PHYSICAL REVIEW E 91, 033003 (2015)

Linear stability of a circular Couette flow under a radial thermoelectric body force

H. N. Yoshikawa,* A. Meyer, O. Crumeyrolle, and I. MutabaziLaboratoire Ondes et Milieux Complexes, UMR 6294 CNRS-Universite du Havre, 53, rue de Prony-76058 Le Havre Cedex, France

(Received 23 December 2014; published 3 March 2015)

The stability of the circular Couette flow of a dielectric fluid is analyzed by a linear perturbation theory. Thefluid is confined between two concentric cylindrical electrodes of infinite length with only the inner one rotating.A temperature difference and an alternating electric tension are applied to the electrodes to produce a radialdielectrophoretic body force that can induce convection in the fluid. We examine the effects of superpositionof this thermoelectric force with the centrifugal force including its thermal variation. The Earth’s gravity isneglected to focus on the situations of a vanishing Grashof number such as microgravity conditions. Dependingon the electric field strength and of the temperature difference, critical modes are either axisymmetric ornonaxisymmetric, occurring in either stationary or oscillatory states. An energetic analysis is performed todetermine the dominant destabilizing mechanism. When the inner cylinder is hotter than the outer one, thecircular Couette flow is destabilized by the centrifugal force for weak and moderate electric fields. The criticalmode is steady axisymmetric, except for weak fields within a certain range of the Prandtl number and of the radiusratio of the cylinders, where the mode is oscillatory and axisymmetric. The frequency of this oscillatory mode iscorrelated with a Brunt-Vaisala frequency due to the stratification of both the density and the electric permittivityof the fluid. Under strong electric fields, the destabilization by the dielectrophoretic force is dominant, leadingto oscillatory nonaxisymmetric critical modes with a frequency scaled by the frequency of the inner-cylinderrotation. When the outer cylinder is hotter than the inner one, the instability is again driven by the centrifugalforce. The critical mode is axisymmetric and either steady under weak electric fields or oscillatory under strongelectric fields. The frequency of the oscillatory mode is also correlated with the Brunt-Vaisala frequency.

DOI: 10.1103/PhysRevE.91.033003 PACS number(s): 47.20.Qr, 44.27.+g

I. INTRODUCTION

The Taylor-Couette flow is a prototype system in non-linear physics and it exhibits a rich variety of bifurcationphenomena (see, e.g., Refs. [1,2]). This flow and its variantsare encountered in many applications, concerning the mass,heat, and momentum transfer in rotating machinery and inprocess industries. Geophysical flows in the atmosphere ofrapidly rotating major planets have also been modelled asTaylor-Couette flows, by dividing the spherical shell of a planetinto superimposed coaxial annular layers [3–5].

In diverse applications, the temperature of the fluid indifferentially rotating cylindrical annuli is not uniform. Thisnonuniformity can induce drastic effects on flow regimes.Effects of a radial temperature gradient on the centrifugal insta-bility have been examined experimentally [6–9]. Convectionrolls inclined with respect to the azimuthal direction have beenobserved when the temperature difference was significant.Controversial results have been reported with regard to theeffects of the heating direction: some observed its influence onthe stability; others did not. Although stability analyses havealso been performed [10–13], they often neglected the thermalvariation of the centrifugal force. Recently, Yoshikawa et al.[14] revisited the problem and confirmed that the stability issensitive to the heating direction when the thermal variation ofcentrifugal force is significant: the flow is more stable under anoutward heating (i.e., the inner cylinder is hotter than the outerone) than under inward heating (i.e., the outer cylinder is hotter

*Present address: Laboratoire J.-A. Dieudonne, UMR 7351 CNRS-Universite Nice Sophia Antipolis, Parc Valrose - 06108, Nice Cedex02, France; Email: harunori@unice.fr

than the inner one). This result can be explained by the fact thatthe density stratification is stable (potentially unstable) in thecentrifugal acceleration field under outward (inward) heating.

The dependence of the stability on the heating directioncan also result from other radial thermal body forces thanthe centrifugal force. Tagg and Weidman [15] considereda magnetic fluid in a vertical Taylor-Couette system withonly the inner cylinder rotating. The thermal variation of themagnetization leads to a thermomagnetic body force density,which can be viewed as a thermal buoyancy force associatedwith a radial magnetic effective gravity. The authors analyzedthe linear stability of a base flow driven both by the innercylinder rotation and by the differential heating under theEarth’s gravity and examined the behavior of the criticalflow parameters as a function of the Grashof number. It wasfound that outward heating destabilized the flow significantlywhen the magnetic body force was of the same order ofmagnitude as the Earth’s gravity force. When the heatingdirection was inverted, the thermomagnetic force yielded animportant stabilizing effect.

Another means to produce a radial thermal body force is analternating radial electric field. The application of an electricfield E to a dielectric fluid results in the electrohydrodynamic(EHD) body force. Its density f is given by

f = ρe E − 1

2E2∇ε + ∇

[1

2ρ

(∂ε

∂ρ

)T

E2

], (1)

where ρe is the free charge density, E is the strength ofthe electric field (E = |E|), and ρ and ε are, respectively,the density and the electric permittivity of the fluid atthe local temperature T [17,18]. The first term, called theelectrophoretic force, reflects the Coulomb forces on the free

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YOSHIKAWA, MEYER, CRUMEYROLLE, AND MUTABAZI PHYSICAL REVIEW E 91, 033003 (2015)

charges. The second term, called the dielectrophoretic (DEP)force, arises from the differential polarization of the fluid. Thethird term represents electrostriction, which can be lumpedwith the pressure in the momentum equation and would notaffect the fluid motion, unless the fluid is compressible orhas mobile boundaries. When the electric field is static oralternating with a low frequency, the electrophoretic forcehas dominant effects on the fluid motion. However, when thefrequency f of the field is high compared to the inverse ofthe fluid relaxation times τν = d2/ν and τκ = d2/κ (d, thecharacteristic length scale of the flow; ν and κ , the kinematicviscosity and the thermal diffusivity of the fluid, respectively),the electrophoretic force is filtered out by viscous and thermalrelaxation and the DEP force becomes dominant.

