Uniform electric field induced lateral migration of a sedimenting drop
Aditya Bandopadhyay1, Shubhadeep Mandal2 and Suman Chakraborty1,2†
1
Advanced Technology Development Center, Indian Institute of Technology Kharagpur, West Bengal- 721302,
India
2
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, West Bengal- 721302,
India
We investigate the motion of a sedimenting spherical drop in the presence of an applied
uniform electric field in an otherwise arbitrary direction in the limit of low surface charge
convection. We analytically solve the electric potential in and around the leaky dielectric
drop, and solve for the Stokesian velocity and pressure fields. We obtain the drop velocity
through perturbations in powers of the electric Reynolds number which signifies the
importance of the charge relaxation time scale as compared to the convective time scale. We
show that in the presence of electric field either in the sedimenting direction or orthogonal to
it, there is a change in the drop velocity only in the direction of sedimentation due to an
asymmetric charge distribution in the same direction. However, in the presence of an electric
field applied in both the directions, and depending on the permittivities and conductivities of
the two fluids, we obtain a non-intuitive lateral migration of drop in addition to the buoyancy
driven sedimentation. These dynamical features can be effectively used for manipulating
drops in a controlled electro-fluidic environment.
Key words: drop, leaky dielectric, sedimentation, electric Reynolds number, migration
__________________________________________________________________
1. Introduction
The study of lateral migration of particles (solid particles, drops, bubbles and
biological cells) is of immense importance because of its practical uses in microfluidic
applications such as sorting and separation of particles, transport of biological cells and so on
(Bhagat et al. 2010; Di Carlo et al. 2007; Chen 2014; Geislinger & Franke 2013). Different
mechanisms that have shown to cause the lateral migration of particles in the presence of
background flows pertain to interface deformability (Haber & Hetsroni 1972; Mortazavi &
Tryggvason 2000; Griggs et al. 2007), fluid inertia (Ho & Leal 2006), non-Newtonian
rheology (Haber & Hetsroni 1971; Aggarwal & Sarkar 2007; Mukherjee & Sarkar 2013), and
Marangoni effects (Hanna & Vlahovska 2010; Pak et al. 2014). Recently, the use of an
external electric field to manipulate particles has been employed due to the ease with which
the electric field can be applied and modulated (McGrath et al. 2014; Velev & Bhatt 2006;
Velev et al. 2003; Zeng & Korsmeyer 2004).
Electrohydrodynamics of drops and bubbles has been studied extensively in the
presence of electric fields. Starting from the pioneering work of Taylor (1966), several
†Email address for correspondence: suman@mech.iitkgp.ernet.in
2
authors have addressed the problem of deformation of neutrally buoyant leaky dielectric
drops in uniform and non-uniform electric fields (Vizika & Saville 2006; Torza et al. 1971;
Supeene et al. 2008; Lanauze et al. 2013; Zhang et al. 2013; Deshmukh & Thaokar 2013;
Thaokar 2012; Lac & Homsy 2007; Feng & Scott 2006; Ward & Homsy 2006; Ward &
Homsy 2003; Ward & Homsy 2001). In these works it was implicitly considered that the flow
field does not affect the charge distribution at the surface and this assumption decouples the
electric field from the flow field. As a non-trivial extension of the previously mentioned
works, Feng (1999) demonstrated that the deformation is significantly altered by the surface
charge convection. From this discussion one would expect that a background flow field may
also significantly affect the drop dynamics. However, there is very little study (Xu & Homsy
2006; Vlahovska 2011; Xu 2007; Mahlmann & Papageorgiou 2009; He et al. 2013) in the
present literature which considers the electrohydrodynamic motion of a single drop or bubble
in the presence of an imposed background flow, which is very common in modern
microfluidic devices. Xu & Homsy (2006) have shown the influence of surface charge
convection on the velocity of a sedimenting leaky dielectric drop in the presence of uniform
axial electric field for nearly spherical drop and small charge relaxation time scale. Later,
Vlahovska (2011) studied the drop dynamics in the presence of imposed shear flow and
uniform electric field, and showed that the surface charge convection alters the rotation rate
of a highly viscous drop. The author also showed that the surface charge convection affects
the shear viscosity and normal stress of a dilute emulsion containing drops.
In the present work, our focus is to bring out the effect of surface charge convection
on the drop motion in the presence of an arbitrarily oriented uniform electric field. We
analytically derive the motion of a sedimenting spherical drop and the associated electric and
flow fields in the presence of the electric field. We first show that the sedimentation velocity
is affected when the applied electric field acts in the direction of sedimentation (longitudinal)
or acts in the orthogonal direction of sedimentation (transverse). Quite remarkably, our
results reveal a non-intuitive transverse migration of drop due to a simultaneous application
of electric field in both longitudinal and transverse directions. The direction of migration
depends on the ratios of the permittivities and conductivities of the two fluids. In sharp
contrast to the case of a uniform axial electric field and contrary to common intuition, our
results reveal the droplet dynamics due to the central role of the coupling between the
direction of the applied electric field and the impending charge convection brought about by
the drop sedimentation. Thus, our analysis extends the axisymmetric considerations by Xu
and Homsy (2006) of a sedimenting drop in the presence of a longitudinal electric field to a
significantly more generalized paradigm, and aims at providing a unified framework for
motion of drops in arbitrary uniform electric field.
2. Problem statement
We consider a Newtonian leaky dielectric drop of radius a suspended in another
Newtonian leaky dielectric fluid sedimenting under the influence of gravity in the positive z
direction with an externally applied electric field ( E ∞ ) in an arbitrary direction (please refer
3
FIGURE 1. (Colour online) Schematic of a drop sedimenting in the presence of an imposed
electric field E∞ ( = Exe x + E y e y + Ez e z ) in an arbitrary direction. The density, viscosity,
permittivity and conductivity of the inner drop are denoted by ρ i , µi , ε i and σ i
respectively; while the same symbols with subscript e denote the properties of the external
fluid.
to figure 1). In the present work, we assume that the Reynolds number is sufficiently small so
as to completely ignore the inertia in the governing equations for fluid flow. It is also
assumed that the interfacial surface tension is sufficiently large to prevent any significant
deformation from sphericity. The center of the spherical coordinate system ( r , θ , φ ) is chosen
at the center of the drop which moves with a, yet unknown, velocity U d .
We non-dimensionalize the length by the drop radius a. In the absence of any applied
electric field, the drop sediments in the z direction with a sedimenting velocity given by the
Hadamard-Rybczynski velocity (Leal 2007) U HR
2
2 ga ( ρ i − ρ e ) ( λ + 1)
, where λ denotes
=
9
µe
( λ + 23 )
