Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes : Numerical Solutions PDF
Preview Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes : Numerical Solutions
Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes Fluid Flow, Heat and Mass Transfer at Bodies of Different Shapes Numerical Solutions Kuppalapalle Vajravelu The University of Central Florida, USA and Swati Mukhopadhyay The University of Burdwan, India AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO Academic Press is an imprint of Elsevier AcademicPressisanimprintofElsevier 125LondonWall,London,EC2Y5AS,UK 525BStreet,Suite1800,SanDiego,CA92101–4495,USA 225WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK ©2016ElsevierLtd.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopying,recording,oranyinformationstorageand retrievalsystem,withoutpermissioninwritingfromthepublisher.Detailsonhowtoseek permission,furtherinformationaboutthePublisher’spermissionspoliciesandour arrangementswithorganizationssuchastheCopyrightClearanceCenterandtheCopyright LicensingAgency,canbefoundatourwebsite:www.elsevier.com/permissions. Thisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythe Publisher(otherthanasmaybenotedherein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchand experiencebroadenourunderstanding,changesinresearchmethods,professionalpractices,or medicaltreatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgein evaluatingandusinganyinformation,methods,compounds,orexperimentsdescribedherein. Inusingsuchinformationormethodstheyshouldbemindfuloftheirownsafetyandthesafety ofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors, assumeanyliabilityforanyinjuryand/ordamagetopersonsorpropertyasamatterofproducts liability,negligenceorotherwise,orfromanyuseoroperationofanymethods,products, instructions,orideascontainedinthematerialherein. ISBN:978-0-12-803733-1 BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressControlNumber:2015939630 ForinformationonallAcademicPresspublications visitourwebsiteathttp://store.elsevier.com/ Preface Fluid mechanics is one of the oldest branches of applied mathematics. It is also the foundationoftheunderstandingofvariousaspectsofscienceandengineering.Awide varietyofmathematicalproblems,appearinginareas asdiverse asfluidmechanics, mechanical engineering, chemical engineering, theoretical physics, and aerospace engineering,havebeensolvedbymeansofanalyticalornumericalmethods.Though analysisofdifferenttypesoffluidflowandheatormasstransferproblemsareavail- ableintheopenliterature,therearestillanumberofgapsthataretobefilledup.There areseveralexcellentbookscoveringdifferentaspectsoffluidflowandheatormass transfer.Yetonestilllooksforasystematicandsequentialanalysisthathelpsinunder- standing thisparticular area ofinterest. Tohelpstudentsandresearchersacquireadeeperunderstandingofthecharacter- isticsoffluidflowandheatandmasstransfer,thismonographaimstopresent,ingen- eral,astudyoftransportphenomena.Itiswellknownthatforexternalflows,theshape oftheobjectinfluencestheflowoveranobject(i.e.,abody)significantly.Asaresult, itaffectstheheatandmasstransfercharacteristics.Inotherwords,thebookaimsto helpreadersdeveloptheirunderstandinginthisparticularfieldwithoutspendinghuge timeinsearchingtheendlessliteratureonthisarea.Tohelpdevelopaclearinsight,we discussseveralflowfeatures.Bymaintainingtheapplicabilityoftheobtainedresults, we alsodiscussseveral cases ofphysical