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Navier-Stokes Equations { 2d case

navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A) navier -StokesEquations{2dcaseSOE32 11/2 FluidMechanicslecture3 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)NSE(A) conservationofmass,momentum. oftenwrittenassetofp de's di erentialform{ uid owatap oint 2dcase,incompressible ow:Continuityequation:@ux@x+@uy@y=0 conservationofmass seenb efore{p otential owNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Momentumequations:@ux@t+ux@ux@x+uy @ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx@uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 +fy (xandycmpts) 3variables,ux,uy,p linkedequations needtosimplfybyconsideringdetailsofprobl emNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)TheNSEare Non-linear{termsinvolvingux@ux@x Partialdi erentialequations{ux,pfunctionsofx,y,t 2ndorder{highestorderderivatives@2ux@x2 Coupled{momentumequationinvolvesp,ux,uyT wowaystosolvetheseequations1 Applytosimplecases{simplegeometry,simple conditions{andreduceequationsuntilwecans olvethem2 Us}}}}}}}}}}}}}

Navier-Stokes Equations {2d case NSE (A) Equation analysis Equation analysis Equation analysis Equation analysis Equation analysis Laminar ow between plates (A) Flow dwno inclined plane (A) Tips (A) NSE (A) conservation of mass, momentum. often written as set of pde's di erential form { uid ow at a point 2d case, incompressible ow :

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Transcription of Navier-Stokes Equations { 2d case

1 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A) navier -StokesEquations{2dcaseSOE32 11/2 FluidMechanicslecture3 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)NSE(A) conservationofmass,momentum. oftenwrittenassetofp de's di erentialform{ uid owatap oint 2dcase,incompressible ow:Continuityequation:@ux@x+@uy@y=0 conservationofmass seenb efore{p otential owNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Momentumequations:@ux@t+ux@ux@x+uy @ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx@uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 +fy (xandycmpts) 3variables,ux,uy,p linkedequations needtosimplfybyconsideringdetailsofprobl emNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)TheNSEare Non-linear{termsinvolvingux@ux@x Partialdi erentialequations{ux,pfunctionsofx,y,t 2ndorder{highestorderderivatives@2ux@x2 Coupled{momentumequationinvolvesp,ux,uyT wowaystosolvetheseequations1 Applytosimplecases{simplegeometry,simple conditions{andreduceequationsuntilwecans olvethem2 Usecomputationalmetho ds{CFD(SOE3212/3) navier -StokesEquations{ 2dcaseNSE(A)EquationanalysisEquationanal ysisEquationanalysisEquationanalysisEqua tionanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)}}}}}}}}}}}}}}}

2 TheNSEare Non-linear{termsinvolvingux@ux@x Partialdi erentialequations{ux,pfunctionsofx,y,t 2ndorder{highestorderderivatives@2ux@x2 Coupled{momentumequationinvolvesp,ux,uyT wowaystosolvetheseequations1 Applytosimplecases{simplegeometry,simple conditions{andreduceequationsuntilwecans olvethem2 Usecomputationalmetho ds{CFD(SOE3212/3) navier -StokesEquations{ 2dcaseNSE(A)EquationanalysisEquationanal ysisEquationanalysisEquationanalysisEqua tionanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx@ux@t changeofuxatap ointNavier-StokesEquations{2dcaseNSE(A)E quationanalysisEquationanalysisEquationa nalysisEquationanalysisEquationanalysisL aminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fxux@ux@x+uy@ux@y transp ort/advectionterm howdo es ow(ux;uy)moveux? non-linearNavier-StokesEquations{2dcaseN SE(A)EquationanalysisEquationanalysisEqu ationanalysisEquationanalysisEquationana lysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx 1 @p@x pressuregradient{usuallydrives owNavier-StokesEquations{2dcaseNSE(A)Equ ationanalysisEquationanalysisEquationana lysisEquationanalysisEquationanalysisLam inar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms:@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fx @2ux@x2+@2ux@y2 viscousterm{e ectofviscosity on ow hasadi usivee ectNavier-StokesEquations{2dcaseNSE(A)Eq uationanalysisEquationanalysisEquationan alysisEquationanalysisEquationanalysisLa minar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Equationanalysis(A)Considerthevari ousterms.}}}}}}}}}}}}}}

