1100隊(duì)題第一屆認(rèn)證杯數(shù)學(xué)

承諾書,參賽隊(duì)員(簽名)(簽名BasicAnalysisofWalkingwithCoffeeinACupXinqiYouHuanXuSouthwesternUniversityofFinanceandEconomicsChengdu,ChinaToevaluatethefactorsthatinfluencethespillageofcoffeeinacup,wemainlyutilizetheknowledgeofliquidsloshingincomputationalfluiddynamics.WeconstructourmodelmostlybasedontheNavier-Stocksequationsthatdescribethemotionoffluidsubstances.Theresultofourmodelisafunctionthatspecifiesthesurfaceelevationabovetheinitialwaterlevel.Indevelopingourmodel,wederiveasystemofpartialdifferentialequationswithLagrangianmethod.Then,wesolvethePDEsbyusingMATLABandacquirethegraphsofthemodel.Also,basedontheequationswesolveearlierinthemodel,weyieldanotherfunctiontodescribesurfaceelevationwithmoredetailedgraphsofthemovement.Thenwenumericallyanalyzefactorsthataffectsurfaceelevationaccordingtothemodel.Byplottinggraphsofcomparingdifferentstates,wegettherelationshipofhowthesefactorsoperate.InTasktwo,wesimulatetheeffectofvibrationbysettinganentrapmenttherobotandobservethesurfaceelevationofcoffeeinthecup.Withexperimentalresultsfromliterature,weacquirethedescriptiveresultoftheimpact.Duringofficebreak,holdingthefreshlymadecoffeeinthehallwaymightbethegreatestpleasureoftheday.Nevertheless,therewouldalwayshavesomecarelesscoworkerstorunintoyourcupofcoffee.Toavoidthechanceofgettingthecoffeespilled,weturntotheliquidsloshingengineeringoffluiddynamics.Influiddynamics,sloshreferstothemovementofliquidinsideanobject.Morespecifically,theliquidmusthaveafreesurfacetoconstituteasloshdynamicsproblem,wherethedynamicsoftheliquidcaninteractwiththecontainertoalterthesystemdynamicssignificantly.Inourcase,thecupisacylindercontainerandcoffeemakestheperfectfree-surfaceliquid.Liquidsloshingengineeringismostlyappliedintheastronomy,whichincludesthepropellantsloshinspacecrafttanksandrockets,andcargosloshinshipsandtruckstransportingliquids.Therehavebeenalotofmethodstoapproachthesloshingproblem.ThemostclassicistheNavier-Stokesequations,whichdescribethemotionoffluidsubstances.WewillimplementNavier-Stokesequationstoderiveourmathematicalmodelandcontrolthesloshingofcoffee.Thefluidisassumedtobehomogenous,continuous,isotropic,inviscidandNewtonianandexhibitsonlylimitedcompressibility.Theinteriorofthecupisfrictionlessandthecontactofthecoffeeandthecupwillnotaffecttheliquidmotion.Thecoffeecupisabstractedtobeaperfectcylindercontainer,withoutanyconcaveorconvexpart.Tosimplifyourmodel,weignoretheinfluenceofairflowandmainlyfocusontheliquidmotionitselfandthevibrations.Ourrobotismovingataconstantspeedinastraightpathwiththeexcitationatthebeginning.Therobotishighlycontinuousandweassumethemovementisnotaffectedbyitsheight,weightorotherhumanfactors.Thetemperatureintheexperimentalenvironmentisconstantandwillnotaffecttheliquid.ThevibrationsobviatingtheprogressofrobotareassumedtobeTheexcitationisassumedtobeharmonic,afterthemotionwasstartedfromtherest.Terminologyand(t,x)：surfaceelevationabovethepoint:angularTask1TheModelTherearevariousmovementsofthecontainingliquid.Typicalexampleishorizontalmovementalongastraightorcurvedpath,verticalmovement,rotationsorcombinationsofthese.Here,weonlystudythehorizontalmotionalongastraightpathcombinedwithrotationintheverticalplanealignedwiththemovementofdirection.Initially,weonlystudythehorizontalmovementwithoutrotationandwilladdtherotationalaccelerationlater.Inthefirstsection,wedivideourmodelintotwoparts.Wegovernthetheflowinthefluidinthefirstpartandthenthefluidflowinahorizontallyacceleratedopencontainer.ThemodelisbasicallyusingtheLagrangianmethod.ModelingofWithintheAccordingtotheNavier-Stokesequations,weexaminethefluidmotionwithinthefluid.Thefirstequationdescribestheconservationofmass(v) Model thedensityofthev=(vx,vy,vz flowvelocityvectorfieldinthe f=(fx,fy,fz theexternalforcevector theviscosityof thefactorofvolume thevorticityofthe potentialofthepotential liquid thediameteroftheliquidinthecylinder gravity thehorizontalaccelerationoftheWehavefourunknownsintheequationoftheconservationofmass.Unlessthedensityofliquidisgivenandtheproblemissingledimensional,wearenotabletosolvethisequationwithoutmoreequations.Therefore,wehavetheconservationofmomentum(v(v)v)p()(v)2v Theequations(1.1)and(1.2)formtheNavier-Stokesequations.Inourcase,wehaveassumedthatthetheliquidisincompressible( andinviscid(.Also,weassumethattheforcefieldhasapotentialthatexcludesforcesgeneratedbyrotationalaccelerationofthecontainer.Thus,withtheconditionsabove,wesolvetheNavier-Stokesequationstogive12pV whereC(t)isanarbitraryfunctionoftime.ThisequationisknownastheBernoulli’sequation.Figure CrosssectionoftheOntheFluidInthisproblem,theboundaryisconsideredtobethewallsofthecoffeecup,bottomandthefreesurfaceofthefluid.Thevelocityatthewalliszeroonthewallsandbottominthenormaldirectionofthewall.Onthefluidsurface therearetwoboundaryconditions:onedynamicandoneThedynamicconditioncanbeexpressedwithBernoulli’s(t,x,y,)