problems. Inselectingspecificproblemstoworkthrough,wehaverestrictedourattentionto thephenomenaoffluidflowandheatormasstransferassuchproblemsintroducea widevarietyofmathematicalproblemsofinterest.Hence,inordertoillustratevarious propertiesandtoolsusefulinanalyzingtheproblems,wehaveselectedrecentresearch inthe area offluid flow and heat ormass transfer. WeappreciatethesupportandmotivationoftheeditorGlynJonesandtheeditorial projectmanagerStevenMathews.WealsoacknowledgetheroleofElsevier(Oxford) formakingthisbookareality.ThankstoMr.SudiptaGhosh(PhDstudentofDr.Swati Mukhopadhyay) for his help in drawing some of the figures. We thank Prof. Mike Taylor for reading the entire manuscript and suggesting some needed changes. The authorsaregratefultoalltheauthorsofthearticleslistedinthebibliographyofthis book. The authors are also very much thankful to their coauthors. Finally, we very much like to acknowledge the encouragement, patience, and support provided by the membersofourfamilies. K.Vajravelu Orlando, Florida S. Mukhopadhyay Burdwan,India 2015 Introduction Air andwater are the mostimportant constituentsof the environment we livein, so thatalmosteverythingwedoisconnectedtothescienceoffluidmechanics.Forexam- ple,theflightofbirdsintheairandthemotionoffishinthewatercanbeexplained from the perspective of fluid mechanics [1]. The designs of airplanes and ships are basedonthetheoryoffluidmechanics.Fluidmechanicsisoneoftheoldestbranches ofappliedmathematics,andthefoundationoftheunderstandingofdifferentaspects of science and engineering [2–5]. From the nineteenth century, the scope of fluid mechanics has steadily broadened, as the study of hydraulics was associated with the growth of the fields of civil engineering and naval architecture. In recent times, thedevelopmentofthedifferentbranchesofengineering,namely,aeronautical,chem- ical,andmechanicalengineering,havegivenadditionalstimulitothestudyoffluid mechanics.Itnowranksasoneofthemostimportantbasicsubjectsnotonlyinapplied mathematicsbutalsoinengineering[6].Now,itisasubjectofwidespreadinterestin almost all fields of engineering as well as in astrophysics, meteorology, physical chemistry, plasma physics,geophysics, biology, and biomedicine[7–9]. Innature,fluidflowoverbodiesoccursfrequentlyandgivesrisetonumerousphys- icalphenomena,forexample,dragforceactingontrees,underwaterpipelines,auto- mobiles,theliftgeneratedbyairplanewings,upwarddraftofrains,dustparticlesin highwinds,andtransportationofredbloodcellsinbloodflow(see,[6]).Sometimes, fluidmovesoverastationarybody,forexample,windblowingoverabuilding,ora bodymovingthroughaquiescentfluidorabusmovingthroughair.Suchmotionsare referred to as flows over bodies or external flows [10]. The shape of the object has profound influence on the flow over a body and thus affects significantly the heat andmasstransfercharacteristics.Flowpastbodiescanbeclassifiedintoincompress- ibleandcompressibleflows.Compressibilityeffectsareneglectedatvelocitiesbelow 360km/hour,andsuchtypesofflowsareknownasincompressibleflows.Inthisbook, we are concerned withincompressible fluidflows [11–17]. Becauseoftherecenthighdemandintheneedforunderstandingandanalyzingthe problemswecomeacrossinscienceandengineering,wefeelthatthereisaneedfora bookofthiskind.Theunderlyingaimofthisbookistopresenttransportphenomena that will help students and researchers in the field of fluid mechanics in acquiring a deeper understanding of the characteristics of flow and heat and mass transfer (see,e.g.,[18]).Obviously,partofthematerialinthebookcanbeconvenientlyused asanintroductorycoursematerialforresearchersworkinginboundarylayertheory, flow,andheatandmasstransfer[19–45].Also,thebookisintendedforgraduatestu- dentsinmathematics,engineering,andinthemathematicalsciences.Inaddition,the materialinthebookmaybeofinteresttoresearchersworkinginphysicalchemistry, soil physics, meteorology, andnanotechnology. x Introduction Thebookisdesignedtoaccommodateseveraltopicsofvaryingemphasis,andthe chapters comprise fairly self-contained material from which one can make various coherentselections.Thereareseveralunderlyingthemesthatbecomeapparentwhen oneexaminestheliteratureonthesubject.Wehopetobringoutclearinsightbydis- cussingseveralflowfeatures.Also,wediscussseveralcasesofphysicalproblemsin general. The outline of the book will be as follows. Chapter 1 deals with the numerical method(s)adoptedintheseworks.InChapters2–4,whichcomprisePartIofthebook, wepresenttheflowpastsurfacesofdifferenttypes,namely,stretching,shrinking,and flat surfaces. This first set of chapters provides explanations intended for general readers and can be directly employed for problems in engineering, applied physics, and other applied sciences. We keep the discussion broad based so as to provide a frameworkforresearchers.Inordertomotivatethereaderandprovideagoodunder- standingofthesubjectmatter,attheendofChapters2–4,weprovidemultipleexam- ples of problemsthat have been solved numerically. InPartIIofthebook,Chapters5–6,weshiftthefocustoconcreteexamplesand problemsrelatedtobluffbodiesinfluidmechanicsandheatandmasstransfer.Here the governingequationsofthe problems are highlynonlineardifferentialequations. TheproblemsconsideredinthisPartwillhelpthereaderunderstandproblemsofphys- icalrelevanceandapplyittotheotherphysicalfieldsofinterest.Wegroupsuchprob- lems into three chapters: general fluid flow past a cylinder in Chapter 5, fluid flow over a sphere in Chapter 6, and finally problems related to flow past a wedge in Chapter7. References [1] YuanSW.Foundationsoffluidmechanics.NewDelhi:Prentice-HallofIndia;1969. [2] Abbas Z, Wang Y, Hayat T, Oberlack M. Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface. Int J Numer Meth Fluid 2009;59: 443–58. [3] AnderssonHI,BechKH,DandapatBS.Magnetohydrodynamicflowofapowerlawfluid overastretchingsheet.IntJNon-LinearMech1992;27:929–36. [4] ArielPD,HayatT,AsgharS.Theflowofanelastico-viscousfluidpastastretchingsheet withpartialslip.ActaMechanica2006;187:29–35. [5] BachokN,IshakA,PopI.Mixedconvectionboundarylayerflowoverapermeablever- tical flat plate embedded in an anisotropic porous medium. Math Problems Eng 2010;659023:http://dx.doi.org/10.1155/2010/659023. [6] BatchelorGK.Anintroductiontofluiddynamics.London:CambridgeUniversityPress; 2006. [7] Bhattacharyya K, Mukhopadhyay S, Layek GC. Slip effects on an unsteady boundary layerstagnation-pointflowandheattransfertowardsastretchingsheet.ChinPhysLett 2011;28(9):094702. [8] BhattacharyyaK,MukhopadhyayS,LayekGC.MHDboundarylayerslipflowandheat transferoveraflatplate.ChinPhysLett2011;28(2):024701. [9] BhattacharyyaK,VajraveluK.Stagnation-pointflowandheattransferoveranexponen- tiallyshrinkingsheet.CommNonlinearSciNumerSimulat2012;17(7):2728–34. Introduction xi [10] Cebeci T, Bradshow P. Momentum transfer in boundary layer. Washington, DC: McGraw-HillHemisphere;1977. [11] ChamkhaAJ,AlyAM,MansourMA.Similaritysolutionforunsteadyheatandmasstrans- fer from