Research ArticleLinear Stability Analysis of Thermal Convection inan Infinitely Long Vertical Rectangular Enclosure inthe Presence of a Uniform Horizontal Magnetic Field

Takashi Kitaura and Toshio Tagawa

Tokyo Metropolitan University 6-6 Asahigaoka Hino-shi 191-0065 Japan

Correspondence should be addressed to Toshio Tagawa tagawa-toshiotmuacjp

Received 23 January 2014 Accepted 2 April 2014 Published 29 April 2014

Academic Editor Robert Spall

Copyright copy 2014 T Kitaura and T Tagawa This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Stability of thermal convection in an infinitely long vertical channel in the presence of a uniformhorizontalmagnetic field applied inthe direction parallel to the hot and cold walls was numerically studied First in order to confirm accuracy of the present numericalcode the one-dimensional computations without the effect of magnetic field were computed and they agreed with a previous studyquantitatively for various values of the Prandtl number Then linear stability analysis for the thermal convection flow in a squarehorizontal cross section under the magnetic field was carried out for the case of Pr = 0025 The thermal convection flow was oncedestabilized at certain low Hartmann numbers and it was stabilized at high Hartmann numbers

1 Introduction

Nuclear fusion energy has received considerable attention asone of the environment-friendly in modern society Headingtowards implementation of the use of nuclear fusion energyexperimental facility called ITER (International Thermonu-clear Experimental Reactor) is currently under constructionin France ITER has a toroidal shape like a donut in which thehigh-temperature plasma enough to induce nuclear fusionreaction is controlled by both operation of magnetic fieldgenerated by the superconducting coils disposed around theplasma and the imposed electric current in the plasma [1]Blankets located close to the plasma side play an importantrole for cooling shielding neutrons and fuel production

Fusion reactor blanket can be classified into the solidblanket using a solid compound of lithium as fuel productionmaterial or liquid blanket using liquid lithium Above allliquid blanket has the advantage of being of relatively simplestructure [2] but on the other hand it has a serious problemcalled the MHD pressure loss [3] The MHD pressure lossobstructs convection of liquid metal as a coolant and itdepends on the direction of themagnetic field and the electric

conductivity of thewall To elucidate that problem researcheson thermal convection under the electromagnetic force havebeen actively conducted not only for the application of fusionreactor blankets but also for crystal growth such as thehorizontal Bridgman method

The effect of the direction of uniform magnetic field onthe natural convection in a cubic enclosure heated from avertical wall and cooled from an opposing vertical wall wasnumerically studied by Ozoe and Okada [5] They showedthat the horizontal magnetic field parallel to the hot and coldwalls (Y-directional magnetic field) has much less influenceon the damping of natural convection than the two otherdirections of magnetic field (X- or Z-directional magneticfield) This finding was confirmed with an experiment using30mm times 30mm times 30mm cubical box filled with molten gal-lium by Okada and Ozoe [6] They noticed the heat transferrate of natural convection under the Y-directional magneticfield indicated not only weak damping of heat transfer rateat high magnetic fields but also slight enhancement of heattransfer rate at low magnetic fields However they were notconvinced by this enhancement of heat transfer

Hindawi Publishing CorporationJournal of FluidsVolume 2014 Article ID 642042 8 pageshttpdxdoiorg1011552014642042

2 Journal of Fluids

After several years Tagawa and Ozoe (1997 1998) [7 8]carried out both three-dimensional computations with thehigher Rayleigh numbers than those by Ozoe and Okada andthe molten gallium experiment using 64mm times 64mm times64mm cubical box for the Y-directional magnetic fieldin order to confirm the existence of enhancement of heattransfer rate at low magnetic fields Due to the exploration ofthe larger Rayleigh numbers both the numerical and experi-mental results showed the enhancement of heat transfer rate(the Nusselt number) when the uniform magnetic field wasapplied in the Y-direction

The enhancement of Nusselt number in the presence oflow magnetic fields has been reported by several othergroups Authie et al [9] carried out both three-dimensionalcomputations and the corresponding experiment using mer-cury for the buoyant convection in a long vertical enclosureTheir experiments indicated that the Nusselt number takesits maximum around the Hartmann number 200ndash250 for thevalues of the Grashof number from 3 times 107 to 15 times 108 Burret al [10] carried out the natural convection for the similarconfiguration under a horizontal magnetic field perpendicu-lar to both the gravity and the applied heat fluxThey showedthat the Nusselt numbers increased from the value of naturalconvection without the magnetic field in the range of theHartmannnumber 100 lt Ha lt 225with a tendency of higherHartmann numbers at larger heat fluxes Zhang et al [11]carried out an experimental study on the flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldThey concluded that the application of a moderate magneticfield destabilizes the global flow and gives rise to transientoscillating flow patterns with predominant frequencies Theelectric conducting fluid motion is usually damped by theuse of static magnetic field The above-mentioned literaturesconcern the enhancement of natural convection heat transferor destabilizing effect of the electric conducting fluid flow atlow Hartmann numbers in the presence of static magneticfield This interesting finding could be related to the flowinstability andor transition

Fujimura and Nagata (1998) [12] studied instability of thenatural convection in an infinitely long vertical rectangularchannel in the presence of a uniform horizontal magneticfield by two-dimensional analysis Kakutani (1964) [13] stud-ied the hydromagnetic stability on the plane laminar flowbetween the parallel plates in the presence of a transversemagnetic field The plane Couette flow is destabilized by themagnetic field in the range ofHartmann number 391 lt Ha lt54

The purpose of this research is to clarify the stability ofparallel flow of thermal convection in an infinitely long verti-cal rectangular channel in the presence of a horizontal mag-netic field

2 Numerical Model

The schematic model considered in this study is shown inFigure 1 The fluid treated in this research is assumed tobe an incompressible Newtonian fluid and the Boussinesqapproximation is employed The boundary conditions are

x

y

zB

2l

2h

g

Hot Cold

Figure 1 Schematic model considered

U W U W

Destabilization

Figure 2 An example of destabilization of the thermal convectionin a long vertical rectangular channel at Pr = 0025 and Gr = 9000

the no-slip condition for all walls and the heated andcooled walls are isothermal and the other two walls are thethermally insulated Governing equations are the equationsof continuity the momentum the energy Ohmrsquos law and theconservation of electric charge We used a finite differencemethod with a staggered mesh system By introducing thetime derivative term into the governing equations we obtainapproximated solutions under discretization using a second-order or fourth-order accurate central difference methodtogether with the use of HS-MAC method for the solutionof the Poisson equations of the pressure and the electricpotential

Journal of Fluids 3

g

Hot Cold

X = 1X = minus1

Figure 3 Simplified calculation model (119860 = ℎ119897 rarr infin)

Table 1 Comparisons of the critical Grashof number and the criticalwavenumber between a reference and the present research at Pr =067 and 75

MethodReference [4] Present

Chebyshev polynomialsexpansion

Fourth-order accuratecentral difference

method with 101 gridsPr 067 75 067 75119896119888 140417 138334 140419 138349Gr119888 503293 491900 503305 491829

The instability of the convection of the infinitely long ver-tical rectangular channel is investigated by using the Grashofnumber which is a dimensionless number representing themagnitude of the influence of the inertial and buoyancyforces Figure 2 shows how the flow of a two-dimensionalvertical rectangular channel becomes unstable When theGrashof number exceeds a certain value the flow structure isdivided into a number of cells In this study the destabilizedflow in the three-dimensional vertical rectangular channelis assumed to be periodic flow in the 119885-axis direction andthen we perform the computation with applying a uniformhorizontal magnetic field perpendicular to the temperaturegradient

3 Verification of the Numerical Code

As the verification of the numerical code preliminary com-putations were performed for the model of the aspect ratio(ℎ119897 rarr infin) without magnetic field and those resultswere compared with previous results [4] The present modeland the numerical results are described in Figures 3 and 4respectively Figure 4 describes the result performed with thenumber of grids 101 using the discretization of a fourth-orderaccurate central difference method

It is noted that this graph shows result of the case of thestanding wave disturbance It is known that for the lowerPrandtl number (Pr lt 1245) [14] the neutral Grashof num-ber in the case of the standing wave disturbance is lowerthan that in the case of the traveling wave disturbance

Table 1 shows the numerical results for comparison of thecritical Grashof number and the critical wavenumber whenair (Pr = 067) and water (Pr = 75) are assumed as test fluidsThe agreement between the referencersquos result and the presentresult is negligibly small

4 Dimensionless Equations

Theflow instability in a long vertical enclosure in the presenceof a magnetic field applied in the Y-direction is investigatedhere In this study the destabilized flow is assumed to beperiodic and stationary in the 119885-axis direction and then thenumerical analyses were performed within a horizontal crosssection Due to the computational resources we focus on thecase that the aspect ratio of the horizontal cross section isunity and the Prandtl number is 0025 which is a typical valueof liquid metals

The dimensionless governing equations are shown below

120597V120597120591 + (V sdot nabla)V = minusnabla119875 + 1Gr

nabla2V + 1Gr

Θe119911 + Ha2

GrJ times e1198610

120597Θ120597120591 + V sdot nablaΘ = 1GrPr

nabla2Θnabla sdot V = 0 nabla sdot J = 0J = minusnablaΦ + V times e1198610

(1)

The dimensionless variables are defined as follows

120591 = 1199051198972 (Gr]) X = x119897 V = kGr]119897

Θ = 119879 minus 1198790Δ119879 119875 = 1199011205880Gr2]21198972 J = j

Gr]1198610 Φ = 120601Gr120590119890]1198610119897 Gr = 119892120573Δ1198791198973

]2

Pr = ]120572 Ha = 1198610119897radic 1205901198901205880] 119860 = ℎ119897

(2)

Since we limit ourselves to the case that the uniformmag-netic field is applied in the Y-direction it holds that e1198610 = e119910

5 Basic Flow

First the flow of the basic state is computed with theassumption that the rectangular enclosure is long enough toneglect recirculation area near the top and the bottom wallsTherefore the basic state of temperature field is in heat con-duction Since the Prandtl andGrashof numbers are includedin the scaling the dimensionless basic flow is independent ofthe Grashof and Prandtl numbers Dimensionless governingequations used in the computation are shown as follows

