ELECTROHYDRODYNAMICS: A REVIEW OF THE

ROLE OF INTERFACIAL SHEAR STRESSES

BY J. R. M~-LCH~Rt A~ G. I. TAYLOR

Massachusetts Institute of Technology, Cambridge, Massachusetts

and Farmfield, Huntingdon Road, Cambridge, England

SCOPE

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.

ELECTRODYNAMIC$

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.

www.annualreviews.org/aronline

Annual Reviews

112 MELCHER & TAYLOR

TABLE I

SUMMARY OF DIFFERENTIAL LAWS, TRANSFORMATIONS AND BOUNDARY

CONDITIONS FOR QUASI-STATIC ELECTRIC FIELD SYSTEM

Differential Laws Transformations Boundaxy Conditionsa

V ~(E = Ia

v.D = q Ib

V.l-b 0q = 0 Ic

D = ¢0E + P Id

E’ = E

Ie

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

Ij

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

Annual Reviews

ELECTROHYDRODYNAMICS 113

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

2.

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

have

dr

q = q0~-t/" o~ ~ = v

5.

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.

HYDRODYNAMICS

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

Annual Reviews

114 MELCHER & TAYLOR

TABLE II

HYDRODYNAMIC EQUATIONS AND BOUNDARY CONDITIONS

Differential Laws Boundary Conditions

D~

~ -b7 = ~g + v. (1"-. + T’)

na

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)

1

F = qg -- -~ .E~W

6.

Equation 6 can be written identically as

1

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.

OBJECTIVES

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

Annual Reviews

ELECTROHYDRODYNAMICS 1 15

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.

STEADY CONVECTION: DC FIELDS

A SIMPLE EXAMPLE

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.

Annual Reviews

1 i 6

MELCHER & TAYLOR

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.

l

This is also the tangential component of E just above the interface (Equa-

tion Ij). Hence, the distribution of potential on the interface is

V0

6(y = b) =

9.

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

11.

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

density

*The components of i are the unit vectors in the coordinate directions.

*[A]---Aa-Ab--the jump of A across the interface.

Annual Reviews

ELECTROHYDRODYNAMICS 1 17 "

T, = eoEvaEx~ = -- eoVd/la

12.

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

model.

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

14.

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

Annual Reviews

118

MELCHER

&

TAYLOR

FIG.

2.

Streak photograph

of

the cellular convection induced

in

an apparatus hav-

ing

the configuration sketched in Figtire

1, 1=24

cm,

b=3.8

cm,

V ~ =2 0

kV and the

fluid is corn oil.

A

comparison of the experimentally determined surface velocity and

the prediction

of

Equation

13 is

shown

in

Figure

3.

velocity, further indicates t hat t he physical arrangement

of

electrodes and

interface

is

successful in producing a nearly uniform electric-shear-stress

distribution over t he interface.

A

plot of t he product

of

observed surface velocity

U

near t he center of

t he apparat us and maximum spacing

a,

as

a

function

of

t he applied voltage

Vo

for three angles of inclination,

is

shown i n Figure 3 where t he solid line is

predicted by Equation

13

evaluated

at

y=b.

For t he maximum velocity

Re

as defined by Equation

14

is

on t he order

of

unity,

so

t hat t he assumption

t hat t he electrical-relaxation process easily keeps up with t he motion

is

not

V,- kV

FIG.

3.

Product

of

maximum electrode spacing and the

interfacial

velocity

u

near

the center

of

the apparatus shown in Figures

1

and

2,

as

a function of the applied

voltage. Liquid

is

corn oil,

e =3.l e O,

~=10-10

mhos/m,

~( =0.055

kg (nis)-' and

1=24

cm,

b

=

3.8

cm.

The

solid

curve, from Equation

1 3

is

Uu

=

~,,V0~b/4pl.

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ELECTROHYDRODYNAM ICS 119

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.

PERIODIC CONVECTION

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

convection.

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,

Annual Reviews

120 MELCHER & TAYLOR

(o)

(b)

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

15.

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ELECTROHYDRODYNAMICS 121

and

where

16.

~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

17.

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

Annual Reviews

122

and

MELCHER & TAYLOR

¢,~ = 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.

where

= ~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.

CONVECTION IN DROPS AND AROUND BUBBLES

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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ELECTROHYDRODYNAM ICs

123

FIG.

5.

Streak photograph

of

periodic cellular convection observed in apparatus

having essentially the fluid and field configuration

of

Figure 4a. The cells have the

spatial period

of

Figure

4b

and velocity-voltage relationship

of

Figure

6

(after Smith

&

Melcher).

spherical coordinates are defined. Electrodes, removed many radii

b

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

Eo

as

r

cos

O-tm,

play a role

analogous t o t hat of t he electrodes and liquids (b) in Figures

1

and

4;

by

virtue of t he conduction current they insure t hat there

is

a

tangential com-

ponent of

E

in t he neighborhood of t he spherical interface.

Once again, convection results because not only

is

there

a

tangential

component

of

E,

but there

is

also

a

surface charge induced on t he interface.

The case illustrated in Figure 7a pertains t o

an

insulating fluid (b)-perhaps

an air bubble-in

a

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

as

a

resistance in series with

a

capacitance (the bubble),

in t urn in series with

a

resistance. Thi s makes reasonable t he polarity of the

induced surface charges indicated

i n

Figure

7a,

and t he surface shear-force

densities

also

sketched there.

Our approach t o describing

a

self-consistent shear flow

is

essentially t he

same as in t he case

of

t he periodic convection, with one exception.

I n

t he

previous case, gravity

is

used t o hold t he normal electric stresses in balance.

