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Electrohydrodynamic settling of drop in uniform electric field: beyond Stokes flow regime

Published online by Cambridge University Press:  24 October 2019

Nalinikanta Behera ,
Shubhadeep Mandal  and
Suman Chakraborty Open the ORCID record for Suman Chakraborty [Opens in a new window]
Nalinikanta Behera
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal-721302, India
Shubhadeep Mandal
Affiliation:
Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, D-37077 Göttingen, Germany
Suman Chakraborty*
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal-721302, India
*
†Email address for correspondence: suman@mech.iitkgp.ac.in
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Abstract

The electrohydrodynamics of a weakly conducting buoyant drop under the combined influence of gravity and a uniform electric field is studied computationally, focusing on the inertia-dominated regime. Numerical simulations are performed for both perfectly dielectric and leaky dielectric drops over a wide range of dimensionless parameters to explore the interplay of fluid inertia and electrical stress to govern the drop shape and charge convection. For perfectly dielectric drops, the fluid inertia alters the drop shape and the deformation behaviour of the drop follows a non-monotonic path. The drop shape at steady state exhibits the transition from oblate to prolate shape on increasing the electric field strength, in sharp contrast to the cases concerning the Stokes flow regime. Similar behaviour is also obtained for leaky dielectric drops for certain fluid properties. For leaky dielectric drops, the fluid inertia also affects the convective transport of charges at the drop surface and thereby alters the drop dynamics. Unlike the Stokes flow regime, where surface charge convection has little effect on the settling speed, the same modifies the drop settling speed quite significantly in the finite inertial regime depending on the combination of electrical conductivity ratio and permittivity ratio. For oblate drops at low capillary number, charge convection alters drop shape, while keeping the nature of deformation unaltered. However, for relatively large capillary number, the oblate drop transforms into a dimpled shape due to charge convection. For all cases, an interesting fact is noticed that under the combined action of electric and inertial forces, the resultant deformation is less than the summation of the deformations caused by individual effects, when inertial effects are strong. These results are likely to provide deep insights into the interplay of various nonlinearities towards altering electrohydrodynamic settling of drops and bubbles.

JFM classification

Drops and Bubbles: Drops and Bubbles
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 881 , 25 December 2019 , pp. 498 - 523
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Copyright
© 2019 Cambridge University Press 

