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OnTheCOMSOLSoftwareAbilityOnStudyingTransitionFlowsForLowPrandtlNumberFluidsH.
Jamai1,A.
Elamari1,M.
ElGanaoui2,F.
S.
Oueslati1,B.
Pateyron2etH.
Sammouda31LETTM,Departementofphysics,F.
S.
T,Tunisia.
2SPCTSUMR6638CNRS,FaculteofSciences,87000Limoges,France.
3ESST-H.
Sousse,Tunisiehanene_jamai@yahoo.
frAbstract:COMSOLindustrialsoftwareisofferinganimportantalternativetohomecodesformodelingandsimulationofcomplexproblemswithincludingcoupledeffectsonHeatandMasseTransfers.
ThepresentworkfocusesonlowPrandtlnumberfluidmeltssubjecttosymmetrybreakingandtransitiontounsteadyregimes.
TheseconfigurationsareforpracticalinterestincrystalgrowthindustrynamelytheBridgmanconfigurationiswidelyusedtoproducesinglecrystalsofcompoundsemi-conductorsand2Denclosuresprovideabasicmodelforstudyinghydrodynamicregimeoccurringonsuchprocesses.
Afirststepinthisworkistostudytheeffectofmagneticfieldonliquidmetalflowandheattransferoccurringinanannularcavity.
Keywords:Naturalconvection,magneto-convection,moltenmetal(fluidsonlowPrandtlnumber),annularcavityNomenclatureΤAspectratioDiensionlessmagneticvector(tesla)GravitationalvectoraccelerationΤ)Cylinderheight(m)ElectricVectordensityΤRatioofraysDimonsionlesspressiontime(s)Differenceofrays(m)TDimensionlesstemperaturr,zDimensionlessradialandaxialcoordinatesu,wRadialandaxialvelocityvectorcomponentsGrecsLettersThermaldiffusivity(m2/s)Thermalexpansioncoefficient(1/k)Density(kg/m3)Dynamicviscosity,(kg/m.
s)ElectricConductivity(S/m)ElectricpotentielleFonction1.
IntroductionLiquidsmetals,inthepresenceofamagneticfield(magneto-convection)havearethesubjectofagreatnumberofresearches.
Theinterestoftheseflowsliesintheirpresenceinmanynaturalandappliedphenomena.
Inthesameway,metallurgicalindustry,coolingenginesinnuclearindustry,crystallinegrowthfortheindustryofthesemiconductorsgenerateseveralquestionsforcontrollingthestabilityoftheseflows[1,2,3].
Theappearanceoftheconvectionduringcrystalgrowthcanproduceinhomogenuitieswhichleadtostriationsandwhichaffectthequalityoftheobtainedcrystals[4].
Inthesecases,theapplicationofmagneticfieldtothethermalconvectionappearsofgreatimportanceforthecontrolofthestabilityoftheseflowsandtheheattransferswhileresulting.
Ourstudyrelatestotheconvectiveflow,ofafluidoflowPrandtlnumberinanannularandverticalcavityformedbytwocoaxialcylinders,differentiallyheatedinthepresenceofaconstantmagneticfield[5,6].
2.
MathematicalModelThesystemconsideredinthispaperisshownschematicallyinfig.
1Boussinesqapproximationisadopted,thatthedensityoffluidisgivenby:001TTT(1)Thedimensionlessequationswhichdescribetheproblem,arerespectivelyequationsof,continuity,conservationofthemomentum,theOhmlawandtheenergyconservationequation:Figure1.
Geometricalconfiguration0uuwrrz(1)22222221PrPr.
.
BruuuvPuuuuuwHajeutrzrrrrrzr(2)222221PrPrPr.
.
BzwwwPwwwuwRaTHajeutrzzrrrz(3)()BjVe(4)22221TTTTTTuwtrzrrrz(6)WithBeistheunitvectorinthedirectionofmagneticfieldB3RagTl,.
.
HalB,andPraretheRayleigh,HartmannandPrandtlnumbers,whicharethedimensionlessnumbersgoverningtheproblem.
TheboundaryconditionsTheboundaryconditionsadoptedfortheresolutionoftheproblemare:HydrodynamicCondition:zerovelocityonthewalls.
u=w=0ThermalCondition:1122:,:,0,1:0rRTTrRTTzTzElectricalconditions:electricalinsulationonthewalls.