4 Journal of Fluids

142

140

138

136

134

120572c

500

480

460

440

Gc

Gc

P

120572c rarr 13827

Gc rarr 4956284

120572c rarr 1344147

Gc rarr 492450

120572

10minus310minus410minus5 10minus2 10minus1 100 101 102

(a) Reference [4]

kc

kcGr c

Grc

142

14

138

136

134

Pr

510

500

490

480

470

460

450

44010minus3 10minus2 10minus1 100 101 102

(b) Present result

Figure 4 Variations of the critical Grashof number and the critical wavenumber for the wide range of the Prandtl number (a) is referredfrom a previous study [4] and (b) indicates the present result

343E minus 02294E minus 02245E minus 02196E minus 02147E minus 02981E minus 03490E minus 03000E + 00minus490E minus 03minus981E minus 03minus147E minus 02minus196E minus 02minus245E minus 02minus294E minus 02minus343E minus 02

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

Wbs

(a)

153E minus 03131E minus 03109E minus 03873E minus 04655E minus 04436E minus 04218E minus 04000E + 00minus218E minus 04minus436E minus 04minus655E minus 04

minus109E minus 03minus131E minus 03minus153E minus 03

minus873E minus 04

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

ΨJ

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

116E minus 02101E minus 02860E minus 03713E minus 03565E minus 03417E minus 03270E minus 03122E minus 03minus256E minus 04minus173E minus 03minus321E minus 03minus468E minus 03minus616E minus 03minus764E minus 03minus911E minus 03

Φbs

(c)

Figure 5 Contourmaps of the basic flow at Ha = 100 (a) vertical velocity119882bs (b) stream line of electric currentΨ119869 and (c) electric potentialΦbs

Journal of Fluids 5

In this study the aspect ratio of the horizontal cross sectionis limited to 119860 = ℎ119897 = 1

1205972119882bs1205971198832 + 1205972119882bs1205971198842 + Θbs +Ha2119869119909bs = 0Θbs = minus119883

120597119869119909bs120597119883 + 120597119869119910bs120597119884 = 0119869119909bs = minus120597Φbs120597119883 minus119882bs

119869119910bs = minus120597Φbs120597119884

(3)

Here the subscript character bs represents the basic stateThe vertical component of velocity 119882 the streamlines ofelectric current densityΨ119895 and the electric potentialΦ of thebasic state at the Hartmann number Ha = 100 are depicted inFigure 5

6 Linear Stability Analysis

Disturbance equations are derived for investigating the flowinstability First of all velocity pressure temperature electricpotential and electric current density are represented withthe sum of the solution of the basic state and the infinitesimaldisturbances [15]

k = kbs + k1015840 119901 = 119901bs + 1199011015840119879 = 119879bs + 1198791015840 120601 = 120601bs + 1206011015840j = jbs + j1015840

(4)

We assume that the infinitesimal disturbances are repre-sented by the normal mode as follows

(k1015840 1199011015840 1198791015840 1206011015840 j1015840) = (k 119901 120601 j) exp (i119886119911 + 120582119905) (5)

Note that 119894 is the imaginary unit 120582 is the complexgrowth rate which is a complex number and 119886 is thewavenumber which is a real number hereThe dimensionlessequations obtained by substituting (4) and (5) to (1) are shownbelow It is known that the natural convection flow in avertical slot becomes unstable for standing wave disturbancerather than for travelling wave disturbance in the range of

Table 2 Variation of the neutral Grashof number for several casesof number of grids at Pr = 0025 119896 = 083 and Ha = 50Number of meshes Gr

2nd-order 4th-order30 times 30 228063 21912250 times 50 217764 21525070 times 70 215361 214276

k

Gr n

077 078 079 08 081 082 083 084 085 086 0873100

3105

3110

3115

3120

3125

3130

Figure 6 Neutral stability curve of the Grashof number at Pr =0025 and Ha = 100

Pr lt 1245 [14]Therefore we assumed that the neutral stablestate of the flow of Pr = 0025 is taken as 120582 = 0

120597120597119883 + 120597120597119884 + 119894119896 = 0119894119882bs119896 = minus 120597120597119883 + 1

Gr( 12059721205971198832 + 12059721205971198842 minus 1198962) + Ha2

Gr(minus119869119911)

119894119882bs119896 = minus120597120597119884 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) 120597119882bs120597119883 + 120597119882bs120597119884 + 119894119882bs119896

= minus119894119896 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) + 1Gr

Θ + Ha2

Gr119869119909

120597119869119909120597119883 + 120597119869119910120597119884 + 119894119896119869119911 = 0 119869119909 = minus120597Φ120597119883 minus 119869119910 = minus120597Φ120597119884 119869119911 = minus119894119896Φ +

(6)

Table 2 shows variation of the critical Grashof number onthe number of grids at the dimensionless wavenumber 119896 =083 and Hartmann number Ha = 50 when discretized with

6 Journal of Fluids

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(a)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(c)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(d)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(e)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(f)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(g)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(h)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(i)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(j)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(k)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(l)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(m)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(n)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(o)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(p)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(q)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(r)

Figure 7 Contour maps of the characteristic function at a marginal state at Pr = 0025 119896 = 083 Gr = 21525 and Ha = 50 (a) real partof (b) imaginary part of (c) real part of (d) imaginary part of (e) real part of (f) imaginary part of (g) real part of (h)imaginary part of (i) real part of Θ (j) imaginary part of Θ (k) real part of 119869119909 (l) imaginary part of 119869119909 (m) real part of 119869119910 (n) imaginarypart of 119869119910 (o) real part of 119869 (p) imaginary part of 119869 (q) real part of Φ and (r) imaginary part of Φ

Journal of Fluids 7

Ha

kc

kc

Gr c

Grc

0 2 4 6 8 10 12068

072

076

08

084

088

2000

2400

2800

3200

3600

4000

Figure 8 Variation of the critical Grashof number and the criticalwavenumber against the Hartmann number at Pr = 0025

a second-order or a fourth-order accurate central differenceConsidering the result of Table 2 and the computational timewe employed 70 times 70 mesh number with the second-orderaccurate central difference For example the neutral stabilitycurve of theGrashof number atHartmann numberHa = 100is depicted in Figure 6 The critical Grashof numbers at thewavenumber 0821 with mesh size 70 times 70 90 times 90 and 120 times120 take the value of 3101 3109 and 3114 respectively So thedependency of themesh size on the result is not so significant

Characteristic functions for various variables at a neutralstate are illustrated in Figure 7The pressure and the potentialdistribution have point symmetry and have reverse signbetween the real and the imaginary parts with respect to119883 =0 and 119884 = 0 On the other hand the velocity temperatureand electric current density distribution have point symmetrywith respect to119883 = 0 and 119884 = 0

The critical Grashof number and the wavenumber plottedagainst the Hartmann number up to 12 are shown in Fig-ure 8 Concerning the application of a horizontal Bridgemancrystal growth Lyubimov et al [16] studied the stability ofthermal convection in a rectangular cross section of an ob-long channel They studied two cases of computation themagnetic field applied in the horizontal direction or in thevertical direction with changing the Prandtl number and theaspect ratio of cross section It should be noted that thepresent study could be compared with the study of Lyubi-mov et al at the case of zero value of Prandtl number (Pr = 0)In the configuration of horizontal Bridgeman crystal growththe plane-parallel flow does not exist in nonzero values ofPrandtl number but in the present study the plane-par-allel flow always exists irrespective of Prandtl number sincethe basic thermal state is independent of Prandtl number Ac-cording to the results of horizontal magnetic field by Lyubi-mov et al the critical Grashof number takes its mini-mum at a certain low Hartmann number and the critical

wavenumber takes its maximum at another low Hartmannnumber This tendency is quite coincident with the findingof the present study This indicates that a moderate magneticfield destabilizes the natural convection of plane-parallel flowbut a strong magnetic field stabilizes the convection

7 Conclusion

In this study we studied the linear stability of thermalconvection in an infinitely long vertical rectangular channelin the presence of a uniform horizontal magnetic field Weobtained some findings First the characteristic function ofthe pressure and the electric potential distribution have pointsymmetry and have reverse sign between the real and theimaginary parts with respect to the center of cross sectionOnthe other hand the velocity temperature and electric currentdensity distribution have point symmetry with respect to thecenter of cross section Second except in the range of theHartmann number of 0 lt Ha lt 8 increase in the Hartmannnumber leads to well stabilization And the critical Grashofnumber takes a minimum value at the Hartmann number ofabout Ha = 45 Third the critical wavenumber increasesat a low Hartmann number but then decreases for furtherincrease in the Hartmann number

Nomenclature

119886 Wavenumber [radm]119860 Aspect ratio (= ℎ119897)1198610 Applied magnetic field [T]e1198610 Unit vector (direction of 1198610)e119910 Unit vector (direction of 119910-axis)e119911 Unit vector (direction of 119911-axis)Gr Grashof number119892 Gravitational acceleration [ms2]ℎ Duct width in the 119910-axis [m]Ha Hartmann number119894 Imaginary unitj Electric current density vector [Am2]J Dimensionless electric current density119869119909bs Dimensionless electric current density of

the basic flow (along 119909-axis)119869119910bs Dimensionless electric current density ofthe basic flow (along 119910-axis)119869119909 Dimensionless electric current density ofthe amplitude (along 119909-axis)119869119910 Dimensionless electric current density ofthe amplitude (along 119910-axis)119869 Dimensionless electric current density ofthe amplitude (along 119911-axis)119896 Dimensionless wavenumber (along 119911-axis)119897 Duct width in the 119909-axis [m]119901 Pressure [Pa]119875 Dimensionless pressure Dimensionless pressure of the amplitude

Pr Prandtl number1198790 Reference temperature [K]119879 Temperature [K]

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

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2 Journal of Fluids

After several years Tagawa and Ozoe (1997 1998) [7 8]carried out both three-dimensional computations with thehigher Rayleigh numbers than those by Ozoe and Okada andthe molten gallium experiment using 64mm times 64mm times64mm cubical box for the Y-directional magnetic fieldin order to confirm the existence of enhancement of heattransfer rate at low magnetic fields Due to the exploration ofthe larger Rayleigh numbers both the numerical and experi-mental results showed the enhancement of heat transfer rate(the Nusselt number) when the uniform magnetic field wasapplied in the Y-direction

