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MicroandNanofluid ConvectionwithMagnetic FieldEffectsforHeatand MassTransferApplications UsingMATLABs Editedby
CHAKRAVARTHULAS.K.RAJU
DepartmentofMathematics,GITAMUniversity,Visakhapatnam,India
ILYASKHAN BasicEngineeringSciencesDepartment,CollegeofEngineering MajmaahUniversity,AlMajma'ah,SaudiArabia
SURESHKUMARRAJUS.
DepartmentofMathematicsandStatistics,KingFaisalUniversity, Hofuf,SaudiArabia
MAMATHAS.UPADHYA
DepartmentofMathematics,KristuJayantiCollege(Autonomous), Bangalore,India
Elsevier
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Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopying,recording,oranyinformationstorageand retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek permission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangements withorganizationssuchastheCopyrightClearanceCenterandtheCopyrightLicensingAgency, canbefoundatourwebsite: www.elsevier.com/permissions .
Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe Publisher(otherthanasmaybenotedherein).
MATLABs isatrademarkofTheMathWorks,Inc.andisusedwithpermission.TheMathWorks doesnotwarranttheaccuracyofthetextorexercisesinthisbook.Thisbook’suseordiscussion ofMATLABs softwareorrelatedproductsdoesnotconstituteendorsementorsponsorshipby TheMathWorksofaparticularpedagogicalapproachorparticularuseoftheMATLABs software.
Notices
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Listofcontributors ix
Abouttheeditors xi
Preface xiii
1.Backgroundtomicro-andnanofluids1
MamathaS.UpadhyaandC.S.K.Raju
References 5
2.Mathematicalmodelingofequationsofcouplestressfluidin respectivecoordinates7
C.S.K.Raju,MamathaS.UpadhyaandIlyasKhan
2.1 Basicflowequations7
2.2 Equationsofmotion8
2.3 Equationsofmotionbystresstensor9
2.3.1 IntheCartesiancoordinatessystem9
2.3.2 Inthecylindricalcoordinatessystem13
2.3.3 Inthesphericalcoordinatessystem16
2.4 Equationsofmotionbyvectorcalculus21
2.4.1 IntheCartesiancoordinatessystem21
2.4.2 Inthecylindricalcoordinatessystem24
2.4.3 Inthesphericalcoordinatessystem26
References 30
3.Mathematicalmodelofsteadyincompressiblenanofluidfor heattransferapplicationsusingMATLABs 31
SathySureshandS.R.Shanthi
3.1 Introduction31
3.2 Problemdescription33
3.3 Methodofsolution36
3.4 AlgorithmandimplementationofMATLABs 38
3.5 Resultsanddiscussion41
3.6 Conclusion 57 References 58
4.Mathematicalmodelforanincompressibleunsteadynanofluidflowwith heattransferapplication59
S.Rajamani,G.Venkatesan,A.SubramanyamReddy,A.K.Shukla,K.Jagadeshkumar andS.Srinivas
4.1 Introduction59
4.2 Formulationoftheproblem61
4.3 Resultsanddiscussion64
5.Mathematicalmodelforincompressibleunsteadynanofluidfluidflowwith heatandmasstransferapplication75 G.P.Ashwinkumar
6.Stefanblowingeffectonnanofluidflowoverastretchingsheet inthepresenceofamagneticdipole91
R.NaveenKumar,R.J.PunithGowda,B.C.PrasannakumaraandC.S.K.Raju
6.1 Introduction
6.2 Mathematicalformulation94
6.2.1 Conditionsandassumptionsofthemodel95
6.2.2 Geometryoffluidflow95
6.2.3 Modelequations95
6.2.4 Nonuniformheatsource/sink96
6.2.5 Magneticdipole96
6.3 Thesolutiontotheproblem97
6.3.1 Expressionofparameters98
6.3.2 Physicalquantitiesofinterest98
6.4 Numericalmethod98
6.4.1 Convergenceanderrortolerance100
6.5 Resultsanddiscussion100
6.5.1 Velocityandthermalprofile100
6.5.2 Concentrationprofile104
6.5.3 Physicalquantitiesofpracticalinterest106