(t,x,y,)2p(t,x,y,)V(t,x,y,) Thekinematicconditionisgiven z y WithahorizontalaccelerationintheSincetheliquidinthecupisfurtherassumedtobeirrotational,thesurfaceelevationabovethepoint isgivenby(t,x).AfterseparateanalysisofFigure2Illustrationofthecoordinatehorizontalandverticalacceleration,wegivethefollowingdescriptionofthefluidflowinthecontainer2(t,x,z) (t,x,z)p(t,x,z)1(t,x,z)2gzu(t)x (t,x)(t,x,z)(t,x)(t,x, (t,0,z)0,(t,a,z)0,(t,x,h) Usingthemethodoftheseparationofvariablesandwiththeassumptionthatpressureisequaleverywhereonthesurface,wehaveaa

(t,xm)

(n1,3,5

cos p p2

Fromthemodel(1.10)and(1.11),wecouldseethattheelevationofsurfaceisrelatedtothediameterofthecontainerandthedepthofliquid,alsothepressureandimplementedhorizontalacceleration.Thisresultmatchesourdailyobservationaswell.Withmoredepthandsmallerdiameter,theangularvelocityislikelytobelarger.Sincethecoffeecupisopenonthetop,tangentialstressisnegligibleduetotherelativelylargerfluiddensitycomparingwiththeairandaccordinglytheonlystressatsuchsurfaceisthenormalpressure.Thesummationoftheforcesnormaltofreesurfacemustbebalancedbytheatmosphericpressure.InthestudyofCelebiandAkyldiz,wehavewherecistheadiabaticspeedof

Thehorizontalacceleration stimulatesthecontainertomove.Inthiscase,wemakethefunctiontobeexponential,i.e. aretheparameterstocontrolthemotionoftheAnalyticInthestudyofMiles[1976],wesolvetheflowproblembasedontheLagrangemechanics.ThefluidismodeledasapotentialflowbytheLaplaceUsingtheseparationofparameters,we (t,x)q(t)(x),(x) 2

Thefunction aregivenbythefollowingnonlineardifferentialalgebraicequations(DAE) (amnqmblmnqlqm)gqncnu,n1, l arefunctionsof istheappliedhorizontalacceleration.Thecoefficientsaregivenby a q1 qq mn lmnl

