Contents lists available at SciVerse ScienceDirect

Acta Astronautica

Acta Astronautica 81 (2012) 563–569

0094-57

http://d

$ Thin Corr

E-m

journal homepage: www.elsevier.com/locate/actaastro

Thermo-electro-hydrodynamic instabilities in a dielectric liquidunder microgravity$

Satish V. Malik, Harunori N. Yoshikawa, Olivier Crumeyrolle, Innocent Mutabazi n

Laboratoire Ondes et Milieux Complexes, UMR 6294, CNRS – Universite du Havre 53, rue de Prony, BP540 – 76058 Le Havre Cedex, France

a r t i c l e i n f o

Article history:

Received 5 March 2012

Received in revised form

22 June 2012

Accepted 28 June 2012Available online 12 October 2012

Keywords:

Electrohydrodynamics

Heat transfer

Convection

65/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.actaastro.2012.06.023

s paper was presented during the 62nd IAC i

esponding author. Tel.: þ33 235217119.

ail address: mutabazi@univ-lehavre.fr (I. Mut

a b s t r a c t

Linear stability analysis of a dielectric fluid confined in a cylindrical annulus of infinite

length is performed under microgravity conditions. A radial temperature gradient and a

high alternating electric field imposed over the gap induce an effective gravity that can

lead to a thermal convection even in the absence of the terrestrial gravity. The

linearized governing equations are discretized using a spectral collocation method on

Chebyshev polynomials to compute marginal stability curves and the critical para-

meters of instability. The critical parameters are independent of the Prandtl number,

but they depend on the curvature of the system. The critical modes are non-axisym-

metric and are made of stationary helices.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A combined action of a radial temperature gradientand an electric field on a dielectric liquid is of greatinterest in both fundamental research and applications.The temperature gradient induces a gradient of thepermittivity E¼ EðTÞ and the latter coupled with theelectric field produces an electric body force [1]. Whenthe electric field is alternating with a high frequency, i.e.,with a time-period much shorter than the charge relaxa-tion time, there is no free charge in the liquid [2] and thedensity of the electric force is given by

fe ¼�1

2E2=Eþ=

1

2rE2 @E

@r

� �T

� �, ð1Þ

where E is the local electric field and r is the liquid density.The first and second terms are called the dielectrophoreticand electrostrictive forces, respectively. As the first termis related to the temperature gradient, it can developthermal convection through a thermo-electro-hydrodynamic

ll rights reserved.

n Cape Town.

abazi).

(THED) instability without any other external force, e.g., theterrestrial gravity. From the fundamental standpoint, thisconvection considered in a cylindrical or spherical geometry isof particular interest. By applying a radial temperaturegradient and electric field, the dielectrophoretic force isaligned in the radial direction and can be deemed as aneffective gravity that we call the electric gravity. Manygeophysical problems can be simulated by this convectionin the dielectric liquid [3,4]. The application of the tempera-ture gradient and the electric field may be used for heattransfer enhancement in dielectric liquids and may yield largereductions in weight and volume of heat transfer systems.This technique is attractive for aerospace cooling systems [5].

In the present paper, we consider the configuration of anannulus filled by a dielectric liquid and subjected to a radialtemperature gradient and a radial alternating electric field.The stability in this configuration was investigated byChandra and Smylie [6] by assuming the microgravityenvironment where the buoyancy associated with theterrestrial gravity was absent. They considered the linearstability to axisymmetric disturbances and found instabilityabove a critical electric Rayleigh number Rac. Chandra andSmylie took into account only disturbances in the velocityand temperature fields in their analysis. No disturbance in