6.6 Conclusions109 References 109
7.Nonlinearunsteadyconvectiononmicroandnanofluidswith Cattaneo-Christovheatflux113 MamathaS.UpadhyaandC.S.K.Raju
Nomenclature 113
7.1 Introduction114
7.2 Problemdevelopments116
7.3 Graphicaloutcomesanddiscussion120
7.4 Conclusions129 References 130
8.Comparisonofsteadyincompressiblemicropolarandnanofluidflow withheatandmasstransferapplications133 SathySuresh,S.R.ShanthiandMamathaS.Upadhya
8.1 Introduction133
8.2 Formulation135
8.3 Entropygeneration139
8.4 Numericalprocedure140
8.5 Resultsanddiscussion141
8.6 Concludingremarks149 References 149
9.Comparisonofunsteadyincompressiblemicropolarandnanofluid flowwithheattransferapplications153
D.Rajkumar,K.Govindarajulu,T.Thamizharasan,A.SubramanyamReddy, K.Jagadeshkumar,S.SrinivasandA.K.Shukla
9.1 Introduction153
9.2 Formulationoftheproblem156
9.3 Resultsanddiscussion160
9.3.1 Velocitydistribution161
9.3.2 Angularmomentumdistribution163
9.3.3 Temperaturedistribution164
9.3.4 Nusseltdistribution164
9.4 Conclusion166 References 166
10.ImplementationofboundaryvalueproblemsinusingMATLAB s 169
MamathaS.UpadhyaandC.S.K.Raju
10.1 IntroductiontoMATLABs 169
10.1.1 Plottingofcurvesandsurfaces169
10.2 Vectorfieldandgradient176
10.2.1 Aim176
10.3 Limitsandcontinuity180
10.3.1 Aim180
10.4 Definiteintegralsandtheirapplications185
10.4.1 Aim185
10.5 Localmaximaandlocalminima189
10.5.1 Aim189
10.6 Lagrange’smultipliersmethod194
10.6.1 Aim194
10.7 Multipleintegrals199
10.7.1 Aim199
10.7.2 Volumeofasolidregion199
10.7.3 Changeofvariables:polarcoordinates199
10.8 Applicationsofderivatives205
10.8.1 Aim205
10.8.2 Maximumandminimumforasinglevariable207
10.9 Casestudy212
10.9.1 Introduction212
10.9.2 Methodology213
10.9.3 MATLABs implementation214
10.9.4 Resultsanddiscussion216
10.9.5 Conclusion220
10.10 Navier StokesequationsolvingusinganODEsolver221
10.11 Solvingtheinitialvalueproblem223
10.12 Solvingtwocouplednonlinearequations224
10.13 Interpretingtheresults228 Furtherreading
Listofcontributors G.P.Ashwinkumar
DepartmentofMathematics,VijayanagaraSriKrishnadevarayaUniversity,Bellary,Karnataka, India
K.Govindarajulu
DepartmentofMathematics,SchoolofAdvancedSciences,VelloreInstituteofTechnology, Vellore,TamilNadu,India
K.Jagadeshkumar
DepartmentofMathematics,SchoolofAdvancedSciences,VelloreInstituteofTechnology, Vellore,TamilNadu,India
IlyasKhan
DepartmentofMathematics,MajmaahUniversity,AlMajma'ah,MajmaahCity,SaudiArabia
R.NaveenKumar
DepartmentofStudiesandResearchinMathematics,DavangereUniversity,Davangere, Karnataka,India
B.C.Prasannakumara
DepartmentofStudiesandResearchinMathematics,DavangereUniversity,Davangere, Karnataka,India
R.J.PunithGowda
DepartmentofStudiesandResearchinMathematics,DavangereUniversity,Davangere, Karnataka,India
S.Rajamani
DepartmentofMathematics,SchoolofAdvancedSciences,VelloreInstituteofTechnology, Vellore,TamilNadu,India
D.Rajkumar
DepartmentofMathematics,SchoolofAdvancedSciences,VelloreInstituteofTechnology, Vellore,TamilNadu,India
C.S.K.Raju
DepartmentofMathematics,GITAMSchoolofScience,GITAMDeemedtobeUniversity, Bengaluru,Karnataka,India
A.SubramanyamReddy
DepartmentofMathematics,SchoolofAdvancedSciences,VelloreInstituteofTechnology, Vellore,TamilNadu,India
S.R.Shanthi
DepartmentofMathematics,CambridgeInstituteofTechnology,Bengaluru,Karnataka,India
A.K.Shukla
