Jul 18, 2012 (6 years and 3 days ago)


Massachusetts Institute of Technology, Cambridge, Massachusetts
and Farmfield, Huntingdon Road, Cambridge, England
Electrohydrodynamics can be regarded as a branch of fluid mechanics
concerned with electrical force effects. It can also be considered as that part
of electrodynamics which is involved with the influence of moving media on
electric fields. Actually, it is both of these areas combined, since many of the
most interesting problems in electrohydrodynamics involve both an effect
of the fluid motion on the fields and an influence of the fields on the motion.
The word "electrohydrodynamics" is relatively new; the area it repre-
sents is not. The related literature is as venerable as that for the subject of
electricity itself. Even more, to generate an engineering interest there is no
need to emphasize the great technological promise of the area, since applica-
tions already form the basis for major industries. But the center of attention
in almost any discussion is the lack of reproducibility in experiments and the
inadequacies of theoretical models. Electrostatic effects in fluids are known
for their vagaries; often they are so extremely dependent on electrical con-
duction that investigators are dlseouraged from carefully relating analytical
models and simple experiments. Yet the foundations of fluid mechanics are
formed from work that relates carefully designed experiments to analytical
models, and we wish to focus attention on electrohydrodynamic research
having this objective. An historical survey of the subject has been given by
Pickard (1) and is not deemed appropriate here.
Laws and approxima~ions.--A summary of the pertinent electrical laws
serves further to define our subject. A salient feature of electrohydrodynamic
interactions is the irrotational nature of the electric field intensity, E. Dy-
namic currents are so small that the magnetic induction is ignorable, and the
appropriate laws are essentially those of electrostatics, as summarized in
Table I.~ Gauss’ law, Equation Ib, relates the free-charge density, q, to the
electric displacement D, while Equation Ic brings in the free-current density
in a dynamic equation that guarantees conservation of charge. As is conven-
t One of the authors, (J. R. M.) acknowledges the support of N.A.S.A. research
grant NGL-22-009-014 #6, and would like to thank Tsen-Chung Cheng for his as-
sistaace in obtaining the data of Figure 3 and the photograph of Figure 2.
z Equations in tables are referenced with the table number as the prefix; e.g.,
Equation Ib is Equation b in Table I.

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Differential Laws Transformations Boundaxy Conditionsa
V ~(E = Ia
v.D = q Ib
V.l-b 0q = 0 Ic
D = ¢0E + P Id
E’ = E
D’ = D If
q’ = q Ig
J’ = J--qv Ih
P’ = P Ii
~ [A]-~Aa--Ab=the jump of A across the interface;
n X [E] = 0
n.fD] = Q
n. [J] + Vz’K I1b
= n-rid - --
Vz~surface Nabla.
tional, the electric displacement is further defined in terms of the polariza-
tion density, P, (Equation Id) with ~0= 8.85X!0-~ in M KS units.
The quasi-static electrical laws of Table I are invariant to a Galilean
transformation (2), which can be used to show that the fields in a primed
frame moving with the velocity ~ are as given by Equations Ie to Ii, in
Table I. The transformations reflect the quasi-static approximation implicit
to the differential laws. Thus, the electric field and current density do not
transform as they do in the magnetohydrodynamic approximation, in which
magnetic induction is essential but net charge is negligible. A boundary hav-
ing the unit normal n directed from region (b) to region (a), supporting
surface-charge density Q and surface-current density K, and having the nor-
mal velocity n- v, is described by conditions Ij to II, which are found by inte-
grating the differential laws over surfaces and volumes that include the
boundary. (2) The surface-current density K of condition II includes con-
tributions from the convection of surface charge, and, if appropriate, con-
tributions due to surface conduction.
Conduction and polarization.--The quasi-static equations of Table I are
written in terms of the macroscopic fields with the effects of material motion
accounted for by constitutive laws. For many purposes, the conduction law
for the fluid at rest takes the form J= J* (q, E). Subject to the assumption
that accelerations do not influence the conduction process, this law holds in
the face of fluid motion if it is evaluated in a frame of reference moving with
the fluid velocity v. That is, with motion we must write J’--J* (q’, E’), and,
in view of Equations Ig and Ih of Table I, the conduction law expressed in
the laboratory frame but with fluid motion becomes
J = J*(q, E) + 1.
Equations Ie and Ii show that if the polarization is a function of E, it is
the same whether viewed from the laboratory frame or in the moving frame
of the fluid. Of course, the assumption implicit in using the transformation
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laws to generalize constitutive laws to the case of material motion is that
accelerational effects on conduction and polarization can be ignored.
Charge relaxation.--Even though electrical conduction in fluids is often
poorly characterized by Ohm’s law, (3) it is evident from recently reported
research that this simplest of all conduction laws can be used to understand
a surprisingly wide range of electrohydrodynamic phenomena. In this re-
view attention is confined largely to this case where
J* = ~E
with the electrical conductivity ~ of a given fluid element constant. In addi-
tion, we will take as the polarization constitutive law simply
D = ~E 3.
where the permittivity ¢ of a moving fluid particle is constant.
In a homogeneous incompressible fluid, where a and ¢ are constants and
V. v = 0, we can make far-reaching conclusions about the distribution of the
free-charge density, q. We combine Equations Ib and Ic with Equations 2
and 3 to obtain
The characteristic lines in (r, t) space are simply the particle lines, hence
q = q0~-t/" o~ ~ = v
where the bulk relaxation time r =¢/~r. Thus, the free-charge density in the
neighborhood of a given fluid particle decays with the relaxation time r.
Moreover, unless a given element of fluid can be traced via a particle line to
a source of charge, it will support no bulk charge density.
Equations of Motion.--We confine ourselves to cases where the mass
density p of a given fluid element is constant; hence the fluid, having a con-
stant viscosity/z and subject to the gravitational acceleration g, has a pres-
sure p and velocity v governed by the equations of Table II. In addition to
the mechanical pressure and viscous stress T’~, there is an electrical force due
to the free-charge density q (the charges that contribute to conduction and
convection currents) and due to polarization. The boundary conditions
(IId to Ill) are found by integrating the conservation of momentum and
mass, Equations IIa-IIc through the interface.
Electrical Forces.--The electrical force on an incompressible fluid can be
correctly written in alternative forms that differ by the gradient of a pres-
sure. This is true because in the differential laws and implied boundary con-
ditions of Table II, the pressure, p, appears only in Equation lib and is
simply redefined by the addition of an electrically induced pressure. Hence,
we ignore electrostriction forces, since they could be of importance only for
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Differential Laws Boundary Conditions
~ -b7 = ~g + v. (1"-. + T’)
v.v = 0 IIc
n[p] = n.[T~+T,] IId
n X [v] = 0 IIe
n.[v] = 0 IIf
dilatational fluid motions, and write the force density in the form due to
Korteweg & Helmholtz in (4, 5)
F = qg -- -~ .E~W
Equation 6 can be written identically as
where the Maxwell stress tensor T~ accounts for not only forces attributable
to free charges, but because ~ =~(r, t), those due to polarization as well.
We are now in a position to set limits on the scope o~ th~s review. The
dyn~mics of fluid s~stems characterized by regions of uniform ohmic conduc-
tivity and permittivity will be highlighted. We have established with Equa-
tion 5 that in the absence of sources of charge that ~re communicated by
mat~rlal convection with th~ volume of interest, the bulk is fr~e of the charg~
density, g. Moreover, because ~ is constant ~n a given reg~on~ it ~s clear from
Equation 6 that the fluid is not coupled to the electric field ~n th~ bulk.
Hence, with the restrictions outlined, ~e review classes o~ motion i~volving
electromechanical coupling at fluid interfaces.
Our obse~atlons should serve to illustrat~ that, ~ a fluid system ~ncludes
interracial regions where electrical parameters suffer discont~nuities~ electro-
mechanical coupling at the interfaces is likely to dominate the resulting elec-
trohydrodynamlcs. Surface interactions are of greater significance in elec-
trohydrodynam~cs than might be expected ~rom much of ordinary hydro-
dynamics. The literature o~ drops and ~ets in electric fields is highly de-
veloped, ~nd relates largely to the dynamics of two-phase systems with inter-
faces stressed by electrical surface forces. Du~ to m~teorological ~nterest,
water and air are often considerS, and these fluids exemplify cases in
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which one fluid is much more highly conducting than the other. Then, if
the relaxation time in the more conducting fluid is short compared to dy-
namical times of interest, the interface can be regarded as perfectly conduct-
ing; it supports no tangential electric stress. Regardless of interracial defor-
mation, surface forces always act perpendicularly to the suriace in this im-
portant class of interaction.
At the opposite extreme, where the fluids in a two-phase system are
considered as perfectly insulating with no free-charge density, the second
term of Equation 6, the polarization-force density, is operative at the inter-
face. Again, as the force-density expression shows, the surface-force density
must act in the direction of --Ve; that is, perpendicular to an interface.
By contrast with these two limiting cases, the physical situations re-
viewed relate mainly to the electrohydrodynamics resulting from electrical
surface shear forces. Thus, our review is confined to a small corner of the
total area of electrohydrodynamics: ohmic fluids and surface interactions
dominated by interfacial electrical shear-force densities. We begin with
steady motions and conclude with stability problems.