The DEP force can arise from a temperature gradientin fluid, as the permittivity varies with the temperature.Its variation can be modelled by a linear relationship ε =εref [1 − e (T − Tref)], where εref is the permittivity at referencetemperature Tref. The coefficient e takes a positive value of theorder of 10−3–10−1 K−1 (e.g., Table I). Substituting ε by thelinear relationship, the DEP force can be written as

−1

2E2∇ε = −eθ∇ εrefE

2

2+ ∇

(eθεrefE

2

2

), (2)

where θ is the temperature deviation: θ = T − Tref. Castingthe first term into the form of −αθ ge, we can regard it asthermal buoyancy associated with an electric effective gravityge [19–21]:

ge = e

αρ∇ εrefE

2

2, (3)

where α is the coefficient of thermal expansion. The secondterm in Eq. (2) can be lumped with the pressure term in the mo-mentum equation. In analogy with the thermal Archimedeanbuoyancy, the thermoelectric buoyancy (−αθ ge) will inducea thermal convection when the following Rayleigh number L

based on ge (= |ge|) becomes larger than a critical value Lc:

L = αθged3

νκ. (4)

Here, θ is the temperature difference in the fluid. When aradial electric field is applied to a fluid in a cylindrical annulus,the electric gravity ge and associated thermoelectric buoyancyare directed radially.

The analogy of the thermomagnetic or thermoelectricbuoyancy forces with the thermal Archimedean buoyancy is

of great interest in geo- and astrophysics, where convectiveflows and stratified shear flows in the central gravity field areof primary importance. Realization of laboratory experimentswith an artificial radial gravity is a key to advance theunderstanding of these flows. Some attempts have been madewith the thermoelectric artificial gravity in annular geometry[20,22,23] and in spherical geometry [24–28]. An experimentwith the thermomagnetic artificial gravity has also beenreported in annular geometry [29].

Theoretical investigations on the thermal convection in theelectric gravity have been carried out in different capacitorgeometries under microgravity conditions. In the annular ge-ometry with stationary cylinders, Chandra and Smylie [20] andTakashima [30] examined the stability of the conductive stateagainst axisymmetric disturbances. The considered geometrieswere small gap annuli: the ratio η of inner to outer cylinderradii was larger than 0.9. Recently, Malik et al. [31] andYoshikawa et al. [21] performed linear stability analyses forannuli with arbitrary gap and for arbitrary perturbations. Theiranalyses showed that the critical mode is a steady spiral.Treating both inward and outward heating, the latter authorsinvestigated exhaustively the effects of the radius ratio η andthe thermoelectric parameter γe = eθ . The direction of theelectric gravity was shown to depend on these two parametersand can be centripetal, centrifugal, or centripetal in an innerlayer and centrifugal in the outer layer. Instability can occuronly either in outward heating where the electric gravity iscentripetal or in inward heating when the electric gravity iscentrifugal. The authors performed an energetic analysis andshowed that the temperature disturbances induce a perturbationcomponent g′

e of the electric gravity, which plays a keyrole in the stabilization of the flow for large values of theradius ratio. This stabilizing effect has also been observedin the analysis on the thermoelectric convection in the planegeometry [32] and explained its quantitative differences fromthe Rayleigh-Benard convection in critical flow parameters aswell as in the heat transfer by developed convective flows.

The circular Couette flow with the inner cylinder rotatingand under the action of thermoelectric body force has beenconsidered by Stiles and Kagan [33]. Their study was focusedon axisymmetric steady perturbations in a small gap geometry.Only outward heating was considered. They obtained steadycritical modes and found that the critical Taylor numberdropped effectively with the temperature difference.

In the present paper, we report a linear stability analysisof the circular Couette flow to arbitrary perturbations when

TABLE I. Properties of some dielectric liquids.

Liquid ν (10−6 m2/s) ρ (kg/m3) α (10−3 K−1) εr e (10−3 K−1) Pr α/e

Acetonitrile 0.439 777 1.38 36.0 155 4.7 0.00890Nitrobenzene 1.52 1198 0.83 34.9 188 19 0.00441Acetone 0.386 785 1.43 19.1 86 4.2 0.0166Chlorobenzene 0.686 1101 0.985 5.61 15.7 7.9 0.0627Cyclohexane 1.17 774 1.22 2.02 1.60 14 0.763Silicone oila 5.0 920 1.08 2.70 1.065 59.9 1.01Silicone oilb 99.2 968 0.96 2.73 3.2 910 0.3

aBaysilone M5.bAfter the properties given in Ref. [16].

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LINEAR STABILITY OF A CIRCULAR COUETTE FLOW . . . PHYSICAL REVIEW E 91, 033003 (2015)

the flow is under the thermoelectric body force. Both inwardand outward heatings are considered. The analysis focuseson the centripetal configuration of the electric gravity, i.e.,the case that is the most relevant to the application in geo-and astrophysics, while some results will also be given forthe other configurations (Sec. V C). Effects of the thermalvariation of the centrifugal force on the stability are examinedwith varying the thermal expansion parameter γa = αθ . Wewill show that, depending on flow parameters, critical statesoccur either steady or oscillatory and as either axisymmetricor nonaxisymmetric modes.

In Sec. II, a mathematical model is presented with the con-ductive base state. Section III is devoted to the formulation ofthe linear stability problem. Critical values of flow parametersand corresponding critical states will be described in Sec. IV:first for a particular case of γa = 0 and then for more generalcases with nonzero γa . The variation of the critical parameterswith the radius ratio and the Prandtl number is considered.Eigenfunctions of different states are presented. Section V isdevoted to analysis of the critical modes: a focus is made onthe energy transfer process to critical modes in different statesand on the origin of the critical frequencies. Section VI givesour concluding remarks.

II. MATHEMATICAL MODEL

We consider a Taylor-Couette system that confines adielectric fluid in the gap d between two concentric cylindricalelectrodes of infinite length (Fig. 1). The radii of the innerand outer electrodes are R1 and R2 (= R1 + d), respectively.The inner cylinder rotates at angular velocity and it iskept at temperature T1, while the outer one is at rest and itis maintained at temperature T2. The annulus is subjected toboth a temperature difference θ = T1 − T2 and an alternatingelectric tension

√2�0 sin(2πf t). The flow is under the action

of the centrifugal force due to the cylinder rotation and thethermoelectric body force due to the electric gravity [Eq. (3)].

