CONF. PRESENTATION SESSION 1: PARTICLES IN FLOWS & FLOW ELECTRIFICATION 1

Effect of Charge Mobility on Electric ConductionDriven Dielectric Liquid FlowMiad Yazdani and Jamal Seyed-Yagoobi, Senior Member, IEEE

Abstract—Electrohydrodynamic (EHD) conduction pumpingis associated with the heterocharge layers of finite thicknessin the vicinity of the electrodes, generated by the process ofdissociation of the neutral electrolytic species and recombinationof the generated ions. EHD conduction generated flow reliesprimarily upon the asymmetry of the electrodes where the flowis always directed toward the specific direction regardless ofthe electrodes polarity. However, the difference in the mobilitiesof positive and negative charges could play an important rolewhen studying the pumping performance. This paper studies theeffects of charge mobility and its difference between the positiveand negative charges on the heterocharge layer structure andgenerated flow. The numerical simulation is conducted for a 2-D rectangular channel with the electrodes embedded against thechannel wall. The net flow generated for the symmetric electrodesis physically illustrated while the impact of mobility differenceon the generated flow for the case of asymmetric electrodes isstudied as well.

Index Terms—EHD Conduction, dissociation and recombina-tion, charge mobility.

NOMENCLATURE

b+/− positive/negative charge mobility coefficientbr mobility ratio, br = b+/b−D charge diffusion constantd channel half heightC0 EHD dimensionless numbere electron chargeE electric field vector|E| electric field vector magnitude

F (ω) I1(2ω)/ωI1(z) modified Bessel function of first type and order onekB Boltzmann universal constantkd dissociation rate constantkr recombination rate constantn negative charge densityn unit normal vectorp positive charge densityP pressureReehd EHD Reynolds numberTsat saturation temperature

u velocity vector

Manuscript received April 17, 2009. This work was financially supportedby the NASA Microgravity Fluid Physics Program.

M. Yazdani is currently PhD student with the Two Phase Flow and HeatTransfer Enhancement Laboratory in Mechanical, Materials and AerospaceEngineering Department, Illinois Institute of Technology, Chicago, IL, 60302(email: myazdan1@iit.edu).

J. Seyed-Yagoobi is IEEE senior member and professor and chair ofthe Mechanical, Materials and Aerospace Engineering Department, IllinoisInstitute of Technology, Chicago, IL, 60302 (email: yagoobi@iit.edu).

uehd EHD reference velocity|u| velocity vector magnitudeV applied electric potentialYw channel wall thicknessα dimensionless charge diffusion constantε absolute electric permittivityη electrode gapµ dynamic viscosity of fluidν kinematic viscosity of fluidξ electrode widthξr ground to HV electrode width, ξr = ξg/ξHV

ω dissociation rate coefficient, ω = [ e3|E|4πεkB

2Tsat]1/2

ρ mass densityσ electric conductivity of fluidφ potential field

SUBSCRIPTS

eq equilibriumg ground electrode

HV high voltage electrode

SUPERSCRIPT

∗ dimensionless variable

I. INTRODUCTION

The EHD pumping phenomena involve the interaction ofelectric fields and flow fields in a dielectric fluid medium.This interaction between electric and flow fields induces thefluid motion through the presence of electric body force. EHDconduction pumping is associated with the heterocharge layersof finite thickness in the vicinity of the electrodes that arebased on the process of dissociation of the neutral electrolyticspecies and recombination of the generated ions [1], [2]. Theconduction term here represents a mechanism for electriccurrent flow in which charged carriers are produced not byinjection from the electrodes, but by dissociation of moleculeswithin the fluid.