Consider a case (6) which has the dual virtues of being easily demon-
strated in the laboratory and easily described mathematically, while ex-
emplifying the nature of electrohydrodynamic shear-stress interactions.
To induce an interfacial electrical shear force, the interface must simul-
taneously support a surface-charge density and a tangential electric-field
intensity. This is accomplished in a simple way with the experiment of
Figure 1, where a shallow, slightly conducting liquid, region (b), fills an in-
sulating container A to the depth b. Electrodes B and C, abutting the right
and left ends of the container, make electrical contact with the liquid to
complete an electrical circuit with the source of potential V0. Thus, one re-
quirement for a shearing-force density at the interface D is provided by the
FIo. 1. Electrode C has the potential V0 relative to electrodes B and F. Surface
charges induced on the interface D act in concert with the field E~, which drives the
conduction currents in the liquid to induce the counterclockwise cellular convection
shown in Figure 2.
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1 i 6
conduction current in the liquid, which insures that there is a tangential
electric field E, at the interface.
For the purpose of fulfilling the second requirement, free charge on the
liquid surface, a third electrode F extends over the interface and is canted
at an angle such that it contacts the interface and electrode at the left and
reaches a height a at the extreme right. The interface assumes a distribution
in electrical potential that varies from V0 at the right to zero at the left.
Because the slanted electrode has zero potential, with a spacing h(x) that
varies essentially linearly with x, there is a surface charge induced on the
interface. Both the potential difference and the spacing vary linearly, and
the surface-charge density therefore tends to be uniform--at least if it is
not redistributed appreciably by the resulting fluid motions. Thus, there is
an electrical shear force on the interface which acts to the left and tends to
produce counter-clockwise cellular convection in the plane of the paper.
There are also normal stresses on the interface, but we assume these are
balanced by gravity and the hydrostatic pressure without a significant al-
teration in the interface geometry.
Quasi-one-dimensional modeL--If the length l of the experiment is large
compared to both a and b, a simple model suffices to make a quantitative
prediction of the fluid convection. For now, we assume that the convection
of charge at the interface gives rise to a current in the x direction that is
negligible compared to conduction current in the bulk of the liquid. That is,
for now the fields are determined as though the liquid were stationary, with
the requirement that n.~E=O on the upper and lower surfaces, conditions
satisfied by the uniform fieldS
V0 .
E~ = -- -- z~ 8.
This is also the tangential component of E just above the interface (Equa-
tion Ij). Hence, the distribution of potential on the interface is
6(y = b) =
where we take the upper electrode and left edge of the liquid as the reference
and ~b is defined such that E= --V~b. It follows that in the region above the
interface the field is approximately
E~ = [4,(y = b)/h(x)] v 10.
We have arranged the experiment such that h(x) =ax/l, so that Equations 9
and 10 give
E° = iuVo/a
The interracial shear-force density is n. [T’] =i,[T~*], Equation IId.’ Thus,
from Equations 8 and 11, the interface is subject to an electrical shear-force
*The components of i are the unit vectors in the coordinate directions.
*[A]---Aa-Ab--the jump of A across the interface.
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T, = eoEvaEx~ = -- eoVd/la
That this expression is negative is consistent with Figure 1. A reversal of the
applied potential polarity has no effect on T, because the signs of both the
surface charge and the tangential field are then reversed.
In the limit where b<<l, the flow in regions of the fluid bulk removed sev-
eral lengths b from the ends can be approximated as being plane: v=v~(y)i,.
The profile is determined by the no-slip condition on the tank bottom v~(0)
=0, the viscous and electrical shear-stress balance at the interface, #Ov~/Oy
= -eoVo~/la, and the condition that net flow in the x direction be zero. Thus,
- ~oVdb
and the profile is as sketched in Figure 1. Of course, the velocity observed is
somewhat less than that given by this expression, since the viscous losses
from the reversal in flow direction near the tank ends are not included in the
Charge transport; electric Reynolds number.--Although the influence ot
the electrical stresses on the fluid is included in the model, it ignores the
reciprocal effect of the motion on the fields. We have assumed that the
convection of charge at the interface in the --x direction is negligible com-
pared to the conduction current through the bulk; i.e., bJ~>>Qv,, or using
J~=aVo/l and Q=~oVo/a,
R, << 1; R~ ~a
The electric Reynolds number R~ is defined by Stuetzer (7) as the ratio of
charge-relaxation time to a time L/v, for the fluid to move a characteristic
length L at the characteristic velocity v,. From Equation 14, in our example,
the length L=ab/l, a combination of lengths, because the component of E
that gives rise to electrical dissipation is not in the same region of space or in
the same direction as the component (above the interface) that represents
much of the energy storage.
In the section on steady, dc-field-induced cellular convection, we will
limit our discussions to cases where R~<<I. However, the ac-field-induced
motions and cases of instability reviewed shortly will include finite-electric-
Reynolds-number effects.
An experiment.--The cellular convection is readily observed in an appa-
ratus having the configuration of Figure 1 by introducing small particles
that are nearly neutrally buoyant. A streak photograph is shown in Figure
2 where the liquid is corn oil (relaxation time ~/~r of about 1 sec) and the
dimensions and voltage are as indicated in the figure caption.
In the photograph of Figure 2, note that the vertical point of flow re-
versal is about 2/a of the distance from the tank bottom to the interface, as
suggested by the plane-flow model, Equation 13. The nearly symmetric
shape of the cell, together with the observation that small particles placed on
the interface at the right traverse most of the length with nearly constant
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Streak photograph
the cellular convection induced
an apparatus hav-
the configuration sketched in Figtire
1, 1=24
V ~ =2 0
kV and the
fluid is corn oil.
comparison of the experimentally determined surface velocity and
the prediction
13 is
velocity, further indicates t hat t he physical arrangement
electrodes and
successful in producing a nearly uniform electric-shear-stress
distribution over t he interface.
plot of t he product
observed surface velocity
near t he center of
t he apparat us and maximum spacing
t he applied voltage
for three angles of inclination,
shown i n Figure 3 where t he solid line is
predicted by Equation
For t he maximum velocity
as defined by Equation
on t he order
t hat t he assumption
t hat t he electrical-relaxation process easily keeps up with t he motion
V,- kV
maximum electrode spacing and the
the center
the apparatus shown in Figures
a function of the applied
voltage. Liquid
corn oil,
e =3.l e O,
~( =0.055
kg (nis)-' and
curve, from Equation
1 3
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well taken for higher voltages. Deviations of the theory and experiment over
the range of voltages shown can be attributed mainly to the viscous end
effects, which are ignored in the simple quasi-one-dimensional model.
It is possible to conceive many variations on the theme of dc-field-in-
duced shear flows. We will concentrate on two further combinations of inter-
facial geometry and field nonuniformity that have well developed and rela-
tively simple analytical descriptions. In this subsection we further indicate
how the application of a nonuniform field at an interface leads to bulk con-
vection, while in the next section the geometric configuration of the interface
leads to a distortion of an initially unlfor’m applied field to secure cellular
The three steps required to analyze a low-electric-Reynolds-number flow
are illustrated in the previous section. First, the fields are computed and,
because Re <<I, effects of the fluid are only geometric. Thus the electric inter-
facial stresses are computed independently of the fluid velocity. Second,
a flow pattern consistent with the distribution of stresses on the interface is
found. This pattern is not coupled to the field in the bulk. Finally, the field-
induced stresses and flow are matched at the interfaces to determine a self-
consistent relationship between the imposed potentials and the flow velocity.
Of course, there are a limited number of situations in which this last step can
be completed in closed form--our reason for discussing the following cases.
A configuration of imposed field and fluid studied by Smith & Melcher
(8, 9) is shown in Figure 4a. A static, spatially periodic distribution of poten-
tial is imposed on a planar electrode A, which also serves as the bottom for a
container filled to a depth b with a slightly conducting liquid (region b).
This layer of liquid is in turn covered by a second fluid, in region (a), which
can, generally, also be slightly conducting. In Figure 4 the upper fluid is
assumed for purposes of discussion to be the less conducting. The mechanism
for creating cellular convection is basically the same as in the case of Figure
1. With the upper fluid less conducting than the one below (having, for
example, zero conductivity) adjacent positions of positive and negative
polarity on the segmented electrode can be thought of as being joined by a
resistance (the lower liquid) in series with a capacitance (the upper liquid)
in series with a resistance (the lower liquid again). Thus, charges induced
the interface have the same sign as the neighboring charges on the seg-
mented electrode. These surface charges are subject to the electric-field in-
tensity, the tangential component of which produces electrical shear forces
on the interface sketched in Figure 4a. Note that the spatial periodicity of
these stresses is such that we expect two cells to form in the length l, with
points of zero interracial velocity on the interface having the same x coor-
dinates as both the peak potential and zero potential on the segmented elec-
trode. These physical considerations serve as a guide in guessing the appro-
priate flow pattern to match the electrical stresses. As in the case of Figure 1,
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FiG. 4. (a) A segmented electrode A is constrained by a static periodic potential
and is in electrical contact with a liquid having depth b and an interface at B. Surface
charges are induced with the polarity shown if the lower fluid is much more highly
conducting than the upper one. (b) The surface charges interact with the imposed
electric-field intensity E to produce the cellular streamlines.
it is again assumed that gravity holds the interface essentially flat, so that
normal stresses are not of interest.