A. Governing equations

We have adopted the electrohydrodynamic Boussinesqapproximation [34], which consists in keeping the fluid

FIG. 1. Geometrical configuration of the problem.

properties independent of temperature, except in the termsthat are responsible for instability mechanisms. These arethe centrifugal force and the dielectrophoretic force. Thegoverning equations are the conservation laws of the mass, themomentum, and the energy and the Gauss’ law for electricity;they can be written as follows:

∇ · u = 0, (5)

∂ u∂t

+ u · ∇u = −∇π + ν∇2u − αθ (ge + gc) , (6)

∂θ

∂t+ u · ∇θ = κ∇2θ, (7)

∇ · (ε E) = 0 with ε = ε2 (1 − eθ ) and E = −∇φ,

(8)

where the electrostatic potential φ and the following general-ized hydrodynamic pressure π have been introduced:

π = p

ρ− eθεrefE

2

2ρ− 1

2

(∂ε

∂ρ

)T

E2. (9)

The reference temperature is set at T2 so that θ = T − T2

and εref = ε2. The last term of the Navier-Stokes Eq. (6)represents two thermal buoyancies: one from the electricgravity and the other from the centrifugal acceleration gc givenby

gc = v2

rer , (10)

where v is the azimuthal velocity and er is the radial unitvector. We call the term (−αθ gc) the centrifugal buoyancy inorder to emphasize its origin in the centrifugal force, althoughit can be either centrifugal or centripetal depending on theheating direction.

We have assumed the electroneutrality of the fluid in Eq. (8).This assumption is valid when the frequency f of the appliedelectric field is high enough to neglect the motion of freecharges in the fluid during a period f −1 and when the size ofthe fluid system is large compared to the electric double layeron the electrodes [32].

We have also assumed that f � τ−1ν ,τ−1

κ for the dominanceof the DEP force over the electrophoretic force. Underthis assumption, only the time-averaged component of thethermoelectric buoyancy can affect the fluid motion and we canreduce the problem with an a.c. electric field to an equivalentproblem with an effective static field. In this time-averageddescription, the boundary conditions on the electrodes read

u = R1 eϕ, θ = θ, φ = �0 at r = R1, (11)

u = 0, θ = 0, φ = 0 at r = R2. (12)

The time-averaged description was found to predict success-fully the behavior of dielectric fluids in experiments [35].A theoretical investigation on the influence of the frequencysuggests that the description is correct when 2πf � 100 τ−1

ν

[36]. This condition was met in an experiment performed inannular geometry [22] with a silicone oil of viscosity ν =10 mm2/s. For the gap width d = 25 mm and the frequencyof applied electric tension f = 50 Hz, the viscous relaxationtime is τν = 62.5 s so that f � 100τ−1

ν = 1.6 Hz.

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YOSHIKAWA, MEYER, CRUMEYROLLE, AND MUTABAZI PHYSICAL REVIEW E 91, 033003 (2015)

B. Base state

For a small temperature difference and a small rotationvelocity, the state of the system will respect its symmetry intime and in space. The velocity U , the temperature �, andthe electric potential � are stationary and depend only on theradial coordinate r: U = V (r)eϕ , � = �(r), and � = �(r).The advective transport vanishes (u · ∇ = 0) so that the basestate has the velocity profile of a circular Couette flow and thetemperature and the electric fields are decoupled from the flowvelocity. From Eqs.(6)–(12), one finds

U(r) =V (r)eϕ with V (r) =

1 − η2

(R2

1

r− η2r

), (13)

�(r) = θ log (r/R2)

log η, �(r) = �0 log (1 − γe�/θ )

log (1 − γe),

(14)

where η = R1/R2. The corresponding electric gravity is [21]Ge = −Geer , with

Ge = eε2�20

αρ(log η)2r3· γ 2

e [1 − γe(�/θ + 1/ log η)]

[log(1 − γe)]2(1 − γe�/θ )3. (15)

The second factor in the last term modifies an inverse cube law(Ge ∝ 1/r3) as a consequence of a competition between theeffects of the geometry curvature and the thermal stratificationof the electric permittivity. This competition brings a variety ofbehavior [21]: the electric gravity is centripetal (i.e., Ge > 0)when γe > log η; it is centrifugal (i.e., Ge < 0) when γe <

log η/(1 + log η); it changes the direction inside the gap whenlog η/(1 + log η) < γe < log η.

III. STABILITY ANALYSIS

We consider the stability of the base stateEqs. (13)–(14) against arbitrary infinitesimal perturbations(u′,v′,w′,π ′,θ ′,φ′), where (u′,v′,w′) are the radial, azimuthal,and axial components of the perturbation velocity, θ ′is the perturbation temperature, π ′ is the perturbationpressure, and φ′ is the perturbation electrostatic potential.We linearize the governing Eqs. (5)–(8), and expand theperturbations into normal modes: (u′,v′,w′,π ′,θ ′,φ′) =(u,v,w,π ,θ ,φ) exp[st + inϕ + ikz]. A tilde over a quantityindicates its complex amplitude. The perturbation growth rates may be complex: s = σ + iω, where ω is the frequency ofthe mode. The axial wavenumber k is a real number, as theperturbations are bounded at infinity. The azimuthal modenumber n takes only integer values.

Introducing the scales d of length, d2/ν of time, ν/d ofvelocity, θ of temperature, and �0 of electric potential,the linearized governing equations become dimensionless andread

1

rD(ru) + inv

r+ ikw = 0, (16)

su + inV

ru = − Dπ + ∇2u − u

r2+ 2

V v

r− 2in

r2v

− L

Pr(−Geθ + �ge,r ) − γa

r(θV 2 + 2�V v),

(17)

sv + inV

rv = − in

rπ + ∇2v − v

r2− uV

r

+ 2in

r2u − (DV )u − L

Pr�ge,ϕ, (18)

sw + inV

rw = −ikπ + ∇2w − L

Pr�ge,z, (19)

sθ + inV

rθ = − (D�) u + 1

Pr∇2θ , (20)

(1 − γe�) ∇2φ − γeD�Dφ − γeD�

(D+ 1

r

)θ

− γe(D2�)θ = 0, (21)

where D = d/dr and ∇2 = D2+ 1rD− n2

r2 − k2. The Taylornumber Ta and the Prandtl number Pr are defined as:

Ta = R1 d

ν

√d

R1, Pr = ν

κ. (22)

In the electric Rayleigh number L defined by Eq.(4), we choosethe base electric gravity at the middle of the gap Ge,0 =Ge(r0) as representative one, where r0 is the dimension-less midgap radius: r0 = (1 + η)/2(1 − η). For a centripetalelectric gravity field (Ge > 0), the Rayleigh number L ispositive in outward heating and negative in inward heating.The perturbation electric gravity (ge,r ,ge,ϕ,ge,z) is relatedto the perturbation electric field resulting from temperaturedisturbances. This perturbation gravity as well as the basegravity have been nondimensionalized with the scale Ge,0. Theexplicit expressions of the dimensionless gravities are given inRef. [21].