Feng and Seyed-Yagoobi [3] conducted an asymptotic anal-ysis to study the effects of flow convection on the EHDconduction pumping and the associated energy transport. TheEHD conduction driven flow rely primarily upon the asym-metry of the electrodes. The electrodes asymmetry results inthe dominance of electric body force and flow generation inone direction. Yazdani and Seyed-Yagoobi [4] showed that theflow generation is always from the narrower electrode towardthe wider electrode regardless of their polarity. They alsoshowed that symmetric charge and body force distributions

CONF. PRESENTATION SESSION 1: PARTICLES IN FLOWS & FLOW ELECTRIFICATION 2

over identical electrodes result in zero net flow generationwhen the charges have identical properties. However, recentexperimental study by Hanaoka et. al [5] revealed that thereexists a net generated flow in the case of symmetric electrodesconfiguration. Assuming that the diffusion has negligible ef-fect on the charge distribution in macro-scales, the chargedistribution symmetry over the identical electrodes could bebroken only if the charges have different mobilities. Thereexist a limited number of publications focusing on the mobilitymeasurements in dielectric liquids. Winkur et. al [6] conductedan experiment to measure the ion mobility in n-hexane and a0.22M solution of biphenyl in n-hexane using a time-of-flighttechnique. This technique was used in several measurementsof the mobility of positive ions in hydrocarbons (see forexample [7] and references therein) with the discrepanciesmainly raised by the methods of measurements. Huang andFreeman [8] measured the mobility of positive ions in liquidhydrocarbons and investigated the effects of density and tem-perature. They claimed that the positive ions mobility does notobey Walden’s rule (b+ ∝ µ−1.0) but rather has a nonlinearrelationship to the viscosity, b+ ∝ µ−1.2±0.1 as found earlierfor ions in ethers. The time of flight method is later usedby Sakamoto and Yamada [9] to study the ion mobilitiesand breakdown processes in dielectric liquids with razor-bladeemitter grid collector used for the electrodes. They reportedthe streaming velocities for positive and negative agents indifferent liquids. Similar technique was used by Casanovas et.al [10] to measure the mobility of positive and negative ionsin polydimethylsiloxane silicone oil.

The current study extends the fundamental study by Yazdaniand Seyed-Yagoobi [4] to illustrate the role of charge mobilityon the performance and characteristics of EHD conductiondriven liquid flow. In particular, the impact of unequal mobili-ties for positive and negative charges on the generated flow forthe case of symmetric electrodes is investigated and physicallyanalyzed. This study is also extended to include the case ofasymmetric electrodes where net flow is expected due to theelectrodes asymmetry even in the case of equal mobilities forpositive and negative carriers.

II. THEORETICAL MODEL

The theoretical model presented here is adopted from themodel developed by Atten and Seyed-Yagoobi [11]. Theirtheoretical model was developed and solved analytically inthe absence of any fluid motion assuming that the positiveand negative charges possess identical mobilities. Jeong et al.[12] applied Atten and Seyed- Yagoobi’s model to a complexelectrode geometry and provided numerical solutions in theabsence of fluid motion as well. The model presented in thissection is based on the assumption that the dielectric liquidis incompressible and single phase with no contribution ofcharge injection from the electrodes’ surfaces. The positiveand negative charge mobilities are different which is the majordeparture of this paper from previous studies. It is assumed thatthe charge diffusion for positive and negative ions is constantand identical.

The schematic of the numerical domain for a symmetricelectrode configuration is presented in Fig. 1. Table I presents

V solid

in/out in/out symmetry

liquid d

Yw

ξg ξHV η

x

y

Fig. 1. Schematic of numerical domain (not to scale). Detail of the electrodesdesign is provided in Table I.

TABLE IDETAILS OF THE ELECTRODES DESIGNS CHARACTERISTICS. ALL

DIMENSIONS ARE NORMALIZED WITH THE CHANNEL HALF HEIGHT,d = 2mm

Electrodes Configuration

Channel wall thickness (mm)

Ground electrode (mm)

HV electrode (mm)

Electrode gap (mm)

Symmetric ξg = 2

(ξ*g = 1)

ξHV =2 (ξ*

HV = 1)

Asymmetric, ξr = 0.35 ξg = 0.7

(ξ*g = 0.35)

ξHV =2 (ξ*

HV = 1)

Asymmetric, ξr = 2.86

Yw = 4 (Yw* = 2) ξg = 2

(ξ*g = 1)

ξHV =0.7 (ξ*

HV = 0.35)

η = 1 (η* = 0.5)

the details of the electrode design for symmetric and asym-

metric electrodes configurations.In steady-state, the system described here is governed by

the following equations.