The enhancement of Nusselt number in the presence oflow magnetic fields has been reported by several othergroups Authie et al [9] carried out both three-dimensionalcomputations and the corresponding experiment using mer-cury for the buoyant convection in a long vertical enclosureTheir experiments indicated that the Nusselt number takesits maximum around the Hartmann number 200ndash250 for thevalues of the Grashof number from 3 times 107 to 15 times 108 Burret al [10] carried out the natural convection for the similarconfiguration under a horizontal magnetic field perpendicu-lar to both the gravity and the applied heat fluxThey showedthat the Nusselt numbers increased from the value of naturalconvection without the magnetic field in the range of theHartmannnumber 100 lt Ha lt 225with a tendency of higherHartmann numbers at larger heat fluxes Zhang et al [11]carried out an experimental study on the flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldThey concluded that the application of a moderate magneticfield destabilizes the global flow and gives rise to transientoscillating flow patterns with predominant frequencies Theelectric conducting fluid motion is usually damped by theuse of static magnetic field The above-mentioned literaturesconcern the enhancement of natural convection heat transferor destabilizing effect of the electric conducting fluid flow atlow Hartmann numbers in the presence of static magneticfield This interesting finding could be related to the flowinstability andor transition

Fujimura and Nagata (1998) [12] studied instability of thenatural convection in an infinitely long vertical rectangularchannel in the presence of a uniform horizontal magneticfield by two-dimensional analysis Kakutani (1964) [13] stud-ied the hydromagnetic stability on the plane laminar flowbetween the parallel plates in the presence of a transversemagnetic field The plane Couette flow is destabilized by themagnetic field in the range ofHartmann number 391 lt Ha lt54

The purpose of this research is to clarify the stability ofparallel flow of thermal convection in an infinitely long verti-cal rectangular channel in the presence of a horizontal mag-netic field

2 Numerical Model

The schematic model considered in this study is shown inFigure 1 The fluid treated in this research is assumed tobe an incompressible Newtonian fluid and the Boussinesqapproximation is employed The boundary conditions are

x

y

zB

2l

2h

g

Hot Cold

Figure 1 Schematic model considered

U W U W

Destabilization

Figure 2 An example of destabilization of the thermal convectionin a long vertical rectangular channel at Pr = 0025 and Gr = 9000

the no-slip condition for all walls and the heated andcooled walls are isothermal and the other two walls are thethermally insulated Governing equations are the equationsof continuity the momentum the energy Ohmrsquos law and theconservation of electric charge We used a finite differencemethod with a staggered mesh system By introducing thetime derivative term into the governing equations we obtainapproximated solutions under discretization using a second-order or fourth-order accurate central difference methodtogether with the use of HS-MAC method for the solutionof the Poisson equations of the pressure and the electricpotential

Journal of Fluids 3

g

Hot Cold

X = 1X = minus1

Figure 3 Simplified calculation model (119860 = ℎ119897 rarr infin)

Table 1 Comparisons of the critical Grashof number and the criticalwavenumber between a reference and the present research at Pr =067 and 75

MethodReference [4] Present

Chebyshev polynomialsexpansion

Fourth-order accuratecentral difference

method with 101 gridsPr 067 75 067 75119896119888 140417 138334 140419 138349Gr119888 503293 491900 503305 491829

The instability of the convection of the infinitely long ver-tical rectangular channel is investigated by using the Grashofnumber which is a dimensionless number representing themagnitude of the influence of the inertial and buoyancyforces Figure 2 shows how the flow of a two-dimensionalvertical rectangular channel becomes unstable When theGrashof number exceeds a certain value the flow structure isdivided into a number of cells In this study the destabilizedflow in the three-dimensional vertical rectangular channelis assumed to be periodic flow in the 119885-axis direction andthen we perform the computation with applying a uniformhorizontal magnetic field perpendicular to the temperaturegradient

3 Verification of the Numerical Code

As the verification of the numerical code preliminary com-putations were performed for the model of the aspect ratio(ℎ119897 rarr infin) without magnetic field and those resultswere compared with previous results [4] The present modeland the numerical results are described in Figures 3 and 4respectively Figure 4 describes the result performed with thenumber of grids 101 using the discretization of a fourth-orderaccurate central difference method

It is noted that this graph shows result of the case of thestanding wave disturbance It is known that for the lowerPrandtl number (Pr lt 1245) [14] the neutral Grashof num-ber in the case of the standing wave disturbance is lowerthan that in the case of the traveling wave disturbance

Table 1 shows the numerical results for comparison of thecritical Grashof number and the critical wavenumber whenair (Pr = 067) and water (Pr = 75) are assumed as test fluidsThe agreement between the referencersquos result and the presentresult is negligibly small

4 Dimensionless Equations

Theflow instability in a long vertical enclosure in the presenceof a magnetic field applied in the Y-direction is investigatedhere In this study the destabilized flow is assumed to beperiodic and stationary in the 119885-axis direction and then thenumerical analyses were performed within a horizontal crosssection Due to the computational resources we focus on thecase that the aspect ratio of the horizontal cross section isunity and the Prandtl number is 0025 which is a typical valueof liquid metals

The dimensionless governing equations are shown below

120597V120597120591 + (V sdot nabla)V = minusnabla119875 + 1Gr

nabla2V + 1Gr

Θe119911 + Ha2

GrJ times e1198610

120597Θ120597120591 + V sdot nablaΘ = 1GrPr

nabla2Θnabla sdot V = 0 nabla sdot J = 0J = minusnablaΦ + V times e1198610

(1)

The dimensionless variables are defined as follows

120591 = 1199051198972 (Gr]) X = x119897 V = kGr]119897

Θ = 119879 minus 1198790Δ119879 119875 = 1199011205880Gr2]21198972 J = j

Gr]1198610 Φ = 120601Gr120590119890]1198610119897 Gr = 119892120573Δ1198791198973

]2

Pr = ]120572 Ha = 1198610119897radic 1205901198901205880] 119860 = ℎ119897

(2)

Since we limit ourselves to the case that the uniformmag-netic field is applied in the Y-direction it holds that e1198610 = e119910

5 Basic Flow

First the flow of the basic state is computed with theassumption that the rectangular enclosure is long enough toneglect recirculation area near the top and the bottom wallsTherefore the basic state of temperature field is in heat con-duction Since the Prandtl andGrashof numbers are includedin the scaling the dimensionless basic flow is independent ofthe Grashof and Prandtl numbers Dimensionless governingequations used in the computation are shown as follows

4 Journal of Fluids

142

140

138

136

134

120572c

500

480

460

440

Gc

Gc

P

120572c rarr 13827

Gc rarr 4956284

120572c rarr 1344147

Gc rarr 492450

120572

10minus310minus410minus5 10minus2 10minus1 100 101 102

(a) Reference [4]

kc

kcGr c

Grc

142

14

138

136

134

Pr

510

500

490

480

470

460

450

44010minus3 10minus2 10minus1 100 101 102

(b) Present result

Figure 4 Variations of the critical Grashof number and the critical wavenumber for the wide range of the Prandtl number (a) is referredfrom a previous study [4] and (b) indicates the present result

343E minus 02294E minus 02245E minus 02196E minus 02147E minus 02981E minus 03490E minus 03000E + 00minus490E minus 03minus981E minus 03minus147E minus 02minus196E minus 02minus245E minus 02minus294E minus 02minus343E minus 02

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

Wbs

(a)

153E minus 03131E minus 03109E minus 03873E minus 04655E minus 04436E minus 04218E minus 04000E + 00minus218E minus 04minus436E minus 04minus655E minus 04

minus109E minus 03minus131E minus 03minus153E minus 03

minus873E minus 04

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

ΨJ

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

116E minus 02101E minus 02860E minus 03713E minus 03565E minus 03417E minus 03270E minus 03122E minus 03minus256E minus 04minus173E minus 03minus321E minus 03minus468E minus 03minus616E minus 03minus764E minus 03minus911E minus 03

Φbs

(c)

Figure 5 Contourmaps of the basic flow at Ha = 100 (a) vertical velocity119882bs (b) stream line of electric currentΨ119869 and (c) electric potentialΦbs

Journal of Fluids 5

In this study the aspect ratio of the horizontal cross sectionis limited to 119860 = ℎ119897 = 1

1205972119882bs1205971198832 + 1205972119882bs1205971198842 + Θbs +Ha2119869119909bs = 0Θbs = minus119883

120597119869119909bs120597119883 + 120597119869119910bs120597119884 = 0119869119909bs = minus120597Φbs120597119883 minus119882bs

119869119910bs = minus120597Φbs120597119884

(3)

Here the subscript character bs represents the basic stateThe vertical component of velocity 119882 the streamlines ofelectric current densityΨ119895 and the electric potentialΦ of thebasic state at the Hartmann number Ha = 100 are depicted inFigure 5

6 Linear Stability Analysis

Disturbance equations are derived for investigating the flowinstability First of all velocity pressure temperature electricpotential and electric current density are represented withthe sum of the solution of the basic state and the infinitesimaldisturbances [15]

k = kbs + k1015840 119901 = 119901bs + 1199011015840119879 = 119879bs + 1198791015840 120601 = 120601bs + 1206011015840j = jbs + j1015840

(4)

We assume that the infinitesimal disturbances are repre-sented by the normal mode as follows

(k1015840 1199011015840 1198791015840 1206011015840 j1015840) = (k 119901 120601 j) exp (i119886119911 + 120582119905) (5)

Note that 119894 is the imaginary unit 120582 is the complexgrowth rate which is a complex number and 119886 is thewavenumber which is a real number hereThe dimensionlessequations obtained by substituting (4) and (5) to (1) are shownbelow It is known that the natural convection flow in avertical slot becomes unstable for standing wave disturbancerather than for travelling wave disturbance in the range of

Table 2 Variation of the neutral Grashof number for several casesof number of grids at Pr = 0025 119896 = 083 and Ha = 50Number of meshes Gr

2nd-order 4th-order30 times 30 228063 21912250 times 50 217764 21525070 times 70 215361 214276

k

Gr n

077 078 079 08 081 082 083 084 085 086 0873100

3105

3110

3115

3120

3125

3130

Figure 6 Neutral stability curve of the Grashof number at Pr =0025 and Ha = 100

Pr lt 1245 [14]Therefore we assumed that the neutral stablestate of the flow of Pr = 0025 is taken as 120582 = 0

