Microscale Flow and Heat Transfer: Mathematical Modelling and Flow Physics PDF
Preview Microscale Flow and Heat Transfer: Mathematical Modelling and Flow Physics
Mechanical Engineering Series Amit Agrawal Hari Mohan Kushwaha Ravi Sudam Jadhav Microscale Flow and Heat Transfer Mathematical Modelling and Flow Physics Mechanical Engineering Series SeriesEditor FrancisA.Kulacki DepartmentofMechanicalEngineering UniversityofMinnesota Minneapolis,Minnesota,USA The Mechanical Engineering Series presents advanced level treatment of topics on the cutting edge of mechanical engineering. Designed for use by students, researchers and practicing engineers, the series presents modern developments in mechanical engineering and its innovative applications in applied mechanics, bioengineering,dynamicsystemsandcontrol,energy,energyconversionandenergy systems,fluidmechanicsandfluidmachinery,heatandmasstransfer,manufacturing science and technology, mechanical design, mechanics of materials, micro- and nano-science technology, thermal physics, tribology, and vibration and acoustics. The series features graduate-level texts, professional books, and research mono- graphsinkeyengineeringscienceconcentrations. Moreinformationaboutthisseriesathttp://www.springer.com/series/1161 Amit Agrawal • Hari Mohan Kushwaha Ravi Sudam Jadhav Microscale Flow and Heat Transfer Mathematical Modelling and Flow Physics 123 AmitAgrawal HariMohanKushwaha DepartmentofMechanicalEngineering DepartmentofMechanicalEngineering IndianInstituteofTechnology,Bombay IndianInstituteofTechnology,Bombay Mumbai,Maharashtra,India Mumbai,Maharashtra,India RaviSudamJadhav DepartmentofMechanicalEngineering IndianInstituteofTechnology,Bombay Mumbai,Maharashtra,India ISSN0941-5122 ISSN2192-063X (electronic) MechanicalEngineeringSeries ISBN978-3-030-10661-4 ISBN978-3-030-10662-1 (eBook) https://doi.org/10.1007/978-3-030-10662-1 LibraryofCongressControlNumber:2019932162 ©SpringerNatureSwitzerlandAG2020 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG. Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Thisbookisdedicatedto pastandcurrentstudentsof“RarefiedGas Dynamics”and“Microfluidics” laboratories,IITBombay, and toourfamilies. Preface This book has been written with the objective of making the readers familiar with the exciting developments on gaseous slip flow and heat transfer in microchannel. A large amount of work is being currently undertaken worldwide in these areas, with numerous potential applications. The subject is therefore topical and also particularly significant as it leads to the question about the validity of the Navier- Stokes equations, which is usually considered sacrosanct. It obviously leads to a subsequentquestion:iftheNavier-Stokesequationsarethemselvesnotvalid,then how do we model the flow and heat transfer? Although the answer to this last question is still not clear, it is important for the fluids and thermal community to first appreciate the limitations of the continuum approach (which leads to the Navier-Stokes equations), to better appreciate the ongoing search for the “beyond Navier-Stokes equations,” and to test some of the available equations for their accuracy,beforethelargerobjectiveoffindingbeyondtheNavier-Stokesequations canbepracticallymet. Thebookisthereforeorganizedintwoparts:thefirstpartisonthegaseousslip flow and heat transfer in microchannel (Chaps. 2 and 3). In the second part, we examinebeyondtheNavier-Stokesequations(Chaps.5–7).Chapter1summarizes the characteristics of microscale flows and provides an introduction to the various modelling approaches available. Chapter 4 is a transition chapter between the two parts, where it is shown that simple extensions of the Navier-Stokes equations are notadequate.Recognizingthatonlyanalyticaltoolwillnotbeadequateforstudying flowsintheslipandtransitionregimes,abriefoverviewtorelevantnumericaland experimental techniques is provided in Chap. 8. Finally, Chap. 9 summarizes our currentunderstandingandprovidessuggestionsforfutureresearchinthissubject. The Knudsen number is the most important parameter, and several known solutionswithKnudsennumberasanadditionalparameterarecompiledinthefirst partofthebook.InterestingobservationsonKnudsenminimaandflowseparation are presented. However, it should become apparent that our understanding of heat transfer at the microscale is not that sound, as several additional effects such as axial conduction, pressure work, conduction in the substrate, viscous dissipation, etc. coexist, but it is virtually not possible to treat all these effects together and vii viii Preface obtainananalyticalsolutionforeventhesimplestproblems.Onlyfewexperimental