3 @ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 +fxfx externalb o dyforces{ {2dcaseNSE(A)EquationanalysisEquationana lysisEquationanalysisEquationanalysisEqu ationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Laminar owb etweenplates(A)Fullydevelop edlaminar owb etweenin niteplatesaty= aWhatdoweexp ectfromthe ow?xy-a+a u=0atwalls Flowsymmetricaroundy=0 FlowparalleltowallsNavier-StokesEquation s{2dcaseNSE(A)EquationanalysisEquationan alysisEquationanalysisEquationanalysisEq uationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@x2+@2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowparalleltowalls{weexp ectuy=0;dpdy=0andux=ux(y) navier -StokesEq uations{2dcaseNSE(A)EquationanalysisEqua tionanalysisEquationanalysisEquationanal ysisEquationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowparalleltowalls{weexp ectuy=0.}}}}}}}

4 Dpdy=0andux=ux(y) navier -StokesEquations{ 2dcaseNSE(A)EquationanalysisEquationanal ysisEquationanalysisEquationanalysisEqua tionanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowfullydevelop ed{nochangeinpro @@t=0;@@x=0 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)@ux@t+ux@ux@x+uy@ux@y= 1 @p@x+ @2ux@y2 @uy@t+ux@uy@x+uy@uy@y= 1 @p@y+ @2uy@x2+@2uy@y2 Flowfullydevelop ed{nochangeinpro @@t=0;@@x=0 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce.}}}}}}}

5 Ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Somomentumequationb ecomes0= 1 dpdx+ d2uxdy2 Integrateonce:ydpdx= duxdy+C1 Butaty=0,duxdy=0(symmetry),soC1= ux+C2 Butaty= a,ux=0,soC2=12a2dpdxNavier-StokesEquatio ns{2dcaseNSE(A)EquationanalysisEquationa nalysisEquationanalysisEquationanalysisE quationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Finalsolutionux(y)=12 y2 a2 dpdx{equationofaparab olaAlso,rememb erthat = @ux@ySofromthisweseethatinthiscase =ydpdxNavier-StokesEquations{2dcaseNSE(A )EquationanalysisEquationanalysisEquatio nanalysisEquationanalysisEquationanalysi sLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Flowdowninclinedplane(A){Flowofliq uiddowninclinedplanehyx uxTakex-comp onentmomentumequation@ux@t+ux@ux@x+uy@ux @y= 1 @p@x+ @2ux@x2+@2ux@y2 +fxNavier-StokesEquations{2dcaseNSE(A)Eq uationanalysisEquationanalysisEquationan alysisEquationanalysisEquationanalysisLa minar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Note.}}}}}}}

6 1 Steady ow2ux(y)only3 Nopressuregradient4fx=gsin Equationb ecomesd2uxdy2= g sin {2dcaseNSE(A)EquationanalysisEquationana lysisEquationanalysisEquationanalysisEqu ationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Note:1 Steady ow2ux(y)only3 Nopressuregradient4fx=gsin Equationb ecomesd2uxdy2= g sin {2dcaseNSE(A)EquationanalysisEquationana lysisEquationanalysisEquationanalysisEqu ationanalysisLaminar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Boundaryconditions: lowersurface{ux(0)=0 upp ersurface{duxdy=0 Solutionux=g sin hy y22 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Boundaryconditions: lowersurface{ux(0)=0 upp ersurface{duxdy=0 Solutionux=g sin hy y22 navier -StokesEquations{2dcaseNSE(A)Equat ionanalysisEquationanalysisEquationanaly sisEquationanalysisEquationanalysisLamin ar owb etweenplates(A)Flowdowninclinedplane(A)T ips(A)Tips(A)MostNSEproblemswillb etime-indep ow,andoneco eeitherpressuredriven(sonoviscousterm)or sheardriven( ,sonopressureterm).}}}}}}}}

7 Thus,mostNSEproblemswillleadtoa2ndorderO DEforavelo citycomp onent(uxoruy)asafunctionofoneco ordinate(xory).Thuswewouldexp ecttointegratetwice,andtoimp osetwob oundaryconditions. Awallb oundaryconditionpro ducesa Afreesurfacepro ducesazerogradientcondition.


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