Electric stresses.--Our introductory remarks make it clear that in the
bulk of the fluids (a) and (b), the electric potential ~b satisfies Laplace’s equa-
tion. In view of the potential constraint at the electrode, the potential dis-
tributions in each fluid must have an x dependence of the form cos (~rx/l).
We take motions as being independent of the z coordinate. To determine the
y dependence of the potential requires four boundary conditions: (a) that
the potential at y= --b is as given in Figure 4a; (b) that the potential must
be continuous at the fluid-fluid interface (Eq. Ij) dp~(y=O)=dpb(y=O);
that the convection of surface charge at the interface is ignored, and so the
normal component of the current must be continuous (Eq. II), o’aOd~’~/Oy(y
=0) =obOdpb/Oy(y= 0); and finally, (d) that the upper fluid is bounded from
above by a sufficiently distant bdundary that ~b-~O as y--~o.
Variable separable solutions having the required x dependence, while
satisfying Laplace’s equation and meeting these boundary conditions, are
¢~a = __ e-,,un cos
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~rV0 V
E0-~--f--L cosn (~-~) q- ~ sinh (~-~)~-’
The electrical shear-force density follows from Equations 7 and Ij
T,, = IT,,’] = ~[eEu] evaluated at y = 0
where the required components of E follow directly from Equations 15 and 16
Convection.~As a further restriction on the electrohydrodynamics, we
assume at the outset that the hydrodynamic Reynolds number based on l
and the peak surface velocity U is small compared to unity, so that the fluid
momentum can be ignored. Then it is appropriate to define a stream func-
tion ~ in the usual way as
. o~ . a~
~ = ~- ~ 18.
Our approximation is the usual limit of creep flow, and ~ satisfies the bi-
harmonic equation (10)
V~V~ = 0 19.
so that v in turn satisfies the’equations of motion IIa-IIc with Dv/Dt~O.
The key to matching the shear-stress conditions is the judicious choice
of the x dependence for the stream function. From the sketches of Figure 4a,
it is clear that we can expect the x component of the velocity to be an odd
function of x with the wavelength l. Thus, we choose # to be of the variable,
separable form
~ =/(y)sh (=~/b); = = 2=b/~
Substitution of Equation 20 into Equation 19 shows that solution in each
fluid region is a linear combination of four solutions. Only two of the solu-
tions in the upper fluid remain finite as y~, thus we are left with six
arbitrary coe~cients, two for solutions in region (a) and four for solutions
region (b), with which to satisfy the boundary conditions.
Owing to gravity, the surface is not deflected enough in the vertical direc-
tion to affect the field distributions. Thus, in our boundary conditions, we
ignore the normal-stress balance at the interface, but instead stipulate that
the normal velocity not only be continuous, but vanish. The boundary con-
ditions are: (a) and (b): that v~(y=0)=0, and vu~(y=0)=0; (¢)
mechanical shear stress balance the surface-force density given by Equations
15-17, [T~]+ T,* = 0 at y = 0 (Eq. I Id); (d) that the tangential component
of velocity be continuous at the interface, [v,](y=0)=0 (Eq. IIe); and,
ally, (e) and ~) that the normal and tangential velocities at the segmented
electrode vanish: v,~(y = -- b) = 0, and v~(y = -- b) =
If we call the tangential velocity in the x-- direction at the interface U
sin (ax/b), the stream functions in the respective fluids are
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¢,~ = Uye-°~’1~ sin (ax/b) 21.
~’ = U{yslnhasinh [a(l + y/b)] - a(y + b) sinh (ay/b)} sin
sinh--- ~ ~ = ~ 22.
The shear-stress boundary condition determines U, which is written in terms
of R = ~/~, S = ~/~ and M = g~/#~ as
U = ~ V~~ 23.
= ~r (sinhS a -- a~). t [2M(sinh2 a --v?) + sinh 2a --
We are successful in satisfying all of the boundary conditions only because
the ~ distribution of stress can be matched exactly to the viscous-stress dis-
tribution. The flow pattern represented by Equations 18, 21, and 22 is shown
~n F~gure 4b.
Note that the velocity U, as given by Equation 23, has the same param-
eter dependence as that found for the canted-electrode experiment of Figure
1. Th~s is particularly evident ~f ~n both cases the upper fluid is a noncon-
ducting gas, so that ~0, #.~0 and ~.~e0. Then Equation 23 becomes:
U = ~(a) (’~) Vo’
with T(a) only a function of geometry. The surface velocity given by Equa-
tion 13 for the canted-plate experiment is identical to this expression if
~(a)~b/a. In the general case, the sign of RS-1 discriminates the sense of
the cellular convection, since T >0.
Experiment.--An experiment with essentially the configuration of Figure
4 gives the streak photograph of Figure 5; the sense of the rotation is as
sketched in Figure 4b, consistent with RS>I. Most of the particles (air
bubbles) entrained to trace the streamlines are in the upper liquid, although
careful examination shows the expected cells in the lower liquid as well.
Data for the experiment are given with Figure 6, which shows quantita-
tive comparison (9) of the measured and predicted peak surface velocity,
as a function of the applied voltage Vo. In this experiment, ~ based on l in
each of the liquids is on the order of 0.01 or less, while the hydrodynamic
Reynolds number is always less than 0.6.
As shown by Taylor (11) electric shear-induced convection can occur
quite naturally in electrified drops and bubbles, for physical reasons closely
related to those responsible for the convection in the configurations of
Figures 1 and 4. For purposes of developing an analytical model, a spherical
drop of vapor void of radius b is shown in Figure 7a, where the appropriate
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Streak photograph
periodic cellular convection observed in apparatus
having essentially the fluid and field configuration
Figure 4a. The cells have the
spatial period
and velocity-voltage relationship
(after Smith
spherical coordinates are defined. Electrodes, removed many radii
from t he
spherical region (b), make electrical contact with t he surrounding fluid (a).
Thus, t he outer fluid, together with t he electrodes t hat constrain t he electric-
field intensity t o be uniform and of magnitude
play a role
analogous t o t hat of t he electrodes and liquids (b) in Figures
virtue of t he conduction current they insure t hat there
tangential com-
ponent of
in t he neighborhood of t he spherical interface.
Once again, convection results because not only
but there
surface charge induced on t he interface.
The case illustrated in Figure 7a pertains t o
insulating fluid (b)-perhaps
an air bubble-in
somewhat conducting liquid.
I n
this extreme limit of t he
general case where fluids (a) and (b) can have arbitrary electrical conductivi-
ties and permittivities, it is appropriate to view t he bubble and its surround-
ing conducting fluid
resistance in series with
capacitance (the bubble),
in t urn in series with
resistance. Thi s makes reasonable t he polarity of the
induced surface charges indicated
i n
and t he surface shear-force
sketched there.
Our approach t o describing
self-consistent shear flow
essentially t he
same as in t he case
t he periodic convection, with one exception.
I n
t he
previous case, gravity
used t o hold t he normal electric stresses in balance.
t he case of t he sphere we find a unique combination
fluid properties
t hat make possible a spherical equilibrium
t he interface. Then
is possible
deduce whether t he drop tends toward a prolate or oblate geometry for
combinations of physical parameters other t han those required for
i li
t he limit of zero electric Reynolds number, t he
solution for t he electric-field intensities
t he classic one,
where t he
field in t he interior of t he drop is uniform, while t hat outside
a superposi-
of a
uniform field and a three-dimensional dipole field. There are four
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- Two-dimensional 1/ /
Theory corrected ~._ /
for side walls
2//~111110 "~
0 5
va- kV
FIG. 6. Peak tangential velocity Uof the interface as a function of applied potential
for the case depicted in Figure 4 and photographed in Figure 5. The liquids are Dow
Corning FS. 1265 below (~b/~0=6.9, ~rb~3.3N10-~, ~0.37) and corn oil above
(e~/~0 = 3.1, ,r, ~ 5 N 10-n, ~ = 0.055). The cells are so~newhat distorted at the top be-
cause of a rigid boundary positioned approximately at the top of the picture in Figure 5.
boundary conditions to be satisfied: (a) that the field be finite at the origin;
(b) that the tangential electric field be continuous at the interface (Eq. Ii),
[Eo](r=b)=O; (c) that the conduction current normal to tl~e interface be
continuous (Eq. II in the limit where K--~O), [~rE~](r=b)=0; and finally,
(d) that E--~Eo(i~cos O--iosinO) as r--~. Thus, the electric potential in each
region is
¢, = - E~ cos 0 r -1- ~---2 R ~
4,~ ~ -- 3E~r cos 0/(2 -I- R)
where again, R=~rb/~r~ and in spherical coordinates
= -- Zr c9~- r
Direct substitution shows that ~b satisfies Laplace’s equation and the neces-
sary boundary conditions. The components of electric surface-force density
]~ = n. [T¢] in the radial and tangential directions follow by direct substitu-
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FIG. 7. (a) Spherical drop or vapor void having radius b and comprising region (b)
immersed in a liquid (region a), which makes electrical contact with electrodes that
impose an electric field that is uniform and of magnitude E0 far from the sphere.
