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