the drop to suspending medium viscosity ratio. Following the arguments by Xu and Homsy
(2006), we choose the characteristic velocity as U HR and correspondingly, the characteristic
H
hydrodynamic stress is obtained as τ ref
= µeU HR a . On similar lines, the electrical stress is
E
2
non-dimensionalized by τ ref
= ε e Eref
with Eref being the characteristic magnitude of the
applied electric field. In the present work, all the other properties are scaled by the properties
of the outside fluid; λ ( = µi µe ) denotes the viscosity ratio of the drop to medium,
R ( = σ i σ e ) denotes the conductivity ratio of the drop to medium and S ( = ε i ε e ) denotes
the permittivity ratio of the drop to medium. Under these considerations, we obtain the
4
following important non-dimensional parameters (Vlahovska 2011; Xu & Homsy 2006):
2
aε e Eref
εU
and ReE = e HR . Here, M is the Mason parameter which signifies the ratio of
M=
µeU HR
aσ e
the electrical stress to the viscous stress; and ReE is the electric Reynolds number which is
the ratio of the charge relaxation timescale to the convective timescale.
2.1. Governing equations and boundary conditions
Throughout our analysis, we present the governing equations and the respective
boundary conditions in a non-dimensional form unless specified otherwise. Under the
consideration of the leaky dielectric model (Melcher & Taylor 1969; Taylor 1966), bulk
fluids are charge free and the electric potentials, ψ i ,e , satisfy the Laplace equation of the form
∇ 2ψ i = 0,
∇ 2ψ e = 0.
(2.1)
The electric potential (ψ i ,e ) can be obtained by solving equation (2.1) by using the following
boundary conditions (Xu & Homsy 2006; Saville 1997):
(e1) specified unperturbed electric field at infinity; at r → ∞, ψ e → ψ ∞ , where E∞ = −∇ψ ∞ ,
(e2) bounded electric field inside the drop,
(e3) continuity of electric potential at the drop interface; at r = 1, ψ i = ψ e ,
(e4) charge conservation at the drop interface; at r = 1, n·( R∇ψ i − ∇ψ e ) = − ReE ∇ s ·( qVs ) ,
where q = n·( S ∇ψ i − ∇ψ e ) is the surface charge distribution, n = e r represents the outward
unit normal vector on the spherical drop interface and Vs is the velocity at the drop interface.
Here ∇ s = ∇ − n ( n·∇ ) is the surface divergence operator.
The flow field satisfies the Stokes equation and the condition of incompressibility of the form
(Happel & Brenner 1981)
∇pi = λ ∇ 2ui , ∇ ⋅ u i = 0,
∇ pe = ∇ 2 u e ,
∇ ⋅ u e = 0,
(2.2)
where ui and pi represent velocity and pressure field inside the drop, respectively, while
subscript e has been used to denote the same quantities outside the drop. One thing to note
here is that pressure pi and pe contain the contribution of the gravitational potential.
In a reference frame attached to the drop centre, the flow field satisfies the following
boundary conditions:
5
(f1) uniform velocity field at infinity; u e ( r → ∞ ) = u ∞ = − U d ,
(f2) velocity and pressure field inside the drop is bounded,
(f3) velocity fields satisfy the following boundary condition at the interface ( r = 1) : ui = u e
and ui ·n = u e ·n = 0 ,
(f4)
tangential
stresses
are
continuous
at
the
drop
interface
( r = 1) ;
n·τ i ·( I − nn ) = n·τ e ·( I − nn ) ,
where τ i ,e represetns the total stress tensor consisting of the hydrodynamic and electrical
effects. Note that in (f4), I − nn represents the surface projection operator.
2.2. Expansion in perturbation of ReE
For problems dealing with migration due to deformation, it is typical to perform a
regular domain perturbation in orders of capillary number which signifies the relative
strength of viscous force as compared to capillary force. However, we would like to focus
only on the effect of surface charge convection on the motion of the drop. Towards this, we
have assumed that the drop remains spherical, which is in line with some previous analysis
(Hanna & Vlahovska 2010; Pak et al. 2014). Here we use electrical Reynolds number ReE as
a perturbation parameter which quantifies the effectiveness of surface convection; at leading
order surface convection is absent. Now, we expand the electric potential, velocity, pressure,
stress fields and the unknown drop velocity in the following regular perturbation form
ψ i ,e = ψ i(,0e) + ReEψ i(,1e) + ReE2ψ i(,2e) + ···,
ui , e = ui(,e) + ReE ui(, e) + ReE2 ui( ,e) + ···,
0
1
2
pi ,e = pi(, e) + ReE pi(,e) + ReE2 pi(,e) + ···,
0
1
2
(2.3)
τ i ,e = τ i(,0e) + ReE τ i(,1e) + ReE2 τ i(,2e) + ···,
U d = U (d ) + ReE U (d ) + ReE2 U (d ) + ···.
0
1
2
Here U (d ) is the drop velocity in the absence of interfacial charge convection and U (d )
0
1
denotes the first contribution of charge convection on the drop velocity.
Considering the above, the governing equations for the electric field at the leading
order are given as:
∇ 2ψ i( 0 ) = 0,
∇ 2ψ e( 0 ) = 0,
subjected to the following boundary conditions
(2.4)
6
at r → ∞, ψ e( 0 ) → ψ ∞
ψ i( 0) bounded for r < 1
(2.5)
at r = 1, ψ i( 0) = ψ e( 0 )
(
)
at r = 1, er · R∇ψ i( 0) − ∇ψ e( 0 ) = 0.
The leading order velocity field and pressure satisfy the following governing equations
∇pi( 0 ) = λ ∇ 2ui( 0) , ∇ ⋅ u(i 0 ) = 0,
∇pe( 0) = ∇ 2u (e0 ) ,
(2.6)
∇ ⋅ u(e0 ) = 0,
which are subjected to the following boundary condition
u(e0 ) ( r → ∞ ) = u(∞0 ) = − U (d0 )
u(i 0) is bounded for r < 1
at r = 1, u(i 0) = u(e0)
(2.7)
at r = 1, u(i 0 ) ·e r = u (e0 ) ·e r = 0
at r = 1, e r ·τ (i 0) ·( I − e r e r ) = e r ·τ (e0) ·( I − e r e r ) .
On similar lines, we obtain the following equations for the first order electric field:
∇ 2ψ i(1) = 0,
(2.8)
∇ 2ψ e(1) = 0,
which are subjected to the, now altered, first order boundary conditions (Feng 1999; Xu &
Homsy 2006)
at r → ∞, ψ e(1) → 0
ψ i(1) bounded for r < 1
at r = 1, ψ i(1) = ψ e(1)
(
)
(2.9)
(
at r = 1, er · R∇ψ i(1) − ∇ψ e(1) = −∇ s · q ( 0 ) Vs( 0 )
(
)
)
where q ( 0) = e r · S ∇ψ i( 0) − ∇ψ e( 0 ) .
It is to be noted that in the boundary condition for the normal electric field, the effect of
surface convection now appears through the redistribution of the surface charge. The first
order velocity field can then be found out using:
∇pi(1) = λ ∇2 ui(1) , ∇ ⋅ ui(1) = 0,
∇pe(1) = ∇ 2u(e1) ,
∇ ⋅ u(e1) = 0,
(2.10)
7
which are subjected to the boundary conditions
u (e1) ( r → ∞ ) = u(∞1) = − U (d1) ,
u (i1) is bounded for r < 1,
at r = 1, u (i1) = u(e1) ,
(2.11)
at r = 1, u(i1) ·e r = u(e1) ·e r = 0,
at r = 1, e r ·τ (i1) ·( I − e r e r ) = er ·τ (e1) ·( I − e r e r ) .
3. Analytical solution in the spherical limit
3.1. First iteration - leading order electric field and flow field
The solution of equation (2.4) which gives the potential distribution inside and outside
the drop is of the form:
∞
(
)
ψ i( 0) = ∑ an( 0, m) cos mφ + aˆn(0,m) sin mφ r n Pn ,m ( cos θ ) ,
n =0
∞
ψ e =ψ ∞ + ∑
(0)
n =0
(3.1)
(
)
(0)
0
bn , m cos mφ + bˆn( ,m) sin mφ r − n −1 Pn , m ( cos θ ) ,
where ψ ∞ = − r ( Ex P1,1 cos φ + E y P1,1 sin φ + Ez P1,0 ) is the unperturbed imposed electric potential
at infinity,
{E , E , E }
x
y
z
are the non-dimensional components of applied electric field and
Pn , m ( cos θ ) is the associated Legendre polynomial of degree n and order m. The unknown
(
coefficients an( 0,m) , aˆn( 0,m) , bn( 0, m) and bˆn( ,0m)
)
can be determined by using the boundary conditions
(equation(2.5)) and subsequently the leading order electric potential can be obtained as
ψ i( 0) = −
 3E cos φ 3E y sin φ 
3 Ez
rP1,0 −  x
+
 rP1,1 ,
2+ R
2+ R 
 2+ R
ψ e(0) = − r ( Ex P1,1 cos φ + E y P1,1 sin φ + Ez P1,0 ) +
Ez ( R − 1)
(2 + R) r2
 Ex ( R − 1) cos φ E y ( R − 1) sin φ  1
+