∇ · u (1)

u · ∇u = −1ρ∇P + ν∇2u + (ρ+ − ρ−)E (2)

The last term on the left hand side of equation. (2) is theelectric body force and should be obtained by solving theMaxwell’s relation and charge transport equations:

∇ · E =(p− n)

ε; E = −∇φ (3)

∇ ·[(

b+p−b−n

)E +

(pn

)u]−D∇2

(pn

)

= kr

(n2

eqF (ω)− pn)

(4)

Here, the assumption of thermodynamic equilibrium for freecharges is utilized to relate the dissociation rate to recombi-nation rate, kdN = krn

2eq where neq stands for the charge

density at equilibrium. Finally, with d, uehd = b+V/d,µ2/ρd2, V/d, and neq, respectively as scaling parameters forlength, velocity, pressure, electric field, and free charges, onecan obtain the nondimensional equations governing the systemas follows:

∇ · u∗ = 0 (5)

(u∗ · ∇) u∗ = −(1

Reehd

2

)∇P ∗

+1

Reehd∇2u∗ + M2C0 (p∗ − n∗) E∗ (6)

∇ · E∗ =1

1 +(

1br

)C0 (p∗ − n∗) , E∗ = −∇φ∗ (7)

CONF. PRESENTATION SESSION 1: PARTICLES IN FLOWS & FLOW ELECTRIFICATION 3

TABLE IISUMMARY OF ELECTROSTATIC AND FLOW BOUNDARY CONDITIONS

HV electrode Ground

electrode Liquid/solid

interface Inlet/outlet Channel

outer wall

0* =u

outin

outin

||

,||**

**

uu

unun

=

∇⋅=∇⋅ -

1*=φ 0*=φ **sφφ =

outin

outin

||

,||**

**

φφ

φφ

=

∇⋅=∇⋅ nn 0*=φ

0

,0*

*

=∇⋅

=

n

p

n 0

,0*

*

=∇⋅

=

p

n

n

( ) 0, ** =∇⋅ npn

( ) ( )( ) ( ) outin

outin

npnp

npnp

|,|,

,|,|,****

****

=

∇⋅=∇⋅ nn

-

∇ ·[(

p∗

−n∗

)E +

(p∗

brn∗

)u]− α∇2

(p∗

brn∗

)=

(C0

brC0

)(F (ω)− p∗n∗) (8)

where Langevine’s approximation for dielectric fluids [13]is used to relate the recombination constant to the liquidproperties; that is: kr = b++b−

ε . The resulting dimensionlesscoefficients in the above equations are defined as follows:

Reehd =b+V

ν, M =

√ε

ρb+2(1 + 1

br

) ,

C0 =σd2

b+εV, α =

D

b+V=

kBT

eV, br =

b+

b−with the charge concentration at equilibrium, neq , defined asneq = σ

b++b−. The liquid is in contact with the channel wall

which is assumed to be solid a insulator. Therefore, with novolumetric electric charges within the solid zone, the Laplaceequation governs the potential field:

∇∗2φ∗s = 0 (9)

The boundary conditions are presented in Table II. Notethat the continuity of potential field is applied across thesolid/liquid interface where the liquid is in contact with thewall while the channel outer surface is grounded. No boundarycondition is required for velocity and electric charges insidethe solid wall since the corresponding equations are not solvedwithin the solid zone. In addition, the zero flux boundarycondition for electric charges across the solid/liquid interfaceis based on the negligibly small double layer thickness whichimplies that no charges cross the solid/liquid interface as thevolumetric electric charge inside the solid zone is assumed tobe zero. In addition, boundary conditions on the electrodessurfaces imply that charges with the same polarity as theelectrodes do not accumulate on the surface of the electrodeswhile no diffusion of the charges with opposite sign exists rightat the surface of the electrode. However, there exists chargediffusion in the upper layers within the heterocharge layer.

