Small deformation theory for two leaky dielectric drops in a uniform electric field

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190517
Author(s):  
Michael Zabarankin
Keyword(s):  
Electric Field ◽  
Deformation Theory ◽  
Small Deformation ◽  
Dielectric Constants ◽  
Dynamics Simulation ◽  
Uniform Electric Field ◽  
Fluid Viscosity ◽  
Ambient Fluid ◽  
Drop Dynamics ◽  
Leaky Dielectric

A small deformation theory for two non-identical spherical drops freely suspended in an ambient fluid and subjected to a uniform electric field is presented. The three phases are assumed to be leaky dielectric (slightly conducting) viscous incompressible fluids and the nonlinear effects of inertia and surface charge convection are neglected. The deformed shapes of the drops are linearized with respect to the electric capillary number that characterizes the balance between the electric stress and the surface tension. When the two drops are sufficiently far apart, their deformed shapes are predicted by Taylor’s small deformation theory—depending on Taylor’s discriminating function, the drops may become prolate, oblate or remain spherical. When the two drops get closer to each other, in addition to becoming prolate/oblate, they start translating and developing an egg shape. (Since there is no net charge, the centre of mass of the two drops remains stationary.) The extent of each of these ‘modes’ of deformation depends on the distance between the drops’ centres and on drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities. The predictions of the small deformation theory for two drops perfectly agree with the existing results of two-drop dynamics simulation based on a boundary-integral equation approach. Moreover, while previous works observed only three types of behaviour for two identical drops—the drops may either become prolate or oblate and move towards each other or become prolate and move away from each other—the small deformation theory predicts that non-identical drops may, in fact, become oblate and move away from each other when the drop-to-ambient fluid conductivity ratios are reciprocal and the drop-to-ambient fluid viscosity ratios are sufficiently large. The presented theory also readily yields an analytical insight into the interplay among different modes of drop deformation and can be used to guide the selection of the phases’ electromechanical properties for two-drop dynamics simulations.

scholarly journals Toroidal drop under electric field: arbitrary drop-to-ambient fluid viscosity ratio

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170379 ◽  
Author(s):  
Michael Zabarankin
Keyword(s):  
Electric Field ◽  
Dielectric Constants ◽  
Uniform Electric Field ◽  
Fluid Viscosity ◽  
Ambient Fluid ◽  
Electric Stress ◽  
Freely Suspended ◽  
Two Phases ◽  
Major Radius ◽  
External Forces

In the absence of external forces, a liquid toroidal drop freely suspended in another fluid shrinks towards its centre. It is shown that if the two phases are slightly conducting viscous incompressible fluids with the drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities to be 1/ R , Q and λ , respectively, then the toroidal drop with volume 4 π /3 and having major radius ρ can become almost stationary when subjected to a uniform electric field aligned with the drop’s axis of symmetry. In this case, Q and electric capillary number Ca E that defines the ratio of electric stress to surface tension, are functions of R , ρ and λ and are found analytically. Those functions are relatively insensitive to λ , and for ρ ≥3, they admit simple approximations, which coincide with those obtained recently for λ =1. Streamlines inside and outside the toroidal drop for the same R and ρ but different λ are qualitatively similar.

Deformation and breakup of a leaky dielectric drop in a quadrupole electric field

2013 ◽  
Vol 731 ◽  
pp. 713-733 ◽  
Author(s):  
Shivraj D. Deshmukh ◽  
Rochish M. Thaokar
Keyword(s):  
Phase Diagram ◽  
Electric Field ◽  
Boundary Element ◽  
Dielectric Medium ◽  
Dielectric Constants ◽  
Boundary Element Methods ◽  
Uniform Electric Field ◽  
Quadrupole Field ◽  
Capillary Stress ◽  
Leaky Dielectric