2+ R
+
2+ R
P1,0
(3.2)
 2 P1,1 .
r
The leading order surface charge distribution at the drop interface is obtained as
q( 0 ) =
3( R − S )
 Ez P1,0 + ( Ex cos φ + E y sin φ ) P1,1  .

R+2 
(3.3)
The general solution of equation (2.6) for the flow field in spherical coordinates can
be obtained by using the Lamb solution (Lamb 1975) which expresses the velocity and
8
pressure in terms of solid spherical harmonics. The leading order velocity and pressure field
inside the drop can be represented as (Pak & Lauga 2014; Happel & Brenner 1981; Hetsroni
& Haber 1970)
∞ 
n+3
n
0
0
0
0
0 
r 2∇pn( ) −
ui( ) = ∑ ∇ × rχ n( ) + ∇Φ (n ) +
rpn( ) ,
2 ( n + 1)( 2n + 3 ) λ
( n + 1)( 2n + 3) λ
n =1 
 (3.4)
(
)
∞
pi( ) = ∑ pn( ) ,
0
0
n =1
where r is the dimensionless position vector of magnitude r ; pn( ) , Φ (n ) and χ n(
0
0
0)
are the
growing solid spherical harmonics of the following form
n
(
)
(
)
(
)
0
0
0
pn( ) = λ r n ∑ An( .m) cos mφ + Aˆ n( ,m) sin mφ Pn , m ,
m =0
n
0
0
0
Φ (n ) = r n ∑ Bn( .m) cos mφ + Bˆn( , m) sin mφ Pn ,m ,
m=0
n
(3.5)
χ n(0) = r n ∑ Cn(0, m) cos mφ + Cˆ n( 0, m) sin mφ Pn ,m .
m= 0
Similarly, the leading order velocity and pressure outside the drop can be represented using
Lamb solution as
u(e ) = u(∞ ) + v(e )
0
0
0
∞ 
n−2
n +1
0
0
0
0
0 
= u (∞ ) + ∑ ∇ × rχ −( n)−1 + ∇Φ (− n)−1 −
r 2∇p−( n)−1 +
rp−( n)−1  ,
2n ( 2n − 1)
n ( 2n − 1)
n =1 

(
∞
)
(3.6)
pe( ) = ∑ p−( n)−1 ,
0
0
n =1
where v (e ) is the disturbance velocity field outside the drop which vanishes at infinity. Here
0
p−( n)−1 , Φ (− n)−1 and χ −( n)−1 are decaying solid spherical harmonics of the following form
0
0
0
n
0
0
0
p−( n)−1 = r − n −1 ∑ A−( n)−1,m cos mφ + Aˆ −( n)−1, m sin mφ Pn , m ,
m=0
(
n
)
0
0
0
Φ (− n)−1 = r − n −1 ∑ B−( n)−1, m cos mφ + Bˆ −( n)−1, m sin mφ Pn , m ,
m=0
n
(
)
χ −( 0n)−1 = r − n −1 ∑ C−(0n)−1, m cos mφ + Cˆ −(0n)−1,m sin mφ Pn,m .
m =0
(
)
In a similar manner u(∞ ) can also be expressed in terms of spherical harmonics as
0
(3.7)
9
u(∞0) =

∞
∑ ∇ × ( rχ ( ) ) + ∇Φ
n =−∞
∞ 0
n

∞( 0 )
n
+

n+3
n
r 2∇pn∞( 0 ) −
rpn∞( 0 ) ,
2 ( n + 1)( 2n + 3)
( n + 1)( 2n + 3)

(3.8)
where pn ( ) , Φ n ( ) and χ n ( ) are solid spherical harmonics of the following form (Hetsroni &
∞ 0
∞ 0
∞ 0
Haber 1970)
2 ( 2n + 3) n n
r ∑ α n( 0,m) cos mφ + αˆ n( 0,m) sin mφ Pn ,m ,
n
m =0
n
1
Φ ∞n ( 0 ) = r n ∑ β n( 0,m) cos mφ + βˆn( 0,m) sin mφ Pn , m ,
n m=0
n
1
r n ∑ γ n( 0,m) cos mφ + γˆn( 0,m) sin mφ Pn , m .
χ n∞( 0) =
n ( n + 1) m =0
(
pn∞( 0 ) =
)
(
)
(
(3.9)
)
0
0
0
0
0
0
The coefficients α n( ,m) , αˆ n( , m) , β n( , m) , βˆn( ,m) , γ n( ,m) and γˆn( ,m) can be related to the unknown drop
velocity at leading order ( U(d ) = U dx( )e x + U dy( ) e y + U dz( ) e z , where e x , e y and e z are the unit
0
0
0
0
vectors in x, y and z directions, respectively) to obtain the non-zero coefficients as (see
appendix A)
(0)
( 0)
(0)
β1,0
= −U dz( 0) , β1,1
= −U dx( 0) , βˆ1,1
= −U dy( 0) ,
(3.10)
where U dx( ) ,U dy( ) and U dz( ) are the component of drop velocity at leading order in x,y and z
0
0
0
directions, respectively.
The
unknown
coefficients
An( 0, m) , Bn( 0, m) , Cn( 0,m) , A−( 0n)−1, m , B−( 0n)−1, m and C−( 0n)−1, m
can
be
determined by applying the boundary conditions given in equation (2.7). One thing to note
here is that it is difficult to deal with the present form of the boundary conditions because
some of the boundary conditions contain the derivative of solid spherical harmonics which
makes the use of orthogonality property of spherical harmonics involved. This complexity
can be avoided by presenting the boundary conditions in the following form (Hetsroni &
Haber 1970; Happel & Brenner 1981)
10
ui(,0r) ⋅ e r  = 0,

s
 v e( 0 ) ⋅ e r  + u(∞0) ⋅ e r  = 0,

s 
s
 ∂ ( 0)
  ∂ ( 0)
  ∂ ( 0)

 r ∂r ui ·e r  =  r ∂r u ∞ ·er  +  r ∂r v e ·er  ,

s 
 

(
)
(
)
(
)
r ⋅ ∇ × u(i 0 )  = r ⋅∇ × u ∞( 0 )  + r ⋅∇ × v(e0 )  ,

s 
s 
s
{ ((
) )}
r·∇ × ( τ ( ) + Mτ ( ) ) ⋅ e 
{
}

{ ((
) )}
r ⋅∇ × r × τ H ( 0 ) + Mτ E ( 0) ⋅ e  = r ⋅∇ × r × τ ( 0) ⋅ e  + r ⋅∇ × r × τ H ( 0 ) + Mτ E ( 0 ) ⋅ e  ,
i
i
r
∞
r
e
e
r

 s
 s 
 s 
H 0
i
E 0
i
{ (
r
s
)}
{(
) }
0
H 0
E 0
= r·∇ × τ ∞( ) ⋅ er  + r·∇ × τ e ( ) + Mτ e ( ) ⋅ e r  ,

s 
s
(3.11)
(
)
where [ Θ]s represents the evaluation of a generic variable, Θ , at the interface of the
spherical drop ( r = 1) . The mathematical expressions of the hydrodynamic and electrical
component of stresses are given by
τ iH ( 0) = λ  − pi( 0 ) I + ∇ui( 0 ) + ∇ui( 0 )  ,


T
τ eH ( 0) =  − pe( 0 ) I + ∇v e( 0 ) + ∇v (e0 )  ,


T
τ ∞H ( 0) =  − p∞( 0 ) I + ∇u(∞0) + ∇u ∞( 0 )  ,


T
2 
1

τ iE ( 0 ) = S Ei( 0 ) Ei( 0 ) − E(i 0 ) I  ,
2


T
2 
1

τ eE ( 0 ) = E(e0) Ee( 0 ) − E(e0) I  .
2


T
(
)
(
)
(
)
(3.12)
( )
( )
Applying these form of boundary conditions (see appendix B), we obtain the coefficients of
the solid spherical harmonics in terms of the unknown drop velocity in the following form
11
( 2n + 3) ( 2n + 1)( 2n − 1) β n( 0,m) + M ( g ni(,0m) − g ne(,m0) )
An , m =
,
n ( 2n + 1)( λ + 1)
( 0)
An( , m)
0
Bn( , m) = −
0
2 ( 2n + 3)
(
M hn ,(m) − hn(, m)
(0)
Cn ,m =
( 0)
A− n −1,m
,
e 0
i 0
)
n ( n + 1) ( λ ( n − 1) + ( n + 2 ) )
,
(
)
( 4n 2 − 1) {( 2n + 1) λ + 2} β ( 0 ) + M (1 − 2n ) g i( 0) − g e( 0 ) 
n ,m
n,m
n,m
,
=−
( n + 1)( 2n + 1)( λ + 1)
( 0)
B− n −1,m = −
( 4n
(
− 1) λβ n( , m) + M g n ,(m) − g n(, m)
0
2
e 0
i 0
2 ( 2n + 1)( n + 1)( λ + 1)
),
C−( n)−1, m = Cn( ,m) ,
0
0
(3.13)
keeping in mind that the following coefficients are non-zero from equation (3.10)
(0)
( 0)
(0)
: β1,0
= −U dz( 0) , β1,1
= −U dx( 0) and βˆ1,1
= −U dy( 0) .
Please
note
that
the
coefficients
g ni ,,em( 0) , gˆ ni ,,em( 0) , hni ,,em( 0) and hˆni ,,em( 0) are obtained from the surface harmonic representation of the
electrical stresses as shown in Appendix B, equation (B5). Similar to the coefficients found in
0
0
0
0
0
0
equation (3.13) the other coefficients Aˆ ( ) , Bˆ ( ) , Cˆ ( ) , Aˆ ( ) , Bˆ ( )
and Cˆ ( )
are obtained
n,m
by replacing β
( 0)
n,m
,g
i (0)
n ,m
and g
e( 0 )
n,m
( 0)
n,m
by βˆ
, gˆ
n,m
i (0)
n ,m
− n −1, m
n,m
and gˆ
e( 0 )
n,m
− n −1, m
− n −1, m
, respectively in equation (3.13). The
complete expressions of velocity and pressure at the leading order are given in Appendix C.
3.2. Leading order drop velocity: force free condition
The sedimenting velocity of a drop is obtained by imposing the force free condition.
At steady state the buoyancy force balances the hydrodynamic drag force on the drop. In a
dimensional form this can be represented at the leading order as (Hetsroni & Haber 1970;
Leal 2007)
4π 3
0
a ( ρ i − ρ e ) ge z − 4π∇ r 3 p−( 2) = 0.
3
(
)
(3.14)
Using the previously prescribed non-dimensional scheme equation (3.14) may be written as
 3λ + 2 
3 ( 0)

