Study on a falling metal drop in a perpendicular magnetic field 2022 Fernando Garzón.pdf

该【Study on a falling metal drop in a perpendicular magnetic field 2022 Fernando Garzón 】是由【贾赦】上传分享，文档一共【8】页，该文档可以免费在线阅读，需要了解更多关于【Study on a falling metal drop in a perpendicular magnetic field 2022 Fernando Garzón 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。FluidDynamicsRevistaMexicanadeF´ısica680306011–8MAY-JUNE2022
Studyonafallingmetaldropinaperpendicularmagneticfield
´
InstitutodeInvestigacion´enCienciasBasicas´yAplicadas,UniversidadAutonoma´delEstadodeMorelos,Mexico.´
´ırez
InstitutodeEnerg´ıasRenovables,UniversidadAutonoma´delMexico,´
Temixco,Morelos,62580Mexico.´

CONACYT-CIDESI,CentroNacionaldeTecnolog´ıasAeronauticas,´
Queretaro,´Qro.,76270,Mexico.´
AFigueroa
CONACYT-CentrodeInvestigacion´enCiencias,UniversidadAutonoma´delEstadodeMorelos,Mexico.´
e-mail:alfi******@
Received24June2021;accepted29October2021
Atheoreticalandexperimentalstudyofafallingmetaldropwhichinteractswithaperpendicularnon-localizedmagneticfieldisaddressed.
Asthemetaldropstraversesthemagneticfield,

Navier--Stokesfor
two--tracking/

throughtheParticleImageVelocimetry.
Keywords:Navier-Stokesequations;particleimagevelocimetry.
DOI:/
-
neticfield[5],whichdemonstratethattherisevelocityof
Magnetohydrodynamics(MHD)isconcernedwiththein-heliumbubbledecreasesmonotonicallywiththeincreaseof

fluidinquestionmustbeelectricallyconductingandnon-numericalsimulationsbymeansoftwo-phaseinterfacetreat-
ferromagneticsuchasmetal-basedfluids,plasmasandelec-ment[6,7].
trolytes[1].TheoreticalmodelsofMHDflowsareformu-Inparticular,thedynamicsofliquidmetaljets/dropsis
latedbycouplingtheNavier-StokesequationsandMaxwell’simportant,forexample,inthedevelopmentofalternative
equations,bothsystemsofequationscoupleviatheLorentzenergysourcessuchasnewplasma-basedfusionreactors
[8],orthedevelopmentofrevolutionary
MHDflowsforindustrialapplicationssuchasheatsinksinthree-dimensionalprintingbasedonliquidmetalgenerated
nuclearreactors,pumps,batteriesandlevitatingliquidmet-bymeltingwhichisprintedusinganink-jettingprocess[9].
,studiesofexoplanetatmospheressuchasthose
OntheinteractionbetweenliquidmetalandmagneticofthegiantexoplanetWASP-76b[10]demonstratethepos-
fields,
flowingthroughachanneltowhichanartificialexternalmag-thisrainfallisaffectedbyinducedMHDeffects.
neticfieldisappliedhavebeenperformed[2–4].Afluidelec-Theeffectonthemotionofaliquidmetaldropfallingin
tricallyconductingthatflowstroughachannelinthepresencepresenceofamagneticfieldwasstudiedfirstby[11].Theau-
ofanexternalmagneticfieldproducedbyamagnetsuffersathorscomputedtheforcepromotedbytheinteractionofthe
magneticbrakinginit’smomentumbyLorentzforce,.,in-inducedelectriccurrentsgeneratedintothedropandtheap-

flowduetherelativemovementtotheappliedmagneticfield,ofthemagneticfieldgradient,theelectromagneticforceis
thustheLorentzforceresultsinaninducedmagneticforceinalwaysintheoppositedirectionofthedropmotion,actingas
theoppositedirectionoffluidflow,,thestudyofafalling
-metaldropwasperformedbyanumericalsimulation[12],
perimentallyinthecaseofarisingsingleheliumbubbleinwhereadropfellintoametalliquidlayerinthe
,´,´
-up,)Hele-
x-)
ofcontrolsignalsandredlinesindicatetheflowofelectricalcurrent.
presenceofimposedverticalmagneticfield,suchresultsre-
vealedaninterestingbehaviouronmetalliquidlayerfora
strongmagneticfield,spreadingwassubstantiallyreduced
andaswellingoffreesurfaceaftercollisionwereobserved.
AninterestinginvestigationispresentedinRef.[13],where
numericalsimulationswereperformedofafallingdropletin
thepresenceofapositiveandnegativemagneticfieldgradi-
ent.
Incontrasttopreviousworks,wepresentatheoreticaland
experimentalstudyofafallingmetaldropundertheeffectof
-upis
rathersimpleandusefulwhichdoesnotrequireexpensive
highspeedcamerassinceitisaliquid-liquidphaseexperi-
,inthenumericalsimulationsarealdis-
0

-
denoteexperimentalmeasurements.