Surface charges, induced as shown for the case in which the outer fluid conducts much
more than the inner one, interact with Eo to create the shear-force density To’. (b)
Flow pattern resulting from field-fluid configuration of (a).
tion of the last three expressions into the relations (see Eq. 7)
rr’ = IT,tel = ½{ [~ETzl -- [~.F.0’]) (r = 2S.
To" = [r,o~] = [eE~Eo](, = b) 29.
The choice of appropriate stream functions with corresponding viscous shear
stresses that can hold the tangential and normal components of the electric
surface-force density in equilibrium at each point on the interface must be
made on the basis of the 0 dependences of T". Observe that T~~ is propor-
tional to a constant term and a term in cos~0, while Toe has the 0 dependence
cos 0 sin 0.
Viscous shear stresses.~In spherical coordinates, it is appropriate to use
the Stokes stream function ~ defined such that (13)
e = ¢ ;~ sin 0
and [or creep flow the equations of motion require
[5 sin00 l o ’ ~ (
The boundary conditions dictate the 0 dependence of the variable separable
solutions that we seek from Equation 31. For now, we consider that the in-
terface is in radial force equilibrium. Then the boundary conditions are
essentially of the same nature as those for the previous case of periodic con-
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vection, except that, because the origin is included in the interior region,
there are now only two solutions of interest in each region, and only four
boundary conditions at the interface must be met. These are: (a) and (b),
that the radial components of velocity vanish at the interface, vra(r=b)
=yr’(r---b)--O; (c) that [vo](r--b)=O; and (d) that To~+[#rO/Or(vo/r)](r
=b)-~O. In this last condition, we evaluate the viscous shear stresses in
spherical geometry (14) and take advantage of the fact that vr vanishes
every point on the interface.
As this last boundary condition is expressed in terms 6f ~b (Eq. 30), ob-
serve that, if the condition is to be satisfied at every angle 0, ~b/sin 0 must
have the same dependence as T0~. Thus we are led to look for solutions of the
~ = f(r) sin~ 0 cos 0 32.
and substitution of this expression into Equation 31 shows thatf(r) has the
form r’~, n= -2, 0, 3, or 5. It follows from the conditions on the flow at the
origin and as r~ ~, that the appropriate linear combinations of solutions are
~ = (Ab*r-~ T Bb~) sin~ 0 cos 0
~ = (Cb~ + Db-3r~) sin~ 0 cos 0 34.
The four constants A, B, C, and D are fully determined by the four boundary
conditions. From either Equation 33 or 34, the velocities at the interface
take the form
vo(r = b) = 2U cos0 sin 35.
where U is the peak velocity. The first three boundary conditions give A
=-B= C=--D= U. Finally, the balance of interfacial shear-force densi-
ties relates the peak surface velocity U to the applied electric-field intensity
to complete the determination of the fluid response:
9~Eo~b(RS - 1)
U=-- 36.
In the case of Figure 7, the drop is highly insulating compared to the
surrounding fluid. Thus, RS < 1 and Equation 36 shows that the convective
response of the fluid is in the direction expected from the sign of the surface
charges. The streamlines of Figure 7b, based on Equations 33 and 34 for the
case RS < 1, are also as would be expected in view of the shear-force densi-
ties sketched in Figure 7a. From Figure 7 or Equation 35, it is evident that
fluid at the interface has its maximum speed at 0= ~/4 and ~5 w/4.
The close relationship between the cellular convection within and around
a spherical drop and the convection produced by a periodic imposed field is
emphasized by a comparison of Figure 4 and 7, or a comparison of Equations
23 and 36. It is not surprising that the sign of (RS-1) determines the sense
the convection, because if a given electric field is applied to an interface, it
is this function of conductivies and permittivities that determines the sign of
the resulting surface-charge density. Of course, as in the cases of Figures 1
and 4, a reversal of the applied potential polarity reverses the sign of both the
tangential electric-field intensity and that of the surface charge; hence, the
dependence of U on the square of E0 is as expected.
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Radial stress balance; oblate versus prolate.--By assuming the interface to
be spherical, we have been able to ignore the radial surface stresses. Never-
theless, they are present and will now be taken into account. The electric
field produces the surface-force density given by Equations 25 to 28. Further
contributions come from [Trr’~], (Equation lib), the viscous part of which
follows from Equations 30, 33, 34 and 36. The pressures pa and pb are deter-
mined by using the known velocity to integrate the equations of motion:
if = IP - 2Uu~b~r-3(3 cos2 0 - 1)
pb = IP - 7Uubb-3r~(3 cos2 0 -- 1) 38.
where FP and IIb are constants. There is also an effective radial-force density
-2T/b because of the surface tension T. Finally, we assume that there is a
force per unit area To cos:0 at our disposal that (for T0>0) tends to elongate
the interface in the direction of the applied field.
The balance of radial stresses thus requires that
T,~ + [Trgn](r = b) -~- + T0cos20 = 39.
It is remarkable that each contribution to Equation 39 is either constant
or proportional to cos: 0. The constant part is satisfied by adjustment of the
relative hydrostatic pressures II. The coefficients of the remaining terms in
cos2 0 sum to zero if the externally applied surface-force density To is ad-
justed to be
To = - 9ebE0~¢/2(2 + R)~
3 (2~ + 3)
~I,= S(R~+ 1) - 2 +-~-(RS- 1) M+--i-- 40,
Thus, if the ratios of the fluid parameters represented by R, S, and M
are adjusted such that ~5=0, the drop can be in steady-state equilibrium,
Further, if ¯ <0, a positive outward-directed surface-force density at the
poles (0=0, 0=Tr) is required to retain the spherical shape, and we conclude
that in the absence of To the drop would decrease its extent in the direction
of E0 (i.e., become an oblate ellipsoid). Similar reasoning shows that for
q~>0, the interface is prolate. The function q~ discriminates between equi-
libria of oblate and prolate geometry.
Experirnent.--Observations of the convection in drops, virtually as de-
scribed, are documented by Allan & Mason (15) and McEwan & DeJong
(11). Experiments are complicated by the need for a neutrally buoyant com-
bination of liquids to obtain a stationary drop, and the tendency of any re-
sidual charge to make the drop migrate. Nevertheless, the photographs of
Figure 8 convincingly show cellular convection streak lines from particles
illuminated over the cross-sections of the drops. The figure caption gives
further information on the experiment.
The model appears to correlate successfully with observations of oblate
and prolate ellipsoidal equilibria (11). Two limiting cases are of particular
interest in this regard. Suppose the drop is highly conducting compared to
the vehicle liquid, so that R--~ ~ (for example, a water drop in insulating
oil). Then eo >0, and the equilibrium geometry is that of a prolate ellipsoid,
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Of course, in this limit the shear stresses make no cont-~ibution because the
electric field acts normal to the interface; hence it is not surprising that the
viscosity ratio M does not play a role. Work on the stability of this ellipsoi-
dal equilibrium of a highly conducting drop justifiably excludes the effects
of the electric shear forces (16, 17).
In the opposite extreme, where a void of gas is suspended in a slightly
conducting liquid, and thus R--~0 and M--~ ~, the discriminating function
becomes ¯ = S- 16/5, and the geometry of the equilibrium depends on the
ratio of permittivities S = e~/~ relative to 16/5.
Steady convection in dc fields, as illustrated by the experiments of Figure
1, 4, and 7, obtains only if there is an electrical-conduction path between the
source of potential and one of the fluids. In the absence of such a path, the
fluids simply polarize, with the electric field confined to the insulating re-
gions and directed perpendicular to interfaces. For example, observe that
in the case of the periodic cellular convection where fluid (b) is adjacent
the electrodes, the interracial velocity vanishes as ~/~a=R-~O (Eq. 23).
Similarly, convection of the spherical interface, represented by Equation 36,
vanishes as the conductivity of fluid (a), which is adjacent to the electrodes
that impose the field E0, becomes small compared to that of the drop (R--* ~).
Interfacial electrical shear stresses can be induced by means of ac fields
without the need for electrical conduction between the source of potential
and the fluids. In electrical terms, the coupling is capacitative and analogous
in many respects to the inductive process by which tlme-varying magnetic
fields couple to the rotor of an induction machine, or to the liquid metal of a
magnetohydrodynamic induction pump. Here we are concerned with ac
electric fields and charge relaxation, rather than with ac magnetic fields and
current diffusion. Early work on rigid-body motions serves not only to give
historical perspective but provides us with a convenient prototype model
for understanding ac-field surface interactions.
The cross-section of a circular cylindrical rotor (b) immersed in a fluid
(a) is shown in Figure 9. We illustrate the effects of an ac field by considering
the consequences of subjecting the rotor and fluid to an electric-field in-
tensity E0, which rotates with the angular velocity w. The rotor, hence the
cylindrical interface, has the angular velocity ~2.
