in an electric field. J. Fluid Mech. 472, 1–27.Google Scholar
Zhou, S., Wang, Z. & Li, B. 2011 Mean-field description of ionic size effects with nonuniform ionic sizes: a numerical approach. Phys. Rev. E 84 (2), 021901.Google Scholar