A spectral boundary integral method for simulating electrohydrodynamic ows in viscous drops Mohammadhossein Firouznia1Spencer H. Bryngelson2and David Saintillan1

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A spectral boundary integral method for simulating electrohydrodynamic

ﬂows in viscous drops

Mohammadhossein Firouznia,1Spencer H. Bryngelson,2and David Saintillan1, ∗

1Department of Mechanical and Aerospace Engineering,

University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA

2School of Computational Science & Engineering, Georgia Institute of Technology,

756 West Peachtree Street NW, Atlanta, GA 30332, USA

(Dated: October 12, 2022)

A weakly conducting liquid droplet immersed in another leaky dielectric liquid can exhibit rich dy-

namical behaviors under the eﬀect of an applied electric ﬁeld. Depending on material properties and

ﬁeld strength, the nonlinear coupling of interfacial charge transport and ﬂuid ﬂow can trigger elec-

trohydrodynamic instabilities that lead to shape deformations and complex dynamics. We present

a spectral boundary integral method to simulate droplet electrohydrodynamics in a uniform electric

ﬁeld. All physical variables, such as drop shape and interfacial charge density, are represented using

spherical harmonic expansions. In addition to its exponential accuracy, the spectral representation

aﬀords a nondissipative dealiasing method required for numerical stability. A comprehensive charge

transport model, valid under a wide range of electric ﬁeld strengths, accounts for charge relaxation,

Ohmic conduction, and surface charge convection by the ﬂow. A shape reparametrization tech-

nique enables the exploration of signiﬁcant droplet deformation regimes. For low-viscosity drops,

the convection by the ﬂow drives steep interfacial charge gradients near the drop equator. This in-

troduces numerical ringing artifacts we treat via a weighted spherical harmonic expansion, resulting

in solution convergence. The method and simulations are validated against experimental data and

analytical predictions in the axisymmetric Taylor and Quincke electrorotation regimes.

I. INTRODUCTION

A wide range of engineering applications involve liquid drops immersed in another ﬂuid while subject to

an applied electric ﬁeld. Some examples include ink-jet printing [1], electrospraying [2], and microﬂuidic

devices and pumps [3]. These systems exhibit rich dynamics due to the electric ﬁeld and ﬂuid ﬂow

coupling. When an interface separating two immiscible ﬂuids is subject to an otherwise uniform electric

ﬁeld, the electric ﬁeld undergoes a jump across the interface due to the mismatch in material properties.

This discontinuity in the electric ﬁeld induces electric stresses that can deform the interface and drive

the ﬂuid into motion.

We are interested in leaky dielectric liquids such as oils, which serve as poor conductors. Unlike

electrolyte solutions where diﬀuse Debye layers aﬀect the system’s dynamics, leaky dielectrics are char-

acterized by the absence of diﬀuse Debye layers [4]. The free charges instead concentrate on the inter-

faces between diﬀerent phases in the system. Consequently, the electric ﬁeld acting on the interfacial

charge creates electric stresses along the normal and tangential directions, which cause deformations

and ﬂuid motion. Surface tension has a stabilizing eﬀect in general, trying to restore the equilibrium

shape. Melcher and Taylor developed a framework for studying electrohydrodynamic phenomena in

leaky dielectric systems, known as the leaky dielectric model (LDM) [5]. Central to their work is a

charge conservation model that describes a balance between Ohmic ﬂuxes from the bulk, interfacial

charge convection, and ﬁnite charge relaxation. It was previously shown that LDM can be derived

asymptotically from electrokinetic models in the limit of strong electric ﬁelds and thin Debye layers

[6,7].

This work focuses on the dynamics of a leaky dielectric drop immersed in another dielectric ﬂuid

under a uniform DC electric ﬁeld. This canonical problem has been a long-standing research problem

in electrohydrodynamics. In his pioneering work, Taylor [8] formulated a small-deformation theory

for an isolated drop based on LDM and could predict oblate and prolate steady shapes depending on

the material properties. While Taylor’s theory shows good agreement with experimental data in the

∗Corresponding author: dstn@ucsd.edu

arXiv:2210.04957v1 [physics.flu-dyn] 10 Oct 2022

limit of vanishing electric capillary number CaE(ratio of electric to capillary forces), the discrepancy is

signiﬁcant at larger values of CaE. Therefore, other researchers attempted to extend Taylor’s work by

accounting for second-order eﬀects in CaE[9], considering spheroidal drops [10,11], including inertial

eﬀects [12] and interfacial charge convection [13–16].