The linearized Eqs. (16)–(21) supplemented with thehomogeneous boundary conditions (u,v,w,θ ,φ) = 0 at thecylinder surfaces form an eigenvalue problem representedformally by

F (s,Ta,L,n,k,η,Pr,γe,γa) = 0. (23)

The Taylor number intervenes in the problem through the baseflow V (∝ Ta). To determine the eigenvalue, Eqs. (16)–(21)are discretised by a Chebyshev spectral collocation method andEq. (23) is transformed into a generalized eigenvalue problemin matrix form. The latter is solved by the QZ decomposition.The highest order of considered Chebyshev polynomials is setfrom 15 to 28 to ensure the convergence, depending on theradius ratio.

IV. RESULTS

Eigenvalues s of the linear stability problem formu-lated in Sec. III are computed for a given parameter set(L,n,η,Pr,γe,γa) at different (k,Ta). A marginal stability curveTa = Ta(k) is determined by seeking the condition that thelargest real part σ of s changes its sign. The minimum ofa marginal curve gives the instability threshold (kn,Tan) ofthe azimuthal mode n. The smallest value of { Tan} and thecorresponding values of (n,kn,ωn) give the critical parameters(Ta,kc,nc,ωc). The critical wavenumber qc of vortices iscomputed on the median cylindrical surface r = r0 from the

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LINEAR STABILITY OF A CIRCULAR COUETTE FLOW . . . PHYSICAL REVIEW E 91, 033003 (2015)

-2000 -1000 0 1000 20000

100

200

300

400

500

Tac

L

η = 0.1

0.3

0.50.9

0.2

(L, Ta )(OA)

(L , Ta )+ +

(a)Regime I

Regime II

Regime III (ONA)

(SA)

- -

Inward heating Outward heating

-2000 -1000 0 1000 20002.4

2.6

2.8

3

3.2

3.4

3.6

qc

OA SA

η = 0.1

0.20.3 0.5 0.9

ONA

(b)Outward heatingInward heating

-2000 -1000 0 1000 20000

1

2

3

4

5

6

ωc

L

(c) η = 0.1

0.2

0.30.5

0.9

η =

0.1

(nc =

1)

0.2

(nc =

1)

0.2

(nc =

2)

0.3

(nc =

1)

0.3

(nc =

2)

| |

FIG. 2. Critical parameters for different radius ratios η (Pr = 10): Taylor number (a), wavenumber of vortices (b), and frequency (c). Thecentrifugal buoyancy is neglected (γa = 0). The thermoelectric parameter is fixed at γe = −0.01 and 0.01 for inward and outward heating,respectively. The critical modes are oscillatory axisymmetric (OA), steady axisymmetric (SA), or oscillatory nonaxisymmetric (ONA). In panel(a), the codimension-2 points are indicated by open and filled circles.

pair (kc,nc) by qc = (k2c + k2

ϕ,c)1/2, where kϕ,c = nc/r0 is theazimuthal wavenumber. We present these critical parametersfor different values of L in two cases: a particular case wherethe centrifugal buoyancy is neglected (i.e., γa = 0) and thegeneral case with the centrifugal buoyancy (i.e., γa �= 0). Inboth cases, we shall distinguish different parameter regimes.The behavior of the critical parameters with varying Pr and theeigenfunctions in the different regimes will also be presented.

A. Case of γa = 0

The case of zero thermal expansion parameter (γa = 0)corresponds to situations where the centrifugal buoyancy issmall compared to the thermoelectric force. This assumptionis reasonable for fluids with a small ratio γa/γe = α/e so thatthe results will be relevant to polar liquids with large coefficiente such as the acetonitrile and the nitrobenzene (Table I).

For a given working fluid (i.e., a fixed value of Pr) in agiven annulus (i.e., a fixed value of η), the critical parametersare determined by varying the electric Rayleigh number L

(i.e., varying the electric tension applied to the electrodes) asshown in Fig. 2. The temperature difference is fixed throughthe thermoelectric parameter γe. This is set at γe = 0.01 foroutward heating where L > 0 and at γe = −0.01 for inwardheating where L < 0. Yoshikawa et al. [21] has shown that, forannulus with η < 0.9, the critical parameters are insensitive toγe as far as γe � 0.01.

When there is no applied electric field (L = 0), the eigen-value problem, Eqs. (16)–(21), reduces to that of the isothermalTaylor-Couette flow. The critical mode is steady axisymmetric(ωc = 0, nc = 0). The known critical wavenumber qc = q isoth

c

and the known critical Taylor number Tac = Taisothc [37] are

recovered. When there is no rotation, i.e., when Ta = 0, theeigenvalue problem, Eqs. (16)–(21), reduces to the problemof the stability of a stationary conductive state under theDEP force. In the centripetal electric gravity, the heating

should be outward for instability [21] so that our system canreach the critical condition only at a positive L for Ta = 0[Fig. 2(a)].

For inward heating (L < 0), the critical Taylor numberdecreases with increasing L [Fig. 2(a)]. Under strong electricfields (i.e., large negative L), the critical mode is oscillatoryaxisymmetric (Regime I). The critical wavenumber variesslightly [Fig. 2(b)]. The frequency |ωc| decreases with L andbehaves as |ωc| ∝ |L|−1/2 at large |L| [Fig. 2(c)]. Above avalue L = L−, the critical mode becomes stationary and thewavenumber jumps to a larger value around q isoth.

c [Fig. 2(b)].This stationary axisymmetric mode is found for weak electricfields (L ∈ [L−, L+]) for both inward and outward heating(Regime II).