a stretching surface embedded in a porous medium with suction/injection and chemicalreactioneffects.ChemEngComm2010;197:846–58. [12] CortellR.Flowandheattransferofafluidthroughaporousmediumoverastretching surface with internal heat generation/absorption and suction/blowing. Fluid Dyn Res 2005;37:231–45. [13] Cortell R. Effects of viscous dissipation and work done by deformation on the MHD flow and heat transfer of a viscoelastic fluid over a stretching sheet. Phys Lett A 2006;357:298–305. [14] CraneLJ.Flowpastastretchingplate.ZAngewMathPhys1970;21:645–7. [15] DandapatBS,SantraB,VajraveluK.Theeffectsofvariablefluidpropertiesandthermo- capillarityontheflowofathinfilmonanunsteadystretchingsheet.IntJHeatMassTrans- fer2007;50:991–6. [16] GuptaPS,GuptaAS.Heatandmasstransferonastretchingsheetwithsuctionorblowing. CanJChemEng1977;55:744–6. [17] HayatT,AbbasZ,SajidM.Seriessolutionfortheupper-convectedMaxwellfluidovera porousstretchingplate.PhysLettA2006;358:396–403. [18] InghamDB,PopI,editors.Transportphenomenainporousmedia.Oxford:Elsevier;2005, vol.III. [19] IshakA.Mixedconvectionboundarylayerflowoveraverticalcylinderwithprescribed surfaceheatflux.JPhysA:MathTheor2009;42:195501. [20] KumaranV,BanerjeeAK,KumarAV,VajraveluK.MHDflowpastastretchingperme- ablesheet.ApplMathComput2008;210:26–32. [21] LiuIC,AnderssonHI.Heattransferinaliquidfilmonanunsteadystretchingsheet.IntJ ThermalSci2008;47:766–72. [22] MukhopadhyayS.Effectofthermalradiationonunsteadymixedconvectionflowandheat transfer over a porous stretching surface in porous medium. Int J Heat Mass Transfer 2009;52:3261–5. [23] MukhopadhyayS.Effectsofsliponunsteadymixedconvectiveflowandheattransferpast astretchingsurface.ChinPhysLett2010;27(12):124401. [24] MukhopadhyayS.HeattransferanalysisforunsteadyMHDflowpastanon-isothermal stretchingsurface.NuclEngDes2011;241:4835–9. [25] MukhopadhyayS.Chemicallyreactivesolutetransferinboundarylayerslipflowalonga stretchingcylinder.FrontChemSciEng2011;5:385–91. [26] Mukhopadhyay S, Vajravelu K. Effects of transpiration and internal heat generation/ absorptionontheunsteadyflowofaMaxwellfluidatastretchingsurface.ASMEJAppl Mech2012;79:http://dx.doi.org/10.1115/1.4006260,044508-1-6. [27] Mukhopadhyay S, Vajravelu K, Van Gorder RA. Flow and heat transfer in a moving fluid over a moving non-isothermal surface. Int J Heat Mass Transfer 2012;55(23-24): 6632–7. [28] Prasad KV, Vajravelu K, Datti PS. Mixed convection heat transfer over a non-linear stretchingsurfacewithvariablefluidproperties.IntJNon-LinearMech2010;45:320–30. [29] PrasadKV,VajraveluK,DattiPS.Theeffectsofvariablefluidpropertiesonthehydro- magnetic flow and heat transfer over a non-linearly stretching sheet. Int J Thermal Sci 2010;49:603–10. [30] PrasadKV,SujathaA,VajraveluK,PopI.MHDflowandheattransferofaUCMfluid overastretchingsurfacewithvariablethermophysicalproperties.Meccanica2012;47: 1425–39. xii Introduction [31] SajidM,AhmadI,HayatT,AyubM.Unsteadyflowandheattransferofasecondgrade fluidoverastretchingsheet.CommNonlinearSciNumerSimulat2009;14:96–108. [32] SchlichtingH.Boundary-layertheory.NewYork:McGraw-Hill;2008. [33] SharidanS,MahmoodT,PopI.Similaritysolutionsfortheunsteadyboundarylayerflow andheattransferduetoastretchingsheet.IntJApplMechEng2006;11:647–54. [34] SiddiquiAM,ZebA,GhoriQK,BenharbitAM.Homotopyperturbationmethodforheat transfer flow of a third grade fluid between parallel plates. Chaos Solitons Fractals 