and direct simulationMonte Carlo(DSMC) data exist,and they do notagree well withthatobtainedfromalternativeapproaches. A good portion of the second part of the book is devoted to deriving and understandingtheBurnettandGradequations.Thesetwosetsofequationsformthe most important “beyond the Navier-Stokes” equations and are generally referred to as “higher-order continuum transport equations.” The study of these equations is important for further development of the subject, as they involve several novel concepts and represent important breakthroughs in the subject. The derivation of these equations has not often been repeated, and it is expected that the stepwise derivationpresentedherewillinvokewidereadership.Specialefforthasbeenmade to make the text readable through the insertion of a large number of figures. The hopeisthatwiththehelpofthisbook,itshouldnowbepossibletoderivetheBurnett andGradequationsinagraduateclass.Afewproblemsaresolvedtoillustratethe typeofsolutionobtainedfromtheseequations.Inthederivationofsomesolutions presentedhere,someminorerrorintheoriginalsourcewasnoticedwhichhasbeen correctedhere. Thefirstpartofthebook(Chaps.1–3alongwithChaps.8and9)shouldappeal toreadersinterestedinunderstandingfundamentalaspectsofmicroscaleflowsand heattransfer,whilethesecondpart(Chaps.4–7)isprimarilyforslightlyadvanced readersinterestedinunderstandingequationsbeyondtheNavier-Stokesequations. Mumbai,India AmitAgrawal Mumbai,India HariMohanKushwaha Mumbai,India RaviSudamJadhav September2018 Acknowledgments Wearegratefultoallthestudentswhohaveworkedinthe“RarefiedGasDynamics” and“Microfluidics”laboratoriesatIITBombay,includingthosewhowereinvolved inpurelytheoreticalornumericalaspectsoftheproblem.Itistheywhoslowlyand patientlyuncoveredthevariousfinepointsofthesubject,whichfinallyculminated intheformofthisbook.WearealsogratefultoProf.K.Muralidhar(IITKanpur) and Prof. F. Kulacki (University of Minnesota), the editors of this series, for first placing their trust in us for writing this book and then patiently waiting for it while we went past several deadlines. The book would not have resulted without thecoaxingofProf.Muralidhartowriteonthissubject.Whilepreparingthebook, wegainedfromthecommentsofProf.AtulSharma(IITBombay). FundingforthislineofinvestigationhasbeenprovidedbytheIndianInstituteof TechnologyBombay(IITB),IndianSpaceResearchOrganisation(ISRO),Depart- mentofScienceandTechnology(DST),andrecentlybytheDepartmentofAtomic Energy (DAE) under its prestigious scheme—DAE-SRC Outstanding Investigator Award,awardedtothefirstauthor. ix Contents 1 IntroductiontoMicroscaleFlowsandMathematicalModelling....... 1 1.1 Introduction............................................................ 2 1.2 ApplicationsofMicroscaleFlows .................................... 3 1.2.1 CoolingofElectronicDevices ............................... 4 1.2.2 Micro-nozzlesandMicro-thruster ........................... 4 1.2.3 BreathAnalyser............................................... 6 1.2.4 MicrodeviceforConductingBloodTest..................... 6 1.3 ClassificationofFlowRegimes....................................... 8 1.4 CharacteristicsofMicroscaleFlows.................................. 9 1.4.1 Rarefaction.................................................... 9 1.4.2 Compressibility ............................................... 10 1.4.3 ThermalCreep ................................................ 11 1.4.4 ViscousDissipation........................................... 12 1.4.5 PropertyVariation............................................. 13 1.4.6 AxialConduction ............................................. 14 1.4.7 ConjugateHeatTransfer...................................... 14 1.5 MathematicalModellingofMicroscaleFlows....................... 15 1.5.1 TheNavier–StokesEquations................................ 16 1.5.2 LimitationsofConventionalEquationsandBoundary Conditions..................................................... 16 1.5.3 ApproachtoModellingMicroscaleFlows................... 18 1.6 RelevanceandScopeoftheBook .................................... 21 2 MicroscaleFlows............................................................. 25 2.1 Introduction............................................................ 25 2.2 GoverningEquationsforFluidFlow ................................. 26 2.2.1 TensorialForm................................................ 26 2.2.2 ConstitutiveRelations ........................................ 28 2.2.3 CompressibleNavier–StokesEquations..................... 30 2.2.4 IncompressibleNavier–StokesEquations ................... 31 xi
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