120597120597119883 + 120597120597119884 + 119894119896 = 0119894119882bs119896 = minus 120597120597119883 + 1

Gr( 12059721205971198832 + 12059721205971198842 minus 1198962) + Ha2

Gr(minus119869119911)

119894119882bs119896 = minus120597120597119884 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) 120597119882bs120597119883 + 120597119882bs120597119884 + 119894119882bs119896

= minus119894119896 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) + 1Gr

Θ + Ha2

Gr119869119909

120597119869119909120597119883 + 120597119869119910120597119884 + 119894119896119869119911 = 0 119869119909 = minus120597Φ120597119883 minus 119869119910 = minus120597Φ120597119884 119869119911 = minus119894119896Φ +

(6)

Table 2 shows variation of the critical Grashof number onthe number of grids at the dimensionless wavenumber 119896 =083 and Hartmann number Ha = 50 when discretized with

6 Journal of Fluids

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(a)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(c)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(d)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(e)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(f)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(g)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(h)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(i)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(j)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(k)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(l)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(m)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(n)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(o)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(p)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(q)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(r)

Figure 7 Contour maps of the characteristic function at a marginal state at Pr = 0025 119896 = 083 Gr = 21525 and Ha = 50 (a) real partof (b) imaginary part of (c) real part of (d) imaginary part of (e) real part of (f) imaginary part of (g) real part of (h)imaginary part of (i) real part of Θ (j) imaginary part of Θ (k) real part of 119869119909 (l) imaginary part of 119869119909 (m) real part of 119869119910 (n) imaginarypart of 119869119910 (o) real part of 119869 (p) imaginary part of 119869 (q) real part of Φ and (r) imaginary part of Φ

Journal of Fluids 7

Ha

kc

kc

Gr c

Grc

0 2 4 6 8 10 12068

072

076

08

084

088

2000

2400

2800

3200

3600

4000

Figure 8 Variation of the critical Grashof number and the criticalwavenumber against the Hartmann number at Pr = 0025

a second-order or a fourth-order accurate central differenceConsidering the result of Table 2 and the computational timewe employed 70 times 70 mesh number with the second-orderaccurate central difference For example the neutral stabilitycurve of theGrashof number atHartmann numberHa = 100is depicted in Figure 6 The critical Grashof numbers at thewavenumber 0821 with mesh size 70 times 70 90 times 90 and 120 times120 take the value of 3101 3109 and 3114 respectively So thedependency of themesh size on the result is not so significant

Characteristic functions for various variables at a neutralstate are illustrated in Figure 7The pressure and the potentialdistribution have point symmetry and have reverse signbetween the real and the imaginary parts with respect to119883 =0 and 119884 = 0 On the other hand the velocity temperatureand electric current density distribution have point symmetrywith respect to119883 = 0 and 119884 = 0

The critical Grashof number and the wavenumber plottedagainst the Hartmann number up to 12 are shown in Fig-ure 8 Concerning the application of a horizontal Bridgemancrystal growth Lyubimov et al [16] studied the stability ofthermal convection in a rectangular cross section of an ob-long channel They studied two cases of computation themagnetic field applied in the horizontal direction or in thevertical direction with changing the Prandtl number and theaspect ratio of cross section It should be noted that thepresent study could be compared with the study of Lyubi-mov et al at the case of zero value of Prandtl number (Pr = 0)In the configuration of horizontal Bridgeman crystal growththe plane-parallel flow does not exist in nonzero values ofPrandtl number but in the present study the plane-par-allel flow always exists irrespective of Prandtl number sincethe basic thermal state is independent of Prandtl number Ac-cording to the results of horizontal magnetic field by Lyubi-mov et al the critical Grashof number takes its mini-mum at a certain low Hartmann number and the critical

wavenumber takes its maximum at another low Hartmannnumber This tendency is quite coincident with the findingof the present study This indicates that a moderate magneticfield destabilizes the natural convection of plane-parallel flowbut a strong magnetic field stabilizes the convection

7 Conclusion

In this study we studied the linear stability of thermalconvection in an infinitely long vertical rectangular channelin the presence of a uniform horizontal magnetic field Weobtained some findings First the characteristic function ofthe pressure and the electric potential distribution have pointsymmetry and have reverse sign between the real and theimaginary parts with respect to the center of cross sectionOnthe other hand the velocity temperature and electric currentdensity distribution have point symmetry with respect to thecenter of cross section Second except in the range of theHartmann number of 0 lt Ha lt 8 increase in the Hartmannnumber leads to well stabilization And the critical Grashofnumber takes a minimum value at the Hartmann number ofabout Ha = 45 Third the critical wavenumber increasesat a low Hartmann number but then decreases for furtherincrease in the Hartmann number

Nomenclature

119886 Wavenumber [radm]119860 Aspect ratio (= ℎ119897)1198610 Applied magnetic field [T]e1198610 Unit vector (direction of 1198610)e119910 Unit vector (direction of 119910-axis)e119911 Unit vector (direction of 119911-axis)Gr Grashof number119892 Gravitational acceleration [ms2]ℎ Duct width in the 119910-axis [m]Ha Hartmann number119894 Imaginary unitj Electric current density vector [Am2]J Dimensionless electric current density119869119909bs Dimensionless electric current density of

the basic flow (along 119909-axis)119869119910bs Dimensionless electric current density ofthe basic flow (along 119910-axis)119869119909 Dimensionless electric current density ofthe amplitude (along 119909-axis)119869119910 Dimensionless electric current density ofthe amplitude (along 119910-axis)119869 Dimensionless electric current density ofthe amplitude (along 119911-axis)119896 Dimensionless wavenumber (along 119911-axis)119897 Duct width in the 119909-axis [m]119901 Pressure [Pa]119875 Dimensionless pressure Dimensionless pressure of the amplitude

Pr Prandtl number1198790 Reference temperature [K]119879 Temperature [K]

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

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ThermodynamicsJournal of

Journal of Fluids 3

g

Hot Cold

X = 1X = minus1

Figure 3 Simplified calculation model (119860 = ℎ119897 rarr infin)

Table 1 Comparisons of the critical Grashof number and the criticalwavenumber between a reference and the present research at Pr =067 and 75

MethodReference [4] Present

Chebyshev polynomialsexpansion

Fourth-order accuratecentral difference

method with 101 gridsPr 067 75 067 75119896119888 140417 138334 140419 138349Gr119888 503293 491900 503305 491829

The instability of the convection of the infinitely long ver-tical rectangular channel is investigated by using the Grashofnumber which is a dimensionless number representing themagnitude of the influence of the inertial and buoyancyforces Figure 2 shows how the flow of a two-dimensionalvertical rectangular channel becomes unstable When theGrashof number exceeds a certain value the flow structure isdivided into a number of cells In this study the destabilizedflow in the three-dimensional vertical rectangular channelis assumed to be periodic flow in the 119885-axis direction andthen we perform the computation with applying a uniformhorizontal magnetic field perpendicular to the temperaturegradient

3 Verification of the Numerical Code

As the verification of the numerical code preliminary com-putations were performed for the model of the aspect ratio(ℎ119897 rarr infin) without magnetic field and those resultswere compared with previous results [4] The present modeland the numerical results are described in Figures 3 and 4respectively Figure 4 describes the result performed with thenumber of grids 101 using the discretization of a fourth-orderaccurate central difference method

It is noted that this graph shows result of the case of thestanding wave disturbance It is known that for the lowerPrandtl number (Pr lt 1245) [14] the neutral Grashof num-ber in the case of the standing wave disturbance is lowerthan that in the case of the traveling wave disturbance

Table 1 shows the numerical results for comparison of thecritical Grashof number and the critical wavenumber whenair (Pr = 067) and water (Pr = 75) are assumed as test fluidsThe agreement between the referencersquos result and the presentresult is negligibly small

4 Dimensionless Equations

Theflow instability in a long vertical enclosure in the presenceof a magnetic field applied in the Y-direction is investigatedhere In this study the destabilized flow is assumed to beperiodic and stationary in the 119885-axis direction and then thenumerical analyses were performed within a horizontal crosssection Due to the computational resources we focus on thecase that the aspect ratio of the horizontal cross section isunity and the Prandtl number is 0025 which is a typical valueof liquid metals

The dimensionless governing equations are shown below

120597V120597120591 + (V sdot nabla)V = minusnabla119875 + 1Gr

nabla2V + 1Gr

Θe119911 + Ha2

GrJ times e1198610

120597Θ120597120591 + V sdot nablaΘ = 1GrPr

nabla2Θnabla sdot V = 0 nabla sdot J = 0J = minusnablaΦ + V times e1198610

(1)

The dimensionless variables are defined as follows

120591 = 1199051198972 (Gr]) X = x119897 V = kGr]119897

Θ = 119879 minus 1198790Δ119879 119875 = 1199011205880Gr2]21198972 J = j

Gr]1198610 Φ = 120601Gr120590119890]1198610119897 Gr = 119892120573Δ1198791198973

]2

Pr = ]120572 Ha = 1198610119897radic 1205901198901205880] 119860 = ℎ119897

(2)

Since we limit ourselves to the case that the uniformmag-netic field is applied in the Y-direction it holds that e1198610 = e119910

5 Basic Flow

First the flow of the basic state is computed with theassumption that the rectangular enclosure is long enough toneglect recirculation area near the top and the bottom wallsTherefore the basic state of temperature field is in heat con-duction Since the Prandtl andGrashof numbers are includedin the scaling the dimensionless basic flow is independent ofthe Grashof and Prandtl numbers Dimensionless governingequations used in the computation are shown as follows

4 Journal of Fluids

142

140

138

136

134

120572c

500

480

460

440

Gc

Gc

P

120572c rarr 13827

Gc rarr 4956284

120572c rarr 1344147

Gc rarr 492450

120572

10minus310minus410minus5 10minus2 10minus1 100 101 102

(a) Reference [4]

kc

kcGr c

Grc

142

14

138

136

134

Pr

510

500

490

480

470

460

450

44010minus3 10minus2 10minus1 100 101 102

(b) Present result

Figure 4 Variations of the critical Grashof number and the critical wavenumber for the wide range of the Prandtl number (a) is referredfrom a previous study [4] and (b) indicates the present result