uidmetalduringtesting,lateritisplacedonascrewlinear
Theexperimentsetupconsistofaliquidmetaldropfallingactuator(sixcentimetersoftravel),thewhichimplementsa
inaHele--steppermotor(nema17)tolinearlymovetheplungerofthe
peredglassgluedtoanO-(V=)iscontrolled
’sdimensionsare230mmlong110mmbyvaryingtheangularvelocityandthenumberofstepsof
,forthispurpose,anArduinoNanomicrocontroller
isverticallyplacedandisopenintheupperside,
ArectangularparallelepipedNeodyniummagnetwithasidemetaldropswithdiameterd=)shows
,-
−axisofthemagnetmeasuresofthecellwerecalculatedsothatitsboundaries
.
,thecellwasfilled
BELL,,
fieldatthecell’,(20◦C).Thedropis
TheformationofliquidmetaldropswascarriedoutbyaneutecticalloyofGallium,IndiumandTin(GaInSn)that
-isliquidatroomtemperature;itsmassdensity,kinematic

STUDYONAFALLINGMETALDROPINAPERPENDICULARMAGNETICFIELD3
3
viscosityandelectricalconductivityareρ1=6360Kg/m,
ν=×10−7m2/sandσ=×106S/m,respec-equationiswritteninaformthatincludesthedifferentfluids
3
(ρ2=1261Kg/m)isseededwithandthesurfacetensionforce:
hollowglassspheres,withanapproximateddiameterof10
∂ρu
µ+∇·ρuu=−∇p+∇·µ(∇u+∇Tu)
∂t
metal(537±13mN/m[14])promotesthattheflowisdomi-Z
natedbythesurfacetensionforceandthedropstakeanap-00β000
+σκnδ(x−x)ds+(ρ−ρf)g+j×B.(2)
proximatelyacircularand/orellipticalshapewithintheHele

Thepressureisdenotedbyp,µisthedynamicviscosity,gis
thedragwiththeglycerin,imagesofthefallingmetaldrop
thegravityvector,ρisthedensityofthecontinuousphase,
wererecordedwithacommonphotographiccameraratherf
σisthesurfacetensioncoefficient,kisthecurvature,jisthe

-
micro-nikkor60mmf/,
fectoftheinterfaceisaccountedintheintegraltermthatrep-
animageanalysisalgorithmwasusedtodetectthepixelsthat
resentsthesurfacetensionforceconcentratedattheboundary
correspondtotheboundaryofthedropintheplaneofthecell
-
(CircularHoughalgorithm).TheParticleImageVelocimetry
terfacethiscanberepresentedastwo-dimensionaldeltafunc-
(PIV)techniquewasusedinordertoobtainvelocityvector
tionδβ(x−x0)=δ(x−x0)δ(y−y0),wherexisthepointat

whichtheequationsareevaluatedandx0isapointinthein-
×,wherethedrophas

×720pix-
fieldisgivenby:
,weobtained24
-¡¢
0=η∇2b+B0·∇u−(u·∇)B0,(3)
formtheanalysis[15];weusedinterrogationareasof32×32
pixelswith50%overlapinthehorizontalandverticaldirec-
whereηisthemagneticdiffusivity,bandB0aretheinduced
tions,
andappliedmagneticfields,,both
datawasobtainedbyadjustinganormaldistributiontothe
theappliedandinducedmagneticfieldaresolenoidalandsat-
%.
isfy:
0
∇·b=0,∇·B=0.(4)
,theinduced
electriccurrentcanbefoundas:
Consideratwodimensionalfluidcomposedbytwo-
i
immisciblefluids(acontinuousphaseandaliquidmetal∇×b=µmj,(5)
droplet)-
tributionofthemagneticfieldismodeledasinRef.[16]thatwhereµ.(1)-
correspondstothedistributionfoundexperimentally,asseen(5)werediscretizedusingameshwith1024×2048grid
−andy−direc-
-
betweenthetwofluidsistakenintoaccountinthemodel,-correctormethodwas
-
normalandtangentialstressesthatariseduetotheinteractionslipwallsthetopandbottomboundaries,andareseparated
-
ityoftheproblem,itwasapproachednumericallybyafinitesolidwallsandarelocated32dropdiametersawayfromeach
volume/front-trackingmethodwhereasinglesetofconserva-other,therefore,thecomputationaldomainissimilarinsize
tionequations[17]aresolvedfortheentiredomain,-
.(3),thezeroboundaryconditionson
ofthenumericalimplementationcanbefoundinRef.[18].bareimposedatallthewallsofthecomputationaldomain.
TheequationsthatdescribethemotionofthedropsandtheThenumericalsimulationswerecarriedoutforsimilarcondi-

withelectromagneticequationsforcomputingtheinducedappliedmagneticfielddistributionisnotconstant,thusthe
magneticfieldwiththelowmagneticReynoldsnumberap-motionoftheconductingdropthoroughthemagneticfield
proximation[1,19].Consideringthatthetwofluidsarein-willpromotetheexistenceoftheinducedmagneticfieldand
compressible,themassconservationisgivenby:consequentlytheinducedcurrentsaccordingtoEqs.(3)and
(5),respectively,whichwillgenerateaLorentzforcethatwill
∇·u=0,(1)brakethemotionofthemetaldrop.

,´,´
productoftheelectricdensitycurrentandtheappliedmag-
~i~002
neticfield(j×B),themagneticforce
02
-
tionEq.(3),whichforthex−yplaneitreads
µ22¶00
1∂bz∂bz∂Bz∂Bz
0=2+2−u−v,(9)
µ0σ∂x∂y∂x∂y
consideringthatthedroponlymovesinthenegativey-direc-
tionandtheinducedmagneticfieldonlydependsthevertical
direction,itgoesas
20