DepartmentofMathematics,SchoolofAdvancedSciencesandLanguages,VIT-Bhopal University,Bhopal,MadhyaPradesh,India
x Listofcontributors
S.Srinivas
DepartmentofMathematics,SchoolofAdvancedSciences,VIT-APUniversity,Amaravati, AndhraPradesh,India
SathySuresh
DepartmentofMathematics,VemanaInstituteofTechnology,Bengaluru,Karnataka,India; DepartmentofMathematics,CambridgeInstituteofTechnology,Bengaluru,Karnataka,India
T.Thamizharasan
DepartmentofMathematics,SchoolofAdvancedSciences,VelloreInstituteofTechnology, Vellore,TamilNadu,India
MamathaS.Upadhya
DepartmentofComputerScience,KristuJayantiCollege(Autonomous),Bengaluru, Karnataka,India
G.Venkatesan
DepartmentofMathematics,SchoolofAdvancedSciences,VelloreInstituteofTechnology, Vellore,TamilNadu,India
Abouttheeditors Dr.ChakravarthulaS.K.Raju, GITAMSchoolofScience, GITAMUniversity,Bengaluru-Campus,Karnataka,India
Dr.C.S.K.Raju worksatGITAMUniversityinIndia.Dr. Raju’sareasofinterestincludemathematicalmodeling,nanoandmicrofluidmodeling,statisticalmechanics,Newtonianand non-Newtonianliquids,andmachinelearningtechniques.He istheauthorofseveralbooksandbookchapters.Healsoacts asaneditorialboardmemberandreviewerforvariousISIand Scopus-indexedpublisherssuchastheAmericanSocietyof MechanicalEngineering,Elsevier,TaylorandFrancis,Wiley,andSpringer.Dr.Raju haspublishedmorethan150researcharticlesandhasaGooglescholarcitationcount of3000andScopuscitationcountof2500.HehasreceivedtheBestResearcher AwardasatokenofappreciationfromtheVITUniversityandIJRULAAssociation. Hehasattended/presentedvariousnational/internationalconferencesasapresenter/ invitedspeakerorresourceperson.Hewaslistedinthetop2%scientist’sdatabaseby theStanfordUniversity,UnitedStates(2020)andalsoin2021aspertheScopusdatabase.Heisaneditorofthisbookandhasalsocontributedsomeofthechapters.
Dr.IlyasKhan receivedhisPhDdegreeinappliedmathematicsfromtheUniversitiTeknologiMalaysia,oneofthe world’sleadinguniversities.Hehasover15yearsofacademic experienceindifferentreputedinstitutionsaroundtheworld. HeiscurrentlyanassociateprofessorwiththeDepartmentof Mathematics,CollegeofScience,Zulfi,MajmaahUniversity, SaudiArabia.Hehaspublishedmorethan700researcharticles indifferentwell-reputedinternationaljournals.Dr.Khanis alsoeditorofanumberofjournalsandarefereeinmorethan 100journals.Dr.Khan’sareasofresearchinterestinclude mathematicalmodeling,analyticalandcomputationalfluiddynamics,biomathematics,andnumericalcomputing.
Dr.SureshKumarRajuS. isoriginallyfromIndiaandcurrentlyworksasanassistantprofessorintheDepartmentof MathematicsandStatistics,CollegeofScience,atKingFaisal University,SaudiArabia.HecompletedhisPhDatUniversiti TeknologiPETRONAS,Malaysia.Hehas13yearsofteaching and8yearsofresearchexperienceatlocalandinternational educationalinstitutions.Hisresearchfocusesontheareasof fluiddynamics,heatandmasstransfer,nanofluids,multiphase flow,mathematicalmodeling,andnumericalanalysis.Hehas publishedresearcharticlesinreputedISIindexedjournalsandpresentedresearcharticlesatinternationalconferencesinvariouscountries.Hehasin-depthknowledgeof theprogramminglanguagesMATLABs andFORTRAN.Inadditiontoteaching andresearch,hewasbeenappointedtosomeadministrativeresponsibilitiessuchasa teamleadforaKeyPerformanceIndicator(KPI)report,curriculumcommitteemember,studentadvisor,examinationcoordinator,andProgrammeOfficerforthe NationalServiceScheme(NSS)Unit.Heisaneditorofthisbookandhascontributed someofthechapters.