III. RESULTS AND DISCUSSIONS

The results presented in this section illustrate the role ofcharge mobility on the performance of EHD conduction pump.The reference values selected to calculate the dimensionlessparameters and br = 1 correspond to the properties of

TABLE IIINUMERICAL VALUES OF DIMENSIONLESS NUMBERS FOR THE BASIC CASE

Dimensionless number Numeric value Reehd 1380 C0 0.011 α 10-6

1.0 br 2.0

4.39 M 3.58

refrigerant R-123 as the working fluid [2]. The values ofthe dimensionless numbers for the base case are presentedin Table III and applied voltage of 10kV. Note that thevalues presented in this paper for br has been arbitrarilyselected due to the lack of measured mobility ratios especiallyfor refrigerant R-123. These results are presented to solelyillustrate the effects of charges mobility on the generated flow.

First we examine the impact of charge mobility for symmet-ric electrodes configuration where no flow is expected whenthe mobility of positive and negative charges is identical. Laterin this section, the role of charge mobility for asymmetricelectrodes is investigated where net flow is generated due tothe electrodes asymmetry in the case of identical mobilities.In addition, the variation of net generated flow over the rangeof mobility ratios for all electrodes configurations is presentedand analyzed. Note that this range

A. Symmetric Electrodes Configuration

The results of electric field, net charge density and electricbody force for symmetric electrodes configuration and identi-cal mobility for positive and negative charges are presentedin Fig. 2. The electrodes symmetry results in symmetricdistribution of electric field and thus, net charge density. Inaddition, the inter-electrode region is characterized by highintensity electric field (see Fig. 2a) which yields in highercharge density in this region (Fig. 2b). Therefore, the resultantelectric body force distribution is identical but in oppositedirections above the electrodes surfaces. The flow streamlinespresented in Fig. 3 confirm zero net generated flow which, ofcourse, is associated with local flow circulations due to thelocal body force distribution.

The symmetric distribution of charges over the surfaces ofidentical electrodes will be broken once different mobilities areassigned for positive and negative charges. This mechanismalong with the distribution of resultant electric body force isillustrated in Fig. 4 for br = 2. Note that this value of mobilityratio has not been specifically measured for R-123 but hasbeen arbitrarily chosen by considering the measurement byCasanovas et. al [10] for purified silicon oil containing C7F14.Here, negative charges have smaller mobility coefficient thanthe positive charges. Therefore, there exists more tendency forpositive charges to migrate toward the electrodes due to theexternal electric field as governed by Eq. (8). The consequenceis higher positive charge density on the surface of the groundelectrode as observed in Fig. 4a which results in the dominance

CONF. PRESENTATION SESSION 1: PARTICLES IN FLOWS & FLOW ELECTRIFICATION 4

x*

y*

9 10 11 12 13-0.5

0

0.5

10.2 0.4 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.6 3 3.4 3.6

| E * |

(a)

x*

y*

9 9.5 10 10.5 11 11.5 12 12.5 13 13.50

0.5

1

-0.16 -0.12 -0.08 -0.04 0 0.04 0.08 0.12 0.16

p * - n *

(b)

x*

y*

9 9.5 10 10.5 11 11.5 12 12.5 13 13.50

0.5

1

0.01 0.05 0.08 0.11 0.14 0.16 0.19 0.22 0.25 0.28 0.31 0.33

| f * |

(c)Fig. 2. Dimensionless contours of (a) electric field, (b) net charge density,and (c) electric body force with br = 1 and ξr = 1.

x*

y*

9 9.5 10 10.5 11 11.5 12 12.5 13 13.50

0.5

1

Fig. 3. Flow streamlines with br = 1 and ξr = 1.

of electric body force directing toward negative x-direction(see Fig. 4b). The flow streamlines displayed in Fig. 5 illustratethe net flow motion toward negative x-direction.