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References

Ajayi, O. O. 1978 A note on Taylor’s electrohydrodynamic theory. Proc. R. Soc. Lond. A 364, 499–507.Google Scholar
Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48, 285–309.Google Scholar
Bandopadhyay, A., Mandal, S., Kishore, N. K. & Chakraborty, S. 2016 Uniform electric-field-induced lateral migration of a sedimenting drop. J. Fluid Mech. 792, 553–589.Google Scholar
Berg, G., Lundgaard, L. E. & Abi-Chebel, N. 2010 Electrically stressed water drops in oil. Chem. Engng Process. 49, 1229–1240.Google Scholar
Bhaga, D. & Weber, M. E. 1981 Bubbles in viscous liquids: shapes, wakes and velocities. J. Fluid Mech. 105, 61–85.Google Scholar
Cano-Lozano, J. C., Bohorquez, P. & Martínez-Bazán, C. 2013 Wake instability of a fixed axisymmetric bubble of realistic shape. Intl J. Multiphase Flow 51, 11–21.Google Scholar
Cimpeanu, R., Papageorgiou, D. T. & Petropoulos, P. G. 2014 On the control and suppression of the Rayleigh–Taylor instability using electric fields. Phys. Fluids 26, 022105.Google Scholar
Dandy, D. S. & Leal, L. G. 1989 Buoyancy-driven motion of a deformable drop through a quiescent liquid at intermediate Reynolds numbers. J. Fluid Mech. 208, 161–192.Google Scholar
Das, D. & Saintillan, D. 2017 A nonlinear small-deformation theory for transient droplet electrohydrodynamics. J. Fluid Mech. 810, 225–253.Google Scholar
Eow, J. S., Ghadiri, M. & Sharif, A. O. 2007 Electro-hydrodynamic separation of aqueous drops from flowing viscous oil. J. Petrol. Sci. Engng 55, 146–155.Google Scholar
Eow, J. S., Ghadiri, M., Sharif, A. O. & Williams, T. J. 2001 Electrostatic enhancement of coalescence of water droplets in oil: a review of the current understanding. Chem. Engng J. 84, 173–192.Google Scholar
Feng, J. Q. 1999 Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number. Proc. R. Soc. Lond. A 455, 2245–2269.Google Scholar
Feng, J. Q. 2010 A deformable liquid drop falling through a quiescent gas at terminal velocity. J. Fluid Mech. 658, 438–462.Google Scholar
Ferrera, C., López-Herrera, J. M., Herrada, M. A., Montanero, J. M. & Acero, A. J. 2013 Dynamical behavior of electrified pendant drops. Phys. Fluids 25, 012104.Google Scholar
Gañán-Calvo, A. M., López-Herrera, J. M., Herrada, M. A., Ramos, A. & Montanero, J. M. 2018 Review on the physics of electrospray: from electrokinetics to the operating conditions of single and coaxial Taylor cone-jets, and AC electrospray. J. Aerosol Sci. 125, 32–56.Google Scholar
Ghasemi, E., Bararnia, H., Soleimanikutanaei, S. & Lin, C. X. 2018 Simulation of deformation and fragmentation of a falling drop under electric field. Powder Technol. 325, 301–308.Google Scholar
Hadamard, J. 1911 Mouvement permanent lent d’une sphère liquide et visqueuse dans un liquide visqueux. C.R. Acad. Sci 152, 1735–1738.Google Scholar
Helenbrook, B. T. & Edwards, C. F. 2002 Quasi-steady deformation and drag of uncontaminated liquid drops. Intl J. Multiphase Flow 28, 1631–1657.Google Scholar
Herrada, M. A., López-Herrera, J. M., Gañán-Calvo, A. M., Vega, E. J., Montanero, J. M. & Popinet, S. 2012 Numerical simulation of electrospray in the cone-jet mode. Phys. Rev. E 86, 026305.Google Scholar
Hu, S. & Kintner, R. C. 1955 The fall of single liquid drops through water. AIChE J. 1, 42–48.Google Scholar
Klee, A. J. & Treybal, R. E. 1956 Rate of rise or fall of liquid drops. AIChE J. 2, 444–447.Google Scholar
Lac, E. & Homsy, G. M. 2007 Axisymmetric deformation and stability of a viscous drop in a steady electric field. J. Fluid Mech. 590, 239–264.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
Liu, L., Tang, H. & Quan, S. 2013 Shapes and terminal velocities of a drop rising in stagnant liquids. Comput. Fluids 81, 17–25.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, 1939–1955.Google Scholar
Magnaudet, J. & Eames, I. 2000 The motion of high-Reynolds-number bubbles in inhomogeneous flows. Annu. Rev. Fluid Mech. 32, 659–708.Google Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2016 Effect of surface charge convection and shape deformation on the dielectrophoretic motion of a liquid drop. Phys. Rev. E 93, 1–21.Google Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2017 The effect of surface charge convection and shape deformation on the settling velocity of drops in nonuniform electric field. Phys. Fluids 29, 012101.Google Scholar
Mandal, S., Sinha, S., Bandopadhyay, A. & Chakraborty, S. 2018 Drop deformation and emulsion rheology under the combined influence of uniform electric field and linear flow. J. Fluid Mech. 841, 408–433.Google Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–146.Google Scholar
Mhatre, S., Vivacqua, V., Ghadiri, M., Abdullah, A. M., Al-Marri, M. J., Hassanpour, A., Hewakandamby, B., Azzopardi, B. & Kermani, B. 2015 Electrostatic phase separation: a review. Chem. Engng Res. Des. 96, 177–195.Google Scholar
Myint, W., Hosokawa, S. & Tomiyama, A. 2006 Terminal velocity of single drops in stagnant liquids. J. Fluid Sci. Technol. 1, 72–81.Google Scholar
Poddar, A., Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2018 Sedimentation of a surfactant-laden drop under the influence of an electric field. J. Fluid Mech. 849, 277–311.Google Scholar
Poddar, A., Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2019 Electrical switching of a surfactant coated drop in Poiseuille flow. J. Fluid Mech. 870, 27–66.Google Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.Google Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.Google Scholar
Raymond, F. & Rosant, J.-M. 2000 A numerical and experimental study of the terminal velocity and shape of bubbles in viscous liquids. Chem. Engng Sci. 55, 943–955.Google Scholar
Rybczynski, W. 1911 Über die fortschreitende Bewegung einer flüssigen Kugel in einem zähen Medium. Bull. Acad. Sci. Cracovie A, 40–46.Google Scholar
Santra, S., Sen, D., Das, S. & Chakraborty, S. 2019 Electrohydrodynamic interaction between droplet pairs in a confined shear flow. Phys. Fluids 31, 032005.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64.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
Sherwood, J. D. 1988 Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, 133–146.Google Scholar
Spertell, R. B. & Saville, D. A. 1974 The roles of electrohydrodynamic phenomena in the motion of drops and bubbles. In Proceedings of the International Colloquium on Drops and Bubbles (ed. Plesset, M.), vol. 1, pp. 106–121. California Institute of Technology.Google Scholar
Sunder, S. & Tomar, G. 2013 Numerical simulations of bubble formation from submerged needles under non-uniform direct current electric field. Phys. Fluids 25, 102104.Google Scholar
Taylor, G. 1966 Studies in Electrohydrodynamics. I. The circulation produced in a drop by electrical field. Proc. R. Soc. Lond. A 291, 159–166.Google Scholar
Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number. J. Fluid Mech. 18, 466–476.Google Scholar
Teh, S.-Y., Lin, R., Hung, L.-H. & Lee, A. P. 2008 Droplet microfluidics. Lab on a Chip 8, 198.Google Scholar
Tripathi, M. K., Sahu, K. C. & Govindarajan, R. 2015a Dynamics of an initially spherical bubble rising in quiescent liquid. Nat. Commun. 6, 6268.Google Scholar
Tripathi, M. K., Sahu, K. C. & Govindarajan, R. 2015b Why a falling drop does not in general behave like a rising bubble. Sci. Rep. 4, 4771.Google Scholar
Wairegi, T. & Grace, J. R. 1976 The behaviour of large drops in immiscible liquids. Intl J. Multiphase Flow 3, 67–77.Google Scholar
Wellek, R. M., Agrawal, A. K. & Skelland, A. H. P. 1966 Shape of liquid drops moving in liquid media. AIChE J. 12, 854–862.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
Yariv, E. & Almog, Y. 2016 The effect of surface-charge convection on the settling velocity of spherical drops in a uniform electric field. J. Fluid Mech. 797, 536–548.Google Scholar