343E minus 02294E minus 02245E minus 02196E minus 02147E minus 02981E minus 03490E minus 03000E + 00minus490E minus 03minus981E minus 03minus147E minus 02minus196E minus 02minus245E minus 02minus294E minus 02minus343E minus 02

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

Wbs

(a)

153E minus 03131E minus 03109E minus 03873E minus 04655E minus 04436E minus 04218E minus 04000E + 00minus218E minus 04minus436E minus 04minus655E minus 04

minus109E minus 03minus131E minus 03minus153E minus 03

minus873E minus 04

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

ΨJ

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

116E minus 02101E minus 02860E minus 03713E minus 03565E minus 03417E minus 03270E minus 03122E minus 03minus256E minus 04minus173E minus 03minus321E minus 03minus468E minus 03minus616E minus 03minus764E minus 03minus911E minus 03

Φbs

(c)

Figure 5 Contourmaps of the basic flow at Ha = 100 (a) vertical velocity119882bs (b) stream line of electric currentΨ119869 and (c) electric potentialΦbs

Journal of Fluids 5

In this study the aspect ratio of the horizontal cross sectionis limited to 119860 = ℎ119897 = 1

1205972119882bs1205971198832 + 1205972119882bs1205971198842 + Θbs +Ha2119869119909bs = 0Θbs = minus119883

120597119869119909bs120597119883 + 120597119869119910bs120597119884 = 0119869119909bs = minus120597Φbs120597119883 minus119882bs

119869119910bs = minus120597Φbs120597119884

(3)

Here the subscript character bs represents the basic stateThe vertical component of velocity 119882 the streamlines ofelectric current densityΨ119895 and the electric potentialΦ of thebasic state at the Hartmann number Ha = 100 are depicted inFigure 5

6 Linear Stability Analysis

Disturbance equations are derived for investigating the flowinstability First of all velocity pressure temperature electricpotential and electric current density are represented withthe sum of the solution of the basic state and the infinitesimaldisturbances [15]

k = kbs + k1015840 119901 = 119901bs + 1199011015840119879 = 119879bs + 1198791015840 120601 = 120601bs + 1206011015840j = jbs + j1015840

(4)

We assume that the infinitesimal disturbances are repre-sented by the normal mode as follows

(k1015840 1199011015840 1198791015840 1206011015840 j1015840) = (k 119901 120601 j) exp (i119886119911 + 120582119905) (5)

Note that 119894 is the imaginary unit 120582 is the complexgrowth rate which is a complex number and 119886 is thewavenumber which is a real number hereThe dimensionlessequations obtained by substituting (4) and (5) to (1) are shownbelow It is known that the natural convection flow in avertical slot becomes unstable for standing wave disturbancerather than for travelling wave disturbance in the range of

Table 2 Variation of the neutral Grashof number for several casesof number of grids at Pr = 0025 119896 = 083 and Ha = 50Number of meshes Gr

2nd-order 4th-order30 times 30 228063 21912250 times 50 217764 21525070 times 70 215361 214276

k

Gr n

077 078 079 08 081 082 083 084 085 086 0873100

3105

3110

3115

3120

3125

3130

Figure 6 Neutral stability curve of the Grashof number at Pr =0025 and Ha = 100

Pr lt 1245 [14]Therefore we assumed that the neutral stablestate of the flow of Pr = 0025 is taken as 120582 = 0

120597120597119883 + 120597120597119884 + 119894119896 = 0119894119882bs119896 = minus 120597120597119883 + 1

Gr( 12059721205971198832 + 12059721205971198842 minus 1198962) + Ha2

Gr(minus119869119911)

119894119882bs119896 = minus120597120597119884 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) 120597119882bs120597119883 + 120597119882bs120597119884 + 119894119882bs119896

= minus119894119896 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) + 1Gr

Θ + Ha2

Gr119869119909

120597119869119909120597119883 + 120597119869119910120597119884 + 119894119896119869119911 = 0 119869119909 = minus120597Φ120597119883 minus 119869119910 = minus120597Φ120597119884 119869119911 = minus119894119896Φ +

(6)

Table 2 shows variation of the critical Grashof number onthe number of grids at the dimensionless wavenumber 119896 =083 and Hartmann number Ha = 50 when discretized with

6 Journal of Fluids

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(a)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(c)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(d)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(e)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(f)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(g)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(h)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(i)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(j)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(k)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(l)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(m)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(n)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(o)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(p)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(q)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(r)

Figure 7 Contour maps of the characteristic function at a marginal state at Pr = 0025 119896 = 083 Gr = 21525 and Ha = 50 (a) real partof (b) imaginary part of (c) real part of (d) imaginary part of (e) real part of (f) imaginary part of (g) real part of (h)imaginary part of (i) real part of Θ (j) imaginary part of Θ (k) real part of 119869119909 (l) imaginary part of 119869119909 (m) real part of 119869119910 (n) imaginarypart of 119869119910 (o) real part of 119869 (p) imaginary part of 119869 (q) real part of Φ and (r) imaginary part of Φ

Journal of Fluids 7

Ha

kc

kc

Gr c

Grc

0 2 4 6 8 10 12068

072

076

08

084

088

2000

2400

2800

3200

3600

4000

Figure 8 Variation of the critical Grashof number and the criticalwavenumber against the Hartmann number at Pr = 0025

a second-order or a fourth-order accurate central differenceConsidering the result of Table 2 and the computational timewe employed 70 times 70 mesh number with the second-orderaccurate central difference For example the neutral stabilitycurve of theGrashof number atHartmann numberHa = 100is depicted in Figure 6 The critical Grashof numbers at thewavenumber 0821 with mesh size 70 times 70 90 times 90 and 120 times120 take the value of 3101 3109 and 3114 respectively So thedependency of themesh size on the result is not so significant

Characteristic functions for various variables at a neutralstate are illustrated in Figure 7The pressure and the potentialdistribution have point symmetry and have reverse signbetween the real and the imaginary parts with respect to119883 =0 and 119884 = 0 On the other hand the velocity temperatureand electric current density distribution have point symmetrywith respect to119883 = 0 and 119884 = 0

The critical Grashof number and the wavenumber plottedagainst the Hartmann number up to 12 are shown in Fig-ure 8 Concerning the application of a horizontal Bridgemancrystal growth Lyubimov et al [16] studied the stability ofthermal convection in a rectangular cross section of an ob-long channel They studied two cases of computation themagnetic field applied in the horizontal direction or in thevertical direction with changing the Prandtl number and theaspect ratio of cross section It should be noted that thepresent study could be compared with the study of Lyubi-mov et al at the case of zero value of Prandtl number (Pr = 0)In the configuration of horizontal Bridgeman crystal growththe plane-parallel flow does not exist in nonzero values ofPrandtl number but in the present study the plane-par-allel flow always exists irrespective of Prandtl number sincethe basic thermal state is independent of Prandtl number Ac-cording to the results of horizontal magnetic field by Lyubi-mov et al the critical Grashof number takes its mini-mum at a certain low Hartmann number and the critical

wavenumber takes its maximum at another low Hartmannnumber This tendency is quite coincident with the findingof the present study This indicates that a moderate magneticfield destabilizes the natural convection of plane-parallel flowbut a strong magnetic field stabilizes the convection

7 Conclusion

In this study we studied the linear stability of thermalconvection in an infinitely long vertical rectangular channelin the presence of a uniform horizontal magnetic field Weobtained some findings First the characteristic function ofthe pressure and the electric potential distribution have pointsymmetry and have reverse sign between the real and theimaginary parts with respect to the center of cross sectionOnthe other hand the velocity temperature and electric currentdensity distribution have point symmetry with respect to thecenter of cross section Second except in the range of theHartmann number of 0 lt Ha lt 8 increase in the Hartmannnumber leads to well stabilization And the critical Grashofnumber takes a minimum value at the Hartmann number ofabout Ha = 45 Third the critical wavenumber increasesat a low Hartmann number but then decreases for furtherincrease in the Hartmann number

Nomenclature

119886 Wavenumber [radm]119860 Aspect ratio (= ℎ119897)1198610 Applied magnetic field [T]e1198610 Unit vector (direction of 1198610)e119910 Unit vector (direction of 119910-axis)e119911 Unit vector (direction of 119911-axis)Gr Grashof number119892 Gravitational acceleration [ms2]ℎ Duct width in the 119910-axis [m]Ha Hartmann number119894 Imaginary unitj Electric current density vector [Am2]J Dimensionless electric current density119869119909bs Dimensionless electric current density of

the basic flow (along 119909-axis)119869119910bs Dimensionless electric current density ofthe basic flow (along 119910-axis)119869119909 Dimensionless electric current density ofthe amplitude (along 119909-axis)119869119910 Dimensionless electric current density ofthe amplitude (along 119910-axis)119869 Dimensionless electric current density ofthe amplitude (along 119911-axis)119896 Dimensionless wavenumber (along 119911-axis)119897 Duct width in the 119909-axis [m]119901 Pressure [Pa]119875 Dimensionless pressure Dimensionless pressure of the amplitude

Pr Prandtl number1198790 Reference temperature [K]119879 Temperature [K]

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

Submit your manuscripts athttpwwwhindawicom

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 Computational  Methods in Physics

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Soft MatterJournal of

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Biophysics

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ThermodynamicsJournal of

4 Journal of Fluids

142

140

138

136

134

120572c

500

480

460

440

Gc

Gc

P

120572c rarr 13827

Gc rarr 4956284

120572c rarr 1344147

Gc rarr 492450

120572

10minus310minus410minus5 10minus2 10minus1 100 101 102

(a) Reference [4]

kc

kcGr c

Grc

142

14

138

136

134

Pr

510

500

490

480

470

460

450

44010minus3 10minus2 10minus1 100 101 102

(b) Present result

Figure 4 Variations of the critical Grashof number and the critical wavenumber for the wide range of the Prandtl number (a) is referredfrom a previous study [4] and (b) indicates the present result

343E minus 02294E minus 02245E minus 02196E minus 02147E minus 02981E minus 03490E minus 03000E + 00minus490E minus 03minus981E minus 03minus147E minus 02minus196E minus 02minus245E minus 02minus294E minus 02minus343E minus 02

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

Wbs

(a)