In

t he case of t he sphere we find a unique combination

of

fluid properties

t hat make possible a spherical equilibrium

of

t he interface. Then

it

is possible

to

deduce whether t he drop tends toward a prolate or oblate geometry for

combinations of physical parameters other t han those required for

a

spherical

eq

II

i li

bri

u

m.

Electrical

slressm-In

t he limit of zero electric Reynolds number, t he

solution for t he electric-field intensities

is

t he classic one,

(12)

where t he

field in t he interior of t he drop is uniform, while t hat outside

is

a superposi-

tion

of a

uniform field and a three-dimensional dipole field. There are four

Annual Reviews

- Two-dimensional 1/ /

o_

6

theory

Theory corrected ~._ /

for side walls

~4//-

°

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 ~

25.

4,~ ~ -- 3E~r cos 0/(2 -I- R)

26.

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-

Annual Reviews

ELECTROHYDRODYNAM ICS 125

(a)

(b)

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-

Annual Reviews

126 MELCHER & TAYLOR

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

form

~ = 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

33.

~ = (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.

10(2

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.

Annual Reviews

ELECTROHYDRODYNAMICS 12 7

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)

37.

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

2T

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,

Annual Reviews

128

MELCHER

&

TAYLOR.

Annual Reviews

ELECTROHYDRODYNAMICS 129

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: AC FIELDS

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.

STEADY ROTATIONS IN ROTATING FIELDS

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)].

Annual Reviews

130 MELCHER & TAYLOR

(a)

+

T;

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

Annual Reviews

ELECTROHYDRODYNAMICS 131

uniform far from the rotor axis, and hence, as r cos (O-wt)--~,

4~a --~ -- Eor cos (~0t -- 0)

41.

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)

44.

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 -- ~)

Q=

45.

7’1 + R~(1 + R)

where

~, = 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~"

T’=

47.

(1+ S)(1 +

lq-R~

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

Annual Reviews

132 MELCHER & TAYLOR

no other torques are present, the balance of torques requires that

~,,Eo~(RS- 1) R~

~2 ----

48.

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.

Annual Reviews

ELECTROHYDRODYNAMICS 13 3

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.

TRAVELING-WAvE- iNDUCED CONVECTION

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.

Annual Reviews

134 MELCHER & TAYLOR

~rode

potential

Re Voexp. j (o~t-~

y

~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

obtain

(Tx~) = ½k~V02ea~baa sinh kb cash kb(RS -- 1)R~/[1 + R~fl]h2~

50.

where

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

ELECTROHYDRODYNAMICS 135

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.

Annual Reviews

136

MELCHER & TAYLOR

SHEAR-INDUCED INSTABILITIES

STEADY ROTATIONS IN a STATIC FIELD

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)

R~S

H0I+R-- = 1; H~ -- E0 ¢~

52.

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

ELECTROHYDRODYNAMICS 13 7

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

number.

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.

INSTABILITY IN THE IN~ERFACIAL PLANE

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

53.

where

/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

Annual Reviews

138

MELCHER & TAYLOR

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~

54.

We expect that this relation would give its best approximation when th’e

Annual Reviews

139

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)]~

55.

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

140

MELCHER

&

TAYLOR

FIG.

14.

Top

view of the experiment

of

Figure

13.

Air bubbles are entrained

i n

fluid injected

in

strips over regions between electrodes;

an

instant after incipience

of

instability, bubbles are carried by the interface to the positions shown. (a)

ke=0.26,

(b)

ka=0.5,

(c)

ka=0.8.

I n

(b),

the elasped time is sufficient that cells are beginning

to

form.

Annual Reviews

ELECTROHYDRODYNAMICS

141

uses energy arguments

(23),

together with

a

judiciously chosen model

for

t he electrohydrodynamic shear stresses, t o predict t he transverse

( z

direc-

tion) wavelengths

of

incipient instability t o within

5

per cent over t he range

of

0.2

<ka

<1.2.

Incipience

on

a curved interface.-Much current interest in t he subject

of

surface electroconvection can be attributed t o

a

paper by Malkus & Veronis

(24)

in which

a

range of electrohydrodynamic phenomena are highlighted

and detailed attention

is

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

of

surface charge induced because of a slight

interfacial curvature. For analytical purposes, t he vertical deflection of t he

interface is approximated

as

[ = P ( ~ - d/2 ) ~.

For t he case shown,

p>O.

Wi t h

t he center of curvature

on

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

16.

Experimentally, it

is

found t hat,

as

t he voltage

is

raised, a threshold

is

reached a t which cells with t he appearance shown in Figure

16

are

set

in

motion. Each cell can be viewed as a rotor in a dc field, with

RS

<

1.

Wi t h

t he assumption

at

t he out set t hat t he upper fluid

is

air, and hence much less

conducting and viscous t han t he liquid below, Malkus

Sr

Veronis develop a

rather complicated eigenvalue theory t hat predicts onset

of

electroconvec-

tion

at

Ne

=

[ 0.33~/pd( l

+

1/S)]4;

11,

=

&v'+

56.

Annual Reviews

142 MELCHER & TAYLOR

>~ -

- ~ °poi

0

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.

i+

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

ELECTROHYDRODYNAMICS 143

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.

INSTABILITY OF SURFACE WAVES

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

important.

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

expected.

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

144 MELCHER & TAYLOR

(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.

SUMMARY REMARKS

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

ELECTROHYDRODYNAMICS 145

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

146 MELCHER & TAYLOR

LITERATURE CITED

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

(1962)

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.

Tech.,

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.

1938)

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

(1966)

21. Ochs, H. T., Traveling-Wave Electro-

hydrodynamic Pumping (M.S.

Thesis, Mass.Inst. Tech., Cam-

bridge, Mas~.,Dept. Elec. Engr.,

1967)

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.,

1968)

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

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