S.V. Malik et al. / Acta Astronautica 81 (2012) 563–569564

the electric field was considered, although the permittivity(and electric field) would have perturbation componentsthrough its temperature dependence. This possible feedbackeffect of the temperature disturbances on the electricpermittivity was taken into account by Takashima [7] byincluding Gauss’s law for the electric field in the linearstability problem. He considered the problem in the sameannular geometry for axisymmetric disturbances. His ana-lysis showed the critical Rayleigh number depends onanother parameter b representing the thermal variation ofthe permittivity. Besides these theoretical stability analysesin the annular geometry, some works exist for the sphericalgeometry [8–10]. Yavorskaya et al. [8] considered a dielec-tric liquid layer between two concentric spherical surfaceswith a radial temperature gradient and a radial electric field.They found instability occurrence and a sensitive depen-dence of the critical Rayleigh number on the radius ratio ofthe two surfaces. Futterer et al. [10] performed a numericalsimulation by a spectral code for cases with and without therotation of the inner surface. They found development ofsteady convection as well as periodic and chaotic behaviorof the flow. When the rotation is absent, they found suddentransition from steady to chaotic flows.

In the literature, some experiments are found thatwere performed in the annular geometry. Chandra andSmylie [6] reported an experiment conducted under theterrestrial gravity condition. They used vertically installedconcentric cylinders with the gap filled by a silicone oil.When Ra exceeded a critical value Rac, they observed therapid increase of the Nusselt number from the unity. Theexperimental Rac agreed well with their theoretical resultunder the microgravity condition. However, there will bea thermal convection due to the terrestrial gravity in thebasic state [11,12]. Further justification is needed toconclude that this Nusselt number behavior is associatedto the TEHD instability. Recently, Sitte et al. [13] con-ducted an experiment in microgravity conditions. Theyvisualized the thermal field in the gap of two cylinders bythe Schlieren technique and observed convection at largeelectric Rayleigh numbers. Their results indicate the non-axisymmetric behavior of flow, which has never beenconsidered in the stability analysis for the annulargeometry.

The present work deals with linear stability to non-axisymmetric as well as axisymmetric disturbances. InSection 2, we present the governing dynamical andelectric equations. The results are given in Section 3. Thelast section is concerned with the discussion andconclusion.

2. Governing equations

We consider an incompressible Newtonian dielectricfluid confined between two coaxial cylinders of radii R1

and R2 ðR1oR2Þ, maintained at temperatures T1 and T2

ðT14T2Þ, respectively. A high frequency alternating elec-tric tension is imposed over the gap. As the liquid does nothave enough time to react to the rapid field variations,only the effective field from the mean value EðrÞ pertainsto the electric force (1).

We adopt a Boussinesq-type approximation: thermalvariation of fluid properties influences the dynamics onlythrough the destabilizing electric body force (1). As to thepermittivity, we assume a linear dependence on thetemperature:

EðTÞ ¼ E2½1�eðT�T2Þ�, ð2Þ

where E2 is the electric permittivity at temperature T2 ande is the thermal coefficient of permittivity. The equationsgoverning the flow subjected to a temperature gradientand an alternating electric field are [1]

= � v¼ 0, ð3aÞ

@v

@tþðv �=Þv¼�

1

r=PþnDvþ1

2

E2e

r E2=T , ð3bÞ

@T

@tþðv �=ÞT ¼ kDT, ð3cÞ

= � ðEEÞ ¼ 0, ð3dÞ

E¼�=f, ð3eÞ

where n is the kinematic viscosity, k is the thermaldiffusivity, v is the velocity. The generalized pressure P

includes the electrostriction component:

P¼ p�1

2r @E

@r

� �T

E2: ð4Þ

We take the gap size d¼ R2�R1 as characteristic lengthand t0 ¼ d2=n as characteristic time as in Takashima [7]; itrepresents the shortest timescale for most of dielectricliquids. The characteristic velocity is given by V ¼ n=d,correspondingly. The temperature is nondimensionalizedby temperature difference DT ¼ T1�T2 between thecylindrical walls. Further the pressure is scaled byP0 ¼ rn2=d2 and the electric field by E0 ¼ Ve=d, where Ve

is the effective value of the potential difference appliedacross the cylinders. The scaling leads to a set of fourdimensionless parameters necessary to specify the flow:the dimensionless curvature d¼ d=R1, the Prandtl numberPr¼ n=k, the thermal variation of permittivity over theimposed temperature difference b¼ eðT1�T2Þ and theelectric Grashoff number:

Gr ¼geaDTd3

n2with ge ¼

E2eV2e

2rad3, ð5Þ

where ge is the electric gravity. The parameter b has beenomitted in [6,8] by assuming either a small coefficient e ora small temperature difference. We will also use later theelectric Rayleigh number Ra¼ GrPr.