When the Taylor number is smaller than Ta+, the electricRayleigh number decreases with decreasing Ta and convergesto the value for stationary electrodes [Fig. 2(a)]. The criticalmode is nonaxisymmetric and oscillatory (Regime III), itsfrequency decreases and vanishes for stationary cylinders(Ta = 0). The critical mode number nc increases with de-creasing Ta. Discontinuities found in the behavior of qc andωc reflects changes in nc [Figs. 2(b) and 2(c)].

The parameter sets (L−,Ta−) and (L+,Ta+) which separatestates of different stability nature are called codimension-2points. At the point (L−,Ta−), both oscillatory and steadyaxisymmetric modes are critical; at the point (L+,Ta+), bothsteady axisymmetric and oscillatory nonaxisymmetric modesare critical. The values of L− and Ta− decrease with η; thevalue of L+ increases with η, while Ta+ approaches zero.Between these codimension-2 points, i.e., in Regime II, thecritical mode consists of a steady axisymmetric vortex patternwith a wavenumber close to its value of isothermal Taylorvortices q isoth.

c . Thermoelectric effects are not strong enoughto change the nature of the Taylor instability, even thoughthe critical Taylor number becomes much smaller than itsisothermal value at large L. In the instability in Regime III, the

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YOSHIKAWA, MEYER, CRUMEYROLLE, AND MUTABAZI PHYSICAL REVIEW E 91, 033003 (2015)

-1000 -500 0 500 1000 15000

20

40

60

80

Tac

L

Regime I

Regime II

Regime IV (OA)

Regime II

(L, Ta )

(L, Ta )- -(a)

| |γa

0.0010

0.01

Inward heating Outward heating

(OA)

(SA) (SA)

**

-+

1500 16000

0.2

0.4

0.6

0.8

1

(L, Ta )++

Regime III

Ta

L

c

(ONA)

-1000 -500 0 500 1000 1500

3.04

3.08

3.12

3.16

γa| |

0.010.0010

1

2

3

4

n = 0n = 0

qc

(b)Outward heatingInward heating

00

-1000 -500 0 500 1000 15000

0.20.40.60.8

11.21.4

ωc

L

(c)00.0010.01

| |γa

n = 423

1

| |

FIG. 3. Critical parameters for different thermal expansion parameters γa (Pr = 60, η = 0.5): Taylor number (a), wavenumber of vortices(b), and frequency (c). The thermoelectric parameter is fixed at γe = −0.01 and 0.01 for inward and outward heating, respectively. The criticalmodes are oscillatory axisymmetric (OA), steady axisymmetric (SA), or oscillatory nonaxisymmetric (ONA).

nonaxisymmetry of the thermoelectric critical mode at Ta = 0is conserved, while the critical mode is oscillatory.

B. General case (γa �= 0)

The critical parameters depend on the thermal expansionparameter γa , in particular under weak electric field (i.e., atsmall |L|). At L = 0 the behavior of the critical parametersbecomes discontinuous (Fig. 3). Indeed, passing throughzero from negative to positive L, we change the imposedtemperature difference T from a finite negative value toa finite positive value. As the centrifugal buoyancy breaksthe symmetry of the stability of the Taylor-Couette flowwith respect to the heating direction [14], we observe thesediscontinuities which are absent when γa = 0.

In inward heating, the qualitative behavior of the criticalparameters are the same as in the case of γa = 0. The criticalmode is oscillatory axisymmetric when L < L− (Regime I)and steady axisymmetric when L ∈ [L−, 0]. We will refer tothe latter regime as Regime II−, because the critical modeshave the same nature as those in Regime II. In Regime II−

the critical Taylor number is lower than in the case of γa = 0.The codimension-2 point (L−,Ta−) separating Regimes I andII− shifts leftward [Fig. 3(a)] when the thermal expansionparameter becomes important.

In outward heating, a new regime (Regime IV) is observedfor weak electric fields (L ∈ [0, L∗]) and large values ofthermal expansion parameter γa (e.g., the case of γa = 0.01in Fig. 3). The critical mode is oscillatory axisymmetric.Regime IV does not exist when γa is small (e.g., the caseof γa = 0.001 in Fig. 3). The critical mode becomes steadyaxisymmetric (Regime II+) beyond a new codimension-2 point(L∗,Ta∗), recovering its nature in Regime II. Further increaseof L results in the transition from the steady axisymmetric tooscillatory nonaxisymmetric critical modes (Regime III) at thecodimension-2 point (L+,Ta+) as in the case of γa = 0.

C. Effects of the Prandtl number Pr

For stationary electrodes, the critical modes are steady andthe Prandtl number has no influence on the critical parameters[21]. This independence can be confirmed by scaling the timeby thermal diffusion time d2/κ , the velocity by κ/d, andthe generalized pressure by κν/d2. The resulting governingequations do not involve the Prandtl number for steady modes.

In the case of γa = 0, the similar argument leads to theindependence of the critical parameters from Pr as long as thecritical modes are steady axisymmetric. Therefore, the Prandtlnumber has no influence on critical values of the Taylor numberTa and the wavenumber q in Regime II. When the critical modeis oscillatory, the critical parameters vary with Pr. In RegimeI, the critical Taylor number decreases significantly when thePrandtl number Pr is increased [Fig. 4(a)]. We found that thecodimension-2 points, (L−,Ta−) and (L+,Ta+), are affectedby Pr. The value of L− increases significantly and is inverselyproportional to Pr at large Pr: L− ∝ −1/Pr [Fig. 4(b)].