The variation of net generated flow with mobility ratiofor different electrodes spacings is presented in Fig. 6. Thedimensionless flow rate at the channel outlet is defined as:

Q∗net =1

uehdd

∫ d

0

uoutdy (10)

The generated flow initially increases and eventually dropswith the mobility ratio. This scenario can be explained asfollows: The initial increase in the generated flow is due tothe more pronounced asymmetric charge distribution over theelectrodes surfaces. However, larger values of mobility ratiosuppress the migration of negative charges toward the HV elec-trode surface to yield lower charge density over the electrodesurface as illustrated in Fig. 7. In addition, the electric field hasa more uniform distribution around the ground electrode sincecharges tend to have a unipolar distribution on the surfaceof the ground electrode as the heterocharge layer on the HVelectrode is inclined to disappear. Therefore, the two edgesof the ground electrode are characterized by high intensity

x*

y*

9 9.5 10 10.5 11 11.5 12 12.5 13 13.50

0.5

1

-0.1 -0.06 -0.03 -0.01 0.01 0.03 0.06 0.1 0.12 0.14

p * - n *

(a)

x*

y*

9 9.5 10 10.5 11 11.5 12 12.5 13 13.50

0.5

1

0.01 0.04 0.07 0.09 0.12 0.15 0.17 0.19 0.22 0.25 0.27 0.29

| f * |

(b)Fig. 4. Dimensionless contours of (a) net charge density and (b) electricbody force with br = 2 and ξr = 1.

x*

y*9 9.5 10 10.5 11 11.5 12 12.5 13 13.5

0

0.5

1

Fig. 5. Flow streamlines with br = 2. and ξr = 1

electric body force in opposite directions as presented in Fig. 7,which adversely impact the generated liquid flow. Also shownin Fig. 6 is the effect of electrode gap on the generatedflow. The flow is expected to increase as the electrode gapis reduced due to the higher electric filed intensity within theinter-electrode region.

B. Asymmetric Electrodes Configuration

The fundamental application of EHD conduction drivenflow is based on the asymmetric electrodes geometry. Asillustrated in Fig. 8, the consequence of electrodes asymmetry

1 2 3 4 5 6−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10

−4

br

Q* ne

t

η* = 1

η* = 0.6

η* = 2

ξr = 1, Re

ehd= 1380, C

0 = 0.011

Fig. 6. Variation of net generated flow with mobility ratio for symmetricelectrodes, ξr = 1.

CONF. PRESENTATION SESSION 1: PARTICLES IN FLOWS & FLOW ELECTRIFICATION 5

x*

y*

9 9.5 10 10.5 11 11.5 12 12.5 13 13.50

0.5

1

-0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

p * - n *

(a)

x*

y*

9 9.5 10 10.5 11 11.5 12 12.5 13 13.50

0.5

1

0.02 0.05 0.08 0.12 0.15 0.18 0.22 0.24 0.26 0.28 0.3

| f * |

(b)Fig. 7. Dimensionless contours of (a) net charge density and (b) electricbody force with br = 6 and ξr = 1.

x*

y*

10 10.5 11 11.5 12 12.5 130

0.5

10.01 0.04 0.06 0.09 0.12 0.15 0.18 0.2 0.22 0.24 0.26 0.3

| f * |

(a)

x*

y*

10 10.5 11 11.5 12 12.5 130

0.5

1

(b)Fig. 8. (a) dimensionless electric body force contours and (b) flow streamlineswith ξr = 0.35 and br = 1.

is the dominance of the electric body force directed towardthe broken symmetry; from the narrower electrode toward thewider electrode causing the net flow generated from left tothe right of the channel. However, the presence of opposingelectric body force results in local reversed flow which remainslimited to the region close to the electrodes.