When the imposed temperature and electric potentialdifferences are small, the electric gravity is collinear withthe temperature gradient, so that there is no source ofvorticity and velocity fields. The basic state will then bepurely conductive without fluid motion: v¼ 0. For aninfinite cylindrical annulus, the basic state is expected tobe axisymmetric and axially invariant, i.e., E¼�ðdF=drÞer

and T ¼ TðrÞ. Thus it is governed by the following non-dimensionalized equations obtained from Eqs. (3b)–(3e):

dP

dr�Gr

dFdr

� �2 dT

dr¼ 0, ð6aÞ

S.V. Malik et al. / Acta Astronautica 81 (2012) 563–569 565

1

r

d

drr

dT

dr

� �¼ 0, ð6bÞ

1

r

d

drrEdF

dr

� �¼ 0, ð6cÞ

with the boundary conditions

Tðr1Þ ¼ 1 and Tðr2Þ ¼ 0, ð6dÞ

Fðr1Þ ¼ 1 and Fðr2Þ ¼ 0, ð6eÞ

where r1 and r2 are the dimensionless radii of the innerand outer cylinders, respectively: r1 ¼ d�1, r2 ¼ 1þd�1.The integration of these equations yields the temperatureand the electric potential of the basic state as

TðrÞ ¼�

logd

1þdr

� �logð1þdÞ

and FðrÞ ¼log½1þbTðrÞ�

logð1�bÞ: ð7Þ

The generalized pressure of the basic flow is obtained bysubstituting these solutions into Eq. (6a):

P¼ Gr

ZdFdr

� �2 dT

drdr: ð8Þ

The stability of the flow can be examined by consider-ing linearized governing equations about the basic statewith respect to infinitesimal perturbation expanded intothe normal mode:

WðrÞexp½ikz�injþst� with W¼ ½u,v,w,p,y,f�T , ð9Þ

where k is the axial wavenumber and n is the azimuthalmode number. In general, the time evolution rate of theperturbation s is complex: s¼ sþ io where s and o arethe temporal growth rate and frequency respectively. Theperturbative velocity components in the radial, azimuthaland axial directions are u, v and w respectively. Theperturbations of the generalized pressure, the tempera-ture and the electric potential are denoted by p, y and f.Substituting the normal mode into the governing equa-tions, subtracting the basic flow and neglecting the non-linear terms in perturbation, we obtain the followinglinearized equations:

1

r

d

drðruÞ�

in

rvþ ikw¼ 0, ð10aÞ

Du�u

r2þ

2in

r2v�

dpdrþGr

dFdr

� �2 dydrþ2

dFdr

dT

dr

dfdr

" #¼ su,

ð10bÞ

Dv�v

r2�

2in

r2uþ

in

rp� in

rGr

dFdr

� �2

y¼ sv, ð10cÞ

Dw�ikpþ ikGrdFdr

� �2

y¼ sw, ð10dÞ

1

PrDy�

dT

dru¼ sy, ð10eÞ

ð1�bTÞDf�bdT

dr

dfdr�b

dFdr

d

drþ

1

r

� �þ

d2Fdr2

" #y¼ 0, ð10fÞ

with the Laplacian

D¼d2

dr2þ

1

r

d

dr�

n2

r2þk2

� �: ð11Þ

In the equations of motion (10), the terms including theGrashof number Gr stem from the dielectric force term in Eq.(3b). The coupling of electric and thermal fields affects liquidmotion. Such coupling is also seen in the terms proportionalto b in Eq. (10f). The temperature dependence of thepermittivity can modify the system behavior through theseterms, as Takashima examined its influence on the stability[7]. These couplings contain the radial temperature gradientdT=dr, the basic electric field ð�dF=drÞ and the gradient ofthe electric field, ð�d2F=dr2

Þ. Since these three gradientsdepend on the curvature of the annulus, the stability of thesystem will be sensitive to the curvature.