In the case of γa �= 0, the critical values of the Taylornumber depends on Pr even for steady axisymmetric modes.In inward heating, the critical Taylor number decreases bothin Regime I and in Regime II− with increasing Pr [Fig. 4(a)].The coordinate L− separating these two regimes is smallerthan in the case of γa = 0 [Fig. 4(b)]. It increases with thePrandtl number at small Pr, reaches a maximum value, andthen decreases for large values of Pr. In outward heating,the critical Taylor number in Regime II+ increases with thePrandtl number [Fig. 4(c)]. In contrast, it is insensitive to Prin Regime IV, when this regime exists. In fact, there existsa minimum value of Prmin below which Regime IV does notexist. The coordinate L∗ of the codimension-2 point betweenRegime IV and Regime II+ vanishes at Prmin and increases forPr > Prmin. The value of Prmin depends on the radius ratio η

as well as on the thermal expansion parameter γa [Fig. 4(d)].The electric Rayleigh number L+ of the codimension-2 pointbetween Regime II+ and Regime III is not sensitive to the

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-1400-1200-1000 -800 -600 -400 -200 0

58

60

62

64

66

68

70

72

74

76

Tac

L

Prγ a0 10 30 100

- 0.01 10 30 100

(L, Ta )- -(a)

0 20 40 60 80 100 120

Pr

-500

-400

-300

-200

-100

0

0.1 0 - 0.001 - 0.010.9 0 - 0.001 - 0.01

γ aη

L-

(b)

0 200 400 600 800 1000120014001600

L

0

10

20

30

40

50

60

70

Pr = 10

Tac

3060

100

(L, Ta )* *

1500 1600 0

0.5

1

1.5

2

L

Tac (L, Ta )+ +

(c)

0 20 40 60 80 100 120

Pr

0

100

200

300

400

500

0.1 0.001 0.010.5 0.001 0.010.9 0.001 0.01

η γa

L*

Prmin

(d)

FIG. 4. Influence of the Prandtl number Pr on the critical parameters: (a) critical Taylor number (η = 0.5), (b) electric Rayleigh numberat the codimension-2 point (L−,Ta−), (c) critical Taylor number (η = 0.5, γa = 0.01), and (d) electric Rayleigh number at the codimension-2point (L∗,Ta∗). For all the results [(a)–(d)], the thermoelectric parameter is set at γe = −0.01 for inward heating and at γe = 0.01 for outwardheating.

variation of Pr, while Ta+ approaches zero with increasing Pr[Fig. 4(c)].

D. Eigenfunctions

The eigenfunctions of different states are illustrated in Fig. 5(first column). The vortices of the critical modes are centeredin the gap for large values of η, while they are shifted inwardwhen η is small. The phase of the temperature perturbationθ ′ relative to the velocity field behaves differently in differentregimes. In steady regimes, the phase is locked: in Regime II−,the temperature θ ′ is in the antiphase of the radial velocity u′; inRegime II+, they are in phase with each other. This means thatin these regimes the flow goes from hot to cold walls inside thepositive perturbation temperature zones, similar to the ordinarythermal convection. In the oscillatory regimes, the phase varieswith the Rayleigh number L. In Regime I, the temperatureadvances (delays) to u′ for positive- (negative-) ωc modesby a phase difference larger than 90◦. The phase differenceincreases with L and reaches 180◦ at the coodimension-2 point(L−,Ta−). In Regime IV, the phase difference is smaller than90◦. It decreases with L and becomes zero at the codimension-2 point (L∗,Ta∗).

V. DISCUSSION

In order to get a better understanding of the roles playedby different forces in the flow stability, we have examinedrelative importance of the corresponding processes in the

energy transfer from the base to perturbation states. We havealso analyzed the frequencies of the oscillatory critical modes,inferring their scales from possible origins of the oscillation.We will make some comments on the flow stability undera centrifugal electric gravity field and the stability when theelectric gravity changes its direction inside the gap.

A. Energy analysis

An evolution equation that governs the density of kineticenergy of the perturbation flow can be derived from thelinearized Navier-Stokes Eqs. (17)–(19) by multiplying themby u′, v′, and w′, respectively, and adding the results. Afterintegrating over the whole gap, averaging over a wavelengthand, for oscillatory modes, averaging over an oscillationperiod, one will find the following equation:

2σK = WTa + WC + WBG + WPG − Dv, (24)

where K is the kinetic energy of the perturbation. The termWTa is the power performed by the centrifugal force, whichis responsible for the Taylor instability, and the term WC

is the power input by the centrifugal buoyancy. The powersWBG and WPG represent the inputs from the thermoelectricbuoyancies associated with the basic electric gravity Ge andthe perturbation electric gravity g′

e, respectively. The lastterm Dv is the viscous energy dissipation rate. At instabilitythresholds (σ = 0), it balances completely the sum of the otherpower terms.

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FIG. 5. (Color online) Eigenfunctions at critical conditions in different regimes (η = 0.5, Pr = 60). The first, second, third, fourth, andfifth columns represent, respectively, (1) perturbation velocity (arrows) and perturbation temperature; (2) the local power wTa; (3) buoyancyassociated with the centrifugal gravity (arrow) and wC ; (4) buoyancy associated with the base electric gravity Ge (arrows) and the powerwBG; (5) buoyancy due to the perturbation electric gravity g′

e (arrow) and the power wPG. The powers are normalized by twice of the kineticenergy K . In the first column, temperature fields are shown without any numerical scale, as they are perturbation fields determined by alinear theory. Black curves are the contours of θ ′ = 0. The values of electric Rayleigh number L used for computation are indicated onthe L axis.

Each term in Eq.(24) is obtained from corresponding localand instantaneous power:

K = 〈K〉 =⟨ |u|′2

2

⟩, (25)

WTa = 〈wTa〉 =⟨−u′v′

(DV − V

r

)⟩, (26)

WC = 〈wC〉 =⟨−γau

′Vr

(θ ′V + 2�v′)⟩ , (27)

WBG = 〈wBG〉 =⟨L

Prθ ′Geu

′⟩, (28)

WPG = 〈wPG〉 =⟨− L

Pr�g′

e · u′⟩, (29)

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Dv = 〈dv〉 = 〈2e : e〉. (30)

The rate-of-strain tensor is designated by e = [∇u′ +(∇u′)T ]/2. Angle brackets mean the integration and averagingoperations.

The power wTa of the centrifugal force is positive (Fig. 5,second column). Since DV − V/r < 0, it implies that theradial and azimuthal perturbation velocities, u′ and v′, arein phase with each other. The maxima of wTa are located at thepoints where the magnitude of the radial velocity is maximum.The power wC of the centrifugal buoyancy depends on thephase of the temperature perturbation with respect to u′, aswell as on the heating direction (Fig. 5, third column). In total,it has positive and negative effects in inward and outwardheating, respectively. The base electric gravity, in contrast,inputs net negative and positive power for inward and outwardheating, respectively (Fig. 5, fourth column), as the electricgravity is opposed to the centrifugal acceleration. The powerwPG of the perturbation electric gravity is small comparedto other energy transfer terms. It is concentrated in an innerregion, while it almost vanishes in the outer region of thegap.