 e z − ∇ r p−2 = 0,
 λ +1 
(
where e z = cos θ e r − sin θ eθ
and
(
(3.15)
)
p−( 02) = r −2  A−( 02,0) P1,0 + A−( 02,1) cos φ + Aˆ−( 02,)1 sin φ P1,1  . Now,


substituting e z and p−( 2) in equation (3.15), we obtain
0
)
12
Non-zero coefficients of ψ i( )
Non-zero coefficients of ψ e( )
(1)
a1,0
)
b−(12,0
(1)
(1) (1)
a2,0
, a2,1
, aˆ2,1
)
) ˆ(1)
b−(13,0
, b−(13,1
, b−3,1
(1)
(1) (1) (1)
(1)
(1) (1)
a3,0
, a3,1
, aˆ3,1 , a3,2 , aˆ3,2
, a3,3
, aˆ3,3
)
) ˆ(1)
) ˆ(1) ˆ (1) ˆ (1)
b−(14,0
, b−(14,1
, b−4,1 , b−(14,2
, b−4,2 , b−4,3 , b−4,3
1
1
TABLE 1. Non-zero coefficients present in first order electric potential.
3λ + 2
, A−( 02,1) = 0, Aˆ−( 02,1) = 0.
2 ( λ + 1)
A−( 02,0) =
(3.16)
We obtain the drop velocity by using equation (3.16) and equation (3.13) of the form
( 0)
U dx =
( 0)
U dy =
( 0)
(
e( 0 )
i( 0 )
M g1,1
− g1,1
3 ( 2 + 3λ )
(
e( 0 )
i( 0 )
M gˆ1,1
− gˆ1,1
U dz = 1 +
3 ( 2 + 3λ )
),
),
(
e( 0 )
i( 0 )
M g1,0
− g1,0
3 ( 2 + 3λ )
(3.17)
).
After solving leading order electric field we obtain g n(, m) = g n,(m) = gˆ n(, m) = gˆ n,(m) = 0 , which gives
i 0
e 0
i 0
e 0
the following drop velocity
U dx( 0) = U dy( 0) = 0, U dz( 0) = 1.
(3.18)
3.3. Second iteration - effect of surface convection on electric field
Having obtained the leading order velocity profile from the previous subsection
(details are given in Appendix C), we may now attempt the solution to equations (2.8) and
(2.10) which are subjected to the boundary conditions (2.9) and (2.11), respectively. The first
effect of charge convection appear in the charge convection boundary condition through the
leading order velocity field and thus affects the first order electric field and, consequently, the
first order velocity field. For convenience, we rewrite the boundary condition here:
(
)
(
) . This differs from the leading order condition in the
right hand side where it is zero, i.e. e ·( R∇ψ ( ) − ∇ψ ( ) ) = 0 .
e r · R∇ψ i( ) − ∇ψ e( ) = −∇ s · q ( ) Vs(
1
1
0
0)
0
r
i
0
e
Given that the potential fields are harmonic functions, the solutions are again given as
13
∞
(
)
ψ i(1) = ∑ an(1,)m cos mφ + aˆn(1,m) sin mφ r n Pn, m ( cos θ ) ,
n =0
∞
(3.19)
(
)
ψ e = ∑ bn, m cos mφ + bn, m sin mφ r
(1)
n =0
(1)
ˆ(1)
− n −1
Pn , m ( cos θ ) .
Hence, it is instructive to first evaluate the right hand side of the charge convection boundary
(
)
condition ∇ s · q ( 0 ) Vs( 0 ) in order to establish the various surface harmonics that it may contain.
It can be recast as (Kim & Karrila 1991)
(
)
(
)
−∇ s · q ( 0 ) Vs( 0) = −2 q ( 0) Vs( 0 ) ·e r −
1 ∂ (0) ( 0)
1 ∂ (0) (0)
q Vs ·eθ −
q Vs ·eφ . (3.20)
sin θ ∂θ
sin θ ∂φ
(
)
(
)
By substituting the form of the velocity field (given in appendix C) and the accumulated
charge (from equation (3.2)) in terms of the leading order electric field, the above expression
can be split into various surface harmonics by making use of the orthogonality of the
spherical
surface
harmonics.
If
we
write
expression
(3.20)
as
∞
n
∑∑ ( Z
n ,m
)
cos mφ + Zˆ n , m sin mφ Pn , m ,
n = 0 m=0
we
obtain
the
following
nonzero
terms:
Z1,0 , Z1,1 , Z 2,0 , Z 2,1 , Z 3,0 , Z 3,1 , Z3,2 , Z 3,3 , Zˆ1,1 , Zˆ2,1 , Zˆ3,1 , Zˆ3,2 and Ẑ3,3 . The complete expressions of
Zs are given in Appendix D. Thus it is expected that the first order electric field contains the
1
1
1
1
non-zero coefficients an( ,m) , aˆn( ,)m , b−( n)−1, m and bˆ−( n)−1, m with n and m given in table 1. Now, we use
rest of the boundary conditions from equation (2.9) and obtain the above unknown
coefficients which are given in Appendix E.
3.4. O ( ReE ) drop velocity
The first order flow field also satisfies Stokes equation (equation (2.10)), so we can
represent
( u( ) , p( ) )
1
i
1
i
and
( u( ) , p( ) )
1
e
1
e
in terms of the solid spherical harmonics
pn(1,)m , Φ (n1,)m , χ n(1, m) , p−(1n)−1,m , Φ (−1n) −1,m and χ −(1n)−1, m using Lamb solution in a similar way as
represented for the leading order solution in section 3.1. The details are worked out in
Appendix F. In tune with the leading order drop velocity estimation, we may proceed in a
similar fashion to evaluate the first order drop velocity
(
)
0 − 4π∇ r 3 p−( 2) = 0.
1
(3.21)
Thus, employing the force-free condition at the first order, we obtain the following
)
)
)
A−(12,0
= 0, A−(12,1
= 0, Aˆ −(12,1
= 0.
This translates into the following higher order velocity:
(3.22)
14
(1)
U dx =
(1)
U dy =
(1)
U dz =
(
e(1)
i (1)
M g1,1
− g1,1
3 ( 2 + 3λ )
(
e(1)
i (1)
M gˆ1,1
− gˆ1,1
3 ( 2 + 3λ )
(
e(1)
i (1)
M g1,0
− g1,0
3 ( 2 + 3λ )
),
),
(3.23)
).
e ,i 1
e ,i 1
Substituting the following g n ,m( ) and gˆ n ,m( ) in equation (3.23), we obtain
U d(1x) = −
3 M ( R − S )( 3R − S + 3) Ex Ez
,
10 ( 3 + 2 R )( R + 2 )2 ( 2 + 3λ )( λ + 1)
U dy(1) = −
3 M ( R − S )( 3R − S + 3) E y Ez
,
10 ( 3 + 2 R )( R + 2 )2 ( 2 + 3λ )( λ + 1)
(1)
U dz
2
3 M ( R − S )( 3R − S + 3) ( Ez + 3)
=−
.
10 ( 3 + 2 R )( R + 2 )2 ( 2 + 3λ )( λ + 1)
(3.24)
(3.25)
Please note that we have made use of the identity E x2 + E y2 + E z2 = 1 .
4. Summary and discussion
The general drop velocity can thus be written as the sum of the drop velocities
obtained from the above two orders as found from equations (3.17) and (3.24).
Mathematically,
U d = U (d0) + ReE U(d1)
2

3 M ( R − S )( 3R − S + 3) ( Ez + 3 ) 
 ez
= 1 − ReE
10 ( 3 + 2 R )( R + 2 )2 ( 2 + 3λ )( λ + 1) 



3 M ( R − S )( 3R − S + 3) Ex Ez 
−  ReE
 ex
10 ( 3 + 2 R )( R + 2 ) 2 ( 2 + 3λ )( λ + 1) 


3 M ( R − S )( 3R − S + 3) E y Ez 
−  ReE
ey
10 ( 3 + 2 R )( R + 2 ) 2 ( 2 + 3λ )( λ + 1) 

(4.1)
If we consider an applied electric field only in the z-direction, i.e. E∞ = ( 0, 0,1) , we obtain
15


