IncompFluidMech Lecture 02 FluidProps

FluidpropertiesanduidforcesAE225IncompressibleFluidMechanicsAniruddhaSinhaDepartmentofAerospaceEngineeringIITBombayWhyuidmechanics ? Mechanics: sciencethatdealswithbothstationaryandmovingbodiesundertheinuenceofforces Statics: dealswithbodiesatrest Dynamics: dealswithbodiesinmotion Fluidmechanics: dealswithbehaviourofuidsatrest(uidstatics)orinmotion(uiddynamics) Alsodealswithinteractionofuidswithsolidsorotheruidsattheboundaries Fluidmechanicsmaybecalleduiddynamicswhentreatinguidsatrestasaspecialcaseofuidsinmotion1 / 47Whatisauid? Auidisanythingthatows;theothercategoryofmatterissolid Solidsarehardandnoteasilydeformed Fluidissoftandeasilydeformed(wecanreadilymovethruair) Althoughfairlydescriptive,thisdenitionisunsatisfactoryforengineers/scientists Closerlookatmolecularstructureismorefruitfulforcategorization Arrangementofmolecules Theirenergy Distancebetweenthem2 / 47Molecularstructureofsolids Denselypackedmoleculesvibratingaboutxedpositions Densepackingresultsinlargeinter-molecularcohesiveforces Solidsmaintaintheirshape&volume Theycanhaveinniteno. offreesurfaces(materialboundaries)3 / 47www.schoolphysics.co.uk/Molecularstructureofliquids Moleculesspacedfartherapart Consequentlyinter-molecularforcesareweaker;moleculeshavegreaterfreedomofmovement Liquidsareeasilydeformedbutnoteasilycompressed(i.e.,changeshapebutnotvolume) Theycanformonefreesurface4 / 47www.physicalgeography.netMolecularstructureofgases Evengreatermolecularspacing Negligiblecohesiveintermolecularforces;moleculeshavegreaterfreedomofmotion Gasesareeasilydeformedandcompressed Theycannotformafreesurface(expandstollanyvolume)5 / 47chemwiki.ucdavis.eduBehaviourofuidsinpresenceofforces Naturalquestioninuidmechanics Solidshavenite(usuallysmall)changeofshapeand/orvolumeunderexternalforces Elastic: regainshapewhenforceisremoved Inelastic: otherwise Exception: Geologicalscales(formationofmountains)largeforces Fluids(bothliquidsandgases)undertheactionofforceshavecontinuouschangeofshapeasparticleskeepslidingovereachother Neverregainshapeonceexternalforceisremoved6 / 47www-mdp.eng.cam.ac.ukDegreeofdetailinstudyinguidmechanics Option#1(Moleculardynamicsapproach): Studythemotionofgazillionsofindividualmolecules Thisisgenerallyintractable Wasteful Dowereallyneedtoknowwhateachmoleculeofthesurroundingairisdoingatanygiveninstanttoknowhowanairplaneperforms? Option#2(Continuumapproach): Consideranextremelysmallelement(volume)ofmatterandstudyitsaveraged-outproperties,alongwithrate-of-changeofselectedpropertiesfromoneelementtoanother Assumption: Propertiesarevaryingcontinuouslyfromoneelementtothenext Corollary: Wecanwritederivativessuchasdp/dx7 / 47Moleculardynamicsapproachtostudyingmechanics Studythemotionofindividualmolecules ApplicabilityofapproachdependsonvalueofKnudsennumber(Kn)Kn =MeanfreepathofmoleculesCharacteristiclength(measureofrelevantdomain) Kn 1: moleculesarerelativelyfarapartcomparedtodomainsize Computationallytractablewhenno. ofmoleculesisfairlysmall Averaging(incontinuumapproach)istrickyduetopaucityofmolecules WhenKn 1,MDisneitherfeasiblenorrelevant Toomanymoleculestokeeptrackofwithinowdomain UselesslevelofdetailFlowbecomescontinuumasdistancebetweenparticlesreducesw.r.t. domainsize8 / 47Deningdensityofmatter Densityofamatteristheamountofmassperunitvolume Wemaydene(spatial-,nottime-)averagedensityofaniteelementofthemattersurroundingapointofinterestwithvolumeV (wedonotspecifythepreciseshapeoftheelement) :=MV=

imiV, mi= Massofi thmolecule Forelementswithsides meanfreepath,willvarygreatlydependingonpreciseshapeandsizeofvolumechosen Forverylargeelementsencompassingregionswithdierentphenomena,willagainvary E.g.: convectionowinakettle Inthiscase,studyofisuseless9 / 47Deningdensityofmatter(contd.)10 / 47LecturenotesofProf. ShevareApplicabilityofcontinuumhypothesis Q:What is thetypicalno. ofairmolecules in amicron-sizedelement atstandardconditions? Ans: About3 107 Largeenoughforrobustdenitionofuidproperties(e.g.,density) Micron-sizedelementstillmeaningfulinstudyofmosteverydayows Distancebetweenairmoleculesis 3 109m Meanfreepathisactually25timeslonger( 8 108m) No. ofmoleculesinacubethatis1meanfreepathoneachsideis15,000(stillalargenumber) Kinetictheoryofgasespredictsthatdensityofthisvolumewilluctuatebyonly0.8%(rootmeansquare) Consideranano-uidicdevicewithwater;inananometer-sizedvolumetherewillbe 30watermoleculesunreliableforestimatingdensity11 / 47Applicabilityofcontinuumhypothesis Q:What is thetypicalno. ofairmolecules in amicron-sizedelement atstandardconditions? Ans: About3 107 Largeenoughforrobustdenitionofuidproperties(e.g.,density) Micron-sizedelementstillmeaningfulinstudyofmosteverydayows Distancebetweenairmoleculesis 3 109m Meanfreepathisactually25timeslonger( 8 108m) No. ofmoleculesinacubethatis1meanfreepathoneachsideis15,000(stillalargenumber) Kinetictheoryofgasespredictsthatdensityofthisvolumewilluctuatebyonly0.8%(rootmeansquare) Consideranano-uidicdevicewithwater;inananometer-sizedvolumetherewillbe 30watermoleculesunreliableforestimatingdensity11 / 47Applicabilityofcontinuumhypothesis Q:What is thetypicalno. ofairmolecules in amicron-sizedelement atstandardconditions? Ans: About3 107 Largeenoughforrobustdenitionofuidproperties(e.g.,density) Micron-sizedelementstillmeaningfulinstudyofmosteverydayows Distancebetweenairmoleculesis 3 109m Meanfreepathisactually25timeslonger( 8 108m) No. ofmoleculesinacubethatis1meanfreepathoneachsideis15,000(stillalargenumber) Kinetictheoryofgasespredictsthatdensityofthisvolumewilluctuatebyonly0.8%(rootmeansquare) Consideranano-uidicdevicewithwater;inananometer-sizedvolumetherewillbe 30watermoleculesunreliableforestimatingdensity11 / 47ChoiceofsizeofuidelementIftheuiddensityisexpectedtochangefrompointtopoint,theelementsizeshouldnotbetoolargeThiswillaecttheresolutionofthepropertyestimate12 / 47Formaldenitionofuidpropertyincontinuum := limV MV, min