Figure 6 shows the behaviors of different terms in Eq.(24).In all the regimes except in Regime III, the power WTa isthe main contribution to the energy transfer from the basestate to perturbations. As inferred from local power inputs(Fig. 5), the power WC by the centrifugal buoyancy is positiveand negative in inward and outward heating, respectively,and the power WBG by the thermoelectric buoyancy has theopposite sign to WC . In Regimes I and IV where the criticalmodes are oscillatory, the net transfer WC + WBG by these twothermal buoyancies is negative, while it is positive in steadyregimes, i.e., Regimes II− and II+. Increasing L in outwardheating, the power WBG becomes the most effective energytransfer process even in Regime II+ at large L and dominatescompletely other transfer processes in Regime III. The powerWPG is always negative as found in the case of stationaryelectrodes [21], confirming the stabilizing effect of the pertur-

FIG. 6. Power terms in Eq. (24) normalized by twice of the kineticenergy K (η = 0.5, Pr = 60; γe = γa = 0.01 for outward heating,γe = γa = −0.01 for inward heating).

bation electric gravity. Its contribution remains smaller thanWTa or WBG.

B. Critical mode frequencies

According to the energetic analysis (Fig. 6), the thermo-electric buoyancy dominates other energy transfer processes inRegime III. On the other hand, the nonaxisymmetric nature ofthe critical modes obtained in the case of stationary electrodespersists in this regime. These observations suggest that theazimuthal base flow transports nonaxisymmetric perturbationsamplified by the thermoelectric buoyancy to induce the criticalmode oscillation. Indeed, the azimuthal phase velocity cφ =−ωc/nc of the perturbations is proportional to the Taylornumber [Fig. 7(a)]. The constant of the proportionality dependson the radius ratio, but it does not depend on the Prandtlnumber.

In inward heating (L < 0), the centripetal electric gravityis directed from outer hot fluid zone to inner cold one.Thermoelectric buoyancy has then a stabilizing effect: it tendsto bring back a deviated fluid parcel to its equilibrium radialposition in the base state. In outward heating, the centrifugalacceleration is directed from hot to cold fluid zones so thatthe centrifugal buoyancy is stabilizing. As mentioned in theprevious section, the net contribution of these two buoyanciesis negative in Regimes I and IV (Fig. 6). This implies thatthe resulting force from the thermal buoyancies acts as arestoring force and would bear a close relation with the criticaloscillatory modes obtained in these regimes.

In fluid with a stable density stratification, the balancebetween the fluid inertia and a restoring force results in wavepropagation. For an oscillatory motion of a fluid parcel with afrequency N and with a small displacement ζ , its inertia perunit volume is estimated by ρN2ζ . The restoring force due tothe radial thermal buoyancy is given by −[dρ(�)/dr] (Ge −Gc)ζ , where Gc is the centrifugal acceleration of the baseflow: Gc = V 2/r . The sound speed has been assumed largeenough to neglect the density variation associated with anisentropic process [38]. The balance between the inertial andrestoring effects yields N2 = (−Ge + Gc) d(log ρ)/dr . Thisis a generalized Brunt-Vaisala frequency, which includes theeffects of both the thermoelectric and centrifugal buoyancies.We can cast this frequency into the following dimensionlessform by scaling it by the viscous time τν :

N =√

2 (1 − η)

(1 + η) log η

[L

Pr− η3 (3 + η)2

2 (1 + η)5γaTa2

], (31)

where Ge and Gc have been estimated at the midgap of theannulus. The right-hand side gives a real value only whenthe sum of the terms in the square brackets is negative:i.e., for inward heating, when the thermoelectric buoyancydominates over the centrifugal one; for outward heating,when the centrifugal buoyancy is the dominant force. Wehave plotted the critical frequency ωc as a function of N fordifferent values of Pr and γa for Regimes I and IV[Fig. 7(b)].When the critical frequency ωc becomes larger than the unity,i.e., when the viscous dissipation is not significant during anoscillation period, the critical frequency is given by ωc = N/3for η = 0.9. The factor of proportionality increases slightly

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YOSHIKAWA, MEYER, CRUMEYROLLE, AND MUTABAZI PHYSICAL REVIEW E 91, 033003 (2015)

0.1 1 10

0.01

0.1

1

cφ

Tac

cφ

Tac

η = 0.4, Pr = 10

η = 0.5, Pr = 0.1Pr = 1

η = 0.6, Pr = 10Pr = 10

η = 0.8, Pr = 0.1

Pr = 10Pr = 1

10

0 1 2 3 4 50

0.4

0.8

1.2

(a)

0.1 1 10

N

0.01

0.1

1

10

ωcωc = N

γaRegime I

Pr 30 0 30 -0.01 60 -0.01 90 -0.01120 0120 -0.01

30 0.01 60 0.01100 0.01

Pr γaRegime IV(b)

FIG. 7. Frequencies of the oscillatory critical modes obtained in Regimes I, III, IV: (a) phase velocity cφ = −ωc/nc in Regime III (γe = 0.01,γa = 0) and (b) frequency ωc in Regimes I and IV (η = 0.9; γe = −0.01 in Regime I, γe = 0.01 in Regime IV).

with the radius ratio η, but it is insensitive to the Prandtlnumber. These observations suggest that oscillatory modesbecome critical in Regimes I and IV by receiving energy fromthe base flow through a resonant mechanism.

C. Stability in other gravity configurations

We have found for centripetal electric gravity fields that thebehavior of the critical mode is closely related to the nature ofthe thermoelectric and centrifugal buoyancies. This is also thecase when the electric gravity is centrifugal, i.e., when Ge < 0.As this configuration can occur only in inward heating (γe <

0), the accelerations Ge and Gc are both directed from coldto hot fluid zones. The associated thermal buoyancies are bothdestabilizing. The critical mode is hence steady, except whenthe azimuthal base flow transports nonaxisymmetric modesamplified by the thermoelectric instability.

