Navier-Stokes Equations

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1、Navier-StokesEquations•Amathematicaldescriptionoffluidmotionunderthefollowingassumptions:–Constantdensity,ρ–Constantviscosity,µ–Continuity(incompressibleflow)∇•u=0CHEM520DerivationofN-S•Derivationbasedonforcebalanceaboutafluidelement:–Netforce=sumoffo

2、rces•Convectiveforce(convectivetransportofmomentum),∇•ρuu•Viscousstressforces(viscousforces),∇•τ•Pressureforces,∇p•Externalforces,e.g.gravity,ρgzy∆z∂∆y∆xρu=−∇•ρuu−∇•τ−∇p+F∂tx∂⇒ρu=−∇•ρuu−∇•τ−∇p+ρg∂tCHEM520DerivationofN-Sτ=−µ∇u•Viscousstresstensorτxx

3、τyxτzxτ=τττxyyyzyτxzτyzτzz•ForNewtonianfluid2∂u2()τ=µ−+∇•uxx∂x3∂vτ=µ−2+2()∇•uyy∂y3∂u2∂w2()3-Dτzz=µ−+3∇•u1-Dτyx=−µ∂z∂y∂u∂vτ=τ=−µ+xyyx∂y∂x∂v∂wτ=τ=−µ+yzzy∂z∂y∂w∂uτzx=τxz=−µ+CHEM520∂x∂zDerivationo

4、fN-SViscousstress(recap)•Stresshasunitofpressure.•Itdescribesthelocalvariationinvelocity.•Viscousforce=Stressxarea•Forcebalanceusingstress:–Viscousforcesbalance(1)convectiveforce,(2)pressureforce,and(3)externalforce.–τ∆A=u•grad(u)∆A+P∆A+F∆A∂u00τ=−µ

5、yx∂yCHEM520DerivationofN-SFromforcebalance:D=−∇•−∇+ρuτpFDtDrruruupFConstantρandµ:ρµ=−∇•−∇−∇+DtNavier-Stokeseqt.rrurDuu2pFρµ=∇−∇+DtInviscidflow,:∇•τ=0Dρu=−∇p+FEuler’seqt.DtCHEM520N-SequationsummaryAssumptions:–Constantdensity,ρ–Constantvis

6、cosity,µ–Continuity(incompressibleflow)∇•u=0RateofchangeofConvectiveViscousPressureExternalmomentum;net=+++forceforceforceforceaccelerationρ∂u+ρu•∇u=µ∇2u−∇p+F∂t⇔ρ∂u=−ρu•∇u+µ∇2u−∇p+F∂tRateofchangeConvectiveViscousPressureExternalofmomentum;forceforcefo

7、rceforceCHEM520netaccelerationN-SEquationin1-DIn1-Dproblem2∂u+u∂u=∂u−dp+ρρµFx∂t∂x∂x2dxRateofchangeofmomentum;netExternalforceacceleratione.g.gravityConvectiveViscousPressureforceforceforceCHEM520DimensionalAnalysis•Why?–Reductionofinformation,•Paramet

8、ersandvariablesinatypicalfluidsystem:P,D,U,ρ,µ.•Parametersandvariablesinatypicalreactivetransportsystemwithouteffectsduetoelectricalfield:T,U,c,ρ,µ,k,D,P,r,abaγ,…•Howaboutwithelectricfieldeffects?Likeelectrochemicalsystems?–Simplifyproblemandf