A variety of computational models have been developed to study drop dynamics under strong electric

ﬁelds at ﬁnite deformations, a problem untractable using analytical theories. In the limit of negligible

inertia, boundary integral equations can be used to formulate and solve the coupled electrohydrody-

namic problem. Sherwood was the ﬁrst to develop a boundary element method for an axisymmetric

drop in an equiviscous system and applied it to capture breakup modes in prolate drops [17]. His

original work was subsequently extended to study drop pair interaction [18] and to cover a wider range

of ﬂuid and electric parameters [19]. These earlier attempts used a simpliﬁed boundary condition for

the electric problem, which neglected transient charge relaxation and interfacial charge convection by

the ﬂow. These two eﬀects have recently been shown to play a signiﬁcant role in drop dynamics and

deformations [16]. Lanauze et al. [20] and Das and Saintillan [21] recently addressed this problem and

developed axisymmetric and three-dimensional boundary element methods based on the full Melcher-

Taylor LDM. The eﬀect of charge convection was speciﬁcally addressed in [21], where it was shown to

be responsible for Quincke electrorotation. These methods, however, were found to lack accuracy and

stability in the regime of strong electric ﬁelds. Other numerical approaches have been used to study

drop electrohydrodynamics, including immersed boundary [22], level set [23,24], and ﬁnite element

methods [25–27]. More recently, ﬁnite element simulations [28,29] were also used to investigate elec-

trohydrodynamic instabilities such as tip and equatorial streaming in drops under strong electric ﬁelds.

These latter techniques all include ﬁnite ﬂuid inertia and, with few exceptions [24], do not treat the

drop surface as a sharp interface.

Improved accuracy within the boundary integral framework can be achieved using spectral methods,

which rely on expansions of the shape and interfacial variables based on spherical harmonics. Such meth-

ods were recently developed to simulate electrohydrodynamics of lipid vesicles [30] and also extended

to the case of individual drops and drop pairs [31–33]. These studies, however, all neglected charge

relaxation and charge convection and were thus restricted to weak electric ﬁelds. Accurately capturing

charge convection is especially challenging as it nonlinearly couples ﬂuid ﬂow and charge transport on

a deformed interface. It can result in spurious aliasing errors with negative consequences for accuracy

and stability. This work addresses this challenge and presents a spectral boundary integral method for

the electrohydrodynamics of deformable liquid drops based on the complete Melcher–Taylor LDM. In-

terfacial charge convection is rigorously accounted for, and dealiasing and reparametrization techniques

are implemented to improve accuracy and stability and enable long-time simulations.

The paper is organized as follows. We deﬁne the problem and discuss the governing equations and

boundary conditions in Sec. II A, along with their non-dimensionalization in Sec. II B. Sec. II C presents

the integral form of the governing equations and boundary conditions used in developing the boundary

integral method. We discuss diﬀerent aspects of the numerical method in Sec. III: the spectral repre-

sentation of all variables in terms of spherical harmonics is discussed in III A, followed by details of the

dealiasing method in Sec. III B. Next, in Sec. III C, we summarize the numerical integration methods

used in this study and correction methods to ensure charge neutrality and incompressibility in Sec. III D.

As explained in Sec. III E, we also use a reparametrization method to improve the numerical stability in

simulations where the drop undergoes signiﬁcant deformations. We test and validate our computational

model by applying it to a wide range of dynamical behaviors, such as the axisymmetric Taylor regime

under weak electric ﬁelds in Sec. IV A, and Quincke electrorotation under stronger electric ﬁelds in

Sec. IV B. We also investigate the dynamics of low-viscosity drops in Sec. IV C, where charge convection

plays an important role. Finally, we discuss our conclusions and possible extensions of our work in

Sec. V.

II. PROBLEM DEFINITION

A. Governing equations

We consider a neutrally buoyant drop of a ﬂuid occupying volume V−immersed in an inﬁnite body

of another ﬂuid V+while subject to a uniform electric ﬁeld E∞=E∞ˆ

ezas depicted schematically

FIG. 1: Problem deﬁnition: a leaky dielectric drop with (σ−, −, µ−) is suspended in another leaky

dielectric ﬂuid with (σ+, +, µ+) and subject to an external electric ﬁeld E∞. The drop deforms and

diverges from its initially spherical shape.

in Fig. 1. The interface Dseparates the two ﬂuid media, and the surface unit normal n(x) is pointed

towards the suspending ﬂuid. Initially, the drop is uncharged and spherical with radius r0. The material

properties, namely the dielectric permittivities, electric conductivities, and dynamic viscosities, are

denoted by (±, σ±, µ±) inside and outside the drop, respectively. Under the Taylor–Melcher leaky

dielectric model [8], any net charge in the system appears on the interface D, and the bulk of the ﬂuids

remain electroneutral. Therefore, the electric potential is harmonic in the bulk:

∇2ϕ±(x)=0,x∈V±.(1)

Far away from the interface, the electric ﬁeld E=−∇ϕtends to the applied electric ﬁeld:

E+→E∞=E∞ˆ

ez,as |x| → ±∞.(2)

While the tangential component of the electric ﬁeld is continuous across the interface, its normal com-

ponent undergoes a jump due to the mismatch in material properties:

n×JEK=0,x∈D. (3)

We deﬁne the operator JFK:=F+−F−as the jump in any variable Facross the interface D. A surface

charge density develops at the interface and follows Gauss’s law,

q(x) = n·JEK,x∈D. (4)

The surface charge evolves due to Ohmic currents from the bulk and convective currents on the interface.

Consequently, it satisﬁes the conservation equation:

∂tq+n·JσEK+∇s·(qu) = 0,x∈D, (5)

where ∇s= (I−nn)· ∇ is the surface gradient operator and uis the ﬂuid velocity.

Neglecting the eﬀect of inertia and gravity, the velocity and pressure ﬁelds satisfy the Stokes and

continuity equations:

µ±∇2u±−∇p±=0,∇ · u±= 0,x∈V±.(6)

The velocity vector is continuous across the interface and vanishes far from it as

Ju(x)K=0,x∈D, (7)

u+(x)→0,as |x|→∞.(8)

The balance of interfacial forces requires that the jump in hydrodynamic and electric tractions across

the interface balance capillary forces:

JfHK+JfEK=γ(∇s·n)n,x∈D. (9)

We neglect Marangoni eﬀects due to variations in surface tension, ∇sγ=0. Hydrodynamic and electric

tractions are expressed in terms of the Newtonian and Maxwell stress tensors, respectively:

fH=n·TH,TH=−pI+µ∇u+∇uT,(10)

fE=n·TE,TE=EE −1

2E2I.(11)

The jump in electric tractions can be decomposed into tangential and normal components as

fE=JEnKEt+1

2J(En2−Et2)Kn=qEt+JpEKn,(12)

where pE=(En2−Et2)/2 is the electric pressure [19]. The ﬁrst term on the right-hand side represents

the tangential electric stresses in leaky dielectrics, and it vanishes when both ﬂuids are either perfect

dielectrics or perfect conductors.

B. Non-dimensionalization

For the system described above, a dimensional analysis yields ﬁve dimensionless groups, three of which

characterize the mismatch of material properties in the drop and the suspending ﬂuid:

R = σ+

σ−,Q = −

+, λ =µ−

µ+.(13)

The limits of λ→0 and ∞correspond to a bubble and a rigid particle, respectively. The remaining

dimensionless groups describe the system’s dynamics and can be obtained by comparing the character-

istic time scales in the problem. First, note that the response of each ﬂuid phase to Ohmic conduction

is characterized by the charge relaxation time:

τ±

c=±

σ±.(14)

The product RQ = τ−/τ +is the ratio of charge relaxation times in two ﬂuids and plays an important

role in the dynamics of the drop [21]. The polarization time for a rigid sphere under an applied electric

ﬁeld is the Maxwell–Wagner relaxation time

τMW =−+ 2+

σ−+ 2σ+=R(Q + 2)

1 + 2R τ+

c,(15)

which provides an approximate timescale for polarization of the drop. The accumulation of free charges

on the interface creates electric forces that drive the ﬂuid into motion on the electrohydrodynamic time

scale

τEHD =µ+

+E2

∞

.(16)

Deformations away from the equilibrium spherical shape relax under the eﬀect of surface tension on the

capillary time scale

τγ=µ+r0

γ.(17)

By taking the ratios of these time scales, the two remaining dimensionless groups can be deﬁned as

CaE=τγ

τEHD

=E2

∞r0

γ.Ma = τEHD

τMW

=µ+

τMW+E2

∞

.(18)

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AspectralboundaryintegralmethodforsimulatingelectrohydrodynamicowsinviscousdropsMohammadhosseinFirouznia,1SpencerH.Bryngelson,2andDavidSaintillan1,1DepartmentofMechanicalandAerospaceEngineering,UniversityofCaliforniaSanDiego,9500GilmanDrive,LaJolla,CA92093,USA2SchoolofComputationalScience&Engineeri...

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A spectral boundary integral method for simulating electrohydrodynamic ows in viscous drops Mohammadhossein Firouznia1Spencer H. Bryngelson2and David Saintillan1

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