In Fig. 8, the critical parameters and different power terms inEq. (24) are shown for centrifugal electric gravity fields. Theelectric Rayleigh number defined by Eq. (4) takes positivevalues, as both γe and Ge are negative. With increasing theelectric field, the critical Taylor number decreases and thecritical wavenumber increases [Fig. 8(a)]. The power WTa

performed by the centrifugal force remains the dominantenergy transfer term until the Rayleigh number L approachesits critical value for stationary electrodes [Fig. 8(b)]. In thelatter situation, the power WBG due to the basic electricgravity is dominant so that the instability is thermoelectricone. The centrifugal electric gravity occurs only at large η

[21], e.g., η should be larger than 0.99 for γe = −0.01. Themode degeneration observed in stationary annuli of large η

[21] occurs in this thermoelectric regime. Nonaxisymmetricmodes, which oscillate due to the base flow transport, aswell as a steady axisymmetric mode are then both critical.The stabilizing effect of the perturbation electric gravity isalso found to be important in the thermoelectric regime, as inRef. [21]. This effect is more significant on small wavenumbermodes and yields large critical wavenumbers at large L.

In inward heating, the electric gravity changes its directionwithin the gap when log η/(1 + log η) < γe < log η. The baseelectric gravity is centrifugal in an outer sublayer of the fluid,where the gravity is directed from cold to hot fluid zonesand destabilizing. However, the base electric gravity is muchsmaller than the stabilizing perturbation electric gravity. Infact, the stability analysis for stationary electrodes [21] showedthat the electric field cannot provoke any instability in thiselectric gravity configuration. The thermoelectric buoyancyhas a net stabilizing effect and oscillatory modes can becritical. Indeed, the critical modes have nonzero frequencywhen the electric field is strong [Fig. 9(a)]. In this oscillatoryregime, energy transfer due to the electric gravity g′

e occursin an inner fluid layer [Fig. 9(b)] and its stabilizing effectdominates the destabilizing centrifugal buoyancy. At small L

the destabilizing effect of the centrifugal buoyancy is dominantand the critical mode is steady. The flow goes in the samedirection as and opposite to the centrifugal acceleration gc

inside, respectively, the cold and hot cores of perturbationtemperature field [Fig. 9(c)].

0 500 1000 1500 2000 2500 3000

L

0

10

20

30

40

3.1

3.2

3.3

3.4

Tac qc

Tac

qc

(a)

0 500 1000 1500 2000 2500 3000L

-40

-20

0

20

40

60

Pow

er te

rms

WTaWC

WBGWPG

(b)

FIG. 8. Critical parameters (a) and power terms in Eq. (24) normalized by twice of the kinetic energy K (b), when the electric gravity Ge

is centrifugal (η = 0.995, Pr = 60, γa = γe = −0.01).

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-14 -12 -10 -8 -6 -4 -2 0

38

39

40

41

42

43

0

1

2

3

4

Tacωc

Tac

ωc

L

qc

qc

,

(a) )c()b(

FIG. 9. (Color online) Critical parameters (a) and eigenfunctions (b, c) when the electric gravity Ge is centripetal (CP) in an inner layer andcentrifugal (CF) in the outer layer (η = 0.99009, Pr = 60, γa = γe = −0.01). In (b) where L = −2.600, the critical mode is oscillatory, whileit is steady in (c) for L = −1.200. Arrows in the first image of (b) and in (c) show perturbation velocity fields. Arrows in the second image of(b) show the buoyancy field associated with the perturbation electric gravity g′

e. The power wPG is normalized by twice of the kinetic energyK . Temperature fields are shown without any numerical scale, as they are perturbation fields determined by a linear theory. Black curves arethe contours of θ ′ = 0.

For applications of the thermoelectric convection to heatexchangers, the present analysis should be extended to includethe Earth’s gravity. We are currently performing this extensionfor vertical Taylor-Couette systems. The resulting problemmay have a lot of similarities to the thermomagnetic convectionstudied by Tagg and Weidman [15] under the Earth’s gravity.

VI. CONCLUSION

A linear stability analysis has been performed for theTaylor-Couette flow of a dielectric fluid in the presence of aradial temperature gradient and a radial dielectrophoretic bodyforce. The latter force arises from the thermal variation of thefluid electric permittivity and has been regarded as thermalbuoyancy associated with an electric effective gravity. Thecritical parameters (the Taylor number Tac, the wavenumberqc, and the frequency ωc) have been determined for differentvalues of the radius ratio, of the temperature difference andof the electric field strength. The thermal buoyancy due tothe centrifugal acceleration is also taken into account in theanalysis through the thermal expansion parameter γa . Differentcritical states have been identified. In inward heating, thecritical mode is oscillatory axisymmetric (Regime I) andsteady axisymmetric (Regime II−) under strong and weakelectric fields, respectively. Energy analysis showed that thecentrifugal force is the dominant mechanism of energy transferfrom the base state to perturbation flow in these regimes. InRegime I, thermal buoyancy has a net restoring effect, whileit is destabilizing in Regime II−. In outward heating, the

critical mode is oscillatory axisymmetric (Regime IV), steadyaxisymmetric (Regime II+), and oscillatory nonaxisymmetric(Regime III) in weak, moderate and strong electric fields,respectively. The dominant energy transfer mechanism is dueto the centrifugal force in Regimes IV and II+ and to thedielectrophoretic force in Regime III. Regime IV is found onlywhen the thermal expansion parameter is significant, withincertain ranges of the Prandtl number and radius ratio. The netcontribution of the thermal buoyancy is stabilizing in RegimeIV, while it is destabilizing in Regime II+.

The frequencies of the oscillatory critical modes obtainedin Regimes I and IV are correlated by the generalized Brunt-Vaisala frequency, which takes into account the restoringeffects of the thermoelectric and centrifugal buoyancies. Thisfinding indicates that in these oscillatory regimes, energy istransferred from the base flow to perturbation flow through aresonant mechanism. In Regime III, the critical nonaxisym-metric modes due to the thermoelectric buoyancy are advectedby the azimuthal base flow. Their critical frequency is scaledby the frequency of the cylinder rotation.

ACKNOWLEDGMENTS

This work has been partly supported by the CNES (CentreNational d’Etudes Spatiales) and the CNRS (Centre Na-tional de la Recherche Scientifique). Authors acknowledgethe French National Research Agency (ANR) for financialsupport through the program Investissements d’Avenir (ANR-10-LABX-09-01 ), LABEX EMC3.

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