Consider for discussion the case where the fluid is much less conducting
FIG. 8. (a) Cross-sectional vie~v of silicone oil drop in mixture of castor oil and
corn oil with electric field applied vertically, as shown in Figure 7a. Particles of
powder entrained in interior of drop show streak lines with the pattern of Figure 7b.
(b) Particles in exterior liquid showing streak liues essentially similar to those
Figure 7b. [after McEwan & DeJong (11)].
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F~(~. 9. A Cylindrical rotor (b) is immersed in a fluid (a) and subjected
imposed electric-field intensity E0 that rotates with the angular velocity ~.
than the cylinder (RS :> 1). Then, if the applied field is static as viewed from
the rotor, it would be shielded from the interior region and oriented per-
pendicular to the interface; there would be no interracial electric shear torces.
By constrast, with the electric field rotating with respect to the cylinder at
the frequency o~-f~, the finite relaxation time for charges to polarize on the
interface comes into play, and in particular, if the period of the field as
measured in the rotating frame of the interface is on the order of the electri-
cal relaxation time (2~r/¢o ~ ~b/ab), surface charges are induced which are not
in spatial phase with the electric-field intensity. As illustrated in Figure 9, if
the field rotates more rapidly than the rotor, there is an effective dipole mo-
ment from the induced charges that lags E0, the imposed electric-field in-
tensity, and a resultant shear surface-force density in the clockwise direc-
tion. By subjecting a fluid system to a time-varying electrical excitation, it is
possible to create finite-relaxation-time effects, even though the flow is i~ the
steady state.
The electromechanica! effect of static and rotating electric fie~ds on
cylindrical and spherical, slightly conducting rotors has been the point of
both theoretical and experimental investigations since the early work of
Arno (18). Rotations induced by dc fields have seen particular and periodic
interest and form the background for the class of instabilities to be discussed
in the next section. An excellent historical review of the subject is given by
Pickard (19), who also discusses torques induced because of the finite time
required for dipoles to relax. This latter effect, not considered here, becomes
significant at much higher frequencies than are usually of interest in electro-
hydrodynamics, but nevertheless deserves more attention in connection with
the electromechanics of fluids.
Rotating fields.--It is a simple matter to give quantitative substance to
our discussion of the rotor dynamics. The imposed electric field is taken as
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uniform far from the rotor axis, and hence, as r cos (O-wt)--~,
4~a --~ -- Eor cos (~0t -- 0)
There are three additional botmdary conditions: (a) that [4~](r=b)=0
I1); (b) that charge be conserved at the interface:
and (c) that the fields be finite at the origin. Note that for the first time
our discussions, we include finite-electric-Reynolds-number effects by re-
taining the convection surface current represented by the term in f/ of
Equation 42.
Fields takd the form of a superimposed dipole and uniform fields in the
exterior fluid and a uniform field in the interior region:
4~ - Re [-- Eor + A~b~] exp j(~ot - O) 43.
4~b = Re rB exp j(~t -- O)
Solution 43 already satisfies condition 41, while Equation 44 is finite at r =0
and substitution in the remaining two boundary conditions determines A
and B. From these solutions it follows that the surface-charge density is
2/~0~[SR - ~] cos (~t -- 0 -- ~)
7’1 + R~(1 + R)
~, = tan-~RE; RE = (~ -- a)(~, +,~)/(~,-t- 46.
The familiar quantity SR-1 once again determines the sign of the induced
surface charge.
To corroborate our introductory discussion, for the case of a conducting
cylinder in an insulating fluid, SR > 1, and if ~o >~2, Equation 45 shows that
3’ is positive so that the axis of the charge distribution in fact lags that of the
applied field as sketched in Figure 9.
Note that RE is an electric Reynolds number composed of the ratio of a
hybrid relaxation time (eant-~b)/(aa+#o) to a transport time. The latter quan-
tity is the time required for a point on the interface to traverse a peripheral
distance b, relative to the frame of the rotating electric field. It is significam
that R~ can be adjusted by controlling the frequency ~0 of the applied field.
Induced torque and rotation.--The electrical torque per unit axial length
of the rotor is bQEo(r=b), integrated over the surface of the cylinder:
4~rEo~,b~(RS- 1) R~"
(1+ S)(1 +
Note that th~s torque has maximum value as RE= 1 and can be positive or
negative, depending on the sign of (RS-1).
If the fluid is of essentially infinite extent, the steady-state viscous torque
per axial length of the rotor is T~= - 4~r#~f~b~, and under the assumption that
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no other torques are present, the balance of torques requires that
~,,Eo~(RS- 1) R~
~2 ----
ua(1 +/~)(1 + s) (1
Of course, the dependence on R~ makes this expression implicit in the angu-
lar rotor velocity ~, but because the electrical frequency w also determines
Re, we are justified in regarding the electric Reynolds number as being in-
dependently controlled.
By contrast with the dc-conduction-driven flows, we now have the pos-
sibility of induced motion even if the outer fluid is highly insulating com-
pared to the rotor. That is, the limit of Equation 48 as R--~ oo does not ap-
proach zero. ~
A graphical representation of the torque balance is given in Figure 10,
where Te and -- T~ are sketched as functions o( ~. Thus, the intersections
between the curves represent the solutions to Equation 48. In the case in
which RS> 1 (a conducting rotor in an insulating fluid, perhaps) the only
equilibrium (i) consists in a positive rotor velocity, less than that of the
field, with the axis of the charge lagging that of the imposed field.
It is possible to have three equilibria for the case RS < 1. For a weak
field, the only equilibrium is (ii), with the rotor and field rotating in opposite
directions and with the charge axis leading the imposed-field axis by more
than 90°. As the field is increased, two positive velocity equilibria are possi-
ble: (iii), with the lower velocity, is unstable because any slight increase
rotor velocity tends to increase the electrical torque and hence to increase
FIG. 10. Electrical torque T¯ and viscous torque --T¯ as functions of the rotor
velocity f~ normalized to r, = (*, +~)/(~ +a~). Intersections represent possible veloci-
ties for steady-state rotation.
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further the rotor velocity; and (iv), with the greater velocity, is stable.
these cases, the rotor and field rotate in the same direction, but the rotor
angular velocity exceeds that of the applied field. Of course, equilibria (i)
and (ii) are stable.
It should beclear that a spherical rotor stressed by a rotating electric
field would be motivated by a torque having a dependence on the physical
parameters similar to that for the cylinder. In fact, much of the early work
relates to spherical rather than cylindrical rotors. Thus, drops and bubbles
under the influence of rotating or traveling fields can be expected not only
to undergo cellular convection, but to suffer rotations as well. As will be
developed in the section on finite-electric-Reynolds-number instability, these
rotations can even be expected for drops and bubbles in dc fields. Naturally,
a rigid-body model for the spherical region would be justified only if it were
composed of a highly viscous liquid (M--~0).
We now consider a case in which each region is occupied by a fluid in a
configuration arranged to give a simple but basic electrohydrodynamlc flow.
The combination of fields and fluids shown in Figure 11 represents a
physical situation strongly suggested by combining the basic interactions
inherent to the experiments of Figures 4 and 9 and investigated by Melcher
(20). As in the case of periodic cellular convection, the interface between two
layers of liquid is stressed by a field from a segmented electrode having a
spatially periodic distribution of potential of wavelength 2~r/le. [~y contrast
with the de case and in a manner suggested by the rotating-field example,
this potential distribution is made to travel in a direction parallel to the in-
terface. Thus, the lower fluid, region (a), can be regarded as the cylindrical
rotor "laid out flat," and the segmented electrode as a means of producing a
field at the interface having essentially the same space-time properties in
linear geometry as the rotating field has in cylindrical geometry.
For discussion purposes, consider the limiting case where the upper fluid
is an insulating gas and the lower one is a slightly conducting liquid. Then
negative charges induced on the electrode by the applied potential in turn
induce image charges of opposite sign on the interface. Because the electrode
charges travel to the right with a velocity such that the field induced in the
frame of the moving interface has a period on the order of the relaxation time,
charges on the inter’face lag their images on the electrodes, as sketched in
Figure 11. Thus, in this case of RS > 1, there is a shear-force density on the
interface acting to the right.
Because the potential wave and its attendant charges on the interface
do not travel with the same velocity U as the interface, the shear stresses at
the interface are pulsating with time at the frequency 2(¢0-k U). In the fol-
lowing we will make the assumption that this frequency is sufficiently high,
compared to characteristic times of fluid-mechanical response, that the fluid
responds only to the time-average electrical shear stress. Thus, the interracial
velocity U is taken at the outset as being independent of x.
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Re Voexp. j (o~t-~
~Segmented electrode with imposed
traveling potential wave
IllllllltllllillllllllltlllJlllllllllllllllt illllltl IIIlllllIIIIIlilllllllllllttlllllllllllfllllllllllllllllttllllllll
+ 7"/71
+//:/./ ÷ ÷ I+
Fro. 11. Cross-section of layers of fluid (a) and (b), with an interface at
Shear forces are induced on the interface by the traveling potential wave imposed on
the segmented electrode.