153E minus 03131E minus 03109E minus 03873E minus 04655E minus 04436E minus 04218E minus 04000E + 00minus218E minus 04minus436E minus 04minus655E minus 04

minus109E minus 03minus131E minus 03minus153E minus 03

minus873E minus 04

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

ΨJ

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

116E minus 02101E minus 02860E minus 03713E minus 03565E minus 03417E minus 03270E minus 03122E minus 03minus256E minus 04minus173E minus 03minus321E minus 03minus468E minus 03minus616E minus 03minus764E minus 03minus911E minus 03

Φbs

(c)

Figure 5 Contourmaps of the basic flow at Ha = 100 (a) vertical velocity119882bs (b) stream line of electric currentΨ119869 and (c) electric potentialΦbs

Journal of Fluids 5

In this study the aspect ratio of the horizontal cross sectionis limited to 119860 = ℎ119897 = 1

1205972119882bs1205971198832 + 1205972119882bs1205971198842 + Θbs +Ha2119869119909bs = 0Θbs = minus119883

120597119869119909bs120597119883 + 120597119869119910bs120597119884 = 0119869119909bs = minus120597Φbs120597119883 minus119882bs

119869119910bs = minus120597Φbs120597119884

(3)

Here the subscript character bs represents the basic stateThe vertical component of velocity 119882 the streamlines ofelectric current densityΨ119895 and the electric potentialΦ of thebasic state at the Hartmann number Ha = 100 are depicted inFigure 5

6 Linear Stability Analysis

Disturbance equations are derived for investigating the flowinstability First of all velocity pressure temperature electricpotential and electric current density are represented withthe sum of the solution of the basic state and the infinitesimaldisturbances [15]

k = kbs + k1015840 119901 = 119901bs + 1199011015840119879 = 119879bs + 1198791015840 120601 = 120601bs + 1206011015840j = jbs + j1015840

(4)

We assume that the infinitesimal disturbances are repre-sented by the normal mode as follows

(k1015840 1199011015840 1198791015840 1206011015840 j1015840) = (k 119901 120601 j) exp (i119886119911 + 120582119905) (5)

Note that 119894 is the imaginary unit 120582 is the complexgrowth rate which is a complex number and 119886 is thewavenumber which is a real number hereThe dimensionlessequations obtained by substituting (4) and (5) to (1) are shownbelow It is known that the natural convection flow in avertical slot becomes unstable for standing wave disturbancerather than for travelling wave disturbance in the range of

Table 2 Variation of the neutral Grashof number for several casesof number of grids at Pr = 0025 119896 = 083 and Ha = 50Number of meshes Gr

2nd-order 4th-order30 times 30 228063 21912250 times 50 217764 21525070 times 70 215361 214276

k

Gr n

077 078 079 08 081 082 083 084 085 086 0873100

3105

3110

3115

3120

3125

3130

Figure 6 Neutral stability curve of the Grashof number at Pr =0025 and Ha = 100

Pr lt 1245 [14]Therefore we assumed that the neutral stablestate of the flow of Pr = 0025 is taken as 120582 = 0

120597120597119883 + 120597120597119884 + 119894119896 = 0119894119882bs119896 = minus 120597120597119883 + 1

Gr( 12059721205971198832 + 12059721205971198842 minus 1198962) + Ha2

Gr(minus119869119911)

119894119882bs119896 = minus120597120597119884 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) 120597119882bs120597119883 + 120597119882bs120597119884 + 119894119882bs119896

= minus119894119896 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) + 1Gr

Θ + Ha2

Gr119869119909

120597119869119909120597119883 + 120597119869119910120597119884 + 119894119896119869119911 = 0 119869119909 = minus120597Φ120597119883 minus 119869119910 = minus120597Φ120597119884 119869119911 = minus119894119896Φ +

(6)

Table 2 shows variation of the critical Grashof number onthe number of grids at the dimensionless wavenumber 119896 =083 and Hartmann number Ha = 50 when discretized with

6 Journal of Fluids

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(a)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(c)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(d)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(e)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(f)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(g)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(h)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(i)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(j)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(k)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(l)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(m)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(n)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(o)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(p)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(q)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(r)

Figure 7 Contour maps of the characteristic function at a marginal state at Pr = 0025 119896 = 083 Gr = 21525 and Ha = 50 (a) real partof (b) imaginary part of (c) real part of (d) imaginary part of (e) real part of (f) imaginary part of (g) real part of (h)imaginary part of (i) real part of Θ (j) imaginary part of Θ (k) real part of 119869119909 (l) imaginary part of 119869119909 (m) real part of 119869119910 (n) imaginarypart of 119869119910 (o) real part of 119869 (p) imaginary part of 119869 (q) real part of Φ and (r) imaginary part of Φ

Journal of Fluids 7

Ha

kc

kc

Gr c

Grc

0 2 4 6 8 10 12068

072

076

08

084

088

2000

2400

2800

3200

3600

4000

Figure 8 Variation of the critical Grashof number and the criticalwavenumber against the Hartmann number at Pr = 0025

a second-order or a fourth-order accurate central differenceConsidering the result of Table 2 and the computational timewe employed 70 times 70 mesh number with the second-orderaccurate central difference For example the neutral stabilitycurve of theGrashof number atHartmann numberHa = 100is depicted in Figure 6 The critical Grashof numbers at thewavenumber 0821 with mesh size 70 times 70 90 times 90 and 120 times120 take the value of 3101 3109 and 3114 respectively So thedependency of themesh size on the result is not so significant

Characteristic functions for various variables at a neutralstate are illustrated in Figure 7The pressure and the potentialdistribution have point symmetry and have reverse signbetween the real and the imaginary parts with respect to119883 =0 and 119884 = 0 On the other hand the velocity temperatureand electric current density distribution have point symmetrywith respect to119883 = 0 and 119884 = 0

The critical Grashof number and the wavenumber plottedagainst the Hartmann number up to 12 are shown in Fig-ure 8 Concerning the application of a horizontal Bridgemancrystal growth Lyubimov et al [16] studied the stability ofthermal convection in a rectangular cross section of an ob-long channel They studied two cases of computation themagnetic field applied in the horizontal direction or in thevertical direction with changing the Prandtl number and theaspect ratio of cross section It should be noted that thepresent study could be compared with the study of Lyubi-mov et al at the case of zero value of Prandtl number (Pr = 0)In the configuration of horizontal Bridgeman crystal growththe plane-parallel flow does not exist in nonzero values ofPrandtl number but in the present study the plane-par-allel flow always exists irrespective of Prandtl number sincethe basic thermal state is independent of Prandtl number Ac-cording to the results of horizontal magnetic field by Lyubi-mov et al the critical Grashof number takes its mini-mum at a certain low Hartmann number and the critical

wavenumber takes its maximum at another low Hartmannnumber This tendency is quite coincident with the findingof the present study This indicates that a moderate magneticfield destabilizes the natural convection of plane-parallel flowbut a strong magnetic field stabilizes the convection

7 Conclusion

In this study we studied the linear stability of thermalconvection in an infinitely long vertical rectangular channelin the presence of a uniform horizontal magnetic field Weobtained some findings First the characteristic function ofthe pressure and the electric potential distribution have pointsymmetry and have reverse sign between the real and theimaginary parts with respect to the center of cross sectionOnthe other hand the velocity temperature and electric currentdensity distribution have point symmetry with respect to thecenter of cross section Second except in the range of theHartmann number of 0 lt Ha lt 8 increase in the Hartmannnumber leads to well stabilization And the critical Grashofnumber takes a minimum value at the Hartmann number ofabout Ha = 45 Third the critical wavenumber increasesat a low Hartmann number but then decreases for furtherincrease in the Hartmann number

Nomenclature

119886 Wavenumber [radm]119860 Aspect ratio (= ℎ119897)1198610 Applied magnetic field [T]e1198610 Unit vector (direction of 1198610)e119910 Unit vector (direction of 119910-axis)e119911 Unit vector (direction of 119911-axis)Gr Grashof number119892 Gravitational acceleration [ms2]ℎ Duct width in the 119910-axis [m]Ha Hartmann number119894 Imaginary unitj Electric current density vector [Am2]J Dimensionless electric current density119869119909bs Dimensionless electric current density of

the basic flow (along 119909-axis)119869119910bs Dimensionless electric current density ofthe basic flow (along 119910-axis)119869119909 Dimensionless electric current density ofthe amplitude (along 119909-axis)119869119910 Dimensionless electric current density ofthe amplitude (along 119910-axis)119869 Dimensionless electric current density ofthe amplitude (along 119911-axis)119896 Dimensionless wavenumber (along 119911-axis)119897 Duct width in the 119909-axis [m]119901 Pressure [Pa]119875 Dimensionless pressure Dimensionless pressure of the amplitude

Pr Prandtl number1198790 Reference temperature [K]119879 Temperature [K]

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 5

In this study the aspect ratio of the horizontal cross sectionis limited to 119860 = ℎ119897 = 1

1205972119882bs1205971198832 + 1205972119882bs1205971198842 + Θbs +Ha2119869119909bs = 0Θbs = minus119883

120597119869119909bs120597119883 + 120597119869119910bs120597119884 = 0119869119909bs = minus120597Φbs120597119883 minus119882bs

119869119910bs = minus120597Φbs120597119884

(3)

Here the subscript character bs represents the basic stateThe vertical component of velocity 119882 the streamlines ofelectric current densityΨ119895 and the electric potentialΦ of thebasic state at the Hartmann number Ha = 100 are depicted inFigure 5

6 Linear Stability Analysis

Disturbance equations are derived for investigating the flowinstability First of all velocity pressure temperature electricpotential and electric current density are represented withthe sum of the solution of the basic state and the infinitesimaldisturbances [15]

k = kbs + k1015840 119901 = 119901bs + 1199011015840119879 = 119879bs + 1198791015840 120601 = 120601bs + 1206011015840j = jbs + j1015840

(4)

We assume that the infinitesimal disturbances are repre-sented by the normal mode as follows

(k1015840 1199011015840 1198791015840 1206011015840 j1015840) = (k 119901 120601 j) exp (i119886119911 + 120582119905) (5)