2008;36:182–92. [35] SoundalgekarVM,TakharHS,VighnesamNV.Thecombinedfreeandforcedconvection flowpastasemi-infiniteplatewithvariablesurfacetemperature.NuclEngDes1988;110: 95–8. [36] Takhar HS, Gorla RSR, Soundalgekar VM. Radiation effects on MHDfree convection flow of a gas past a semi-infinite vertical plate. Int J Numer Meth Heat Fluid Flow 1996;6:77–83. [37] TsaiR,HuangKH,HuangJS.Flowandheattransferoveranunsteadystretchingsurface withanon-uniformheatsource.IntCommHeatMassTran2008;35:1340–3. [38] VafaiK,TienCL.Boundaryandinertiaeffectsonflowandheattransferinporousmedia. IntJHeatMassTransfer1981;24:195–204. [39] VajraveluK,PrasadKV,Chiu-OnNg.Unsteadyflowandheattransferinathinfilmof Ostwald–deWaeleliquidoverastretchingsurface.CommNonlinearSciNumerSimulat 2012;17:4163–73. [40] Van Gorder RA, Vajravelu K. Hydromagnetic stagnation point flow of a second grade fluidoverastretchingsheet.MechResComm2010;37:113–8. [41] VanGorderRA,SweetE,VajraveluK.Nanoboundarylayersoverstretchingsurfaces. CommNonlinearSciNumerSimulat2010;15:1494–500. [42] VanGorderRA,VajraveluK,AkyildizFT.Existenceanduniquenessresultsforanon- linear differential equation arising in viscous flow over a nonlinearly stretching sheet. ApplMathLett2011;24:238–42. [43] Van Gorder RA, Vajravelu K. Convective heat transfer in a conducting fluid over a permeablestretchingsurfacewithsuctionandinternalheatgeneration/absorption.Appl MathComput2011;217:5810–21. [44] VleggaarJ.Laminarboundary-layerbehavioroncontinuousacceleratingsurfaces.Chem EngSci1977;32:1517–25. [45] WangCY.Flowduetoastretchingboundarywithpartialslip—anexactsolutionofthe Navier-Stokesequations.ChemEngSci2002;57:3745–7. 1 Numerical methods The governing equations of fluid flow problems are generally of a nonlinear and boundary value type. Usually, the exact solutions of the boundary value problems (BVPs) are very difficult to obtain, and so we have to use numerical methods. For some special classes of flow problems, a set of partial differential equations aretransformedintoasetofordinarydifferentialequationswiththehelpofsimilarity variables.Thetransformed(final)equationsthencanbesolvedanalyticallyornumer- ically. The procedure for finding a numerical solution for a BVP is generally more difficultthanthatofaninitialvalueproblem(IVP).Anumberofmethodscanbeused to solve linear BVPs. The method of differences is useful in such cases. But this methodcannotbeusedfornonlinearequations.Othermethodscanbeusedtoobtain linearlyindependentsolutions,whichcanthenbecombinedinsuchawaythatthey satisfytheboundaryconditions(Mukhopadhyay[1]).Forsuchproblems,thediffer- ence method can be adapted. The most popular numerical method is the shooting method. The shooting method can be used for both linear and nonlinear problems. Because the convergence of the method depends on a good initial guess, there is noguaranteethatthemethodwillconverge.Butthemethodiseasytoapply,andwhen itdoesconverge,itisusuallymoreefficientthanothermethods(Mukhopadhyay[1], MukhopadhyayandLayek[2]).Moreover,theshootingmethodgivesmoreaccurate resultsif guess values (slopes) are chosencorrectly. ThebasicideaofashootingmethodistoreplacetheBVPbysomeIVPwherethe slopeattheinitialpointisobviouslyunknown.Wecanguessthisunknownquantity, and then, usingiteration, the guess valuecan beimproved. The shooting method consists ofthe following steps: 1. transformationofthegivenBVPtoanIVP, 2. findingasolutionoftheIVP,and 3. findingasolutionofthegivenBVP. (cid:1) (cid:3) Letusconsideranonlinearsecond-orderdifferentialequationy==¼f x,y,y= with the boundary conditions yðaÞ¼y and yðbÞ¼y . 