Dr.MamathaS.Upadhya has21yearsofteachingexperience,combiningacademicandpragmaticapproaches.Sheis presentlyassistantprofessorattheDepartmentofMathematics, KristuJayantiCollege,Bengaluru,India.Shehas45publicationsthatareSCI-indexed,withreputedpublishersincluding theAmericanSocietyofMechanicalEngineering,Elsevier, TaylorandFrancis,Wiley,andSpringer.Sheisactivein researchasareviewerinmanyinternationalandnationalpeerreviewedjournals.Herresearchinterestsincludeareassuchas dustyfluid,nanofluids,non-Newtonianfluids,hybridnanofluids,andfuzzylogic.Shehasgiveninvitedtalksinvariousfacultydevelopmentprogrammes(FDPs)andseminars.Herexpertiseincludesafairknowledgeinsoftware suchasSPSSR,MATLABs ,andMaxima.Sheisaneditortothisbookandhasalso contributedsomeofthechapters.
Preface Thisbookprovidesreaderswithdetailsonthevariousapplicationsofmicro-and nanofluidflowandheatandmasstransfer.Differentnumericalmethodshavebeen employedtofindthesolutionstogoverningequationsandtheresultssimulatedusing MATLABs .SolvingtheboundarylayerequationsinMATLABs isdiscussedin detail.
ThefirstchaptergivesdetailedexplanationsregardingNewtonianandnonNewtonianfluid,micropolarfluid,andnanofluid,alongwiththeirapplications.In Chapter2,thebasicgoverningequationsofmotionforcouplestressfluidin Cartesian,cylindrical,andsphericalcoordinatesareexplainedindetail.
In Chapter3,theflowandheattransferofnanofluidsoverastretchingrotating diskarediscussed.Theflow-governingequationsaresolvednumericallyusingthe Runge Kutta-basedshootingmethodandimplementationinMATLABs is describedindetail.Theinfluenceofuniformmagneticfields;stretchingstrengthparameters;thermalbuoyancy;thermalradiationonaxial,tangential,andradialvelocities; andheattransferisalsodiscussed.
Chapter4 drawinsightsintothemathematicalmodelingforanincompressible unsteadynanofluidflowoveraninclinedplane.Heatandmasstransferapplicationsare discussed.Inthischapter,theconceptofhybridnanofluidisexplainedalso.Amathematicalmodelfortwo-dimensionalincompressibleMHD,unsteadynanofluidflow alonganelongatedsheetwithheatandmasstransfer,anditsapplicationsarereported in Chapter5.Theinfluenceofamagneticfield,chemicalreaction,viscousdissipation, andnonuniformheatsource/sinkparametersonthermal,concentration,andvelocity fieldsofthenanofluidaredescribed.
Chapter6 elaboratesontheStefanblowingeffectonnanofluidflowoverastretchingsheetinthepresenceofamagneticdipole.Theflow-governingequationsare numericallysolvedusingtheRunge Kutta Fehlberg(RKF-45)technique,along withtheshootingmethodandtheinfluenceofdimensionlessparametersonconcentration,thermal,andvelocitygradientsisportrayed. Chapter7 dealswithacomparativestudyofunsteadymicro-andnanofluidflowandheattransferconsidering nonlinearflowandCattaneo Christovheatfluxoverastretchingsheet.Nonlinear densityvariationandthepurposeofintroducingCattaneo Christovheatfluxare elaboratedon.
Chapter8 providesacomparativediscussiononheatandmasstransferinmicropolarandnanofluidflowonacurvedstretchingsheet,withdetailspresentedonentropy generation.Velocityandthermalslipareintroducedin Chapter9,andacomparative
studyonunsteadyincompressiblemicropolarandnanofluidflowandheattransfer overpermeableinclinedstretchingsheetispresentedalso.
Chapter10 providesadetailexplanationoftheimplementationofboundaryvalue problemsusingMATLABs .Severalexamplesarepresentedinthebookwhichhelp thereadertounderstandflowproblemsandtheirapplications.Numericalsolutionsare obtainedbythemimplementinginMATLABs .Theuser(Bachelor’s,Master’s,and PhDstudents;universityteachers;andalsoresearchcentersinanumberoffields)will thusbeabletoencountersuchsystemsinconfidence.