Note that 119894 is the imaginary unit 120582 is the complexgrowth rate which is a complex number and 119886 is thewavenumber which is a real number hereThe dimensionlessequations obtained by substituting (4) and (5) to (1) are shownbelow It is known that the natural convection flow in avertical slot becomes unstable for standing wave disturbancerather than for travelling wave disturbance in the range of

Table 2 Variation of the neutral Grashof number for several casesof number of grids at Pr = 0025 119896 = 083 and Ha = 50Number of meshes Gr

2nd-order 4th-order30 times 30 228063 21912250 times 50 217764 21525070 times 70 215361 214276

k

Gr n

077 078 079 08 081 082 083 084 085 086 0873100

3105

3110

3115

3120

3125

3130

Figure 6 Neutral stability curve of the Grashof number at Pr =0025 and Ha = 100

Pr lt 1245 [14]Therefore we assumed that the neutral stablestate of the flow of Pr = 0025 is taken as 120582 = 0

120597120597119883 + 120597120597119884 + 119894119896 = 0119894119882bs119896 = minus 120597120597119883 + 1

Gr( 12059721205971198832 + 12059721205971198842 minus 1198962) + Ha2

Gr(minus119869119911)

119894119882bs119896 = minus120597120597119884 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) 120597119882bs120597119883 + 120597119882bs120597119884 + 119894119882bs119896

= minus119894119896 + 1Gr

( 12059721205971198832 + 12059721205971198842 minus 1198962) + 1Gr

Θ + Ha2

Gr119869119909

120597119869119909120597119883 + 120597119869119910120597119884 + 119894119896119869119911 = 0 119869119909 = minus120597Φ120597119883 minus 119869119910 = minus120597Φ120597119884 119869119911 = minus119894119896Φ +

(6)

Table 2 shows variation of the critical Grashof number onthe number of grids at the dimensionless wavenumber 119896 =083 and Hartmann number Ha = 50 when discretized with

6 Journal of Fluids

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(a)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(c)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(d)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(e)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(f)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(g)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(h)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(i)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(j)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(k)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(l)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(m)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(n)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(o)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(p)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(q)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(r)

Figure 7 Contour maps of the characteristic function at a marginal state at Pr = 0025 119896 = 083 Gr = 21525 and Ha = 50 (a) real partof (b) imaginary part of (c) real part of (d) imaginary part of (e) real part of (f) imaginary part of (g) real part of (h)imaginary part of (i) real part of Θ (j) imaginary part of Θ (k) real part of 119869119909 (l) imaginary part of 119869119909 (m) real part of 119869119910 (n) imaginarypart of 119869119910 (o) real part of 119869 (p) imaginary part of 119869 (q) real part of Φ and (r) imaginary part of Φ

Journal of Fluids 7

Ha

kc

kc

Gr c

Grc

0 2 4 6 8 10 12068

072

076

08

084

088

2000

2400

2800

3200

3600

4000

Figure 8 Variation of the critical Grashof number and the criticalwavenumber against the Hartmann number at Pr = 0025

a second-order or a fourth-order accurate central differenceConsidering the result of Table 2 and the computational timewe employed 70 times 70 mesh number with the second-orderaccurate central difference For example the neutral stabilitycurve of theGrashof number atHartmann numberHa = 100is depicted in Figure 6 The critical Grashof numbers at thewavenumber 0821 with mesh size 70 times 70 90 times 90 and 120 times120 take the value of 3101 3109 and 3114 respectively So thedependency of themesh size on the result is not so significant

Characteristic functions for various variables at a neutralstate are illustrated in Figure 7The pressure and the potentialdistribution have point symmetry and have reverse signbetween the real and the imaginary parts with respect to119883 =0 and 119884 = 0 On the other hand the velocity temperatureand electric current density distribution have point symmetrywith respect to119883 = 0 and 119884 = 0

The critical Grashof number and the wavenumber plottedagainst the Hartmann number up to 12 are shown in Fig-ure 8 Concerning the application of a horizontal Bridgemancrystal growth Lyubimov et al [16] studied the stability ofthermal convection in a rectangular cross section of an ob-long channel They studied two cases of computation themagnetic field applied in the horizontal direction or in thevertical direction with changing the Prandtl number and theaspect ratio of cross section It should be noted that thepresent study could be compared with the study of Lyubi-mov et al at the case of zero value of Prandtl number (Pr = 0)In the configuration of horizontal Bridgeman crystal growththe plane-parallel flow does not exist in nonzero values ofPrandtl number but in the present study the plane-par-allel flow always exists irrespective of Prandtl number sincethe basic thermal state is independent of Prandtl number Ac-cording to the results of horizontal magnetic field by Lyubi-mov et al the critical Grashof number takes its mini-mum at a certain low Hartmann number and the critical

wavenumber takes its maximum at another low Hartmannnumber This tendency is quite coincident with the findingof the present study This indicates that a moderate magneticfield destabilizes the natural convection of plane-parallel flowbut a strong magnetic field stabilizes the convection

7 Conclusion

In this study we studied the linear stability of thermalconvection in an infinitely long vertical rectangular channelin the presence of a uniform horizontal magnetic field Weobtained some findings First the characteristic function ofthe pressure and the electric potential distribution have pointsymmetry and have reverse sign between the real and theimaginary parts with respect to the center of cross sectionOnthe other hand the velocity temperature and electric currentdensity distribution have point symmetry with respect to thecenter of cross section Second except in the range of theHartmann number of 0 lt Ha lt 8 increase in the Hartmannnumber leads to well stabilization And the critical Grashofnumber takes a minimum value at the Hartmann number ofabout Ha = 45 Third the critical wavenumber increasesat a low Hartmann number but then decreases for furtherincrease in the Hartmann number

Nomenclature

119886 Wavenumber [radm]119860 Aspect ratio (= ℎ119897)1198610 Applied magnetic field [T]e1198610 Unit vector (direction of 1198610)e119910 Unit vector (direction of 119910-axis)e119911 Unit vector (direction of 119911-axis)Gr Grashof number119892 Gravitational acceleration [ms2]ℎ Duct width in the 119910-axis [m]Ha Hartmann number119894 Imaginary unitj Electric current density vector [Am2]J Dimensionless electric current density119869119909bs Dimensionless electric current density of

the basic flow (along 119909-axis)119869119910bs Dimensionless electric current density ofthe basic flow (along 119910-axis)119869119909 Dimensionless electric current density ofthe amplitude (along 119909-axis)119869119910 Dimensionless electric current density ofthe amplitude (along 119910-axis)119869 Dimensionless electric current density ofthe amplitude (along 119911-axis)119896 Dimensionless wavenumber (along 119911-axis)119897 Duct width in the 119909-axis [m]119901 Pressure [Pa]119875 Dimensionless pressure Dimensionless pressure of the amplitude

Pr Prandtl number1198790 Reference temperature [K]119879 Temperature [K]

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Fluids

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(a)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(b)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(c)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(d)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(e)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(f)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(g)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(h)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(i)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(j)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(k)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(l)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(m)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(n)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(o)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(p)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(q)

X

Y

0

0

05

05

1

1

minus05

minus05

minus1minus1

(r)

Figure 7 Contour maps of the characteristic function at a marginal state at Pr = 0025 119896 = 083 Gr = 21525 and Ha = 50 (a) real partof (b) imaginary part of (c) real part of (d) imaginary part of (e) real part of (f) imaginary part of (g) real part of (h)imaginary part of (i) real part of Θ (j) imaginary part of Θ (k) real part of 119869119909 (l) imaginary part of 119869119909 (m) real part of 119869119910 (n) imaginarypart of 119869119910 (o) real part of 119869 (p) imaginary part of 119869 (q) real part of Φ and (r) imaginary part of Φ

Journal of Fluids 7

Ha

kc

kc

Gr c

Grc

0 2 4 6 8 10 12068

072

076

08

084

088

2000

2400

2800

3200

3600

4000

Figure 8 Variation of the critical Grashof number and the criticalwavenumber against the Hartmann number at Pr = 0025

a second-order or a fourth-order accurate central differenceConsidering the result of Table 2 and the computational timewe employed 70 times 70 mesh number with the second-orderaccurate central difference For example the neutral stabilitycurve of theGrashof number atHartmann numberHa = 100is depicted in Figure 6 The critical Grashof numbers at thewavenumber 0821 with mesh size 70 times 70 90 times 90 and 120 times120 take the value of 3101 3109 and 3114 respectively So thedependency of themesh size on the result is not so significant

Characteristic functions for various variables at a neutralstate are illustrated in Figure 7The pressure and the potentialdistribution have point symmetry and have reverse signbetween the real and the imaginary parts with respect to119883 =0 and 119884 = 0 On the other hand the velocity temperatureand electric current density distribution have point symmetrywith respect to119883 = 0 and 119884 = 0

The critical Grashof number and the wavenumber plottedagainst the Hartmann number up to 12 are shown in Fig-ure 8 Concerning the application of a horizontal Bridgemancrystal growth Lyubimov et al [16] studied the stability ofthermal convection in a rectangular cross section of an ob-long channel They studied two cases of computation themagnetic field applied in the horizontal direction or in thevertical direction with changing the Prandtl number and theaspect ratio of cross section It should be noted that thepresent study could be compared with the study of Lyubi-mov et al at the case of zero value of Prandtl number (Pr = 0)In the configuration of horizontal Bridgeman crystal growththe plane-parallel flow does not exist in nonzero values ofPrandtl number but in the present study the plane-par-allel flow always exists irrespective of Prandtl number sincethe basic thermal state is independent of Prandtl number Ac-cording to the results of horizontal magnetic field by Lyubi-mov et al the critical Grashof number takes its mini-mum at a certain low Hartmann number and the critical

wavenumber takes its maximum at another low Hartmannnumber This tendency is quite coincident with the findingof the present study This indicates that a moderate magneticfield destabilizes the natural convection of plane-parallel flowbut a strong magnetic field stabilizes the convection

7 Conclusion

In this study we studied the linear stability of thermalconvection in an infinitely long vertical rectangular channelin the presence of a uniform horizontal magnetic field Weobtained some findings First the characteristic function ofthe pressure and the electric potential distribution have pointsymmetry and have reverse sign between the real and theimaginary parts with respect to the center of cross sectionOnthe other hand the velocity temperature and electric currentdensity distribution have point symmetry with respect to thecenter of cross section Second except in the range of theHartmann number of 0 lt Ha lt 8 increase in the Hartmannnumber leads to well stabilization And the critical Grashofnumber takes a minimum value at the Hartmann number ofabout Ha = 45 Third the critical wavenumber increasesat a low Hartmann number but then decreases for furtherincrease in the Hartmann number