Traveling-wave fields and induced shear stresses.--With the assumption
that the interface is moving with a constant velocity, it is a straightforward
matter to determine the fields, even including the effect of the convection on
the interracial charge distribution. The potentials in each of the fluids take
the traveling-wave form
¢ = Re(A~,b sinh ky + B=,b cash ky) exp j(~t - kx) 49.
with the four constants Aa, Ab, Ba, and Bb determined by the boundary con-
ditions. These conditions require: (a) that 4~=(y=a)=Re Vo expj(oat--kx);
(b) that 4~b(y= -b)=0; (c) that [4~](y =0)=0; and finally, (d) that
conserved at the interface, [aEu]+(O/Ot-b UO/Ox) [~Eu] =0. An example of
the electric-field distribution is given in Figure 11, where for the case shown
b--~ ~ and o-~ = O.
With the fields determined from the known potential distribution, the
electrical shear-force density, E~’~[eEu] is evaluated and time-averaged to
(Tx~) = ½k~V02ea~baa sinh kb cash kb(RS -- 1)R~/[1 + R~fl]h2~
R~ = (o,-- k U)h/Y,
Z =~r~ cash ka sinh kb+a& sinh ka cash kb
A=e~ cash ka sihn kb+e~ sinh ka cash kb
Annual Reviews

Note the similarity of this result and the expression for the rotor torque,
Equation 47, particularly as it depends on RE and (RS-1).
Flow equilibrium and experiments.--Consider the case where the response
of the fluid is in plane Couette flow. Then the viscous shear stresses combine
to give an effective surface-force density T~---- U~a/a+l~b/b), and the
balance of viscous and electrical shear stresses requires
(T,) + T~ =
This last expression takes the same form as Equation 48 in its dependence on
R~, ~, and RS-1. Thus, if we simply think of the lower fluid as being the cyl-
inder "laid out" in plane geometry, it is clear that the steady-state equilibria
of Figure 10 pertain equally well to the system of two fluid layers. Charges
are distributed over one wavelength on the interface essentially as they are
on the periphery of the cylinder.
In the case of a gas over a liquid, RS > 1 and the fluid travels in the same
direction as the wave, but with less velocity. Experiments (20, 21) verify
this prediction as well as the dependence of the interface velocity on the
applied frequency and peak voltage. As an example, Figure 12 shows the
measured dependence of U on the driving frequency o~. In this experiment,
the potential-wave velocity greatly exceeds that of the fluid. The solid curve
has the predicted frequency dependence, but is normalized to the peak ampli-
tude of the data. Differences between the absolute observed and predicted
velocities in this experiment are on the order of 5 to 30 per cent, depending
on a and b relative to the length of an electrode segment (16 ram). Further
particulars are given in the figure caption.
One case where RS < 1 consists in placing the electrodes under the insulat-
ing bottom of a channel filled with a layer of slightly conducting liquid. In
the terminology of Figure 11, this is essentially equivalent to making region
(b) an insulating gas and region (a) a slightly conducting liquid. So far as
analysis is concerned, it is irrelevant that the system is turned upside down.
In this configuration of electrodes covered by a liquid and then a gas, the
liquid interface is experimentally observed to travel in a direction opposite
to that of the traveling wave (21), as is consistent with the negative velocity
equilibrium (ii) of Figure 10.
The stable flow equilibrium (iv) with RS < 1 and the interface moving in
the same direction and faster than the traveling wave must generally be
established by using external means to impel the fluid. At the same time,
equilibrium flow (iv) requires a minimum voltage before the viscous stresses
are balanced by the electrical shear stresses. If by dint of external forces the
fluid reaches the velocity (iii) in Figure 10b, with the required threshold
applied voltage, it continues to accelerate until it reaches the stable equili-
rium (iv). Thus, an important exception to the need for an external starting
mechanism is the limiting case in which o~--~0 so that the unstable equilib-
rium coincides with RE=0. Such instabilities, in which an initially static
iquid spontaneously establishes the equilibria (iv) or (ii) of Figure 10b,
the subject of the next section.
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The physical mechanism basic to a class of instabilities found with RS < 1
can be described in terms of the rotor of Figure 9, now with the applied elec-
tric-field intensity constant (~ = 0). In Figure 10b, the torque curve e passes
through the origin of the rotor-velocity axis. For small applied fields, the
only equilibrium is with the rotor stationary (at the origin in Figure 10b)
and that one is stable. As the field is increased to a level such that the slope
of the Te curve exceeds that of the -- T* curve, the equilibrium at the origin
becomes unstable, and stable equilibria (ii) and (iv) are possible. The rotor
will spontaneously reach a steady-state rotation in either direction, if (from
Equation 48)
H0I+R-- = 1; H~ -- E0 ¢~
This expression gives the threshold condition for an instability that
would not arise were it not for the influence of the convection on the charge
distribution (finite-electric-Reynolds-number effects). With RS < 1, surface
charges induced by E0 have a dipole moment anti-parallel to the direction of
E0. The induced charges are carried in the direction of the rotation by the
interface and this resulting deflection of the induced dipole moment gives
rise to a torque that tends further to increase the rotational velocity [see
sketch of Figure 10b for equilibrium (iii)].
I 1 1 I I
0 i 2 3 4 5 6
FIG. 12. Surface velocity as a function of traveling wave frequency for the system
of Figure 11, with region (a) air, and region (b) Monsanto Aroclor 1232 (~5.73
vb~ I0-° mhos/m). The channel takes the re-entrant form of a race track of width 5.1
cm, length 0.886 m equal to the wavelength, and depth b--1.6 cm.
Annual Reviews

The condition of Equation 52 for spontaneous rotation represents the
point at which the electrical torque attributable to those charges induced by
the motion competes with the viscous torque. Thus, by its analogy with the
Hartmann number of magnetohydrodynamics, H~ is the electric Hartmann
Observations on rotations of dielectric spheres and cylinders immersed
in slightly conducting fluids stressed by dc fields have been recorded pe-
riodically since Quincke’s observations (22). An indication of the early,
well as the more recent, literature on this subject is given by Pickard (19).
The conditions for incipient instability predicted by Equation 52, and also
the predicted parameter dependence of the steady rotational equilibrium
velocity, have been well documented, at least for certain well-behaved fluids.
We confine our further attention here to electrohydrodynamic instabilities
which, as with the rotor, have a physical basis in the competing processes
of charge convection and viscous dissipation.
When the spatially periodic, imposed potential is static, the fluid inter-
face of Figure 11, like the rotor of Figure 9, can be unstable. Of course, the
interface between the fluids, unlike the surface of the rotor, is not constrained
to rigid-body motions. Even so, we can obtain insights into the nature of
electroconvective instabilities by further considering the implications of
Equation 50. This expression, derived under the assumption that the fluid
responds with a uniform interfaclal velocity U to the average stress, is based
on an approximation that is excellent for a rapidly traveling wave, but highly
questionable in the limit
In a manner analogous to that used in obtaining Equation 52 from Equa-
tion 48, we take the limit of Equation 51 where ¢o--~0, and then require that
the rate at which < T’ > increases with U, in the neighborhood of a static
equilibrium U=0, just equals that with which --T~ increases with U. The
resulting condition for spontancaus translation of the interface, either to the
right or left, is
Hgrl[ + Rtanhkacothkb] = 1; H~=kVo
/x = [ak sinh kb cosh kb ]i/[1 + a/b M ]~ (cosh ka sinh kb)
Note the similarity of this expression and Equation 52 for the rotor. The
same fundamental processes are at work in each case of instability.
With the imposed spatially periodic potential static, it is clear that the
configuration of Figure 11 has the same ingredients as used to produce
periodic cellular convection (Fig. 4). In fact, if an attempt is made experi-
mentally to demonstrate the periodic roll-cell convection described in con-
nection with Figure 4, and a sufficiently nonconducting liquid system is used
that Equation 53 is approximately satisfied, the dominant cellular motions
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observed are likely not to be the expected ones in the x-y plane, but rather
to be in the x-~ plane of the interface. Conditions similar to Equation 53,
involving the electric Hartmann number, would determine whether or not
the cellular convection with the spherical interface of Figure 7 could be es-
tablished without other dominating motions, such as the rotation of the
drop or bubble.
We now discuss experiments in which these charge-convection instabili-
ties in the plane of an interface have been studied. In the first, Jolly (23)
used a configuration very similar to the one just described to produce cellular
motion in the plane of an interface by means of a nonuniform field. In the
second, Malkus & Veronis (24) studied similar rotations but produced by the
curvature of the interface in an essentially uniform imposed tangential field.
Incipience in a nonuniform field.--An experiment which gives graphic
evidence of electroconvective instability in the plane of an interface is shown
in Figure 13. Electrodes of alternate polarity are spaced along the bottom
of an insulating container. A layer of slightly conducting liquid then covers
these electrodes to a depth a. For the present discussion, the upper region
(b) is air, although as long as RS < 1 it might be a second liquid.
Motions of the interface are observed by entraining strips of bubbles in
the regions of the interface above the lines of abutment between electrodes.