2j

j nl 1( nl q11

cna0xndx

an

g, almnClmnajlmnDjlmn(aman)2DjmiDlnThemodelsgivenin(1.11),(1.12)and(1.13)arebasedonseparationofvariablesanddescribesthesurfaceshapebyS(t,x)

c(t)cosn isthesurfaceelevationattimetinthehorizontalpositionxrelativetothecup.Thebehaviorofsurfacemovementisthengivenbycn(t).Inourmode,weassumethetimeintervalis andthespaceintervalis Theaveragecoffeecupis0.0762mwidewhichgives508datapointsthedurationofthesimulationis20swhichgives1000datapoints.Therefore,wecouldreplacetheparametersandhave withk=0,1,…,999andcoeffiieaintegralc(t) s(t,x) a c(kh)

S(kh,lh)

n l0

NumericalFirstofall,wedeterminesomeparametersoftheAveragecoffeecupisapproximatelywiththevolumeof12ozandthemugisofthesizeboth31/2feetwideand31/2feetdeep.Inourproblem,tofacilitatecalculation,weassumethedepthofcoffeeis10cm andthewidthisalso8cm .Forthegravityacceleration,weassumethatitequalsthestandardgravity .ThespeedofsoundisalsoassumedtobeSolutionsforNavier-Stokesequationsareabundantnowadays.Fromthepreviousliteratureofcomparingdifferentmethods,wechoosetheFiniteElementMethod(FEM)forthenumericalapproach.FEMisdesignedtofindapproximatesolutionstopartialdifferentialequationsandtheirsystems.InapplyingFEM,wemainlyutilizethetoolboxIFISS(IncompressibleFlowIterativeSolutionSoftware),designedbyD.J.Silvester,H.C.Elman,A.Ramage[2011].IPISSisacollectionofover290matlabfunctionsandincludesalgorithmsfordiscretizationbyFEM.Withinthetoolbox,fourPDEsaretreated:thePoissonequation,andsteady-stateversionsofconvection-diffusionequation,Stokesequations,andNavier-Stokesequations.Obviouslywewillusethefourthsystem.Thepackageallowsustostudytheeffectofdifferentparametersonthecontourofthefluidandtheestimatederroroftheanalysis.Fortheanalyticapproach,wewillplotthegraphsofequations(1.11)and(1.14)andobservethechangeofliquidsurface.Toimplementrandomvibrations,werandomlyaddaccelerationstoourrobotstosimulatetheeffect.FirstwepresentthesolutionoftheNavier-StokesequationsusingMATLABtoolboxIFISS.Figure3SolutionoftheNavier-StokesFigure4EstimatederroroftheTheliquidhorizontalvelocityisbouncingaroundaspecificvalueandestimatederrorismoreconcentratedaroundzero.Therefore,weclaimthatthesolutionisvalidandacceptable.Thebackandforthpropertyofliquidmotionmeetsourgoalofconductingthesimulation.Thecoreofthemodelistoplotthegraphofsurfaceelevationwithrespecttotimeanddeterminehowcouldcontroltheparametersnottomaketheliquidspilloutofthecontainer.Andsolvingmodel(1.14)and(1.15)givesusmoreinsightofthat.FollowingisthesolutionFigure5MotionofsurfaceFactorModel(1.11)specifiesthatparameters arerelativetothesurfaceelevation.HereareourresultsbyusingMATLAB.Firstweexaminetheinfluenceofthechangein.Byincreasingtheangularvelocity,surfaceelevationremainsthesameasinitialcondition.Sincetheaccelerationdoesnotchange,Nevertheless,itseffectontheliquidsurfacelastsmuchlongerthantheinitialcondition.Thisispartlyduetolargervorticity,longertheperiod.Figure6Effectofthechange onsurfaceFigure7Effectofthechangeinratioh/aFigure8Effectofthechangeinratioh/aonthesurfaceFigure9Effectofthechangeinhorizontalaccelerationonsurface isthefunctionofthe ,wewouldliketoseehowtheinthisratioinfluences.Figure7explainsthatthechangeintheratiodoesnothaveasignificanteffectatthebeginning.Butlaterthedisparitybecomesgreaterandgreater.Whenthediametersremainthesameanddepthofliquidlarger,seemstobesmaller.Inotherway,whendepthsofliquidarethesameanddiametersofcontainersmaller,thiswouldalsohavethesameeffect.Forthesurfaceelevation,thechangeofratioalsoappearstobesignificantaccordingtoFigure8.Withhigherratio,surfaceelevationexperiencesasurgeatthebeginningofmovement.Thepatternsofdifferentstatesarealmostthesameexcepttheamplitude.Eventually,weexaminehowthechangeinhorizontalaccelerationaffectsthesurfaceelevation.Thepatternsofdifferentstatesalsoresembleeachother.Andwithgreateracceleration,theliquidsurfacerisesmuchhigher.However,afterthefirstspike,twolinestendtohavethesameamplitudeandpattern.Task2RobotTosimulatetheprocessofwalkingwithacupofcoffee,wetheoreticallyuserobot.Forthepurposeofconcentratingonhowtoavoidcoffeespilling,weselectivelyleavealonethedetailsoftheconfigurationofrobotandassumeitisinthetheoreticalperfectcondition.Accordingtodailyexperience,inmostcases,thecoffeespillswhensomeonerunintoyouandyouhavetostop.Therefore,wesimulatethiseffectbycommandingrobottostopattime.Herewepresentthenumericalsimulationresultsofthiseffect.FromFigure10,wecouldseethatafterentrapment,waterlevelrisessignificantly.Ourgoalistocontrolthewaternottooverthebrinkofcup.Soclearlyforthisgoal,weneedtofillinlesscoffee.Buthowmuchisenough?HerewepresentthenumericalresultfromourmodelinFigure11.Tosimulatetheentrapment,weaddanaccelerationinanoppositedirection.Weknowfrombasicphysics,objectcannotstopimmediatelyiftheaccelerationbecomeszero,onlywhenthevelocityiszero.Thus,weimplementarelativelylargerinverseaccelerationtomaketherobotstopatonce,whichisalsothesameprocessaspeopledowhenconfrontingthesituation.Inourmodel,weonlycalculatethewaterlevelatonesideofthecup.FromFigure11,itcouldbeseenthatatt=1.5s,thesurfaceelevationdivesimmediatelyaftertheimpact.Itisobviousthatifonesideisdeependown,theothersideoftheliquidwouldenjoyasurge.Figure NumericalsimulationofFigure11SurfaceElevationaftertheSincethesurfaceelevationisonlyrelatedtothewidth,depth,vorticityandaccelerationandwecannotchangethewidthanddepthofthecupatthegivensituation,theonlywaytocontrolthecoffeenottospilloutistocontroltheacceleration.Nevertheless,duetoourroughworkinthistask,wedidnotgettheexactnumericalsolutiontothisproblem.Butwedoknowthatthestrategyistocontrolthevorticitywhenrobotturnstoanotherdirectionandtherotationalaccelerationtoavoidthecollision.Whethercoffeewillspilloutofacupisconcernedwiththediametersofthecup,depthofcoffeeinthecup,angularvelocityandhorizontalacceleration.Greaterangularvelocity,ratioh/aandhorizontalaccelerationbothleadtohigherchanceofspillingoutofthecup.Fortherobotsimulationpart,afterentrapment,thesurgeofcoffeeissignificantandthestrategyforcontrollingthesurfaceelevationistocontrolthevorticityandrotationalacceleration.WeaknessandOurmodelishighlytheoreticalandwebelieveinempiricalexperiment,thesituationmightnotmatchwitheachother.Weassumethattheviscositytobezeroinmodel(1.11)tomakethecalculationeasier,whichisnotprevalentinrealsituation.Liquidviscosityisusuallysmallbutdefinitelybiggerthanzero.WeonlyconsiderthehorizontalaccelerationandeliminatetherotationalInsolvingthemodel,weusetheMATLABtoolboxIFISSdevelopedbyH.C.ELMAN.IFISSsolvesNavierStokesequationinfivespecificexamples,inourcase,wechoosetheoption“flowovertheplate”,whichisthemostrelativetoourmodel.Theexampl