M ( R − S )( 3R − S + 3)
6
U d = 1 − ReE
 ez ,
5 ( 3 + 2 R )( R + 2 ) 2 ( 2 + 3λ )( λ + 1) 

(4.2)
which corroborates with the drop sedimentation velocity obtained by Xu and Homsy (2006).
It is interesting to note one particular aspect of the drop velocity obtained above.
When an electric field is applied only in the x-direction, we still get a change in the
sedimentation velocity while there is no lateral migration velocity. However, when the
applied electric field has all three components as non-zero, i.e. E∞ = ( Ex , E y , Ez ) , not only
does the sedimentation get affected but there is also a lateral migration observed in both the x
and y directions. Towards discussing the physical reason behind the observations, we
consider the following three cases: (a) E∞ = ( 0, 0,1) (longitudinal electric field), (b)
(
E∞ = (1, 0, 0 ) (transverse electric field) and (c) E∞ = 1
2 , 0,1
)
2 (both longitudinal and
transverse fields).
4.1. Longitudinal electric field
Under the assumption of a neutrally buoyant droplet, advancing the problem
considered by Taylor (1966), Feng (1999) showed that the consideration of surface charge
convection led to alterations in the drop deformation. It is to be noted that despite charge
convection, for a neutrally buoyant drop, the charge distribution remains anti-symmetric
along the equatorial plane. The consequence of this is that there is no extra drag acting on the
drop and consequently the drop remains stationary. The problem of a drop with a different
density than the suspending fluid in the presence of longitudinal applied electric field was
then considered by Xu and Homsy (2006) who highlighted the pivotal role of the surface
charge convection in breaking the anti-symmetry about the equatorial plane thus giving rise
to a net drag on the drop. This can be seen from figure 2 which is derived using the
expressions in section 3 and setting E z = 1 while setting the other components of the electric
field to be zero. The choice of the parameters used in figure 2 is mentioned in the figure
caption. Figure 2(a) depicts the leading order charge distribution which is anti-symmetric
about the equatorial plane. However, it can be clearly seen from figure 2(b) that the surface
convection for a sedimenting drop does lead to a breaking in the anti-symmetric structure of
the charge distribution. It is precisely this asymmetric charge distribution that results in the
alteration in the sedimentation velocity.
16
Meridional plane
(a)
(b)
Equatorial plane
E∞ = ( 0, 0,1)
FIGURE 2. (Colour online) Contours depicting the variation of the (a) q( ) and (b)
0
q( ) + ReE q ( ) for a longitudinal electric field (case I). The parameters employed are
0
1
R = 0.5 , S = 2 , M = 2 , λ = 0.5 and ReE = 0.2 .
4.2. Transverse electric field
A striking result obtained in equation (4.1) is that even if there is an applied electric
field in transverse direction, the influence of charge convection is solely affecting the
sedimentation velocity without causing any migration in the lateral direction. To understand
this from the view point of symmetries in the charge distribution, we appeal to figure 3. It is
seen from figure 3(a) that the absence of surface convection leads to an anti-symmetric
structure about the meridional plane and maintains a symmetry about the equatorial plane. In
the presence of surface convection due to sedimentation, however, it is observed that the antisymmetric structure is preserved about the meridional plane, but the symmetry is broken
about the equatorial plane. This leads to a net force in the longitudinal direction without
affecting the motion in the transverse direction, leading to an alteration in the sedimentation
velocity.
17
(a)
(b)
E∞ = (1, 0, 0 )
FIGURE 3. (Colour online) Contours depicting the variation of the (a) q( ) and (b)
0
q( ) + ReE q ( ) for a transverse electric field (case II). The parameters employed are
0
1
R = 0.5 , S = 2 , M = 2 , λ = 0.5 and ReE = 0.2 .
4.3. Combined longitudinal and transverse electric field
We depict the charge distribution due to surface convection in figure 4 where it can be
seen that the combined nature of the applied electric field causes an asymmetry about both
the equatorial and meridional plane leading to a force in both the longitudinal and transverse
directions respectively. The findings are also seen in the expressions for the droplet velocity
(
by setting E∞ = 1
2 , 0,1
2
)
in equation (4.1). A transverse or longitudinal applied
electric field alone is insufficient to cause any lateral migration.
1 
 1
E∞ = 
,0,

2
 2
FIGURE 4. (Colour online) Contours depicting the variation of q( ) + ReE q ( ) for a
0
1
combined longitudinal and transverse electric field (case III). The parameters employed
are R = 0.5 , S = 2 , M = 2 , λ = 0.5 and ReE = 0.2 .
18
5. Conclusions
We conclude by remarking on the direction of lateral velocity of drop in the presence
of both longitudinal and transverse electric field. Depending on the combination of the
direction of applied electric field and the ratios of permittivities and conductivities, the
direction of the motion is decided. Towards this, figure 5 depicts the two regimes of the
transverse velocity of drop in x direction (U dx ) in the R − S plane considering
(
E∞ = 1
2 , 0,1
)
2 . We identify two regime boundaries given by R = S and S = 3R + 3 ,
where the former case is tantamount to a perfect dielectric case with zero circulation while
the latter case denotes the zero-lateral-migration line for a leaky dielectric drop (in
accordance with equation (4.1) for U dx = 0 ).
FIGURE 5. (Colour online) Different regimes of lateral velocity of drop (U dx ) in R − S
plane. The parameters employed are M = 2 , λ = 0.5 and ReE = 0.2 .
It is to be noted that in the above case, the lateral migration depends on the product of
E x and E z . As a result, the migration velocity, U dx , remains unaltered by the reversal of
direction of E∞ . In conclusion, we have demonstrated that the arbitrary nature of the uniform
electric field is responsible for the lateral migration of an otherwise longitudinally
sedimenting drop for small Re E . The asymmetry in the charge distribution about the
meridional plane is caused due to the simultaneous action of the electric field in both the
longitudinal and transverse directions. The impending sedimentation velocity is attributed to
19
the tendency of the drop to migrate laterally while sedimenting. The analytical results
presented here are expected to have a broad appeal to scientists and engineers alike who are
concerned with processes pertaining to electrohydrodynamic drop motion and manipulation.
Appendix A. Derivation of the relation between the coefficients present in the solid
spherical harmonics present in u(∞ ) with the component of drop velocity U (d )
0
0
The coefficients present in u(∞ ) can be related to the cartesian components of drop
0
velocity U dx( ) , U dy( ) and U dz( ) . At first we obtain the dot product of unit radial vector with u(∞ )
0
0
0
0
as (Hetsroni & Haber 1970)
∞
n
∞
n
u(∞ ) ⋅ e r = ∑ ∑ α n( , m) r n +1 + β n( ,m) r n −1 + α −( n)−1, m r − n + β −( n)−1r n − 2 Pn , m cos mφ


n =1 m = 0
0
0
+ ∑∑ αˆ n ,m r
( 0)
0
n +1
0
ˆ (0)
+ β n ,m r
n −1
0
(0)
+ αˆ − n −1, m r
−n
ˆ ( 0)
+ β − n −1, m r
n =1 m = 0
n− 2
(A1)
Pn , m sin mφ .