Inthedifferentchaptersofthebook,notonlyarethebasicideasofthemethods broadlydiscussed,butalsotheyarepracticallysolvedusingtheproposed methodology.
Backgroundtomicro-andnanofluids MamathaS.Upadhya1 andC.S.K.Raju2
1DepartmentofComputerScience,KristuJayantiCollege(Autonomous),Bengaluru,Karnataka,India
2DepartmentofMathematics,GITAMSchoolofScience,GITAMDeemedtobeUniversity,Bengaluru,Karnataka,India
Today’sresearchersarefascinatedbybreakthroughsintechnologies,andthistrendwill continueinthefuture.Asfluidsareasubstantialconstituentoftheuniverse,theyhave drawnsignificantattentionfromengineersandresearcherstomodifytheirvarious properties.Numerousfluidsencounteredinengineeringandindustrialprocessespossessnon-Newtonianfluidcharacteristics,forexample,moltenplastics,pulps,polymers, liquidmetals,nuclearfuelslurries,mercuryamalgams,lubricationbyheavyoil,etc. Fluidflowinthemicroscalebehavesdifferentlyfromthatinthemacroscale.Thereare situationswheretheNavier Stokesequation,whichisderivedfromtheclassicalcontinuum,isincapableofexploringthemicroscalefluidtransportphenomena.Thisis because,whenthechannelsizeiscomparedtothemolecularsize,thespinningof moleculeswhichisobservedinmoleculardynamicssimulationsisfoundtoinfluence significantlytheflowfield.Thiseffectofmolecularspinisnotconsideredinthe Navier Stokesequations.Thecomplexnatureofthesefluidshasforcedresearchersto inventconstitutivemodels.Inrecentyears,studiesrelatedtomicropolarfluidhave receivedgreatlyincreasedattention.Eringen(Eringen,1972;Eringen,1966)wasfirst toestablishedmicrocontinuumtheoryconsideringmicropolar,microstretch,and micromorphic(3M)theory.In3Mtheory,eachparticlehasafinitesizeandmicrostructurethatcanrotateanddeformindependently,regardlessofthemotionofthe centroidoftheparticle.Theformulationofmicropolarfluidtheoryhasadditional degreesoffreedom,gyration,todeterminetherotationofthemicrostructure.Thus, thebalancelawofangularmomentumisprovidedforsolvinggyration.Thisequation introducesthemechanismtotakeintoaccounttheeffectofmolecularspin.Thus, micropolartheoryisaverygoodalternativeapproachtosolvingmicroscalefluid dynamicsandismuchmorecomputationallyefficientthanmoleculardynamicssimulations.Underamicroscopicview,onecanseethatthemicromotionofrigidfluid elementsisrandomlyoriented(orspherical)withtheirpeculiarspinsandmicrorotationsinmicropolarfluid.Amicropolarfluidmodelhasbeenfoundusefulinthestudy offlowsofpaints,ferrofluids,exoticlubricants,colloidalsuspensions,liquidcrystals, polymericfluids,additivesuspensions,bodyfluids,bloodflows,flowsincapillaries, microchannels,andturbulentshearflows.Thepresenceofsmokeordust,particularly
inagas,couldalsobemodeledusingmicropolarfluiddynamics.Physically,micropolarfluidrepresentsafluidcontainingrandomlyorientedparticlessuspendedinaviscousmedium.MicropolarfluidandNewtonianfluiddifferoverthenumberof viscositycoefficients.Theformerhavingsixcoefficientsof3viscosity,namely α, β , γ , λ, μ,and κ,whilethelatterhasonlyonecoefficientofviscosity,forexample, μ Thegoverningequationsinthevectorfieldsareasfollows(AbdEl-Aziz,2013):
where D Dt isthematerialtimederivative, D representsthedeformationtensor with D 5 1 2 Vk;l 1 Vl ;k , Φ isthedissipationfunctionofmechanicalenergypermass unit, E thespecificinternalenergy, q theheatflux, V thevelocityvector, ρ thedensityofthefluid, μ thedynamicviscosity, ω themicrorotationvector, f thebodyforce vector, p thethermodynamicpressure, j themicroinertiadensity, l thebodycouple vector, λ thesecond-orderviscositycoefficient, κ thevortexviscosity(orthemicrorotationviscosity)coefficientand αv, β v,and γ arethespingradientcoefficients, respectively.