0jnThegoverningequationsalongwiththeboundaryconditionsaresolvednumericallyusingtheCOMSOLcodewhichbasedonthefiniteelementmethod,allowingacouplingbetweenthedynamic,thermalandelectricproblems.
Totestandassessgridindependenceofthenumericalscheme,variousmeshesareexamined.
Veryfinenouniformgridsclosetothewallsareadopted.
Weassumethatthesolutionisconvergedwhentheerrorislessthan10-6.
3.
TheeffectofanaxialmagneticfieldInthispart,themagneticfieldissupposedaccordingtothezdirectionandweneglecttheelectrictermintheforceofLaplace.
WepresentonFigure2,theStreamlines,theisothermsandvelocityprofilesforRa=104andtheaspectratioA=1forvariousvaluesoftheHartmannnumber.
Intheabsenceofmagneticfield(Ha=0,Fig.
2(a)),theflowexhibitsasimplecirculatingpatternrisingalongthehotwallanddescendingalongthecoldwallofthecavity.
Itisinterestingtonotethatasthestrengthofthemagneticfieldincreases(Ha=40),thecentralstreamlinesareelongatedhorizontallyandthetemperaturestratificationinthecorediminishes(Fig.
2(b)).
AsHartmannnumberisfurtherincreased(Ha=100),theisothermsarealmostparallelandarenearlyconductionlike(Fig.
2(c))andthisisduetothesuppressionofconvectionbythemagneticfield.
Alsoitcanbeseenthattheflowfieldbecomesbicellularasstrengthofthemagneticfieldisincreased.
Itcanalsobeseenfromthefigure3(aandb)thattheflowoscillationinshallowcavityissuppressedmoreeffectivelybytheaxialmagneticfieldthantheradialmagneticfield.
'r'zBguruzu2TR1R2T1T2'r(a)Ha=0(b)Ha=40(c)Ha=100theStreamlinestheisothermvelocityprofilesFigure2:dynamics,thermalsFieldsandvelocityfromvariousvaluesofHartmann,A=1,k=2,Ra=104and=/2.
Thisbehaviorisconsistentwiththefactthatthemagneticfieldsuppressestheflowmoreeffectivelywhenthemagneticfieldisimposedperpendiculartothedirectionoftheflow.
1.
01.
52.
0-20-15-10-5051015wrHa=0Ha=20Ha=40Ha=60Ha=80Ha=100Ha=200-a-0,00,51,0-20-1001020uzHa=0Ha=20Ha=40Ha=60Ha=80Ha=100Ha=200-b-Figure3:Vertical(a)andhorizontal(b)velocityatmidheightfor,A=1,k=2,Ra=104and=/2.
4.
HeatTransferQuantitativeheattransferresultsarepresentedintermsofaverageNusseltnumber.
TheeffectofmagneticfieldontheheattransferrateisshowninFig.
4.
Asexpected,foragivenvalueofRaitcanbeseenthatNuisadecreasingfunctionofHa.
ThisisduetothefactthatwiththeincreaseinHartmannnumbertheconvectionisprogressivelyreducedbythemagneticdrag,resultinginalowerheattransfer.
102103104105106110__NuRaHa=020406080100Figure4:LocalheattransferratefordeferentvaluesofHaandRa5.
ConclusionThenumericalresultsindicatethatthemagneticfieldsuppressestheconvectiveflowandeliminatestheflowoscillations.
QuantitativeresultsarepresentedintermsofthelocalandaverageNusseltnumber.
TheheattransferrateincreaseswithradiiratioanddecreaseswiththeHartmannnumbers.
Thedirectionofmagneticfieldplaysanimportantroleinsuppressingtheconvectiveflows.
Themagneticfieldismoreeffectivewhenitisperpendiculartothedirectionoftheprimaryflow.
Thisphenomenonhasaseriousimplicationonthedesignofmagneticsystemsforstabilizingorweakeningtheconvectiveeffects.
Thenumericalresultsofthepresentstudyareingoodagreementwiththeexistingnumericalstudies,intheabsenceofmagneticfield.
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