The impact of unequal mobility coefficients for positive andnegative charges on the electric body force and flow field forξr = 0.35 is depicted in Fig. 9. As discussed earlier, withbr > 1, the positive charge density over the surface of theground electrode surpasses the negative charge density over theHV electrode. For the given configuration, this will enhancethe opposing body force on the edge of the ground electrodeto increase the circulating flow (see Fig. 9b) and lower thegenerated flow.

Figure 10 presents the variation of generated flow withmobility ratio. The immediate drop of the flow rate for br > 1is due to the increase in the opposing electric body force onthe edge of the narrower (ground) electrode. However, the

x*

y*

10 10.5 11 11.5 12 12.5 130

0.5

10.01 0.04 0.08 0.1 0.14 0.18 0.22 0.24 0.26 0.3 0.34 0.38

| f * |

(a)

x*

y*

10 10.5 11 11.5 12 12.5 130

0.5

1

(b)Fig. 9. (a) dimensionless electric body force contours and (b) flow streamlinesξr = 0.35 and br = 2.

1 2 3 4 5 62

4

6

8

10

12

14x 10

−4

br

Q* ne

t

ξr = 0.35, Re

ehd= 1380, C

0 = 0.011

η* = 0.6

η* = 1η* = 2

Fig. 10. Variation of net generated flow with mobility ratio for Asymmetricelectrodes, ξr = 0.35.

course is eventually reversed for larger values of mobilityratios per our earlier discussion. Noticeably, the upward trendstarts at smaller mobility ratios and leads to a slightly largerflow rate for large mobility ratios when the gap betweenthe electrodes expands. This is because the electric fieldencounters less variation across the electrodes edges when thegap is larger yielding in earlier transition to ”pseudo-unipolar”charge distribution over the electrode surface.

Finally, before concluding this study, we present the varia-tion of net flow with mobility ratio for the case of ξr = 2.86where the broken symmetry requires the flow to be generatedfrom right to the left. As observed in Fig. 11, the flow increaseswith the mobility ratio as the direct consequence of promotedbody force on the right edge of the ground electrode. However,the transition to a symmetric distribution over the electrodesurface occurs at much larger mobility ratios since the mobilecharge carriers are now migrating toward the wider electrode.The threshold at which the trend shifts is seen to occur farbeyond br = 6 and therefore is not presented here.

CONF. PRESENTATION SESSION 1: PARTICLES IN FLOWS & FLOW ELECTRIFICATION 6

1 2 3 4 5 6−3

−2.5

−2

−1.5

−1

−0.5x 10

−3

br

Q* ne

t

η* = 2

η* = 1η* = 0.6

ξr = 2.86, Re

ehd= 1380, C

0 = 0.011

Fig. 11. Variation of net generated flow with mobility ratio for asymmetricelectrodes, ξr = 2.86.

IV. CONCLUSIONSThe effect of unequal mobility ratios of positive and neg-

ative charge carriers (br = b+/b− > 1) on the performanceof EHD conduction pumping was investigated. Net flow wasobserved for the case of symmetric electrodes and br > 1which is attributed to the non-symmetric charge distributionover the electrodes. The net flow decreased for large mobilityratios as the migration of negative charges is suppressed toresult in relatively uniform charge and body force distributionover the surface of ground electrode. The impact of chargemobility for asymmetric electrodes was analyzed as well. Withbr > 1, the flow was adversely impacted compared to thecase of identical mobilities when it was generated from theground electrode toward the HV electrode, but it was positivelyaffected by the unequal mobility coefficients when it wasgenerated from the HV electrode toward the ground electrode.

ACKNOWLEDGMENT

This work was financially supported by the NASA Micro-gravity Fluid Physics Program.

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