2The Navier-Stokes and Euler Equations

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2.TheNavier–StokesandEulerEquations–FluidandGasDynamics

Fluidandgasdynamicshaveadecisiveimpactonourdailylives.Therearethe nedropletsofwaterwhichsprinkledowninourmorningshower,thewaveswhichwefaceswimmingorsur ngintheocean,theriverwhichadaptstothetopographybyformingawaterfall,theturbulentaircurrentswhichoftendisturbourtransatlantic ightinajetplane,thetsunami1whichcanwreckanentireregionofourworld,theathmospheric owscreatingtornados2andhurricanes3,thelive-giving owofbloodinourarteriesandveins4…Allthese owshaveagreatcomplexityfromthegeometrical,(bio)physicaland(bio)mechanicalviewpointsandtheirmathematicalmodelingisahighlychallengingtask.

Clearly,thedynamicsof uidsandgasesisgovernedbytheinteractionoftheiratoms/molecules,whichtheoreticallycanbemodeledmicroscopically,i.e.byindividualparticledynamics,relyingonagrandHamiltonianfunctiondependingon3Nspacecoordinatesand3Nmomentumcoordinates,whereNisthenumberofparticlesinthe uid/gas.NotethattheNewtonianensembletrajectorieslivein6Ndimensionalphasespace!FormostpracticalpurposesthisisprohibitiveanditisessentialtocarryoutthethermodynamicBoltzmann–Gradlimit,which–undercertainhypothesisontheparticleinteractions–givestheBoltzmannequationofgasdynamics(seeChapter1onkineticequations)fortheevolutionoftheeffectivemassdensityfunctionin6-dimensionalphasespace.

Undertheassumptionofasmallparticlemeanfreepath(i.e.inthecolli-siondominatedregime)afurtherapproximationispossible,leadingtotime-dependentmacroscopicequationsinpositionspaceR3,referredtoasNavier–StokesandEulersystems.Thesesystemsofnonlinearpartialdifferentialequa-tionsareabsolutelycentralinthemodelingof uidandgas ows.

Formore(precise)informationonthismodelinghierarchywereferto[3].TheNavier–Stokessystem5waswrittendowninthe19thcentury.ItisnamedaftertheFrenchengineerandphysicistClaude–LuisNavier6andtheIrishmath-ematicianandphysicistGeorgeGabrielStokes7.

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7/http://www.spc.noaa.gov/faq/tornado/http://www.nhc.noaa.gov/http://iacs.ep .ch/cmcs/NewResearch/vascular.php3http://www.navier–/http://www-groups.dcs.st-and.ac.uk/～history/Mathematicians/Navier.htmlhttp://www-groups.dcs.st-and.ac.uk/～history/Mathematicians/Stokes.html