工程流体力学(英文版)第二章.pdf

ChapterTwoFluidStatics(WhatisFluidStaticsGeneralRulesoffluidatrest,andtheirengineeringapplication.fluidatrestfluidinequilibriumEquilibrium(a=0)relativeequilibrium(a=0)CharacteristicofFluidatrestu=0du=00dudyτμ==Contents2.1Staticalpressureintensityanditscharacteristic2.2DifferentialEquationofFluidEquilibrium2.3PressureDistributionintheStaticFluid2.4PressureMearurements2.5FluidinRelativeEquilibriumFluid.2.6FluidStaticForceonPlaneandCurvedArea2.2.1DefinitionofstaticalpressureintensityNormalforceactingoverperunitareaofastaticfluid2.1Staticalpressureintensityanditscharacteristic
0lim
nAFpA→=Unit:PaorN/m22.2.1Characteristic1direction2magnitudeThepressureatapointinafluidatrestisthesameinalldirections.Ithasnothingtodowiththenormaldirectionoftheactingsurface.NegativeNormalForcePositive--PullingShearing

√√√√Thereisonlycompressivestress(orpressure)inafluidatrest,andthedirectionofpressureisthesameasthedirectionofinwardnormallineofactingpoint.Fluidatrestcannotbearpullingforcebecauseofthetrendstoflow.xyznpppp===(1)SelectatriangularprismelementOABC,dx,dy,dzpx,py,pz,pnarepressureintensityactedontherespectivesurface.Thesurfacepressure:1dd2xpyz⋅1dd2ypxz⋅1dd2zpxy⋅dnpA⋅,,,nisnormaldirectionofinclinedsurfaceABC.yxz0pzpypxpndydxdzCABConsiderforcecomponentsinxdirection:Surfaceforces:Massforces(2)Forceanalysis:OAC:OAB:OBC:ABC:1dd2xpyz⋅1dd2ypxz⋅1dd2zpxy⋅dnpA⋅1ddd6xfxyzρ⋅1ddd6yfxyzρ⋅1ddd6zfxyzρ⋅,,(3)Equationoffluidinequilibrium0F=
11cos(,)026xnxpdydzpdAnxfdxdydzρ⋅−⋅+⋅=yxz0pzpypxpndydxdzCABSimilarlyWhentheelementshrinkstoapointo,dx→0thusTheresultsshowthatthepressuresareindependentofdirectionbecauseisarbitrary.Hencethepressureatapointonastaticfluidisthesameinalldirections.n
n
n
And:1dcos(,)dd2Anxyz=So:111ddddddd0226xnxpyzpyzfxyzρ⋅−⋅+⋅=1d03xnxppfxρ−+⋅=yxz0pzpypxpndydxdzCABxnpp=xyznpppp===(),,ppxyz=11cos(,)026xnxpdydzpdAnxfdxdydzρ⋅−⋅+⋅=1DifferentialEquationsofaFluidinEquilibrium------EulerEquilibriumEquations2PressureDifferenceEquation3ForcePotentialFunction2SurfaceofEqualPressure2.2DifferentialEquationofFluidEquilibrium2.2.1DifferentialEquationsofaFluidinEquilibrium------EulerEquilibriumEquations2.2DifferentialEquationsofaFluidinEquilibriumConsiderthesixsurfacesofinfinitesimalelementinequilibriumfluid.Itssidesaredx,dy,dz.Assumethepressureatthecenteroftheelementisp(x,y,z)=p.2.2DifferentialEquationofFluidEquilibriumyxz•dydxdz2.2DifferentialEquationsofaFluidinEquilibriumConsiderforcecomponentsinydirectionMassforcesyxz•dydxdzSurfaceforces:d2Bpyppy∂=−∂d2Cpyppy∂=+∂d()dd2pypxzy∂−∂d()dd2pypxzy∂−+∂left:right:dddyfxyzρThereisFy=0inydirectionbecausetheelementisinequilibrium:dd()dd()ddddd022ypypypxzpxzfxyzyyρ∂∂−−++=∂∂()200''()()()'()2fxfxxfxfxxxΔΔΔ+=+++⋅⋅⋅DifferentialEquationsofaFluidinEquilibriumEulerEquilibriumEquationscondition:or2.2DifferentialEquationsofaFluidinEquilibriumdd()dd()ddddd022ypypypxzpxzfxyzyyρ∂∂−−++=∂∂10ypfyρ∂−=∂101010xyzpfxpfypfzρρρ∂−=∂∂−=∂∂−=∂1grad0fpρ−=
EquilibriumandrelativeequilibriumCompressibleandincompressibleflowPhysicalMeaning:Forthefluidinequilibrium,surfaceforcecomponentspermassfluidareequaltomassforcecomponentspermassfluid.Pressurevariationrateinaxesdirections)areequaltomassforcecomponentsperunitvolumeinaxesdirectionsρfx,ρfy,ρfz)respectively.zpypxp∂∂∂∂∂∂,,or2.2DifferentialEquationsofaFluidinEquilibrium101010xyzpfxpfypfzρρρ∂−=∂∂−=∂∂−=∂1grad0fpρ−=
2.2.2PressureDifferenceEquation(GeneralDifferentialEquationsofaFluidinEquilibrium)101010xyzpfxpfypfzρρρ∂−=∂∂−=∂∂−=∂ddddppppxyzxyz∂∂∂=++∂∂∂1ddd(ddd)xyzpppfxfyfzxyzxyzρ∂∂∂++=++∂∂∂p=p(x,y,z)Multiplyeveryequationinequationgroup(1)withdx,dy,dzrespectively,thenaddthem:thetotaldifferentialofpressureis:2.2DifferentialEquationsofaFluidinEquilibriumd(ddd)xyzpfxfyfzρ=++2.2.3ForcePotentialFunction()xyzdpfdxfdyfdz=++ρIfthedensityisaconstant:Defineaforcepotentialfunction:p=−πρ()xyzdpfdxfdyfdzρ=++()pdddxdydzxyyππππρ ∂∂∂=−=−−−∂∂∂xyzfxfyfzπππ∂=−∂∂=−∂∂=−∂2.2.4EquipressureSurfaceEquipressureSurfaceisasurfacethatthepressureofeverypointinliquidisequal.Commonequipressuresurfacesarefreeliquidsurfaceandinterfaceoftwounmixedfluidsinequilibrium.massforceofanypointontheequipressuresurfaceinequilibriumfluidisperpendiculartotheequipressuresurface.KinescopeCartoon2.2DifferentialEquationsofaFluidinEquilibriumd0p=ddd0xyzfxfyfz++=0fdr⋅=
Importantcharacterofequipressuresurface:xyxffifjfkdrdxidyjdzk
=++=++







()xyzdpfdxfdyfdz=++ρProvingConsiderafluidparticleMontheequipressuresurface,drisadifferentialdistanceontheequipressuresurface.AssumetheunitmassforceoftheparticleisAttheequipressuresurfaceinequilibriumfluid:drdxidyjdzk=++



()0xyzdpfdxfdyfdzρ=++=Massforceisperpendiculartodrdrislinevectorontheequipressuresurface0fdr∴⊥=
Massforceisperpendiculartoequipressuresurfacexyzdrisxyxffifjfk=++



0fdr⋅=
2.3.1BasicEquationofStaticFluidunderGravity1.BasicEquationGeneraldifferentialequationofafluidinequilibrium()xyzdpfdxfdyfdzρ=++0xyzfffg===−0dpgdzdzdpgρρ=−+=cρ=pzCgρ+=ABABppzzggρρ+=+(uniform,impressiblefluid,undertheactionofgravity)2.3PressureDistributi