Nomenclature

119886 Wavenumber [radm]119860 Aspect ratio (= ℎ119897)1198610 Applied magnetic field [T]e1198610 Unit vector (direction of 1198610)e119910 Unit vector (direction of 119910-axis)e119911 Unit vector (direction of 119911-axis)Gr Grashof number119892 Gravitational acceleration [ms2]ℎ Duct width in the 119910-axis [m]Ha Hartmann number119894 Imaginary unitj Electric current density vector [Am2]J Dimensionless electric current density119869119909bs Dimensionless electric current density of

the basic flow (along 119909-axis)119869119910bs Dimensionless electric current density ofthe basic flow (along 119910-axis)119869119909 Dimensionless electric current density ofthe amplitude (along 119909-axis)119869119910 Dimensionless electric current density ofthe amplitude (along 119910-axis)119869 Dimensionless electric current density ofthe amplitude (along 119911-axis)119896 Dimensionless wavenumber (along 119911-axis)119897 Duct width in the 119909-axis [m]119901 Pressure [Pa]119875 Dimensionless pressure Dimensionless pressure of the amplitude

Pr Prandtl number1198790 Reference temperature [K]119879 Temperature [K]

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Fluids 7

Ha

kc

kc

Gr c

Grc

0 2 4 6 8 10 12068

072

076

08

084

088

2000

2400

2800

3200

3600

4000

Figure 8 Variation of the critical Grashof number and the criticalwavenumber against the Hartmann number at Pr = 0025

a second-order or a fourth-order accurate central differenceConsidering the result of Table 2 and the computational timewe employed 70 times 70 mesh number with the second-orderaccurate central difference For example the neutral stabilitycurve of theGrashof number atHartmann numberHa = 100is depicted in Figure 6 The critical Grashof numbers at thewavenumber 0821 with mesh size 70 times 70 90 times 90 and 120 times120 take the value of 3101 3109 and 3114 respectively So thedependency of themesh size on the result is not so significant

Characteristic functions for various variables at a neutralstate are illustrated in Figure 7The pressure and the potentialdistribution have point symmetry and have reverse signbetween the real and the imaginary parts with respect to119883 =0 and 119884 = 0 On the other hand the velocity temperatureand electric current density distribution have point symmetrywith respect to119883 = 0 and 119884 = 0

The critical Grashof number and the wavenumber plottedagainst the Hartmann number up to 12 are shown in Fig-ure 8 Concerning the application of a horizontal Bridgemancrystal growth Lyubimov et al [16] studied the stability ofthermal convection in a rectangular cross section of an ob-long channel They studied two cases of computation themagnetic field applied in the horizontal direction or in thevertical direction with changing the Prandtl number and theaspect ratio of cross section It should be noted that thepresent study could be compared with the study of Lyubi-mov et al at the case of zero value of Prandtl number (Pr = 0)In the configuration of horizontal Bridgeman crystal growththe plane-parallel flow does not exist in nonzero values ofPrandtl number but in the present study the plane-par-allel flow always exists irrespective of Prandtl number sincethe basic thermal state is independent of Prandtl number Ac-cording to the results of horizontal magnetic field by Lyubi-mov et al the critical Grashof number takes its mini-mum at a certain low Hartmann number and the critical

wavenumber takes its maximum at another low Hartmannnumber This tendency is quite coincident with the findingof the present study This indicates that a moderate magneticfield destabilizes the natural convection of plane-parallel flowbut a strong magnetic field stabilizes the convection

7 Conclusion

In this study we studied the linear stability of thermalconvection in an infinitely long vertical rectangular channelin the presence of a uniform horizontal magnetic field Weobtained some findings First the characteristic function ofthe pressure and the electric potential distribution have pointsymmetry and have reverse sign between the real and theimaginary parts with respect to the center of cross sectionOnthe other hand the velocity temperature and electric currentdensity distribution have point symmetry with respect to thecenter of cross section Second except in the range of theHartmann number of 0 lt Ha lt 8 increase in the Hartmannnumber leads to well stabilization And the critical Grashofnumber takes a minimum value at the Hartmann number ofabout Ha = 45 Third the critical wavenumber increasesat a low Hartmann number but then decreases for furtherincrease in the Hartmann number

Nomenclature

119886 Wavenumber [radm]119860 Aspect ratio (= ℎ119897)1198610 Applied magnetic field [T]e1198610 Unit vector (direction of 1198610)e119910 Unit vector (direction of 119910-axis)e119911 Unit vector (direction of 119911-axis)Gr Grashof number119892 Gravitational acceleration [ms2]ℎ Duct width in the 119910-axis [m]Ha Hartmann number119894 Imaginary unitj Electric current density vector [Am2]J Dimensionless electric current density119869119909bs Dimensionless electric current density of

the basic flow (along 119909-axis)119869119910bs Dimensionless electric current density ofthe basic flow (along 119910-axis)119869119909 Dimensionless electric current density ofthe amplitude (along 119909-axis)119869119910 Dimensionless electric current density ofthe amplitude (along 119910-axis)119869 Dimensionless electric current density ofthe amplitude (along 119911-axis)119896 Dimensionless wavenumber (along 119911-axis)119897 Duct width in the 119909-axis [m]119901 Pressure [Pa]119875 Dimensionless pressure Dimensionless pressure of the amplitude

Pr Prandtl number1198790 Reference temperature [K]119879 Temperature [K]

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Fluids

Dimensionless velocity of the amplitude (along119909-axis)

v Velocity vector [ms]V Dimensionless velocity vector Dimensionless velocity of the amplitude (along119910-

axis)119882bs Dimensionless velocity of the basic flow (along 119911-axis) Dimensionless velocity of the amplitude (along 119911-axis)120572 Thermal diffusivity [m2s]120573 Volumetric thermal expansion coefficient [Kminus1]Θbs Dimensionless temperature of the basic flowΘ Dimensionless temperature of the amplitude120582 Complex growth rate [rads]

] Kinematic viscosity [m2s]1205880 Fluid density at reference temperature [kgm3]120590119890 Electric conductivity [Ωminus1mminus1]120601 Electric potential [V]Φbs Dimensionless electric potential of the basic flowΦ Dimensionless electric potential of the amplitude

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] Y Shimomura ldquoThe new ITERrdquo Journal of Plasma and FusionResearch vol 79 no 9 pp 949ndash961 2000

[2] C P C Wong S Malang M Sawan et al ldquoAn overview ofdual coolant Pb-17Li breeder first wall and blanket conceptdevelopment for the US ITER-TBMdesignrdquo Fusion Engineeringand Design vol 81 no 1ndash4 pp 461ndash467 2006

[3] T Tanaka T Muroga T Shikama et al ldquoElectrical insulatingproperties of ceramic coatingmaterials for liquid Li blanket sys-tem under irradiationrdquo Journal of Plasma and Fusion Researchvol 83 no 4 pp 391ndash396 2007

[4] The society of Fluid Mechanics of Japan Handbook of FluidMechanics (Japanese) The society of Fluid Mechanics of JapanTokyo Japan 2002

[5] H Ozoe and K Okada ldquoThe effect of the direction of the exter-nal magnetic field on the three-dimensional natural convectionin a cubical enclosurerdquo International Journal of Heat and MassTransfer vol 32 no 10 pp 1939ndash1954 1989

[6] K Okada and H Ozoe ldquoExperimental heat transfer rates ofnatural convection of molten gallium suppressed under anexternalmagnetic field in either theX Y or Z directionrdquo Journalof Heat Transfer vol 114 no 1 pp 107ndash114 1992

[7] T Tagawa and H Ozoe ldquoEnhancement of heat transfer rate byapplication of a static magnetic field during natural convectionof liquid metal in a cuberdquo Journal of Heat Transfer vol 119 no2 pp 265ndash271 1997

[8] T Tagawa and H Ozoe ldquoEnhanced heat transfer rate measuredfor natural convection in liquid gallium in a cubical enclosureunder a static magnetic fieldrdquo Journal of Heat Transfer vol 120no 4 pp 1027ndash1032 1998

[9] G Authie T Tagawa and RMoreau ldquoBuoyant flow in long ver-tical enclosures in the presence of a strong horizontal magneticfield Part 2 Finite enclosuresrdquo European Journal of MechanicsBFluids vol 22 no 3 pp 203ndash220 2003

[10] U Burr L Barleon P Jochmann and A Tsinober ldquoMagnetohy-drodynamic convection in a vertical slot with horizontal mag-netic fieldrdquo Journal of Fluid Mechanics no 475 pp 21ndash40 2003

[11] C Zhang S Eckert and G Gerbeth ldquoThe flow structure of abubble-driven liquid-metal jet in a horizontal magnetic fieldrdquoJournal of Fluid Mechanics vol 575 pp 57ndash82 2007

[12] K Fujimura andM Nagata ldquoDegenerate 1 2 steady state modeinteraction-MHD flow in a vertical slotrdquo Physica D NonlinearPhenomena vol 115 no 3-4 pp 377ndash400 1998

[13] T Kakutani ldquoThe hydromagnetic stability of themodified planeCouette flow in the presence of a transverse magnetic fieldrdquoJournal of the Physical Society of Japan vol 19 no 6 pp 1041ndash1057 1964

[14] K Fujimura and J Mizushima ldquoNonlinear equilibrium solu-tions for travelling waves in a free convection between verticalparallel platesrdquo European Journal of Mechanics BFluids vol 10no 2 pp 25ndash31 1991

[15] T P Lyubimova D V Lyubimov V A Morozov R V ScuridinH B Hadid and D Henry ldquoStability of convection in a hori-zontal channel subjected to a longitudinal temperature gradientPart 1 Effect of aspect ratio and Prandtl numberrdquo Journal ofFluid Mechanics vol 635 pp 275ndash295 2009

[16] D V Lyubimov T P Lyubimova A B Perminov D Henry andH B Hadid ldquoStability of convection in a horizontal channelsubjected to a longitudinal temperature gradient Part 2 Effectof amagnetic fieldrdquo Journal of FluidMechanics vol 635 pp 297ndash319 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of