Theconstitutiveequations,givingthestresstensor τ kl andthecouplestresstensor Mkl,aregivenby:
where δ kl and εklm arethemetrictensorandcovariant ε symbol.
Thematerialconstantsmustsatisfythefollowinginequalities,derivedfromthe Clausius Duheminequality:
Eqs.(1.1) (1.4) representtheconservationofmass,linearmomentum,angular momentum,andenergy.For κ 5 αv 5 β v 5 γ 5 0andvanishing l and f,microrotation ω becomeszero,and Eq.(1.2) reducestotheclassicalNavier Stokesequations.Also, wenotethatfor κ 5 0,thevelocity V andmicrorotation ω arenotcoupledandthe microrotationdoesnotaffecttheglobalmotion.
Abbas,Malik,andNadeem(2020) studiedmicropolarhybridnanomaterialflow overRigasurface. Nawaz,Elmoasry,andAlebraheem(2020),usingthe Cattaneo Christovmodel,studiedthethermalnatureofmicropolarfluids.They noticedthatheatdissipatedinamono-nanofluidislowerthanforahybridnanofluid. Al-Hanaya,Sajid,Abbas,andNadeem(2020) studiedtheinfluenceofmultiwalled carbonnanotubes(MWCNTs)andsingle-walledcarbonnanotubes(SWCNTs)on micropolarhybridnanofluidflowonacurvedsurfaceandnotedthatmicrorotation improveswithvolumetricfraction. ReddyandFerdows(2021) investigatedthethermalandspeciesradiationinfluenceinmicropolaranddustyfluidflowacrossaparaboloidrevolution. NabweyandMahdy(2021) investigatednumericallythenatural convectionofmicropolaranddustparticlesduetopermeableconeconsideringthe nonlineartemperature. Kaneez,Alebraheem,Elmoasry,Saif,andNawaz(2020) numericallyinvestigatedthetransportofenergyandmomentainmicropolarfluid withsuspendeddustandnanoparticles. Abdelmalek,Nawaz,andElmasry(2020) studiedtheimpactofdustparticlesandnanoparticlesinheattransferinafluidwithmicrorotationandthermalmemoryeffects.
The21stcenturyhasbeenaccompaniedbyasignificantincreaseinenergyconsumptionbecauseofrapidgrowthofindustriesandmassivegrowthofthepopulation. Theeffectivedeploymentofenergyisnecessarytopreserveandbettermanageenergy resources.Intensifyingheattransferperformanceandreducingenergylosseshas becomeanincrediblygreatchallengetothehigh-technologyindustrialsectors. Nanoscienceandnanotechnologyareanticipatedtoplayamajorroleinrevitalizing theconventionalandemergingrenewableenergyindustries.Nanofluidisacolloidal defermentofnano-sized(diameterlessthan100nm)solidparticlesinbasefluidthat hasbroughtarevolutionarychangeinheattransferproperty.Experimentalresults (Keblinski,Eastman,&Cahill,2005)haveprovedthatnanofluidexhibitshigherthermalconductivitycomparedtothebasefluid.Nanofluidhasseveraladvantages,suchas higherstabilityofcolloidalsuspension,lowerpumpingpowerthatisessentialto
achievethecorrespondingheattransfer,andasuperiorlevelofcontrolforthermodynamicsandtransferpropertiesbyalteringtheparticlematerial,size,shape,andconcentration(Choi&Eastman,1995;Saidur,Leong,&Mohammad,2011).Experimental studiesby BuongiornoandHu(2005) illustratedthatforeffectiveheatenhancement, nanofluidrequiresonly5%volumetricfractionofnanoparticles.Nanoparticlesare madefromseveralmaterials,suchascarbonnanotubes,metals(Cu,Ag,Au)oxide ceramics(Al2O3,CuO),carbideceramics(SiC,TiC),nitrideceramics(AlN,SiN), semiconductors(TiO2,SiC),andcompositematerialssuchasnanoparticle core polymershellcompositesoralloyednanoparticles.Nanofluidsincludenumerous practicalapplications,withexamplesincludingprocessindustries(foodanddrink, materialsandchemicals,oilandgas,detergency,paperandprinting,textiles),nanofluid coolant(vehiclecooling,electronicscooling,etc.),medicalapplications(safersurgery bycooling,cancertherapy,anddrugdelivery),etc.