As u(∞ ) = − U(d ) , by taking the dot product of both the sides with unit radial vector and using
0
0
equation (A1), we obtain
∞
n
∑ ∑ α ( ) r
0
n +1
n,m
n =1 m = 0
∞
+ β n( ,0m) r n −1 + α −( 0n)−1, m r − n + β −( 0n)−1,m r n − 2 Pn ,m cos mφ
n
+ ∑ ∑ αˆ n( 0, m) r n +1 + βˆn( ,0m) r n −1 + αˆ −( 0n)−1,m r − n + βˆ−( 0n)−1, m r n −2 Pn ,m sin mφ
(A2)
n =1 m = 0
= − U dx( 0 ) cos φ P1,1 + U dy( 0 ) sin φ P1,1 + U dz( 0) P1,0  .
Now, comparing both sides of equation (A2) we obtain the non-zero α and β in the
following form
(0)
( 0)
( 0)
β1,0
= −U dz( 0) , β1,1
= −U dx( 0) , βˆ1,1
= −U dy( 0) .
For other values of n and m, the coefficients α (
0)
(A3)
and β ( ) vanish. The coefficient γ ( ) can be
0
0
obtained by taking r ⋅∇ × operation over both sides of u(∞ ) = − U(d ) . We obtain
0
r ⋅ ∇ × u(∞ ) = r ⋅ ∇ × U (d ) = 0,
0
0
0
(A4)
20
where r ⋅∇ × u(∞ ) can be expressed as (Hetsroni & Haber 1970)
0
∞
n
{
}
{
}
r ⋅ ∇ × u(∞0 ) = ∑ ∑  γ n( 0,m) r n + γ −( 0n)−1, m r − n −1 cos mφ + γˆn( 0, m) r n + γˆ−( 0n)−1,m r − n −1 sin mφ Pn , m . (A5)


n =1 m = 0
Now, comparing equations (A4) and (A5), we obtain γ n( , m) = γˆn( ,m) = γ −( n)−1, m = γˆ−( n)−1, m = 0 for all
0
0
0
0
values of n and m. Similar relations can also be obtained for O ( ReE ) .
Appendix B. Implementation of boundary conditions to obtain the solid spherical
harmonics present in the Lamb solution
The implementation of the velocity and stress boundary conditions are summarized
here. To apply the boundary conditions given in equation (3.11), at first one has to express
u( 0 ) ⋅ e r  ,  r ∂ u( 0) ·e r  and r ⋅∇ × u ( 0 )  in terms of solid spherical harmonics in the


 s  ∂r

s
s
(
)
following form (Happel & Brenner 1981)
∞ 
n
(0)
( 0) 
 u ( 0) ⋅ e r  = ∑ 
p
+
n
Φ
,
n
n

 s n =−∞  2 N ( 2n + 3)

 µ
∞ 
n ( n + 1)
 ∂ (0)

(0)
(0) 
r
u
·
e
=
p
+
n
n
−
1
Φ
(
)

∑
r 
n
n ,
 ∂r
 s n =−∞  2 N µ ( 2n + 3)

(
)
(B1)
∞
r ⋅ ∇ × u(0 )  = ∑ n ( n + 1) χ n(0 ) .

 s n = −∞
{ ((
) )}
H 0
0
To satisfy the shear stress balance, we have to obtain r ⋅∇ × r × τ ( ) + Mτ E ( ) ⋅ e r  and

 s
{(
) }
r·∇ × τ H ( 0 ) + Mτ E ( 0 ) ⋅ e  . Among these terms the hydrodynamic stress components can
r 

s
be obtained as
∞ 
n 2 ( n + 2 ) ( 0) 
( 0)
 r ⋅ ∇ × r × τ H ( 0) ⋅ e  = − N
2
n
−
1
n
n
+
1
Φ
+
pn  ,
(
)
(
)
r
µ ∑ 
n
 s

N
2
n
+
3
(
)
n =−∞ 

µ

( (
))
(
∞
)
r ⋅ ∇ × τ H ( 0 ) ⋅ e r  = N µ ∑ ( n − 1) n ( n + 1) χ n( 0 ) .

s
n =−∞
E (0)
The radial traction vector, τ i
(B2)
(B3)
·e r and τ e ( ) ·e r , arising from the Maxwell stress tensor arising
E 0
due to the zeroth order electric field are expressed as
21
 (0) 2 1 ( 0) 2 
 (0) 2 1 ( 0) 2 
 Ei ,r − 2 Ei ,r 
 Ee, r − 2 E e, r 

 E( 0)


τ iE ( 0 ) ·e r = S 
Ei(,0r ) Ei(,0θ)
Ee(,0r) Ee(,0θ)
 , τ e ·er = 
.




Ei(,0r ) Ei(,0φ)
Ee(,0r) Ee(,0φ)








( )
( )
(B4)
We express the electrical stress components at the drop interface in terms of surface
harmonics as
{ (
)}
∞
)}
∞
r ⋅ ∇ × r × τ E (0) ⋅ e  = ∑ g i (0) cos mφ + gˆ i(0 ) sin mφ P ,
i
r
n ,m
n ,m

 s n =0 n ,m
{ (
(
)
r ⋅ ∇ × r × τ E (0) ⋅ e  = ∑ g e(0 ) cos mφ + gˆ e(0) sin mφ P ,
i
r
n ,m
n ,m

 s n =0 n ,m
(
r·∇ × τ

E (0)
i
∞
(
∞
(
⋅ er  = ∑ h
 s n =0
)
i( 0 )
n ,m
(
)
(B5)
)
i 0
cos mφ + hˆn(,m) sin mφ Pn ,m ,
r·∇ × τ eE (0 ) ⋅ e r  = ∑ hne,(m0 ) cos mφ + hˆne,(m0) sin mφ Pn ,m ,

 s n =0
(
)
)
i 0
i 0
i 0
i 0
e 0
e 0
e 0
e 0
where g n(, m) , gˆ n(, m) , hn(, m) , hˆn(,m) , g n(,m) , gˆ n,(m) , hn,(m) and hˆn,(m) are the coefficients that can be obtained
after solving electrical potential distribution. The nonzero coefficients obtained from the
leading order electric potential (equation (3.2)) are:
g
i (0)
2,0
=
i 0
gˆ 2,1( ) =
g 2,1( ) =
e 0
9 ( E y2 + E x2 − 2 E z2 ) S
( R + 2)
−18 E y Ez S
( R + 2)
2
e 0
gˆ 2,(2 ) = −
2
9Ex E y R
( R + 2)
2
,g
i (0)
=−
, gˆ 2,2
−18 R Ez E x
( R + 2)
2
=−
9Ex E y S
( R + 2)
, g 2,(2 ) = −
e0
i( 0 )
2,1
18SEz E x
( R + 2)
( )
2 , g 2,0 =
e0
2
2
9 ( − E y + Ex ) R
2
2
( R + 2)
2
,g
i( 0 )
2,2
2
2
9 ( Ex − E y ) S
=−
,
2 ( R + 2)2
9 ( E x2 + E y2 − 2 Ez2 ) R
( R + 2)
, gˆ 2,1( ) =
e0
2
(B6)
−18 E y E z R
( R + 2)
,
2
,
.
Appendix C. Velocity and pressure field at leading order
The velocity and pressure fields inside the drop is given by

5 (0)   1 ( 0)
2 (0) 
 1 ( 0)
u(i 0) = ∇ Φ1( 0 ) + Φ (20 ) + r 2 ∇ 
p1 +
p2  − r 
p1 +
p2   ,
42λ
21λ
 5λ
  10λ


(
(0)
( 0)
)
( 0)
pi = p1 + p2 ,
where the solid harmonics are of the following form
(C1)
22
( )
Φ1( ) = rB1,0
P1,0 ,
0
0
(
)
(
)
0
(0)
(0)
(0)
( 0)
( 0)
Φ (2 ) = r 2  B2,0
P2,0 + B2,1
cos φ + Bˆ2,1
sin φ P2,1 + B2,2
cos 2φ + Bˆ 2,2
sin 2φ P2,2  ,


( )
p1( ) = λ rA1,0
P1,0 ,
0
0
(
)
(
(C2)
)
0
(0)
( 0)
( 0)
(0)
(0)
cos φ + Aˆ2,1
sin φ P2,1 + A2,2
cos 2φ + Aˆ 2,2
sin 2φ P2,2  .
p2( ) = λ r 2  A2,0
P2,0 + A2,1


0
0
0
0
The coefficients A( ) , Aˆ ( ) , B( ) and B̂ ( ) present in the above solid harmonics can be obtained
using equation (3.13) of the following form
5
63
63
63
(0)
(0)
(0)
, A2,0
= − (1 − 3Ez2 ) Λ, A2,1
= Ex Ez Λ , A2,2
= ( E x2 − E y2 ) Λ ,
λ +1
5
5
20
63
63
1
9
(0)
(0)
( 0)
= E y Ez Λ, Aˆ2,2
= E x E y Λ , B1,0
=
, B2,0
= (1 − 3Ez2 ) Λ,
5
10
2 ( λ + 1)
20
(0)
A1,0
=−
(0)
Aˆ 2,1
(0)
B2,1
=−
(C3)
9
9
9
9
(0)
( 0)
(0)
E x E z Λ , B2,2
= − ( Ex2 − E y2 ) Λ, Bˆ 2,1
= − E y Ez Λ , Bˆ2,2
= − E x E y Λ,
10
40
10
20
where the parameter Λ =
M (R− S)
( λ + 1)( R + 2 )
2
.
Similarly, the velocity and pressure fields external to the drop are given by
u(e ) = −U (d ) + v(e )
0
0
0
∞