As to boundary conditions, perturbation velocity, tem-perature and electric potential must vanish on cylindricalsurfaces:

u¼du

dr¼ v¼w¼ y¼f¼ 0 at r¼ r1, r2: ð12Þ

3. Results

The governing Eqs. (10a)–(10f) is discretized usingcollocation method [14,15] and considered only at finitenumber of points rj inside the gap (j¼ 1;2, . . . ,N�1). N isthe highest order of considered Chebyshev polynomials.These points are related to the Chebyshev–Gauss–Lobattocollocation points xj ¼ cosðjp=NÞ by rj ¼ ðxjþ1Þ=2þd�1.The discretized governing equations and the boundaryconditions results in a generalized eigenvalue problem:

LW¼ sMW, ð13Þ

where the operators L andM contains the characteristicson the basic state.

The eigenvalue problem (13) is solved in order toobtain the eigenvalue relation F ðPr,Gr,Z,n,k,s,oÞ ¼ 0 forthe parameter ranges Pr 2 ½1;200� and Z 2 ½0:1,0:9�. Mar-ginal stability curves Gr ¼ GrðkÞ is computed for differentvalues of n. The critical mode ðkc ,nc ,ocÞ is then deter-mined from the curves. Fig. 1 shows few marginalstability curves for different values of the curvature dand a fixed value of the Prandtl number Pr¼7. The lowestminimum belongs to the marginal stability curve withn¼ nca0, i.e., the critical mode is non-axisymmetric.Moreover, it is observed that the critical modes arestationary (oc ¼ 0). Values of the critical parameters fromFig. 1 are given in Table 1. The values of kc and nc implythe convection rolls of the critical mode are made ofhelices which are inclined to the azimuthal direction byan angle c¼ tan�1½2nd=kð2þdÞ�. For d¼ 0:111, c¼ 57:31.

The variation of the critical Grashof number withPrandtl number is elucidated in Fig. 2 for d¼ 0:25. It isfound that the critical values of Gr vary inversely with Pr

for a given curvature d. This means that critical Rayleighnumber Rac is constant. In the figure, the values of thiscritical Rayleigh number Rac are also plotted showing theinstability threshold is characterized by a Rac independent

Fig. 1. Marginal stability curves for axisymmetric (n¼0) and non-axisymmetric (n¼ nca0) perturbations for Pr¼7, b¼ 0:01 and various values of

curvature d: (a) d¼ 0:111, (b) d¼ 0:25, (c) d¼ 1 and (d) d¼ 4.

S.V. Malik et al. / Acta Astronautica 81 (2012) 563–569566

of Pr. The independence from the Prandtl number is alsofound in the critical axial wavenumber kc and the criticalazimuthal mode number nc (Fig. 3). It is also seen that kc isalways around 1.5 and becomes smaller with increasing d,in contrast with nc which decreases substantially with d.The influence of the curvature d on the critical Rayleigh

number Rac is shown in Fig. 4. The value of Rac decreaseswith d, i.e., the curvature lowers the instability threshold.This would be related to the fact that the electric field andtemperature gradient take locally large values at the innercylindrical surface for large curvatures: the dielectro-phoretic body force becomes consequently important.

S.V. Malik et al. / Acta Astronautica 81 (2012) 563–569 567

4. Discussion

The obtained results are different from those reported inChandra and Smylie [6] and Takashima [7] who consideredonly axisymmetric disturbances. Therefore, the criterion

Table 1Critical parameters and the marginal curve minima of axisymmetric

mode for different curvatures (Pr¼7, b¼ 0:01).