Theyear2020hasseentremendousaccelerationtowardtheadoptionandresearch intonanofluids.Numerousresearchers(Ahmed,Saleem,Nadeem,&Khan,2020; Anwar,Rafique,Misiran,Shehzad,&Ramesh,2020;Dogonchi,Waqas,Seyyedi, Hashemi-Tilehnoee,&Ganji,2020;Gopal,Naik,Kishan,&Raju,2020;Kumar, Sood,Raju,&Shehzad,2019;Rashid,Hayat,&Alsaedi,2019;Rostami,Dinarvand, &Pop,2018;Saleem,Nadeem,Rashidi,&Raju,2019;Shehzad,Reddy, VIjayakumari,&Tlili,2020;Sheikholeslami,Arabkoohsar,&Jafaryar,2020;Tariq, Hussain,Sheikh,Afaq,&Ali,2020;Turkyilmazoglu,2020a;Turkyilmazoglu,2019; Turkyilmazoglu,2020b;Waqas,Shehzad,Hayat,Khan,&Alsaedi,2019)havedemonstratedthatnanofluidshavebetterthermalperformance.Attractiveandpromising thermophysicalpropertiesofsolidnanoparticlesinbasefluid(nanofluid)haveencouragedresearcherstoanalyzetheirinfluenceinseveralareas.Themainapplicationof nanofluidsisasaheatexchanger,insolarcells,electronics,food,medicine,etc. Advantagesofnanofluidsandacceleratingresearchtowardflow,heat,andmasstransportphenomenahavemotivatedresearcherstowarddevelopinghybridanddihybrid nanofluids.Ifthemixtureiscomprisedofmorethanonetypeofnanoparticleitis knownasa “hybridnanofluid.” Ifthemixturecomprisesmorethanonetypeofnanoparticleandmorethanonefluiditisdescribedas “dihybridnanofluid.” Lietal. (2020) indicatedthatthemainadvantageofahybridnanofluidisgreaterconservation ofenergy,moreefficiency,andincreasedenergysaving. WainiandIshak(2020) studiedtheoutcomeoftranspirationonhybridnanofluidflowforuniformshearflow overastretchingsheetandobservedthatanimprovementinvolumefractionsofcoppernanoparticlescausesanenhancementinheattransfer. Raju,Upadhya,andSeth (2020) observedthatAl2O3 andgraphene-basedhybridnanofluidloweredthewall frictionrate. Kumar,Sandeep,Sugunamma,andAnimasaun(2020) observedthatthe proportionofheattransferisgreaterinahybridferrofluidthaninferrofluid. Shehzad (2020) investigatedthethermalperformanceofhybridnanofluidbyconsidering
sphericalparticles. HuminicandHuminic(2020) foundthattheemploymentof hybridnanofluidinminichannels,microchannels,andcavitieswasabetteralternative thantraditionalthermalsystems.
Thisstudyinvestigatedthenumericalsolutionstoanumberofproblemsforsteady, unsteady,laminar,andincompressibleflowofbothmicropolarfluidsandnanofluids. Thebodycouplesareneglectedinthecaseofmicropolarfluidflow.Similaritytransformationsareusedtohandlethegoverningpartialdifferentialofmotiontotransform themintoordinarydifferentialequations.Further,theresultingboundaryvalueproblemissolvedbyusingappropriatenumericaltechniqueswhicharestraightforward, easytoprogram,andeconomical.