1
1 0 

0
0
0
0
= − ( cos θ e r − sin θ eθ ) + ∑ ∇ Φ (−2) + Φ (−3) − r 2∇p−( 2) + r  2 p−( 2) + p−( 3)  ,
2
2


n =1 
(
)
(C4)
pe( ) = p−( 2) + p−( 3) ,
0
0
0
The solid harmonics are of the following form
1 (0)
1
B−2,0 P1,0 , p−( 02) = 2 A−( 02,0) P1,0 ,
2
r
r
1
0)
Φ(−03) = 3  B−( 03,0) P2,0 + B−( 03,1) cos φ + Bˆ −( 03,1) sin φ P2,1 + B−( 03,2) cos 2φ + Bˆ −( 3,2
sin 2φ P2,2  , (C5)


r
1
0)
p−( 03) = 3  A−( 3,0
P2,0 + A−( 03,1) cos φ + Aˆ −( 03,1) sin φ P2,1 + A−( 03,2) cos 2φ + Aˆ −( 03,2) sin 2φ P2,2  .


r
Φ (−02) =
(
(
)
)
(
(
)
)
0
0
0
0
The coefficients A( ) , Aˆ ( ) , B( ) and B̂ ( ) present in the above solid harmonics can be obtained
using equation (3.13) of the following form
23
A−( 2,0) =
0
2 + 3λ
9
18
9
0
0
0
, A−( 3,0) = − (1 − 3Ez2 ) Λ , A−( 3,1) = E x E Λ, A−( 3,2) = ( E x2 − E y2 ) Λ ,
2 ( λ + 1)
5
5
10
λ
18
9
3
0
0
0
0
Aˆ −( 3,1) = E y E z Λ , Aˆ−( 3,2) = E x E y Λ , B−( 2,0) =
, B−( 3,0) = − (1 − 3Ez2 ) Λ ,
5
5
4 ( λ + 1)
10
(C6)
3
3
3
3
0
0
0
0
B−( 3,1) = Ex Ez Λ , B−( 3,2) = ( E x2 − E y2 ) Λ , Bˆ −( 3,1) = E y E z Λ , Bˆ −( 3,2) = E x E y Λ.
5
20
5
10
Appendix D. Complete expressions of Zs present at the right hand side of charge
conservation equation at O ( ReE )
The complete expressions of the non-zero Z n , m are given as
2
Z1,0
Z 2,0 = −
2
2
54 M ( R − S ) Ez
=
,
25 ( R + 2 )3 ( λ + 1)
3( R − S ) Ez
,
( R + 2 )( λ + 1)
54 M ( R − S ) E x
Z1,1 =
,
25 ( R + 2 )3 ( λ + 1)
54 M ( R − S ) E y
Zˆ1,1 =
,
25 ( R + 2 )3 ( λ + 1)
3 ( R − S ) Ex
,
2 ( R + 2 )( λ + 1)
3 ( R − S ) Ey
Zˆ 2,1 = −
,
2 ( R + 2 )( λ + 1)
Z 2,1 = −
2
2
2
Z 3,0
216 M ( R − S ) Ez
216 M ( R − S ) E x
216 M ( R − S ) E y
=−
, Z 3,1 = −
, Zˆ3,1 = −
,
3
3
25 ( R + 2 ) ( λ + 1)
25 ( R + 2 ) ( λ + 1)
25 ( R + 2 )3 ( λ + 1)
Z 3,2
2
2
54 M ( R − S ) E z ( E x − E y )
=
,
3
25
( R + 2 ) ( λ + 1)
2
2
108 M ( R − S ) E x E y Ez
Zˆ 3,2 =
,
25 ( R + 2 )3 ( λ + 1)
2
2
2
2
9 M ( R − S ) Ex ( E x − 3E y ) ˆ
9 M ( R − S ) E y ( 3E x − E y )
Z 3,3 =
, Z 3,3 =
.
3
3
25
25
( R + 2 ) ( λ + 1)
( R + 2 ) ( λ + 1)
2
2
(D1)
Appendix E. The O ( ReE ) electric field
The electric potential at O ( ReE ) is given by
(
)
(
)
(1)
(1)
(1)
(1)
(1)
(1)
ψ i(1) = r  a1,0
P1,0 + a1,1
cos φ + aˆ1,1
sin φ P1,1  + r 2  a2,0
P2,0 + a2,1
cos φ + aˆ2,1
sin φ P2,1 

(


)
(
)
(1)
(1)
(1)
(1)
(1)
+ r 3  a3,0
P3,0 + a3,1
cos φ + aˆ3,1
sin φ P3,1 + a3,2
cos 2φ + aˆ3,2
sin 2φ P3,2

(1)
(1)
+ a3,3
cos 3φ + aˆ3,3
sin 3φ P3,3  ,

(
)

(E1)
24
ψ e(1) =
1  (1)
1
1)
1)
1)
1
1
b P + b−( 2,1
cos φ + bˆ−( 2,1
sin φ P1,1  + 3 b−( 3,0
P2,0 + b−( 3,1) cos φ + bˆ−( 3,1) sin φ P2,1 
2  −2,0 1,0
 r 

r
1
1)
1)
1
1)
1)
+ 4 b−( 4,0
P3,0 + b−( 4,1
cos φ + bˆ−( 4,1) sin φ P3,1 + b−( 4,2
cos 2φ + bˆ−( 4,2
sin 2φ P3,2
r 
(
)
(
(
)
(
(
)
)
)
1)
1)
+ b−( 4,3
cos 3φ + bˆ−( 4,3
sin 3φ P3,3  ,

(E2)
where the non-zero coefficients present in the above expressions can be given in terms of
Z n , m in the following form
Z1,0
Z
Z
Z 2,1
Z 3,0
Z 3,1
(1)
(1)
(1)
(1)
(1)
, a1,1
= 1,1 , a2,0
= 2,0 , a2,1
=
, a3,0
=
, a3,1
=
,
2+R
2+ R
3 + 2R
3 + 2R
4 + 3R
4 + 3R
Z
Z 3,3
Zˆ
Zˆ 2,1
Zˆ3,1
Zˆ
(1)
(1)
(1)
(1)
(1)
(1)
a3,2
= 3,2 , a3,3
=
, aˆ1,1
= 1,1 , aˆ2,1
=
, aˆ3,1
=
, aˆ3,2
= 3,2 ,
4 + 3R
4 + 3R
2+R
3 + 2R
4 + 3R
4 + 3R
ˆ
Z 3,3
(1)
)
(1)
(1)
)
(1)
(1)
)
(1)
)
(1)
aˆ3,3
=
, b−(12,0
= a1,0
, b−(12,1) = a1,1
, b−(13,0
= a2,0
, b−(13,1) = a2,1
, b−(14,0
= a3,0
, b−(14,1
= a3,1
,
4 + 3R
)
(1)
)
(1) ˆ(1)
(1) ˆ (1)
(1) ˆ(1)
(1) ˆ (1)
(1) ˆ (1)
(1)
b−(14,2
= a3,2
, b−(14,3
= a3,3
, b−2,1 = aˆ1,1
, b−3,1 = aˆ2,1
, b−4,1 = aˆ3,1
, b−4,2 = aˆ3,2
, b−4,3 = aˆ3,3
.
(1)
=
a1,0
(E3)
Appendix F. Non-zero solid harmonics of O ( ReE ) flow field
To apply the boundary conditions given in equation (2.11) in a similar manner to
 ∂ (1)

1
(1) 

equation (3.11), first one has to express u( ) ⋅ e r  ,  r

 s  ∂r u ·e r  and r ⋅∇ × u  s in terms
s
(
)
of solid spherical harmonics in the following form (Happel & Brenner 1981)
∞ 
n
(1)
(1) 
 u (1) ⋅ e r  = ∑ 
p
+
n
Φ
n
n ,

 s n =−∞  2 N ( 2n + 3)

µ

∞ 
n ( n + 1)
 ∂ (1)

(1)
(1) 
r
u
·
e
=
p
+
n
n
−
1
Φ
(
)

∑
r
n
n ,
 ∂r

n =−∞ 
 2 N µ ( 2n + 3)
s

(
)
(F1)
∞
r ⋅ ∇ × u(1)  = ∑ n ( n + 1) χ n(1) .

 s n = −∞
{ ((
) )}
H 1
1
To satisfy the shear stress balance, we have to obtain r ⋅∇ × r × τ ( ) + Mτ E ( ) ⋅ e r  and

 s
{(
) }
r·∇ × τ H (1) + Mτ E (1) ⋅ e  . Among these terms the hydrodynamic stress components can
r 

s
be obtained as
25
∞ 
n2 ( n + 2 ) (1) 
(1)
r ⋅ ∇ × r × τ H (1) ⋅ e  = − N
pn  ,
r
µ ∑  2 ( n − 1) n ( n + 1) Φ n +

 s
N µ ( 2n + 3)
n =−∞ 


( (
))
(
(F2)
∞
)
r ⋅ ∇ × τ H (1) ⋅ er  = N µ ∑ ( n − 1) n ( n + 1) χ n(1) .