AbstractThe deformation and breakup of a leaky dielectric drop suspended in a leaky dielectric medium subjected to a quadrupole electric field are studied. Analytical (linear and nonlinear asymptotic expansions in the electric capillary number, $C{a}_{Q} $, a ratio of electric to capillary stress) and numerical (boundary element) methods are used. A complete phase diagram for the drop deformation in the $R$–$Q$ plane is presented, where $R$ and $Q$ are the non-dimensional ratios of the resistivities and dielectric constants, respectively, of the drop and the medium phase. The prolate and oblate deformations are mapped in the phase diagram, and the flow contours are also shown. The large deformation and breakup of a drop at higher $C{a}_{Q} $ are analysed using the boundary element method. Several non-trivial shapes are observed at the onset of breakup of a drop. A prolate drop always breaks above a certain critical value of $C{a}_{Q} $. In the oblate deformation cases, breakup as well as steady shapes are observed at a higher value of $C{a}_{Q} $. A detailed study of prolate and oblate deformation tendencies due to the normal and tangential electric stresses and the countervailing role of viscous stresses is presented. The circulation inside a drop is found to be more intense for a quadrupole field as compared with a uniform electric field. More intense internal circulations can lead to enhanced mixing characteristics and will have implications in microfluidic devices.

Removal of Particles From the Surface of a Droplet

2008 ◽  
Author(s):  
N. Aubry ◽  
P. Singh ◽  
S. Nudurupati ◽  
M. Janjua
Keyword(s):  
Dielectric Constant ◽  
Electric Field ◽  
Strong Electric Field ◽  
Dielectric Constants ◽  
The Other ◽  
Uniform Electric Field ◽  
Drop Deformation ◽  
Ambient Fluid ◽  
Dielectrophoretic Force ◽  
Different Types

We present a technique to concentrate particles on the surface of a drop, separate different types of particles, and remove them from the drop by subjecting the drop to a uniform electric field. The particles are moved under the action of the dielectrophoretic force which arises due to the non-uniformity of the electric field on the surface of the drop. Experiments show that depending on the dielectric constants of the fluids and the particles, particles aggregate either near the poles or near the equator of the drop. When particles aggregate near the poles and the dielectric constant of the drop is greater than that of the ambient fluid, the drop deformation is larger than that of a clean drop. In this case, under a sufficiently strong electric field the drop develops conical ends and particles concentrated at the poles eject out by a tip streaming mechanism, thus leaving the drop free of particles. On the other hand, when particles aggregate near the equator, it is shown that the drop can be broken into three major droplets, with the middle droplet carrying all particles and the two larger sized droplets on the sides being free of particles. The method also allows us to separate particles for which the sign of the Clausius-Mossotti factor is different, making particles of one type aggregate at the poles and of the second type aggregate at the equator. The former are removed from the drop by increasing the electric field strength, leaving only the latter inside the drop.

scholarly journals Liquid toroidal drop under uniform electric field

2017 ◽  
Vol 473 (2202) ◽  
pp. 20160633 ◽  
Author(s):  
Michael Zabarankin
Keyword(s):  
Surface Tension ◽  
Electric Field ◽  
Dielectric Constants ◽  
Uniform Electric Field ◽  
Normal Velocity ◽  
Spherical Drop ◽  
Electric Stress ◽  
Freely Suspended ◽  
Major Radius ◽  
External Forces

The problem of a stationary liquid toroidal drop freely suspended in another fluid and subjected to an electric field uniform at infinity is addressed analytically. Taylor’s discriminating function implies that, when the phases have equal viscosities and are assumed to be slightly conducting (leaky dielectrics), a spherical drop is stationary when Q =(2 R 2 +3 R +2)/(7 R 2 ), where R and Q are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, Ca E , that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (Ca E =0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some Ca E >0. This work finds Q and Ca E such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radius ρ and volume 4 π /3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The found Q and Ca E depend on R and ρ , and for large ρ , e.g. ρ ≥3, they have simple approximations: Q ∼( R 2 + R +1)/(3 R 2 ) and Ca E ∼ 3 3 π ρ / 2   ( 6  ln  ⁡ ρ + 2  ln ⁡ [ 96 π ] − 9 ) / ( 12  ln  ⁡ ρ + 4  ln ⁡ [ 96 π ] − 17 )   ( R + 1 ) 2 / ( R − 1 ) 2 .