As the voltage V, is raised, there is a threshold at which the interface begins
to swim in its own plane. The appearance of the interface as motions begin is
shown in Figure 14, and eventually the interface establishes a pattern of
cellular convection. The smaller the depth, the shorter the initial wave-
length of instability and the smaller and more numerous the cells. Figure 14
shows how the wavelength increases with the depth. For (ka) larger than
about 1, cells formed from one pair of electrodes begin to interact with those
from another pair; the patterns on the interface are not so regular in ap-
pearance, and the interface begins to move as a wh61e, interacting with the
boundaries of the container.
Even if available, an exact analysis of the conditions for incipient insta-
bility is more complicated than is appropriate in this review. However, by
making suitable approximations we can establish that the physical phe-
nomenon shown in Figure 14 is of the same nature as gives rise to spontane-
ous rotations of the rotor. We have arranged the coordinate system of
Figure 13 so that the instability experiment can be regarded as a special
case of the arrangement shown in Figure 11. Thus, Equation 53 gives the
condition under which the interface is unstable to uniform translations. In
the case of the instability experiment we have the limit b-~ ¢o, M--~ 0o and
R--~0. In addition the imposed potential is a square-wave function of x,
which we approximate by considering the fundamental Fourier component;
Vo--4V,/~r. The resulting condition of incipient instability is
Hg ~- coshka/~/-~; HE = 4kV~r " i~-~-a,vt/~a~
We expect that this relation would give its best approximation when th’e
Annual Reviews

top view
FIG. 13. A layer of liquid makes contact with strip electrodes having alternately
the constant potentials _+ V,. Cellular motions in the plane of the interface are estab-
lished by instability and appear as sketched in the top view and as shown by the
photographs of Figure 14.
characteristic motions of the interface overlap several electrodes; i.e., for
large depths, a. In any case, it seems likely that it is an upper bound on the
voltage required to produce instability, because it accounts for only one of
many possible motions the interface can execute. The predictions of Equa-
tion 54 are compared to the experimental measurements of the voltage re-
quired for incipient instability in Figure 15. As expected, the simple theory
appears to provide an upper bound on V,.
To obtain a lower bound on the voltage for instability, it is reasonable to
postulate that if at any point on the interface the electrical stress is a more
rapidly increasing function of surface velocity than is the viscous stress, the
fluid begins to move. Again, the interracial motions are taken as uniform, so
that the electric stress can be calculated in a manner analogous to that used
in determining Equation 50. Now, the stress used in Equation 51 is evaluated
at the coordinate origin in Figure 13. Following the same procedure as with
the average stress, the condition for incipient instability using this "point-
stress" approximation is
HE = cosh ka/2 [ka(1 + tanh ka/S)]~
The voltage predicted by this expression, also plotted in Figure 15, tends
to bound the measured values from below. At depths less than a ~ 1 cm, the
higher harmonics in the Fourier expansion make an essential contribution
to the prediction of Vs, and if these are included in the analysis (23), the
point-stress prediction remains below the measured value even for small
depths. Yet a third model, based on representing the individual cells as
fluid-mechanical rotors, is successful in predicting the incipience to within
20 per cent over the range 0.2 <ka > 1. For the present purposes, our approxi-
mate analysis supports the physical basis given for the instabilities. Jolly
Annual Reviews

view of the experiment
Air bubbles are entrained
i n
fluid injected
strips over regions between electrodes;
instant after incipience
instability, bubbles are carried by the interface to the positions shown. (a)
I n
the elasped time is sufficient that cells are beginning
Annual Reviews

uses energy arguments
together with
judiciously chosen model
t he electrohydrodynamic shear stresses, t o predict t he transverse
( z
tion) wavelengths
incipient instability t o within
per cent over t he range
a curved interface.-Much current interest in t he subject
surface electroconvection can be attributed t o
paper by Malkus & Veronis
in which
range of electrohydrodynamic phenomena are highlighted
and detailed attention
given t o cellular interfacial motions akin t o those
j ust described. The experiment sketched in Figure 16 combines an essenti-
ally uniform electric field imposed tangential t o an interface with a poten-
tially unstable distribution
surface charge induced because of a slight
interfacial curvature. For analytical purposes, t he vertical deflection of t he
interface is approximated
[ = P ( ~ - d/2 ) ~.
For t he case shown,
Wi t h
t he center of curvature
t he side of fluid (b), t he lower fluid must be t he
more conducting t o be consistent with t he potentially unstable distribution
of equilibrium surface charges shown in Figure
Experimentally, it
found t hat,
t he voltage
raised, a threshold
reached a t which cells with t he appearance shown in Figure
motion. Each cell can be viewed as a rotor in a dc field, with
Wi t h
t he assumption
t he out set t hat t he upper fluid
air, and hence much less
conducting and viscous t han t he liquid below, Malkus
Veronis develop a
rather complicated eigenvalue theory t hat predicts onset
[ 0.33~/pd( l
Annual Reviews

>~ -
- ~ °poi
0 0
0 1.0 2.0
depth a - cm
FIG. 15. Voltage for incipience of instability in the plane of the interface. The ap-
paratus is sketched in Figure 13, and the appearance of the interface is shown in
Figure 14. Theoretical approximations are Equations 54 and 55. The fluid is corn oil,
~ra=9.5X10 -n mhos/m, ~=3.1 ~o and/~a=5.46X10 -~ kg(ms)-~.
Note the similarity between this expression and Equation 55. The new in-
gredient in Equation 56 is the surface curvature, represented by ~d.
Using pinene, it is found that Equation 56 predicts a threshold voltage
75 per cent of that observed. This order of difference is expected since the
theory is based on the assumption that the electrode boundaries do not re-
tard the shear flow, but that rather, at these walls, the fluid is free to slip.
side view
top view
F~G. 16. Plane, parallel electrodes having the dc p~teutial difference V, induce the
potentially unstable distribution of ~urface charges shown on a slightly curved inter-
face between fluids (a) and (b) ; it is assumed that RS (1. Cellular convection in the
plane of the interface is akin to the spontaneons rotor rotations.
Annual Reviews

Malkus & Veronis also predict the transverse (~. direction) wavelength for
incipient instability. In the best case, again using pinene, the predicted
wavelength is twice the measured wavelength. This discrepancy is also
attributable to effects from the transverse boundaries.
In the electroconvective forms of instability so far reviewed, the geom-
etry of the interface remains essentially unaltered by the motion. The
effects of interracial shears on the vertical motions of an interface are still
another avenue of research--one that can only be touched on here. We wish
to emphasize the close tie between the physical processes discussed in the
previous sections and the dynamics of gravity-capillary waves coupled to
electric fields under circumstances where effects of finite relaxation time are
Overstability of the rotor in a dc field.--With the vertical interracial mo-
tions and their attendant gravity and capillary forces, new modes of poten-
tial-energy storage are added. Again, the rotor of Figure 9 is a convenient
vehicle for establishing insights into the dynamical phenomena that can be
By way of simulating the effects of capillary and gravity forces, which
depend on the geometry of the interface and not its velocity, we consider the
case in which the rotor is not only subject to viscous and electrical torques,
but to a spring torque as well. This torque, unlike others we have con-
sidered, depends on the absolute angular position of the rotor, 0(t).
analysis of the dynamics about a static equilibrium follows the same lines as
outlined in the section on steady rotations. For present purposes, we note
that if the relaxation time r = (~-I-~b)/(ga-bgb) is short compared to dynami-
cal times of interest we are justified in using the torque expression, Equation
47, in the limit where o~-~0 and f~=dO/dt is small. We define I and K as the
moment of inertia and the torsional spring constant of a unit length of the
rotor and write the torque equation of motion:
I-~+d~0 4~rb’~ E1 He’(1._~__~__R_~- RS)’]A ~ + KO = 0 57.
Here we have assumed that motions are slow enough to justify use of the
steady-state viscous-torque expression Tv=-4~rl~f~b ~. Thus, the motions
are those of a torsional pendulum having a damping constant that is positive
or negative depending on the magnitude of He. In fact, Equation 52 now
discriminates between a negative and positive damping coefficient; between
damping and overstability. Overstabilities of the rotor constrained by a
torsional spring are found to be in reasonable agreement with this simple
model, at least in a restricted number of fluids (9).
On the basis of the rotor dynamics, it is not surprising that electrical
shear forces can conspire to overstabilize a configuration which, like the
capillary-gravity wave system, behaves in an oscillatory or wave-like fashion
in the absence of the field.
Annual Reviews

(b) Y1
...:......:..:...:!...:.:...:.................’........ ....
FIG. 17. Electrical shear stresses induced on interface of gravity-capillary wave system
lead to overstability.
Overstability of an interface.--Melcher & Schwarz (25) show that electrical
shear forces can also lead to overstability on the interface of a gravity-
capillary wave system, depicted in Figure 17. Here motions are in the plane
of the paper rather than in the plane of the interface. In general, this situa-
tion is extremely complex, but consider as a limiting case gas (b) over
liquid (a), as with the rotor, in the limit where r ~ (~q-~b)/(~a-k~b) is
compared to dynamical times of interest. Then, if the liquid has a small
viscosity, in the sense that for surface perturbations of the form ~ cos kx
exp st, s and k satisfy the relation s))k~tz/p, it can be shown that the interface
is overstable as
H~ ~ 1 58.
where He is defined by Equation 52.