s
n =−∞
(F3)
The radial traction vectors τ i ( ) ·e r and τ e ( ) ·e r arising at O ( ReE ) from the Maxwell stress
E1
E1
tensor are expressed as
 ( 0 ) (1) 1
( 0 ) (1)
( 0 ) (1)
( 0) (1)
 Ei ,r Ei , r − 2 2 Ei ,r Ei , r + 2 Ei ,θ Ei ,θ + 2 Ei ,φ Ei ,φ

E1
0
1
1
0
τ i ( ) ·e r = S 
Ei(, r ) Ei(,θ) + Ei(, r) Ei(,θ)

0
1
1
0

Ei(, r ) Ei(,φ) + Ei(, r) Ei(,φ)

(
(
(
)
)
 ( 0) (1) 1
( 0 ) (1)
( 0) (1)
( 0 ) (1)
 Ee ,r Ee, r − 2 2 Ee, r Ee, r + 2 Ee ,θ Ee,θ + 2 Ee,φ Ee ,φ

E1
0
1
1
0
τ e ( ) ·e r = 
Ee(, r) Ee( ,θ) + Ee( ,r) Ee( ,θ)

0
1
1
0

Ee( ,r) Ee(,φ) + Ee(, r) Ee( ,φ)

(
(
(
)
)
)

,



(F4)
)

.



(F5)
We express the electrical stress components at the drop interface in terms of surface
harmonics as
{ (
)}
∞
)}
∞
r ⋅ ∇ × r × τ E (1) ⋅ e  = ∑ g i (1) cos mφ + gˆ i(1) sin mφ P ,
i
r
n ,m
n ,m

 s n = 0 n ,m
{ (
(
)
r ⋅ ∇ × r × τ E (1) ⋅ e  = ∑ g e(1) cos mφ + gˆ e(1) sin mφ P ,
e
r
n ,m
n ,m

 s n = 0 n ,m
(
r·∇ × τ

E (1)
i
∞
⋅ er  = ∑ h
 s n =0
)
∞
(
e(1)
n ,m
(
)
(F6)
)
e1
cos mφ + hˆn ,(m) sin mφ Pn ,m ,
r·∇ × τ eE (1) ⋅ e r  = ∑ hne,(m1) cos mφ + hˆne,(m1) sin mφ Pn ,m ,

 s n =0
(
)
(
)
i1
i1
i1
i1
e1
e1
e1
e1
where g n(, m) , gˆ n(, m) , hn(, m) , hˆn(, m) , g n(, m) , gˆ n(,m) , hn,(m) and hˆn,(m) are the coefficients that can be obtained
after solving electrical potential distribution. The nonzero coefficients obtained after solving
for the O ( ReE ) electric potential (equation (3.2)) are given in table 2. The complete
expressions of the terms given in table 2 are extremely cumbersome to write down here. We
provide here the expressions necessary for determining the drop velocity:
26
g ni (,1m) and gˆ ni(,1m)
g ne(, m1) and gˆ ne,(m1)
hni(,1m) and hˆni(,1m)
hne,(m1) and hˆne,(m1)
()
g1,0( ) , g1,1( ) , g 2,0
, g 2,1( ) ,
g1,0( ) , g1,1( ) , g 2,0( ) , g 2,1( ) ,
()
()
h2,0
, h2,1( ) , h2,2
, h3,0( ) ,
e (1)
e (1)
e (1)
e(1)
h1,0
, h1,1
, h2,0
, h2,1
,
()
()
()
, g3,0
, g3,1( ) , g3,2
,
g 2,2
g 2,2( ) , g3,0( ) , g 3,1( ) , g3,2( ) ,
i1
i1
i1
i1
h3,1( ) , h3,2( ) , h3,3( ) , hˆ2,1( ) ,
e (1)
e (1)
e (1)
e(1)
h2,2
, h3,0
, h3,1
, h3,2
,
()
()
()
g 4,0
, g 4,1( ) , g 4,2
, g 4,3
,
g 4,0( ) , g 4,1( ) , g 4,2( ) , g 4,3( ) ,
i (1) ˆ i (1) ˆ i (1) ˆi (1)
hˆ2,2
, h3,1 , h3,2 , h3,3
e (1) ˆ e (1) ˆ e (1) ˆ e (1)
h3,3
, h1,1 , h2,1 , h2,2 ,
()
()
g 4,4
, gˆ1,1( ) , gˆ 2,1( ) , gˆ 2,2
,
g 4,4( ) , gˆ1,1( ) , gˆ 2,1( ) , gˆ 2,2( ) ,
i1
i (1)
i1
i (1)
gˆ 3,1( ) , gˆ 3,2
, gˆ 4,1( ) , gˆ 4,2
,
e1
e1
e1
e1
gˆ 3,1( ) , gˆ 3,2( ) , gˆ 4,1( ) , gˆ 4,2( ) ,
i (1)
i (1)
gˆ 4,3
, gˆ 4,4
e1
e1
gˆ 4,3( ) , gˆ 4,4( )
i1
i1
i1
i1
i1
i1
i1
i1
i1
i1
i1
i1
i1
i1
i1
i1
e1
e1
e1
e1
e1
e1
e1
e1
e1
e1
e1
e1
e1
e1
e1
i1
i1
i1
i1
e (1) ˆ e (1) ˆ e(1)
hˆ3,1
, h3,2 , h3,3
e1
i1
i1
i1
i1
e1
e1
e1
e1
TABLE 2. Non-zero g n(, m) , gˆ n(, m) , hn(, m) , hˆn(, m) , g n(, m) , gˆ n(,m) , hn,(m) and hˆn,(m) obtained from the O ( ReE )
electric field.
i (1)
g1,0
=
3S
(1)
(1)
(1)
3Ex a2,1
+ 3E y aˆ2,1
+ 2 E z a2,0
,
5( R + 2)
(
)
i (1)
g1,1
=−
3S
(1)
(1)
E x a2,0
− 3Ez a2,1
,
5( R + 2)
i (1)
gˆ1,1
=−
3S
(1)
−3E z aˆ2,1
+ E y a2(1,0) ,
5( R + 2)
e (1)
1,0
g
(
)
(
)
9 ( R + 1)
(1)
=
3E y bˆ−(13,1) + 2b3,0
E z + 3Ex b−(13,1) ,
5 ( R + 2)
(
)
e (1)
g1,1
=
9 ( R + 1)
)
Ex ,
3E z b−(13,1) − b−(13,0
5 ( R + 2)
e (1)
gˆ1,1
=
9 ( R + 1)
(1)
3E z bˆ−(13,1) − E y b3,0
.
5 ( R + 2)
(
(F7)
)
(
)
Now, we obtain the coefficients of the solid spherical harmonics in terms of the unknown
drop velocity at O ( ReE ) as:
An( , m) =
1
( 2n + 3)  2n + 1 2n − 1 β (1) + M g i(1) − g e(1)  ,
(
)(
) n,m
( n ,m n ,m ) 
n ( 2n + 1)( λ + 1) 
(1)
Bn , m = −
An(1, m)
2 ( 2n + 3)
,
(F8)
(F9)
27
(
M hn ,(m) − hn(,m)
(1)
Cn ,m =
A−(1n)−1,m = −
e1
i1
)
n ( n + 1) ( λ ( n − 1) + ( n + 2 ) )
,
(F10)
1
( 4n 2 − 1) {( 2n + 1) λ + 2} β n(1, m) + M (1 − 2n ) g ni(,1m) − g ne(, 1m)  ,

( n + 1)( 2n + 1)( λ + 1) 
(
)
(F11)
(1)
B− n −1,m = −
( 4n
2
(
− 1) λβ n( , m) + M g n ,(m) − g n(,m)
1
e1
2 ( 2n + 1)( n + 1)( λ + 1)
C−(1n)−1, m = Cn(1, )m ,
i1
),
(F12)
(F13)
(1)
(1)
(1)
= −U dz(1) , β1,1
= −U dx(1) and βˆ1,1
= −U dy(1) . Similarly, the coefficients Aˆn(1, m) , Bˆn(1, m) , Cˆ n(1,m) ,
where β1,0
Aˆ−(1n)−1,m , Bˆ −(1n)−1, m and Cˆ −(1n)−1, m can be obtained by replacing β n(1, m) , g ni (,1m) and g ne(, 1m) by βˆn(1, m) , gˆ ni (,1m) and
gˆ ne(, m1) respectively in the above expressions.
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