scholarly journals Electrohydrodynamics of viscous drops in strong electric fields: numerical simulations

2017 ◽  
Vol 829 ◽  
pp. 127-152 ◽  
Author(s):  
Debasish Das ◽  
David Saintillan
Keyword(s):  
Numerical Simulations ◽  
Electric Fields ◽  
Small Deformation ◽  
Dielectric Liquid ◽  
Uniform Electric Field ◽  
Liquid Drops ◽  
Wide Range ◽  
Dielectric Model ◽  
Leaky Dielectric Model ◽  
Leaky Dielectric

Weakly conducting dielectric liquid drops suspended in another dielectric liquid and subject to an applied uniform electric field exhibit a wide range of dynamical behaviours contingent on field strength and material properties. These phenomena are best described by the Melcher–Taylor leaky dielectric model, which hypothesizes charge accumulation on the drop–fluid interface and prescribes a balance between charge relaxation, the jump in ohmic currents from the bulk and charge convection by the interfacial fluid flow. Most previous numerical simulations based on this model have either neglected interfacial charge convection or restricted themselves to axisymmetric drops. In this work, we develop a three-dimensional boundary element method for the complete leaky dielectric model to systematically study the deformation and dynamics of liquid drops in electric fields. The inclusion of charge convection in our simulations permits us to investigate drops in the Quincke regime, in which experiments have demonstrated a symmetry-breaking bifurcation leading to steady electrorotation. Our simulation results show excellent agreement with existing experimental data and small-deformation theories.

Transient Electrohydrodynamic Manipulation of Particles on the Surface of a Drop

2016 ◽  
Author(s):  
Edison C. Amah ◽  
Ian S. Fischer ◽  
Pushpendra Singh
Keyword(s):  
Electric Field ◽  
Applied Electric Field ◽  
Control Parameter ◽  
Transient Behavior ◽  
Uniform Electric Field ◽  
Electrohydrodynamic Flow ◽  
Dielectric Model ◽  
Leaky Dielectric Model ◽  
Leaky Dielectric

In our previous studies we have shown that particles adsorbed on the surface of a drop can be concentrated at its poles or equator by applying a uniform electric field. This happens even when the applied electric field is uniform; the electric field on the surface of the drop is nonuniform, and so particles adsorbed on the surface are subjected to dielectrophoretic (DEP) forces. In this study, we use leaky dielectric model to model the transient behavior of particles at low electric field frequencies. We show that the frequency of the electric field is an important control parameter that under certain conditions can be used to collect particles at the poles or the equator, and to move them from the poles to the equator. The speed with which particles move on the surface depends on the strength of the electrohydrodynamic flow which diminishes with increasing frequency.

Electrohydrodynamics of Drops and Vesicles

2019 ◽  
Vol 51 (1) ◽  
pp. 305-330 ◽  
Author(s):  
Petia M. Vlahovska
Keyword(s):  
Electric Field ◽  
Electric Fields ◽  
Lipid Membranes ◽  
Uniform Electric Field ◽  
Dielectric Fluids ◽  
Weakly Conducting ◽  
Strong Electric Fields ◽  
Leaky Dielectric ◽  
Quincke Rotation ◽  
Experimental And Theoretical Studies

The 1969 review by J.R. Melcher and G.I. Taylor defined the field of electrohydrodynamics. Fifty years on, the interaction of weakly conducting (leaky dielectric) fluids with electric fields continues to yield intriguing phenomena. The prototypical system of a drop in a uniform electric field has revealed remarkable dynamics in strong electric fields such as symmetry-breaking instabilities (e.g., Quincke rotation) and streaming from the drop equator. This review summarizes recent experimental and theoretical studies in the area of fluid particles (drop and vesicles) in electric fields, with a focus on the transient dynamics and extreme deformations. A theoretical framework to treat the time evolution of nearly spherical shapes is provided. The model has been successful in describing the dynamics of vesicles (closed lipid membranes) in an electric field, highlighting the broader range of applicability of the leaky dielectric approach.

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