Curvature Critical mode Axisymmetric mode

d kc nc Grc kmin Grmin

(a) 0.111 1.69 25 1075 3.12 1077

(b) 0.25 1.62 12 526.7 3.12 530.6

(c) 1 1.53 4 171.3 3.14 183.5

(d) 4 1.41 2 71.37 3.25 98.31

Fig. 2. Variation of critical Grashof number Grc and critical Rayleigh

Fig. 3. Variation of the critical mode with the Prandtl number Pr for differe

azimuthal mode number nc.

given by these authors to predict the instability has to beextended to non-axisymmetric perturbations.

The independence of the critical parameters from thePrandtl number is also found in the ordinary Rayleigh–Benard problem, in which a liquid layer between twohorizontal plates is heated from below. The critical value ofthe Rayleigh number Ra0 ¼ gaDT 0d03=kn (DT 0, temperaturedifference; d0, depth of the layer; g, gravitational acceleration)is 1708 and the critical wavenumber k0c is 3.117, independentof Pr [16]. The main differences of the TEHD instabilityproblem considered in the present work from this Ray-leigh–Benard problem is that the electric gravity dependson electric force and geometry, while the gravity is constantin the latter problem. As a consequence, the critical para-meters in the present problem depend sensitively on thecurvature (Figs. 3 and 4). Such a sensitive dependence isalso found in the TEHD instability problem in a sphericalgeometry [8]. In the ordinary Rayleigh–Benard problem, the

number Rac with Prandtl number Pr for d¼ 0:25 and b¼ 0:01.

nt values of curvature d. (a) Critical axial wavenumber kc. (b) Critical

Table 2

Few values of the critical electric potential Ve for the Baysilones silicone

oil M5 (I) and the 1-nonanol (II).

Liquid d Rac Ve (kV) ge (m/s2)

I 0.11 7785 6.60 4.46

1 1206 2.60 0.691

4 501 1.67 0.287

II 0.11 7254 3.38 13.8

1 1192 1.37 2.27

4 498 0.89 0.951

I. r¼ 920 kg=m3, n¼ 5� 10�6 m2=s, E=E0 ¼ 2:7, e¼ 1:065� 10�3 (25 1C).

II. r¼ 829 kg=m3, n¼ 14:2� 10�6 m2=s, E=E0 ¼ 8:83, e¼ 3:03� 10�2

(20 1C).

Fig. 4. Variation of critical electric Rayleigh number Rac with the curvature d.

S.V. Malik et al. / Acta Astronautica 81 (2012) 563–569568

behavior above the criticality depends on Pr, e.g., the occur-rence of zigzag, cross-roll and bimodal instabilities found inthe nonlinear regime is influenced by Pr [16]. One mayexpect such dependence on the Prandtl number in the TEHDinstability.

According to our results, under microgravity condi-tions, it is easier to trigger critical modes for an annuluswith a large curvature, i.e., a system of cylindrical surfaceswith largely different radii. In Table 2, values of theelectric tension Ve required to provoke the TEHD instabil-ity are shown for different curvatures in two dielectricliquids. The corresponding electric gravity ge is alsopresented for the gap d¼ 5 mm. Considered liquids arethe Baysilones silicone oil M5 and the 1-nonanol, whichwere used in the GeoFlow experiments [3,4] and shouldbe used in future experiments with annular geometries inmicrogravity conditions. The imposed temperature differ-ence DT is assumed to be 51. The critical parametersðkc ,nc ,RacÞ for both liquids take similar values for the samecurvature d, as these parameters are independent of Pr.Small differences in values of Rac are due to the differencein the thermal variation of the permittivity b. It is seenthat Ve at large curvature is substantially lower than Ve atsmall curvature. By choosing the geometry of d¼ 1, forexample, one can lower Ve by a factor 0.4 from the valuefor d¼ 0:111.