0 1 Atfirst,wesety=¼pandp=¼fðx,y,pÞwithy¼y atx¼a.Actually,theequation 0 isrewrittenintermsofafirst-ordersystemoftwounknownfunctions.Becausethese equationsarenonlinear,wecannotgetthesolutionbysuperpositionprinciple.Inorder tointegratetheabovesystemasanIVP,werequireavalueforpatx¼athatisy/(a). Generally, we take a guess for p(a) and use it to obtain a numerical solution. Thencomparingthecalculatedvalueforyatx¼bwiththegivenboundarycondition y¼y at x¼bandadjustingtheguessvalue,p(a),wegetabetterapproximationfor 1 thesolution.Thederivativeatx¼agivesthetrajectoryofcomputedsolution.Thatis why, it is called shooting method. Basically, we seek a solution that also satisfies the relation yðbÞ¼y . A suitable solution can be found by successively refining the 1 interval.Withthehelpoflinearinterpolationorotherroot-findingmethods,itisalso FluidFlow,HeatandMassTransferatBodiesofDifferentShapes.http://dx.doi.org/10.1016/B978-0-12-803733-1.00001-6 ©2016ElsevierLtd.Allrightsreserved. 4 FluidFlow,HeatandMassTransferatBodiesofDifferentShapes possible to improve the obtained solution. To start the integration, the initial slope, thatis,y/(a),isrequired,andtheshootingmethoddependsonthechoiceofthevalue of y/(a). To explain the method, we consider the equation 1 f===ðηÞ+ fðηÞf==ðηÞ¼0; (1.1) 2 along with the boundary conditions fð0Þ¼0, f=ð0Þ¼0; (1.2) f=ð1Þ¼1; (1.3) which isa third-ordernonlinear BVP. Now, the key factor is to choose an appropriate and suitable finite value of η as η!1,sayη .Hereη¼η correspondstotheedgeoftheboundarylayer.Forcom- 1 1 putationalpurposes,η istobechosenarbitrarilylargerthantheboundarylayerthick- 1 ness.Themostimportantfactoroftheshootingmethodistochooseanappropriatefinite valueofη .Inordertodetermineη fortheBVPstatedbyequations(1.1)–(1.3),we 1 1 startwithsomeinitialguessvalueαthatmustbedeterminedsothattheresultingsolu- tions yield the prescribed value f=¼1atη ¼η for some particular set of physical 1 parameters (Mukhopadhyay and Layek [2]). We therefore guess at the initial slope, andaniterativeprocedureissetupforconvergencetothecorrectslope.Anormally betterapproximationtoαcannowbeobtainedbythefollowinglinearinterpolation formula: f=ðη Þ(cid:2)f=ðα ,η Þ α ¼α +ðα (cid:2)α Þ 1 0 1 ; 2 0 1 0 f=ðα ,η Þ(cid:2)f=ðα ,η Þ 1 1 0 1 whereα ,α aretwoguessesattheinitialslopef//(0)andf=ðα ,η Þ,f=ðα ,η Þarethe 0 1 0 1 1 1 values of f/ at η¼η . We now integrate the differential equation using the initial 1 valuesfð0Þ¼0,f=ð0Þ¼0,andf==ð0Þ¼α toobtainf=ðα ,η Þ.Usinglinearinterpo- 3 2 1 lationbasedonα ,α ,wecanobtainanextapproximationα .Thisprocessisrepeated 1 2 3 until convergence is obtained. The convergence depends on a good initial guess (Conte andBoor[3]). Thesolution procedure is repeated with another large value of η until two suc- 1 cessive values of f//(0) differ only by the specified significant digit. The last value of η is finally chosen to be the most appropriate value of the limit η . The value 1 1 ofη maychangeforanothersetofphysicalparameters,ifinvolvedintheproblem. 1 Oncethefinitevalueofη isdetermined,thentheintegrationiscarriedout.Wecom- 1 pare the calculated value for f/ at η¼15 (say) with the given boundary condition f=ð15Þ¼1,andtheestimatedvalueisadjustedusingthesecantmethodtogetabetter approximationfor the solution(Mukhopadhyay [4]).
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