Overstabilities on an interface satisfying the above restrictions are ob-
served in an apparatus having essentially the geometry of Figure 16, except
that the equilibrium interface is flat. For a typical experiment, hexane doped
with ethyl alcohol has properties S~2.5, ~a~3.2X10-~, a~3X10-~ (ohm-
rn) -~. With an electrode spacing d~4.7 cm. spontaneous oscillations of the
interface are just perceptible with an applied voltage of 14 kv. The instabil-
ity appears as a standing wave with points of constant phase perpendicular
to the applied electric field. The condition that He~ 1 predicts that E0 for
incipient instability is 2.2X10~ V/m, while the measured value is 3.0X10~
Vim. In view of the accuracy with which the electrical-conduction process
can be described, this agreement is as good as could be expected.
When the fluid properties and allowed wavelengths are not so circum-
scribed as outlined here, the condition for inc~plent instability is consider-
ably more complicated than simply H,~ 1. (25) Nevertheless, the basic
mechanism of overstability is a salient feature of the dynamics.
Our review is restricted to the dynamics of fluids having uniform electri-
cal properties--to cases where the electromechanlcal coupling is confined to
interfaces. Even more, the theme is interracial shear effects. Confined as this
class of electrohydrodynamics may seem, it is clear that we have only begun
to form a picture of the dynamics that are possible when there is an influence
of the motion on the field, as well as an effect of the field on the fluid. This
Annual Reviews

review starts with cases where the former coupling can be ignored, then in-
troduces the finite electric Reynolds number to represent the effect of con-
vection on the charge distribution. We have used the Hartmann number as
an aid in recognizing the connection between various interracial motions
attributable to the competing processes of electroconvection and viscous
shear. Thus, we begin with a simple case that has an easily presented model
and end with cases that are somewhat more complicated.
In conclusion We should ask, where are the scientific and engineering
implications of this developing area? Our list of applications can only be
representative of the spectrum of interests; the literature of electrohydro-
dynamics connected with each is larger than our own brief list of references.
Dielectrophoretic forces, represented by the second term in Equation 6,
are being applied to fluid mechanics problems ranging from the separation of
living and dead cells (26) to the orientation of cryogenic liquid propellants
the zero-gravity environment of space (27). As pointed out in the introduc-
tion, in almost all cases this class of electrohydrodynamics is representable by
coupling at interfaces. Even though the effects of free charge are usually
undesirable in dielectrophoretic interactions, they must be understood. Thus,
in this area electrical surface-shear effects are essential to answer such ques-
tions as, can a cryogenic propellant be oriented indefinitely in adc field?
Two-phase heat transfer in an electric field (28), another area of electro-
hydrodynamics, is dominated by electromechanical coupling at interfaces.
Shearing effects can be important in determining interl~acial stability or in
providing a mechanism similar to that of blowing for shearing the fluid
from a surface.
Imaging on liquid interfaces by means of charged particles is being devel-
oped and demands careful attention to the effects of electrical shear stresses
(29). The formation of charged liquid particles, with a multitude of applica-
tions including image reproduction and space propulsion (30), clearly in-
volves electrical relaxation and shear effects (31, 32). From the basic view-
point, there are a host of fluid systems in which the electric Reynolds number
is large, even at modest applied-field strengths. Thus, in studies of electrical
conduction, as well as other physical phenomena at interfaces, the electro-
hydrodynamic contributions to measurements can hardly be ignored. It is
now becoming obvious to many (33) that it makes no sense to study mecha-
nisms of current conduction in slightly conducting liquids without paying
heed to the electrohydrodynamics.
Finally, let us recognize that an interface is a particular kind of dis-
continuity in the electrical and mechanical properties of a fluid. Studies of
surface dynamics have clear implications for coupling in the bulk of fluids.
Consider by analogy the connection between gravity surface waves repre-
sented by a discontinuity in density and the internal dynamics of fluids hav-
ing distributed density gradients. Many of the types ot" interactions we have
reviewed are being studied in the context of electrohydrodynamic bulk
interactions (34).
Annual Reviews

I. Pickard, W. F., in Progress in Di-
electrics, 1-39 (Academic Press,
New York, 334 pp., 1965)
2. Woodson, H. H., Melcher, J. R., in
Electromechanical Dynamics: Part
I, Discrete Systems, 251-317 (John
Wiley & Sons, Inc. New York,
N. Y., 894 pp., 1968)
3. Watson, P. K., Charbaugh, A. H., in
Progress in Dielectrics, Vol. IV,
201-46, (Birks, J. B., Hart, J.,
Eds., Academic Press, Inc., New
York, 309 pp., 1962)
4. Stratton, J. A., Electromagnetic Theory,
137-40 (McGraw-Hill, New York,
N. Y., 615 pp. 1941)
5. Penfield, P. A., Jr., Haus, H. A.,
Eleclrodynamlcs of Moving Media,
65-72 (M.I.T. Press, Cambridge,
Mass., 276 pp., 1967)
6. Private communication, G. I. Taylor
to J. R. Melcher, Nov. 23, 1966
7. Stuetzer, O. M., Phys. Fluids, 5, 534-44
8. Smith, C. V., Jr., Melcher, J. R.,
Phys. Fluids, 10, 2315-22 (1967)
9. Smith, C. V., Jr., Steady Shear-Induced
Electrohydrodynamic Flows (Doc-
toral Thesis, Mass. Inst.
Cambridge, Mass., Dept.Elec.
Engr., 1968)
10. Sneddon, I. N., Fourier Transforms,
269-270 (McGraw-Hill, New York,
542 pp., 1951)
11. Taylor, G. I., P~oc. Royal Soc. A, 291,
159-66 (1966)
12. Fano, R. M., Chu, L. J., Adler, R. B.,
Electromagnetic Fields, Energy, and
Forces, 150-53 (John Wiley & Sons,
Inc., New York, 520 pp., 1960)
13. Goldstein, S., Modern Developments in
Fluid Dynamics, 114-115 (Oxford
University Press, Oxford, 702 pp.
14. Bird, R. B., Stewart, W. E. Lightfoot,
E. N., Transport Phenomena, 90
(John Wiley & Sons, Inc., New
York, 780 pp., 1960)
15. Allan, R. S., .Mason, S. G., Proc. Roy.
Soc., A, 267, 383-97 (1964)
16. Taylor, G. I. Proc. Roy. Sot., A, 280,
383-97 (1964)
17. Sample, S. B., Static and Dynamic Be-
havior of Liquid Drops in Electric
Fields (Doctoral Thesis, Univ. of
Illinois, Urbana, Illinois, Dept. of
Etee. Engr., 1965)
18. Arno, R., Rendi. Alti Reale Accad.
Lineci, 1, (2), 284 (1892)
19. Pickard, W. F., Nuovo Cimento, 21,
316-32 (1961)
20. Melcher, J. R., Phys. Fluids, 9, 1548-55
21. Ochs, H. T., Traveling-Wave Electro-
hydrodynamic Pumping (M.S.
Thesis, Mass.Inst. Tech., Cam-
bridge, Mas~.,Dept. Elec. Engr.,
22. Quincke, G., Ann. Phys. Chemic, 59,
417-85 (1896)
23. Jolly, D. C., Cellular Electroconvective
Instability in a Fluid Layer (M.S.
Thesis, Mass. Inst. Tech., Cam-
bridge, Mass., Dept. Elec. Engr.,
24. Malkus, W. V. R., Veronis, G., Phys.
Fluids, 4, 13-23 (1961)
25. Meleher, J. R., Schwarz, W. J., Jr.,
Interracial Relaxation Overstability
in a Tangential Electric Field
(Mass. Inst. Tech., Center Space
Res. Rept. CSR TR 68-2, Cam-
bridge, Mass., 1968)
26. Crane, J. S., Pohl, H. A., J. Electro-
chem. Sot., 115, 584-86 (1968)
27. Meleher, J. R. Hurwitz, M., J. Space-
craft 6~" Rockets, 4, 864-81 (1967)
28. Choi, H. Y., Trans. ASME J. Heat
Transfer, 90, 98-102 (1968)
29. Poritsky, FI., in Developments in Me-
chanics, 2, Part I, 145-72 (Ostrach,
S., Scanlan, R. H., Eds., Pergamon
Press, 830 pp., 1965)
30. Hendricks, C. D., J. Colloid Sci., 17,
249-59 (1962)
31. Carson, R. S., Hendricks, C. D.,
Natural Pulsations in Electrical
Spraying of Liquids (Paper 64-
675, Am. Inst. Aeron. Astronaut.
Fourth Elec. Propulsion Conf.,
Philadelphia, Penn., 1964)
32. Hogan, J. J., Carson, R. S., Schneider,
J. M., Hendricks, C. D., AIAA J.,
4, 1460-1461 (1964)
33. Lewis, T. J., Seeker, P. E., 1966 Annual
Report: Conference on Electrical In-
sulation and Dielectric Phenomena,
87-94 (Nat. Acad. Sci., Washing-
ton, D. C., 199 pp., 1968)
34. Melcher, J. R., Firebaugh, M. S.,
Phys. Fluids, 10, 1178-85 (1967)
Annual Reviews