5. Conclusion

We have conducted the linear stability analysis of thedielectric fluid confined between two coaxial cylindersand subjected to a radial temperature gradient and a

radial alternating electric field. The instability threshold ischaracterized by the critical Rayleigh number Rac. Thevalue of Rac as well as the critical mode ðkc ,ncÞ areindependent of the Prandtl number Pr, while they dependsensitively on the curvature d. The critical modes arestationary helices (o¼ 0, nca0). When the curvature d islarge, the basic state is more unstable to non-axisym-metric disturbances than to axisymmetric ones.

Acknowledgments

This work has been partly supported by the CNES(Centre National d’Etudes Spatiales), the CNRS (CentreNational de la Recherche Scientifique) and the FEDER(Fonds Europeen de Developpement Regional).

References

[1] L.D. Landau, E.M. Lifshitz, Electrodynamics of Continuous Media,2nd ed., Landau and Lifshitz Course of Theoretical Physics, vol. 8,Butterworth–Heinemann, 1984.

[2] T.B. Jones, Electrohydrodynamically enhanced heat transfer inliquids—a review, Adv. Heat Transfer 14 (1979) 107–148.

[3] B. Futterer, M. Gellert, T. von Larcher, C. Egbers, Thermal convectionin rotating spherical shell: an experimental and numericalapproach within GeoFlow, Acta Astronaut. 62 (4–5) (2008)300–307.

[4] B. Futterer, N. Dahley, S. Koch, N. Scurtu, C. Egbers, From viscousconvective experiment ‘GeoFlow I’ to temperature-dependent visc-osity in ‘GeoFlow II’—fluid physics experiments on-board ISS forthe capture of convection phenomena in Earth’s outer core andmantle, Acta Astronaut. 71 (2012) 11–19.

[5] J.S. Paschkewitz, D.M. Pratt, The influence of fluid properties onelectrohydrodynamic heat transfer enhancement in liquids underviscous and electrically dominated flow conditions, Exp. Therm.Fluid Sci. 21 (4) (2000) 187–197.

[6] B. Chandra, D.E. Smylie, A laboratory model of thermal convectionunder a central force field, Geophys. Fluid Dyn. 3 (1972) 211–224.

[7] M. Takashima, Electrohydrodynamic instability in a dielectric fluidbetween two coaxial cylinders, Q. J. Mech. Appl. Math. 33 (1) (1980)93–103.

[8] I.M. Yavorskaya, N.I. Fomina, Y.N. Belyaev, A simulation of central-symmetry convection in microgravity conditions, Acta Astronaut.11 (3–4) (1984) 179–183.

[9] J.E. Hart, G.A. Glatzmaier, J. Toomre, Space-laboratory and numer-ical simulations of thermal convection in a rotating hemisphericalshell with radial gravity, J. Fluid Mech. 173 (1986) 519–544.

[10] B. Futterer, C. Egbers, N. Dahley, S. Koch, L. Jehring, First identifica-tion of sub- and supercritical convection patterns from ‘GeoFlow’,the geophysical flow simulation experiment integrated in FluidScience Laboratory, Acta Astronaut. 66 (2010) 193–200.

[11] M. Ali, P.D. Weidman, On the stability of circular Couette flow withradial heating, J. Fluid Mech. 220 (1990) 53–84.

S.V. Malik et al. / Acta Astronautica 81 (2012) 563–569 569

[12] A. Bahloul, I. Mutabazi, A. Ambari, Codimension 2 points in the flowinside a cylindrical annulus with a radial temperature gradient, Eur.Phys. J. AP 9 (2000) 253–264.

[13] B. Sitte, J. Immohr, O. Heinrichs, R. Maier, C. Egbers, H. Rath, Rayleigh–Benard convection in dielectrophoretic force fields, in: 12th Interna-tional Couette Taylor Workshop, Evanston, IL USA, 2001.

[14] S.V. Malik, A.P. Hooper, Three-dimensional disturbances in channelflows, Phys. Fluids 19 (2007) 052102.

[15] C. Canuto, M. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methodsin Fluid Dynamics, Springer-Verlag, New York, 1988.

[16] P.G. Drazin, W.H. Reid, Hydrodynamic Stability, 2nd ed. CambridgeUniversity Press, 2004.