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Mathematicalmodelingofequations ofcouplestressfluidinrespective coordinates C.S.K.Raju1,MamathaS.Upadhya2 andIlyasKhan3
1DepartmentofMathematics,GITAMSchoolofScience,GITAMDeemedtobeUniversity,Bengaluru,Karnataka,India
2DepartmentofComputerScience,KristuJayantiCollege(Autonomous),Bengaluru,Karnataka,India
3DepartmentofMathematics,MajmaahUniversity,AlMajma'ah,MajmaahCity,SaudiArabia
2.1Basicflowequations Thetheoryofcouplestressesdoesnotcontainthemicrostructureinthefluidasthe intrinsicangularmomentumandthekineticenergyofspindensityarenotconsidered. Theforemostconcernofcouplestressesistointroducethelength-dependenteffect, whichisnotpresentinclassicaltheoriesforNewtonianornonpolarfluids.Thebasic flowequationsrepresentingthecouplestresseshavebeenpresentedin Chang-Jianand Chen(2008), MurthyandNagaraju(2009), RamanaiahandSarkar(1979), Soundalgekar(1971),and Stokes(1966,1984) andareasfollows:
Thecontinuityequation
Cauchy’sfirstlawofmotion,
Cauchy’ssecondlawofmotion
where ρ isthefluiddensity, Vi arethevelocitycomponent, aj isthecomponentsof acceleration, Tij isthesecond-orderstresstensor, mij isthesecond-ordercouplestress tensor, f j representsthebodyforceperunitvolume, lj representsthebodymoment perunitvolume,and ejik isthethird-orderalternatingpseudotensor. Also,theconstitutiveequationforthepolarfluidisassumedtohavetheform
Thequantities ψ and μ aretheviscositycoefficients, ξ and ξ 0 arethecouplestress viscositycoefficients,and dij and kij arekinematicvariables.
2.2Equationsofmotion Tofindtheequationsofmotionforthecouplestressfluid,theconstitutiveequationhas beenutilizedtotransformCauchy’slawofmotionintermsofvelocityandtoobtain thejointformofthesymmetricandantisymmetricpartofthetensor,whichisgivenby:
Inwhichthefirsttermsignifiesthesymmetricpartofthestresstensorthatistransformedfrom Eq.(2.4) into:
FromCauchy’ssecondlaw
Bysimplification,itbecomes
Using m 5 mii , Eq.(2.5) transformsto mqk 5
qk ,alongwith kik 5 ω k;i ,whichreduces Eq.(2.9) into
as ω q;q 5 0.
However ω 5 1 2 r 3 V, ω i;j isthespintensor, m isthetrace. Usingthefollowingexpressions
itconverts Eq.(2.10) into
Sothat TA ij
;i sincetheterm eijk m ;ki mustbezero.Finally, since
Substitutionfrom Eq.(2.7) and Eq.(2.12) inCauchy’sfirstlawofmotion,
InGibbsnotation(GB),itcanbemodifiedas:
Forincompressiblefluids r V 5 0.Andifthebodyforce f andthebodymoment l areabsent,theequationsofmotionreduceto
Howeverby Eqs.(2.4)and(2.12),thestresstensortakestheform
Inthisthesis,thefluidisdeliberatedtobeincompressibleandalsothebodyforce andthebodymomentarenotconsidered. drr representsthedivergenceofvelocity thatcanbereservedastheequationofcontinuityandcanbeequatedtozero,forsimplicity.Also,theterm eijk m ;ki mustbezero, Eq.(2.17) reducesto
2.3Equationsofmotionbystresstensor
Thissectionisdevotedtoevaluatingthestresstensorforthecouplestressfluidinthe Cartesian,cylindrical,andsphericalcoordinatessystems.
2.3.1IntheCartesiancoordinatessystem Atfirst,allthecomponentsofthe “rateofstraintensor,” dij ,and “rateofspintensor” ω k aredeterminedwith V 5 ut ; x; y; z ðÞ; vt ; x; y; z ðÞ; wt ; x; y; z ðÞ ðÞ and i; j 5 1; 2; 3
where1; 2; 3denotesthe x; y; z componentsofvelocityanddifferentiationwith respectto x; y; z.
Since, dij 5 1 2 Vj ;i 1 Vi;j
dij 5
:
Þ And ω ij 5 eijk ω k 5 1 2 vj ;i vi;j
ω ij 5 eijk
TheKroneckerdeltaisdefinedas
Andthealternatingpseudotensorisdefinedas
eijk 5
8 < :
1 ; if i; j ; k isacyclicpermutationof1; 2; 3
1 ; if i; j ; k isananticyclicpermutationof1; 2; 3 0 ; ifanytwoof i; j ; k areequal
Thestresstensormatrixcanbegivenby:
Eq.(2.18) istransformedtogivetheshearcomponent
Bysettingthevaluesfrom Eqs.(2.20) and (2.23) in Eq.(2.25),theshearstress componentsareasfollows:
