7. Mechanics of fluids

 

 

 

We now apply the general principles described in the preceding chapters to specific problems.  In this chapter, we give a short introduction to the governing equations for fluids, and give solutions to some simple boundary value problems.

7.1 Summary of governing equations for fluids

 

Fluids have (by definition) the following properties:

(1)   A fluid has no natural reference configuration, so all governing equations are expressed in spatial form.  The lack of reference configuration also means that the response of the fluid can only depend on kinematic variables that characterize motion only of the current configuration MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  i.e. the velocity gradient, and measures derived from it.

(2)   A fluid at rest can support no shear stress MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the stress state in a stationary fluid is just a hydrostatic pressure, which may vary with position.

 

The central problem in a fluid mechanics problem is generally to determine the velocity distribution v(y,t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacaGGOaGaaCyEaiaacYcacaWG0b Gaaiykaaaa@356B@ , Cauchy stress distribution σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaaaaa@3434@  and (sometimes) temperature θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321E@ , as functions of position and time.  The fluid is characterized by the following physical quantities:

* The mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@

* The specific internal energy ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLbaa@320F@

* The specific entropy s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohaaaa@3160@

* The specific Helmholtz free energy ψ=εθs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iabew7aLjabgkHiTi abeI7aXjaadohaaaa@387E@

* A stress response function σ ij = σ ^ ij (θ,ρ,kinematic and internal variables) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcuaHdpWCgaqcamaaBaaaleaacaWGPbGaamOAaaqa baGccaGGOaGaeqiUdeNaaiilaiabeg8aYjaacYcacaqGRbGaaeyAai aab6gacaqGLbGaaeyBaiaabggacaqG0bGaaeyAaiaabogacaqGGaGa aeyyaiaab6gacaqGKbGaaeiiaiaabMgacaqGUbGaaeiDaiaabwgaca qGYbGaaeOBaiaabggacaqGSbGaaeiiaiaabAhacaqGHbGaaeOCaiaa bMgacaqGHbGaaeOyaiaabYgacaqGLbGaae4CaiaabMcaaaa@5C16@

* A heat flux response function q i = q ^ i (θ,ρ,kinematic and internal variables) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadghadaWgaaWcbaGaamyAaaqabaGccq GH9aqpceWGXbGbaKaadaWgaaWcbaGaamyAaaqabaGccaGGOaGaeqiU deNaaiilaiabeg8aYjaacYcacaqGRbGaaeyAaiaab6gacaqGLbGaae yBaiaabggacaqG0bGaaeyAaiaabogacaqGGaGaaeyyaiaab6gacaqG KbGaaeiiaiaabMgacaqGUbGaaeiDaiaabwgacaqGYbGaaeOBaiaabg gacaqGSbGaaeiiaiaabAhacaqGHbGaaeOCaiaabMgacaqGHbGaaeOy aiaabYgacaqGLbGaae4CaiaabMcaaaa@589E@

 

Body forces: The fluid is subjected to an external body force b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaamyAaaqabaaaaa@3269@  per unit mass.

 

Its motion is characterized by the usual deformation measures

*  Velocity Gradient L ij = v i y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0ZaaSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaadMga aeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGQbaabeaaaaaaaa@3B66@

*  Strain rate decomposition L ij = D ij + W ij D ij =( L ij + L ji )/2 W ij =( L ij L ji )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaamiramaaBaaaleaacaWGPbGaamOAaaqabaGccqGH RaWkcaWGxbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey ypa0JaaiikaiaadYeadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey4k aSIaamitamaaBaaaleaacaWGQbGaamyAaaqabaGccaGGPaGaai4lai aaikdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGxbWa aSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iaacIcacaWGmbWaaS baaSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadYeadaWgaaWcbaGa amOAaiaadMgaaeqaaOGaaiykaiaac+cacaaIYaaaaa@718B@

*  Vorticity ω i = ijk v k y j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGPbaabeaaki abg2da9iabgIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqaaOWa aSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaey OaIyRaamyEamaaBaaaleaacaWGQbaabeaaaaaaaa@3FFC@

*  Vorticity-Spin relation ω=2dual(W) ω i = ijk W jk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM8acqGH9aqpcaaIYaGaamizaiaadw hacaWGHbGaamiBaiaacIcacaWHxbGaaiykaiaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaeqyYdC3aaSbaaSqaaiaadMgaaeqaaOGa eyypa0JaeyOeI0IaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4Aaa qabaGccaWGxbWaaSbaaSqaaiaadQgacaWGRbaabeaaaaa@5B5E@

* Velocity-acceleration relations

 

a i = v i t | x k =const = v i y k y k t + v i t | y i =const = L ik v k + v i t | y i =const =( D ik + W ik ) v k + v i t | y i =const = 1 2 y i ( v k v k )+2 W ik v k = v i t | y k =const + 1 2 y i ( v k v k )+ ijk ω j v k ijk a k y j = ω i t | x=const D ij ω j + v k y k ω i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamyyamaaBaaaleaacaWGPbaabe aakiabg2da9maaeiaabaWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeyOaIyRaamiDaaaaaiaawIa7amaaBaaale aacaWG4bWaaSbaaWqaaiaadUgaaeqaaSGaeyypa0Jaam4yaiaad+ga caWGUbGaam4CaiaadshaaeqaaOGaeyypa0ZaaSaaaeaacqGHciITca WG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaa leaacaWGRbaabeaaaaGcdaWcaaqaaiabgkGi2kaadMhadaWgaaWcba Gaam4AaaqabaaakeaacqGHciITcaWG0baaaiabgUcaRmaaeiaabaWa aSaaaeaacqGHciITcaWG2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaey OaIyRaamiDaaaaaiaawIa7amaaBaaaleaacaWG5bWaaSbaaWqaaiaa dMgaaeqaaSGaeyypa0Jaam4yaiaad+gacaWGUbGaam4Caiaadshaae qaaOGaeyypa0JaamitamaaBaaaleaacaWGPbGaam4AaaqabaGccaWG 2bWaaSbaaSqaaiaadUgaaeqaaOGaey4kaSYaaqGaaeaadaWcaaqaai abgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG 0baaaaGaayjcSdWaaSbaaSqaaiaadMhadaWgaaadbaGaamyAaaqaba WccqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGH 9aqpdaqadaqaaiaadseadaWgaaWcbaGaamyAaiaadUgaaeqaaOGaey 4kaSIaam4vamaaBaaaleaacaWGPbGaam4AaaqabaaakiaawIcacaGL PaaacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaey4kaSYaaqGaaeaada WcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGH ciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaadMhadaWgaaadbaGaam yAaaqabaWccqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqa baaakeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 Uaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiabgkGi 2cqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaaiikai aadAhadaWgaaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadUga aeqaaOGaaiykaiabgUcaRiaaikdacaWGxbWaaSbaaSqaaiaadMgaca WGRbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccqGH9aqpdaab caqaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaaaO qaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaamyEamaaBaaa meaacaWGRbaabeaaliabg2da9iaadogacaWGVbGaamOBaiaadohaca WG0baabeaakiabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaWaaSaa aeaacqGHciITaeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaa aakiaacIcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaamODamaaBaaa leaacaWGRbaabeaakiaacMcacqGHRaWkcqGHiiIZdaWgaaWcbaGaam yAaiaadQgacaWGRbaabeaakiabeM8a3naaBaaaleaacaWGQbaabeaa kiaadAhadaWgaaWcbaGaam4AaaqabaaakeaacqGHiiIZdaWgaaWcba GaamyAaiaadQgacaWGRbaabeaakmaalaaabaGaeyOaIyRaamyyamaa BaaaleaacaWGRbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaam OAaaqabaaaaOGaeyypa0ZaaqGaaeaadaWcaaqaaiabgkGi2kabeM8a 3naaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiW oadaWgaaWcbaGaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadoha caWG0baabeaakiabgkHiTiaadseadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeqyYdC3aaSbaaSqaaiaadQgaaeqaaOGaey4kaSYaaSaaaeaa cqGHciITcaWG2bWaaSbaaSqaaiaadUgaaeqaaaGcbaGaeyOaIyRaam yEamaaBaaaleaacaWGRbaabeaaaaGccqaHjpWDdaWgaaWcbaGaamyA aaqabaaaaaa@08C0@

 

 

and is governed by the standard Conservation laws:

 

  Mass Conservation ρ t | x=const +ρ D kk =0or    ρ t | y=const + ρ v i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcqaHbp GCaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahIhacqGH 9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGHRaWkcq aHbpGCcaWGebWaaSbaaSqaaiaadUgacaWGRbaabeaakiabg2da9iaa icdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaab+gacaqGYbGaae iiaiaabccacaqGGaWaaqGaaeaadaWcaaqaaiabgkGi2kabeg8aYbqa aiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaaCyEaiabg2da9i aadogacaWGVbGaamOBaiaadohacaWG0baabeaakiabgUcaRmaalaaa baGaeyOaIyRaeqyWdiNaamODamaaBaaaleaacaWGPbaabeaaaOqaai abgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaeyypa0JaaGim aaaa@6AFF@

  Linear momentum conservation σ ji y j +ρ b i =ρ( v i y k v k + v i t | y i =const ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeq4Wdm3aaSbaaS qaaiaadQgacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGa amOAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaaleaacaWGPb aabeaakiabg2da9iabeg8aYnaabmaabaWaaSaaaeaacqGHciITcaWG 2bWaaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaale aacaWGRbaabeaaaaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaey4k aSYaaqGaaeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaa qabaaakeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaadMha daWgaaadbaGaamyAaaqabaWccqGH9aqpcaWGJbGaam4Baiaad6gaca WGZbGaamiDaaqabaaakiaawIcacaGLPaaaaaa@5C04@

  Energy conservation ρ ε t | x=const = σ ij D ij q i y i +q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaeiaabaWaaSaaaeaacqGHci ITcqaH1oqzaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaa hIhacqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccq GH9aqpcqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamiramaa BaaaleaacaWGPbGaamOAaaqabaGccqGHsisldaWcaaqaaiabgkGi2k aadghadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgUcaRiaadghaaaa@51CC@

 

Finally the constitutive law must satisfy the Entropy Inequality: σ ij D ij 1 θ q i θ y i ρ( ψ t +s θ t )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTmaa laaabaGaaGymaaqaaiabeI7aXbaacaWGXbWaaSbaaSqaaiaadMgaae qaaOWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgkHiTiabeg8aYnaabmaabaWaaSaaae aacqGHciITcqaHipqEaeaacqGHciITcaWG0baaaiabgUcaRiaadoha daWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi2kaadshaaaaacaGLOa GaayzkaaGaeyyzImRaaGimaaaa@5714@

 

Transformations under observer changes:

*  Velocity Gradient L * =QL Q T + Q ˙ Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahYeadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaCitaiaahgfadaahaaWcbeqaaiaadsfaaaGccqGH RaWkceWHrbGbaiaacaWHrbWaaWbaaSqabeaacaWGubaaaaaa@3A66@

*  Stretch rate D * =QL Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahseadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaCitaiaahgfadaahaaWcbeqaaiaadsfaaaaaaa@36AF@

*  Spin W * =QW Q T + Q ˙ Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahEfadaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaC4vaiaahgfadaahaaWcbeqaaiaadsfaaaGccqGH RaWkceWHrbGbaiaacaWHrbWaaWbaaSqabeaacaWGubaaaaaa@3A7C@

*  Cauchy Stress σ * =Qσ Q T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaho8adaahaaWcbeqaaiaacQcaaaGccq GH9aqpcaWHrbGaaC4WdiaahgfadaahaaWcbeqaaiaadsfaaaaaaa@37AB@

 

 

7.2 General Form for Constitutive equations for fluids: 

 

We now list the general form of the constitutive equations for a fluid that are consistent with frame indifference and the entropy inequality.  In practice, most problems of interest are approximated using one of several special cases of the general equations.  These will be listed separately.

 

To be consistent with frame indifference and the laws of thermodynamics, the specific free energy, internal energy, Helmholtz free energy, stress response function and heat transfer function must have the forms

* Specific internal energy ε= ε ^ (ρ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLjabg2da9iqbew7aLzaajaGaai ikaiabeg8aYjaacYcacqaH4oqCcaGGPaaaaa@3A4B@

* Specific entropy s= s ^ (ρ,θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpceWGZbGbaKaacaGGOa GaeqyWdiNaaiilaiabeI7aXjaacMcaaaa@38ED@

* Specific Helmholtz free energy ψ= ψ ^ (ρ,θ)=εθs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iqbeI8a5zaajaGaai ikaiabeg8aYjaacYcacqaH4oqCcaGGPaGaeyypa0JaeqyTduMaeyOe I0IaeqiUdeNaam4Caaaa@40E1@

* Stress response function σ ij = σ ^ ij (θ,ρ, D ij )= π ^ eq (ρ,θ) δ ij + σ ^ ij vis (ρ,θ, D ij ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccqGH9aqpcuaHdpWCgaqcamaaBaaaleaacaWGPbGaamOAaaqa baGccaGGOaGaeqiUdeNaaiilaiabeg8aYjaacYcacaWGebWaaSbaaS qaaiaadMgacaWGQbaabeaakiaacMcacqGH9aqpcqGHsislcuaHapaC gaqcamaaBaaaleaacaWGLbGaamyCaaqabaGccaGGOaGaeqyWdiNaai ilaiabeI7aXjaacMcacqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqa aOGaey4kaSIafq4WdmNbaKaadaqhaaWcbaGaamyAaiaadQgaaeaaca WG2bGaamyAaiaadohaaaGccaGGOaGaeqyWdiNaaiilaiabeI7aXjaa cYcacaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacMcaaaa@61F7@  

    with σ ^ ij vis (ρ,θ,0)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaajaWaa0baaSqaaiaadMgaca WGQbaabaGaamODaiaadMgacaWGZbaaaOGaaiikaiabeg8aYjaacYca cqaH4oqCcaGGSaGaaGimaiaacMcacqGH9aqpcaaIWaaaaa@3FD9@ .  Here, π eq MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabec8aWnaaBaaaleaacaWGLbGaamyCaa qabaaaaa@3431@  can be interpreted as the pressure that would be measured in the fluid with density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@  and temperature θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321E@  when at rest.   This decomposition is given a-priori, and is constructed to ensure that the stress state in a fluid at rest supports zero shear stress.

* Heat flux response function q i = q ^ i ( θ,ρ, θ y i , D ij ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadghadaWgaaWcbaGaamyAaaqabaGccq GH9aqpceWGXbGbaKaadaWgaaWcbaGaamyAaaqabaGcdaqadaqaaiab eI7aXjaacYcacqaHbpGCcaGGSaWaaSaaaeaacqGHciITcqaH4oqCae aacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiaacYcacaWG ebWaaSbaaSqaaiaadMgacaWGQbaabeaaaOGaayjkaiaawMcaaaaa@4651@

It is useful also to introduce the specific heat at constant volume c v (θ,ρ)= ε ^ θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamODaaqabaGcca GGOaGaeqiUdeNaaiilaiabeg8aYjaacMcacqGH9aqpdaWcaaqaaiab gkGi2kqbew7aLzaajaaabaGaeyOaIyRaeqiUdehaaaaa@3F4F@

 

Then, the equilibrium pressure π ^ eq MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbec8aWzaajaWaaSbaaSqaaiaadwgaca WGXbaabeaaaaa@3441@ , free energy, entropy and specific heat capacity are related by

π ^ eq = ρ 2 ψ ^ ρ s ^ = ψ θ π ^ eq θ = ρ 2 s ^ ρ π ^ eq =θ π ^ eq θ + ρ 2 ε ^ ρ c v =θ 2 ψ ^ θ 2 c v ρ = θ ρ 2 2 π ^ eq θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafqiWdaNbaKaadaWgaaWcbaGaam yzaiaadghaaeqaaOGaeyypa0JaeqyWdi3aaWbaaSqabeaacaaIYaaa aOWaaSaaaeaacqGHciITcuaHipqEgaqcaaqaaiabgkGi2kabeg8aYb aacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7ceWGZbGbaKaacqGH9aqpcqGHsi sldaWcaaqaaiabgkGi2kabeI8a5bqaaiabgkGi2kabeI7aXbaaaeaa daWcaaqaaiabgkGi2kqbec8aWzaajaWaaSbaaSqaaiaadwgacaWGXb aabeaaaOqaaiabgkGi2kabeI7aXbaacqGH9aqpcqGHsislcqaHbpGC daahaaWcbeqaaiaaikdaaaGcdaWcaaqaaiabgkGi2kqadohagaqcaa qaaiabgkGi2kabeg8aYbaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UafqiWdaNbaKaadaWgaaWcbaGaamyzaiaadghaaeqaaOGaeyypa0Ja eqiUde3aaSaaaeaacqGHciITcuaHapaCgaqcamaaBaaaleaacaWGLb GaamyCaaqabaaakeaacqGHciITcqaH4oqCaaGaey4kaSIaeqyWdi3a aWbaaSqabeaacaaIYaaaaOWaaSaaaeaacqGHciITcuaH1oqzgaqcaa qaaiabgkGi2kabeg8aYbaaaeaacaWGJbWaaSbaaSqaaiaadAhaaeqa aOGaeyypa0JaeyOeI0IaeqiUde3aaSaaaeaacqGHciITdaahaaWcbe qaaiaaikdaaaGccuaHipqEgaqcaaqaaiabgkGi2kabeI7aXnaaCaaa leqabaGaaGOmaaaaaaGccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVpaalaaabaGaeyOaIyRaam4yamaaBa aaleaacaWG2baabeaaaOqaaiabgkGi2kabeg8aYbaacqGH9aqpcqGH sisldaWcaaqaaiabeI7aXbqaaiabeg8aYnaaCaaaleqabaGaaGOmaa aaaaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiqbec8a WzaajaWaaSbaaSqaaiaadwgacaWGXbaabeaaaOqaaiabgkGi2kabeI 7aXnaaCaaaleqabaGaaGOmaaaaaaaaaaa@D9F4@

In addition, the viscous stress and heat flux functions must satisfy

σ ij vis (ρ,θ, D ij ) D ij 0 q i ( ρ,θ, θ y i ) θ y i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaDaaaleaacaWGPbGaamOAaa qaaiaadAhacaWGPbGaam4CaaaakiaacIcacqaHbpGCcaGGSaGaeqiU deNaaiilaiaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykai aadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyyzImRaaGimaiaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaamyCamaaBaaaleaacaWGPbaabeaakmaabmaa baGaeqyWdiNaaiilaiabeI7aXjaacYcadaWcaaqaaiabgkGi2kabeI 7aXbqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaaGccaGL OaGaayzkaaWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG5b WaaSbaaSqaaiaadMgaaeqaaaaakiabgwMiZkaaicdaaaa@6EE1@

 

We next summarize the reasoning that leads to these conclusions:

 

·         Frame indifference shows that the free energy function and the stress and heat transfer response functions depend only on the symmetric part of the velocity gradient (i.e. the stretch rate), and must be isotropic functions.  To see this, note that frame indifference requires that

ψ ^ (ρ,θ,L)= ψ ^ (ρ,θ,QL Q T + Q ˙ Q T ) Q ik σ ^ kl (ρ,θ,L) Q jl = σ ^ ij (ρ,θ,QL Q T + Q ˙ Q T ) Q ik q ^ k ( θ,ρ,θ,L )= q ^ i ( θ,ρ,Qθ,QL Q T + Q ˙ Q T ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafqiYdKNbaKaacaGGOaGaeqyWdi NaaiilaiabeI7aXjaacYcacaWHmbGaaiykaiabg2da9iqbeI8a5zaa jaGaaiikaiabeg8aYjaacYcacqaH4oqCcaGGSaGaaCyuaiaahYeaca WHrbWaaWbaaSqabeaacaWGubaaaOGaey4kaSIabCyuayaacaGaaCyu amaaCaaaleqabaGaamivaaaakiaacMcaaeaacaWGrbWaaSbaaSqaai aadMgacaWGRbaabeaakiqbeo8aZzaajaWaaSbaaSqaaiaadUgacaWG SbaabeaakiaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiilaiaahYeaca GGPaGaamyuamaaBaaaleaacaWGQbGaamiBaaqabaGccqGH9aqpcuaH dpWCgaqcamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGOaGaeqyWdi NaaiilaiabeI7aXjaacYcacaWHrbGaaCitaiaahgfadaahaaWcbeqa aiaadsfaaaGccqGHRaWkceWHrbGbaiaacaWHrbWaaWbaaSqabeaaca WGubaaaOGaaiykaaqaaiaadgfadaWgaaWcbaGaamyAaiaadUgaaeqa aOGabmyCayaajaWaaSbaaSqaaiaadUgaaeqaaOWaaeWaaeaacqaH4o qCcaGGSaGaeqyWdiNaaiilaiabgEGirlabeI7aXjaacYcacaWHmbaa caGLOaGaayzkaaGaeyypa0JabmyCayaajaWaaSbaaSqaaiaadMgaae qaaOWaaeWaaeaacqaH4oqCcaGGSaGaeqyWdiNaaiilaiaahgfacqGH his0cqaH4oqCcaGGSaGaaCyuaiaahYeacaWHrbWaaWbaaSqabeaaca WGubaaaOGaey4kaSIabCyuayaacaGaaCyuamaaCaaaleqabaGaamiv aaaaaOGaayjkaiaawMcaaaaaaa@92A7@

for all proper orthogonal tensors Q(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahgfacaGGOaGaamiDaiaacMcaaaa@3394@ .  Recall that L=D+W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahYeacqGH9aqpcaWHebGaey4kaSIaaC 4vaaaa@34D2@ .  If we choose Q ˙ =ΩQ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahgfagaGaaiabg2da9iaahM6acaWHrb aaaa@3460@  with Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahM6aaaa@319D@  an arbitrary skew tensor such that Q(0)=I MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahgfacaGGOaGaaGimaiaacMcacqGH9a qpcaWHjbaaaa@352D@ , then Q ˙ Q T =Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqahgfagaGaaiaahgfadaahaaWcbeqaai aadsfaaaGccqGH9aqpcaWHPoaaaa@3570@ , and applying frame indifference at time t=0 then gives

ψ ^ (ρ,θ,L)= ψ ^ (ρ,θ,D+W+Ω) σ ^ ij (ρ,θ,L)= σ ^ ij (ρ,θ,D+W+Ω) q ^ i ( θ,ρ,θ,L )= q ^ i ( θ,ρ,θ,D+W+Ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafqiYdKNbaKaacaGGOaGaeqyWdi NaaiilaiabeI7aXjaacYcacaWHmbGaaiykaiabg2da9iqbeI8a5zaa jaGaaiikaiabeg8aYjaacYcacqaH4oqCcaGGSaGaaCiraiabgUcaRi aahEfacqGHRaWkcaWHPoGaaiykaaqaaiqbeo8aZzaajaWaaSbaaSqa aiaadMgacaWGQbaabeaakiaacIcacqaHbpGCcaGGSaGaeqiUdeNaai ilaiaahYeacaGGPaGaeyypa0Jafq4WdmNbaKaadaWgaaWcbaGaamyA aiaadQgaaeqaaOGaaiikaiabeg8aYjaacYcacqaH4oqCcaGGSaGaaC iraiabgUcaRiaahEfacqGHRaWkcaWHPoGaaiykaaqaaiqadghagaqc amaaBaaaleaacaWGPbaabeaakmaabmaabaGaeqiUdeNaaiilaiabeg 8aYjaacYcacqGHhis0cqaH4oqCcaGGSaGaaCitaaGaayjkaiaawMca aiabg2da9iqadghagaqcamaaBaaaleaacaWGPbaabeaakmaabmaaba GaeqiUdeNaaiilaiabeg8aYjaacYcacqGHhis0cqaH4oqCcaGGSaGa aCiraiabgUcaRiaahEfacqGHRaWkcaWHPoaacaGLOaGaayzkaaaaaa a@8121@

Then, selecting W=Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahEfacqGH9aqpcqGHsislcaWHPoaaaa@3470@  shows that ψ ^ (ρ,θ,L)= ψ ^ (ρ,θ,D) σ ^ ij (ρ,θ,L)= σ ^ ij (ρ,θ,D) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeI8a5zaajaGaaiikaiabeg8aYjaacY cacqaH4oqCcaGGSaGaaCitaiaacMcacqGH9aqpcuaHipqEgaqcaiaa cIcacqaHbpGCcaGGSaGaeqiUdeNaaiilaiaahseacaGGPaGaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlqbeo8aZzaajaWaaSbaaSqaaiaadM gacaWGQbaabeaakiaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiilaiaa hYeacaGGPaGaeyypa0Jafq4WdmNbaKaadaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaaiikaiabeg8aYjaacYcacqaH4oqCcaGGSaGaaCiraiaa cMcaaaa@6E0B@ Finally, with arbitrary Q, we see that

ψ ^ (ρ,θ,D)= ψ ^ (ρ,θ,QD Q T ) Q ik σ ^ kl (ρ,θ,D) Q jl = σ ^ ij (ρ,θ,QD Q T ) Q ik q ^ k ( θ,ρ,θ,D )= q ^ i ( θ,ρ,Qθ,QD Q T ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGafqiYdKNbaKaacaGGOaGaeqyWdi NaaiilaiabeI7aXjaacYcacaWHebGaaiykaiabg2da9iqbeI8a5zaa jaGaaiikaiabeg8aYjaacYcacqaH4oqCcaGGSaGaaCyuaiaahseaca WHrbWaaWbaaSqabeaacaWGubaaaOGaaiykaaqaaiaadgfadaWgaaWc baGaamyAaiaadUgaaeqaaOGafq4WdmNbaKaadaWgaaWcbaGaam4Aai aadYgaaeqaaOGaaiikaiabeg8aYjaacYcacqaH4oqCcaGGSaGaaCir aiaacMcacaWGrbWaaSbaaSqaaiaadQgacaWGSbaabeaakiabg2da9i qbeo8aZzaajaWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacIcacqaH bpGCcaGGSaGaeqiUdeNaaiilaiaahgfacaWHebGaaCyuamaaCaaale qabaGaamivaaaakiaacMcaaeaacaWGrbWaaSbaaSqaaiaadMgacaWG RbaabeaakiqadghagaqcamaaBaaaleaacaWGRbaabeaakmaabmaaba GaeqiUdeNaaiilaiabeg8aYjaacYcacqGHhis0cqaH4oqCcaGGSaGa aCiraaGaayjkaiaawMcaaiabg2da9iqadghagaqcamaaBaaaleaaca WGPbaabeaakmaabmaabaGaeqiUdeNaaiilaiabeg8aYjaacYcacaWH rbGaey4bIeTaeqiUdeNaaiilaiaahgfacaWHebGaaCyuamaaCaaale qabaGaamivaaaaaOGaayjkaiaawMcaaaaaaa@876A@

This implies that the free energy and stress and heat transfer response functions are all isotropic functions.

 

Not everyone agrees with the argument that constitutive behavior must be independent of spin.  This conclusion relies on the assumption that the response functions are invariant to changes between reference frames that are rotating relative to one another.   If invariance is required only to transformations between frames that have a fixed relative velocity and orientation, spin can enter in the constitutive behavior.   Some models of turbulent flow do include spin.

 

·         The free energy inequality provides the remaining conclusions.   Take the time derivative of the free energy function and substitute into the free energy inequality to see that

σ ij D ij 1 θ q i θ y i ρ( ψ ρ ρ ˙ + ψ θ θ ˙ + ψ D ij D ˙ ij +s θ t )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTmaa laaabaGaaGymaaqaaiabeI7aXbaacaWGXbWaaSbaaSqaaiaadMgaae qaaOWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgkHiTiabeg8aYnaabmaabaWaaSaaae aacqGHciITcqaHipqEaeaacqGHciITcqaHbpGCaaGafqyWdiNbaiaa cqGHRaWkdaWcaaqaaiabgkGi2kabeI8a5bqaaiabgkGi2kabeI7aXb aacuaH4oqCgaGaaiabgUcaRmaalaaabaGaeyOaIyRaeqiYdKhabaGa eyOaIyRaamiramaaBaaaleaacaWGPbGaamOAaaqabaaaaOGabmiray aacaWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgUcaRiaadohadaWc aaqaaiabgkGi2kabeI7aXbqaaiabgkGi2kaadshaaaaacaGLOaGaay zkaaGaeyyzImRaaGimaaaa@6DF2@

(time derivatives are all with fixed x). Note that

σ ij D ij = π ^ eq (ρ,θ) D ii + σ ^ ij vis (ρ,θ, D ij ) D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iab gkHiTiqbec8aWzaajaWaaSbaaSqaaiaadwgacaWGXbaabeaakiaacI cacqaHbpGCcaGGSaGaeqiUdeNaaiykaiaadseadaWgaaWcbaGaamyA aiaadMgaaeqaaOGaey4kaSIafq4WdmNbaKaadaqhaaWcbaGaamyAai aadQgaaeaacaWG2bGaamyAaiaadohaaaGccaGGOaGaeqyWdiNaaiil aiabeI7aXjaacYcacaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaaki aacMcacaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@58D1@

and also recall that from mass conservation ρ ˙ +ρ D kk =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeg8aYzaacaGaey4kaSIaeqyWdiNaam iramaaBaaaleaacaWGRbGaam4AaaqabaGccqGH9aqpcaaIWaGaaGPa VlaaykW7caaMc8oaaa@3E12@ .  Collecting coefficients of rate quantities we see that

( ρ 2 ψ ρ π ^ eq (ρ,θ) ) D ii + σ ^ ij vis (ρ,θ, D ij ) D ij ψ D ij D ˙ ij 1 θ q i θ y i ρ( ψ θ +s ) θ t 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaeqyWdi3aaWbaaSqabeaaca aIYaaaaOWaaSaaaeaacqGHciITcqaHipqEaeaacqGHciITcqaHbpGC aaGaeyOeI0IafqiWdaNbaKaadaWgaaWcbaGaamyzaiaadghaaeqaaO Gaaiikaiabeg8aYjaacYcacqaH4oqCcaGGPaaacaGLOaGaayzkaaGa amiramaaBaaaleaacaWGPbGaamyAaaqabaGccqGHRaWkcuaHdpWCga qcamaaDaaaleaacaWGPbGaamOAaaqaaiaadAhacaWGPbGaam4Caaaa kiaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiilaiaadseadaWgaaWcba GaamyAaiaadQgaaeqaaOGaaiykaiaadseadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaeyOeI0YaaSaaaeaacqGHciITcqaHipqEaeaacqGHci ITcaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaaaaGcceWGebGbaiaa daWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyOeI0YaaSaaaeaacaaIXa aabaGaeqiUdehaaiaadghadaWgaaWcbaGaamyAaaqabaGcdaWcaaqa aiabgkGi2kabeI7aXbqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaa qabaaaaOGaeyOeI0IaeqyWdi3aaeWaaeaadaWcaaqaaiabgkGi2kab eI8a5bqaaiabgkGi2kabeI7aXbaacqGHRaWkcaWGZbaacaGLOaGaay zkaaWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG0baaaiab gwMiZkaaicdaaaa@87D8@

This inequality must hold for all possible D ij , D ˙ ij , θ ˙ ,θ/ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae qaaOGaaiilaiqadseagaGaamaaBaaaleaacaWGPbGaamOAaaqabaGc caGGSaGafqiUdeNbaiaacaGGSaGaeyOaIyRaeqiUdeNaai4laiabgk Gi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaa@4145@ , which can all be independently prescribed.   It follows immediately that

ψ D ij =0s= ψ θ 1 θ q i θ y i 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeqiYdKhabaGaey OaIyRaamiramaaBaaaleaacaWGPbGaamOAaaqabaaaaOGaeyypa0Ja aGimaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8Uaam4Caiabg2da9iabgkHiTmaalaaa baGaeyOaIyRaeqiYdKhabaGaeyOaIyRaeqiUdehaaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVpaalaaabaGaaGymaaqaaiabeI7aXbaacaWGXbWaaS baaSqaaiaadMgaaeqaaOWaaSaaaeaacqGHciITcqaH4oqCaeaacqGH ciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiabgsMiJkaaicdaaa a@7434@

Next, note that

( ρ 2 ψ ρ π ^ eq (ρ,θ) ) D ii + σ ^ ij vis (ρ,θ, D ij ) D ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaeqyWdi3aaWbaaSqabeaaca aIYaaaaOWaaSaaaeaacqGHciITcqaHipqEaeaacqGHciITcqaHbpGC aaGaeyOeI0IafqiWdaNbaKaadaWgaaWcbaGaamyzaiaadghaaeqaaO Gaaiikaiabeg8aYjaacYcacqaH4oqCcaGGPaaacaGLOaGaayzkaaGa amiramaaBaaaleaacaWGPbGaamyAaaqabaGccqGHRaWkcuaHdpWCga qcamaaDaaaleaacaWGPbGaamOAaaqaaiaadAhacaWGPbGaam4Caaaa kiaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiilaiaadseadaWgaaWcba GaamyAaiaadQgaaeqaaOGaaiykaiaadseadaWgaaWcbaGaamyAaiaa dQgaaeqaaOGaeyyzImRaaGimaaaa@5E49@

and replace D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@333A@  with α D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjaadseadaWgaaWcbaGaamyAai aadQgaaeqaaaaa@34D9@  with α>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjabg6da+iaaicdaaaa@33C9@  it follows that

( ρ 2 ψ ρ π ^ eq (ρ,θ) ) D ii + σ ^ ij vis (ρ,θ,α D ij ) D ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaeqyWdi3aaWbaaSqabeaaca aIYaaaaOWaaSaaaeaacqGHciITcqaHipqEaeaacqGHciITcqaHbpGC aaGaeyOeI0IafqiWdaNbaKaadaWgaaWcbaGaamyzaiaadghaaeqaaO Gaaiikaiabeg8aYjaacYcacqaH4oqCcaGGPaaacaGLOaGaayzkaaGa amiramaaBaaaleaacaWGPbGaamyAaaqabaGccqGHRaWkcuaHdpWCga qcamaaDaaaleaacaWGPbGaamOAaaqaaiaadAhacaWGPbGaam4Caaaa kiaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiilaiabeg7aHjaadseada WgaaWcbaGaamyAaiaadQgaaeqaaOGaaiykaiaadseadaWgaaWcbaGa amyAaiaadQgaaeqaaOGaeyyzImRaaGimaaaa@5FE8@

We can now let α0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg7aHjabgkziUkaaicdaaaa@34AE@  and since σ ^ ij vis (ρ,θ,0)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeo8aZzaajaWaa0baaSqaaiaadMgaca WGQbaabaGaamODaiaadMgacaWGZbaaaOGaaiikaiabeg8aYjaacYca cqaH4oqCcaGGSaGaaGimaiaacMcacqGH9aqpcaaIWaaaaa@3FD9@  we see that

π ^ eq (ρ,θ)= ρ 2 ψ ρ σ ^ ij vis (ρ,θ, D ij ) D ij 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbec8aWzaajaWaaSbaaSqaaiaadwgaca WGXbaabeaakiaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiykaiabg2da 9iabeg8aYnaaCaaaleqabaGaaGOmaaaakmaalaaabaGaeyOaIyRaeq iYdKhabaGaeyOaIyRaeqyWdihaaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7cuaHdpWCgaqcamaaDaaale aacaWGPbGaamOAaaqaaiaadAhacaWGPbGaam4CaaaakiaacIcacqaH bpGCcaGGSaGaeqiUdeNaaiilaiaadseadaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaaiykaiaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eyyzImRaaGimaaaa@7FAF@

 

·         The remaining identities follow from these results, together with the definition ψ=εθs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iabew7aLjabgkHiTi abeI7aXjaadohaaaa@387E@  .  This implies that

dψ dt = ψ ρ ρ ˙ + ψ θ θ ˙ = π eq ρ ˙ ρ 2 s θ ˙ dψ dt = dε dt dθ dt sθ ds dt dε dt = π eq ρ ˙ ρ 2 +θ s ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaaSaaaeaacaWGKbGaeqiYdKhaba GaamizaiaadshaaaGaeyypa0ZaaSaaaeaacqGHciITcqaHipqEaeaa cqGHciITcqaHbpGCaaGafqyWdiNbaiaacqGHRaWkdaWcaaqaaiabgk Gi2kabeI8a5bqaaiabgkGi2kabeI7aXbaacuaH4oqCgaGaaiabg2da 9iabgkHiTiabec8aWnaaBaaaleaacaWGLbGaamyCaaqabaGcdaWcaa qaaiqbeg8aYzaacaaabaGaeqyWdi3aaWbaaSqabeaacaaIYaaaaaaa kiabgkHiTiaadohacuaH4oqCgaGaaiaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVdqaamaalaaabaGaamizaiabeI8a5bqaaiaadsga caWG0baaaiabg2da9maalaaabaGaamizaiabew7aLbqaaiaadsgaca WG0baaaiabgkHiTmaalaaabaGaamizaiabeI7aXbqaaiaadsgacaWG 0baaaiaadohacqGHsislcqaH4oqCdaWcaaqaaiaadsgacaWGZbaaba GaamizaiaadshaaaGaeyO0H49aaSaaaeaacaWGKbGaeqyTdugabaGa amizaiaadshaaaGaeyypa0JaeyOeI0IaeqiWda3aaSbaaSqaaiaadw gacaWGXbaabeaakmaalaaabaGafqyWdiNbaiaaaeaacqaHbpGCdaah aaWcbeqaaiaaikdaaaaaaOGaey4kaSIaeqiUdeNabm4Cayaacaaaaa a@8A71@

In addition, since

π eq = ρ 2 ψ ρ s= ψ θ ρ 2 2 ψ ρθ = π eq θ = ρ 2 s ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabec8aWnaaBaaaleaacaWGLbGaamyCaa qabaGccqGH9aqpcqaHbpGCdaahaaWcbeqaaiaaikdaaaGcdaWcaaqa aiabgkGi2kabeI8a5bqaaiabgkGi2kabeg8aYbaacaaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadohacq GH9aqpcqGHsisldaWcaaqaaiabgkGi2kabeI8a5bqaaiabgkGi2kab eI7aXbaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlabgkDiElaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqaHbp GCdaahaaWcbeqaaiaaikdaaaGcdaWcaaqaaiabgkGi2oaaCaaaleqa baGaaGOmaaaakiabeI8a5bqaaiabgkGi2kabeg8aYjabgkGi2kabeI 7aXbaacqGH9aqpdaWcaaqaaiabgkGi2kabec8aWnaaBaaaleaacaWG LbGaamyCaaqabaaakeaacqGHciITcqaH4oqCaaGaeyypa0JaeyOeI0 IaeqyWdi3aaWbaaSqabeaacaaIYaaaaOWaaSaaaeaacqGHciITcaWG ZbaabaGaeyOaIyRaeqyWdihaaaaa@A579@

whence

ψ ρ = ε ρ θ s ρ π eq ρ 2 = ε ρ + θ ρ 2 π eq θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeqiYdKhabaGaey OaIyRaeqyWdihaaiabg2da9maalaaabaGaeyOaIyRaeqyTdugabaGa eyOaIyRaeqyWdihaaiabgkHiTiabeI7aXnaalaaabaGaeyOaIyRaam 4CaaqaaiabgkGi2kabeg8aYbaacqGHshI3daWcaaqaaiabec8aWnaa BaaaleaacaWGLbGaamyCaaqabaaakeaacqaHbpGCdaahaaWcbeqaai aaikdaaaaaaOGaeyypa0ZaaSaaaeaacqGHciITcqaH1oqzaeaacqGH ciITcqaHbpGCaaGaey4kaSYaaSaaaeaacqaH4oqCaeaacqaHbpGCda ahaaWcbeqaaiaaikdaaaaaaOWaaSaaaeaacqGHciITcqaHapaCdaWg aaWcbaGaamyzaiaadghaaeqaaaGcbaGaeyOaIyRaeqiUdehaaaaa@644E@

The definitions of free energy and heat capacity also show that

ψ θ = ε θ sθ s θ ε θ =c=θ s θ =θ 2 ψ θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaeqiYdKhabaGaey OaIyRaeqiUdehaaiabg2da9maalaaabaGaeyOaIyRaeqyTdugabaGa eyOaIyRaeqiUdehaaiabgkHiTiaadohacqGHsislcqaH4oqCdaWcaa qaaiabgkGi2kaadohaaeaacqGHciITcqaH4oqCaaGaeyO0H49aaSaa aeaacqGHciITcqaH1oqzaeaacqGHciITcqaH4oqCaaGaeyypa0Jaam 4yaiabg2da9iabeI7aXnaalaaabaGaeyOaIyRaam4CaaqaaiabgkGi 2kabeI7aXbaacqGH9aqpcqGHsislcqaH4oqCdaWcaaqaaiabgkGi2o aaCaaaleqabaGaaGOmaaaakiabeI8a5bqaaiabgkGi2kabeI7aXnaa CaaaleqabaGaaGOmaaaaaaaaaa@66C8@

Furthermore

c ρ =θ 2 θ 2 ψ ρ = θ ρ 2 2 π eq θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaam4yaaqaaiabgk Gi2kabeg8aYbaacqGH9aqpcqGHsislcqaH4oqCdaWcaaqaaiabgkGi 2oaaCaaaleqabaGaaGOmaaaaaOqaaiabgkGi2kabeI7aXnaaCaaale qabaGaaGOmaaaaaaGcdaWcaaqaaiabgkGi2kabeI8a5bqaaiabgkGi 2kabeg8aYbaacqGH9aqpcqGHsisldaWcaaqaaiabeI7aXbqaaiabeg 8aYnaaCaaaleqabaGaaGOmaaaaaaGcdaWcaaqaaiabgkGi2oaaCaaa leqabaGaaGOmaaaakiabec8aWnaaBaaaleaacaWGLbGaamyCaaqaba aakeaacqGHciITcqaH4oqCdaahaaWcbeqaaiaaikdaaaaaaaaa@5724@

 

 

7.3 Special cases of constitutive equations for fluids

 

The following special cases of constitutive equations are frequently used in fluid mechanics:

 

* An Elastic fluid (also known as a barotropic fluid or Eulerian fluid) has free energy and stress response independent of temperature

ψ= ψ ^ (ρ) σ ij = π eq (ρ) δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iqbeI8a5zaajaGaai ikaiabeg8aYjaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8Uaeq4Wdm3aaSbaaSqaaiaadM gacaWGQbaabeaakiabg2da9iaaykW7cqGHsislcqaHapaCdaWgaaWc baGaamyzaiaadghaaeqaaOGaaiikaiabeg8aYjaacMcacqaH0oazda WgaaWcbaGaamyAaiaadQgaaeqaaaaa@598F@

Its heat capacity is zero, and heat flow is not usually considered in applications that use this model. Either the free energy, or the pressure, must be fit to experiment.   The two are related through the equations listed in the preceding section.

 

* An Ideal Gas has free energy and stress response function

ε= c v θ= p (γ1)ρ ψ= c v θθ( c v logθRlogρ s 0 ) σ ij =p δ ij =ρRθ δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLjabg2da9iaadogadaWgaaWcba GaamODaaqabaGccqaH4oqCcqGH9aqpdaWcaaqaaiaadchaaeaacaGG OaGaeq4SdCMaeyOeI0IaaGymaiaacMcacqaHbpGCaaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlabeI8a5jabg2da9iaadogadaWgaaWcba GaamODaaqabaGccqaH4oqCcqGHsislcqaH4oqCdaqadaqaaiaadoga daWgaaWcbaGaamODaaqabaGcciGGSbGaai4BaiaacEgacqaH4oqCcq GHsislcaWGsbGaciiBaiaac+gacaGGNbGaeqyWdiNaeyOeI0Iaam4C amaaBaaaleaacaaIWaaabeaaaOGaayjkaiaawMcaaiaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8a ZnaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcqGHsislcaWGWb GaeqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iabgkHi Tiabeg8aYjaadkfacqaH4oqCcqaH0oazdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@9044@

where R is the gas constant, c v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamODaaqabaaaaa@3277@  is the specific heat capacity (a constant), and s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohadaWgaaWcbaGaaGimaaqabaaaaa@3246@  is an arbitrary constant.  Ideal gases are also characterized by the specific heat at constant pressure c p = c v +R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamiCaaqabaGccq GH9aqpcaWGJbWaaSbaaSqaaiaadAhaaeqaaOGaey4kaSIaamOuaaaa @3753@  and the ratio γ= c p / c v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNjabg2da9iaadogadaWgaaWcba GaamiCaaqabaGccaGGVaGaam4yamaaBaaaleaacaWG2baabeaaaaa@37EA@ .  An ideal gas is inviscid.  Pressure, temperature and density in an ideal gas are related by the equation of state p=ρRθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacqGH9aqpcqaHbpGCcaWGsbGaeq iUdehaaa@36B0@ .  Note that R in these equations is the individual gas constant, which has units of Joules/kg/Kelvin, and varies from one gas to another.   It can be computed from the universal gas constant R u =8.314J/mol/K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfadaWgaaWcbaGaamyDaaqabaGccq GH9aqpcaaI4aGaaiOlaiaaiodacaaIXaGaaGinaiaadQeacaGGVaGa amyBaiaad+gacaWGSbGaai4laiaadUeaaaa@3CFB@  using the relation R= R u /m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacqGH9aqpcaWGsbWaaSbaaSqaai aadwhaaeqaaOGaai4laiaad2gaaaa@35F1@  where m is the molecular weight of the gas (the mass of one mole of the gas).

 

* A Compressible Linear (Newtonian) Viscous Fluid has free energy and stress response function

ψ= ψ ^ (ρ,θ) σ ij =( π eq (ρ,θ)κ(ρ,θ) D kk ) δ ij +2η(ρ,θ)( D ij D kk δ ij /3) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iqbeI8a5zaajaGaai ikaiabeg8aYjaacYcacqaH4oqCcaGGPaGaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlabeo8aZn aaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcaaMc8UaeyOeI0Ia aiikaiabec8aWnaaBaaaleaacaWGLbGaamyCaaqabaGccaGGOaGaeq yWdiNaaiilaiabeI7aXjaacMcacqGHsislcqaH6oWAcaGGOaGaeqyW diNaaiilaiabeI7aXjaacMcacaWGebWaaSbaaSqaaiaadUgacaWGRb aabeaakiaacMcacqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaOGa ey4kaSIaaGOmaiabeE7aOjaacIcacqaHbpGCcaGGSaGaeqiUdeNaai ykaiaacIcacaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHi TiaadseadaWgaaWcbaGaam4AaiaadUgaaeqaaOGaeqiTdq2aaSbaaS qaaiaadMgacaWGQbaabeaakiaac+cacaaIZaGaaiykaaaa@80AD@

where κ,η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeQ7aRjaacYcacqaH3oaAaaa@3476@  are functions of density and temperature that must be fit to experiment.  They are known as the bulk and shear viscosity of a fluid, respectively.

 

* A Compressible nonlinear (non-Newtonian) Viscous Fluid has free energy and stress response function

ψ= ψ ^ (ρ,θ) σ ij = π eq (ρ,θ) δ ij + η 1 ( I 1 , I 2 , I 3 ,ρ,θ) δ ij + η 2 ( I 1 , I 2 , I 3 ,ρ,θ) D ij + η 3 ( I 1 , I 2 , I 3 ,ρ,θ) D ik D kj MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaeqiYdKNaeyypa0JafqiYdKNbaK aacaGGOaGaeqyWdiNaaiilaiabeI7aXjaacMcaaeaacqaHdpWCdaWg aaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaGPaVlabgkHiTiabec 8aWnaaBaaaleaacaWGLbGaamyCaaqabaGccaGGOaGaeqyWdiNaaiil aiabeI7aXjaacMcacqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqaaO Gaey4kaSIaeq4TdG2aaSbaaSqaaiaaigdaaeqaaOGaaiikaiaadMea daWgaaWcbaGaaGymaaqabaGccaGGSaGaamysamaaBaaaleaacaaIYa aabeaakiaacYcacaWGjbWaaSbaaSqaaiaaiodaaeqaaOGaaiilaiab eg8aYjaacYcacqaH4oqCcaGGPaGaeqiTdq2aaSbaaSqaaiaadMgaca WGQbaabeaakiabgUcaRiabeE7aOnaaBaaaleaacaaIYaaabeaakiaa cIcacaWGjbWaaSbaaSqaaiaaigdaaeqaaOGaaiilaiaadMeadaWgaa WcbaGaaGOmaaqabaGccaGGSaGaamysamaaBaaaleaacaaIZaaabeaa kiaacYcacqaHbpGCcaGGSaGaeqiUdeNaaiykaiaadseadaWgaaWcba GaamyAaiaadQgaaeqaaOGaey4kaSIaeq4TdG2aaSbaaSqaaiaaioda aeqaaOGaaiikaiaadMeadaWgaaWcbaGaaGymaaqabaGccaGGSaGaam ysamaaBaaaleaacaaIYaaabeaakiaacYcacaWGjbWaaSbaaSqaaiaa iodaaeqaaOGaaiilaiabeg8aYjaacYcacqaH4oqCcaGGPaGaamiram aaBaaaleaacaWGPbGaam4AaaqabaGccaWGebWaaSbaaSqaaiaadUga caWGQbaabeaaaaaa@8C30@

where I 1 , I 2 , I 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMeadaWgaaWcbaGaaGymaaqabaGcca GGSaGaamysamaaBaaaleaacaaIYaaabeaakiaacYcacaWGjbWaaSba aSqaaiaaiodaaeqaaaaa@36FE@  are the three invariants of D ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@333A@ , and   η 1 , η 2 , η 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeE7aOnaaBaaaleaacaaIXaaabeaaki aacYcacqaH3oaAdaWgaaWcbaGaaGOmaaqabaGccaGGSaGaeq4TdG2a aSbaaSqaaiaaiodaaeqaaaaa@3997@  are three functions that must be fit to experiment.

 

 

Heat Flux: For all these cases, in problems where heat flow is considered, it is most common to set

q i =k(ρ,θ) θ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadghadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcqGHsislcaWGRbGaaiikaiabeg8aYjaacYcacqaH4oqCcaGG PaWaaSaaaeaacqGHciITcqaH4oqCaeaacqGHciITcaWG5bWaaSbaaS qaaiaadMgaaeqaaaaaaaa@418E@

where k is the thermal conductivity (often taken to be constant).

 

 

 

7.4 Special forms of field equations for viscous fluids

 

The flow of a viscous fluid is governed by

(i)                 The mass balance equation

(ii)               The linear momentum balance equation

(iii)             The constitutive law for a compressible, or incompressible viscous fluid

(iv)             In problems with heat transfer, the energy conservation equation.

together with any relevant boundary conditions.  For calculations, it is convenient to combine (ii) and (iii) to obtain a single equation relating the static pressure and velocity, with the following results

 

* The Compressible Navier-Stokes Equation is a convenient form of the linear momentum balance equations for a compressible, linear viscous fluid.

p y i +2 y j η(ρ,θ)( D ij D kk δ ij /3 )+ρ b i =ρ d v i dt | x=const p= π eq (ρ,θ)κ(ρ,θ) D kk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaeyOaIyRaamiCaa qaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaey4kaSIa aGOmamaalaaabaGaeyOaIylabaGaeyOaIyRaamyEamaaBaaaleaaca WGQbaabeaaaaGccqaH3oaAcaGGOaGaeqyWdiNaaiilaiabeI7aXjaa cMcadaqadaqaaiaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaey OeI0IaamiramaaBaaaleaacaWGRbGaam4AaaqabaGccqaH0oazdaWg aaWcbaGaamyAaiaadQgaaeqaaOGaai4laiaaiodaaiaawIcacaGLPa aacqGHRaWkcqaHbpGCcaWGIbWaaSbaaSqaaiaadMgaaeqaaOGaeyyp a0JaeqyWdi3aaqGaaeaadaWcaaqaaiaadsgacaWG2bWaaSbaaSqaai aadMgaaeqaaaGcbaGaamizaiaadshaaaaacaGLiWoadaWgaaWcbaGa aCiEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaaki aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadchacq GH9aqpcqaHapaCdaWgaaWcbaGaamyzaiaadghaaeqaaOGaaiikaiab eg8aYjaacYcacqaH4oqCcaGGPaGaeyOeI0IaeqOUdSMaaiikaiabeg 8aYjaacYcacqaH4oqCcaGGPaGaamiramaaBaaaleaacaWGRbGaam4A aaqabaaaaa@94CE@

 

* For the case of Density independent viscosity this equation can be simplified to

1 ρ π eq y i + η ρ 2 v i y j y j +( κ ρ 2η 3ρ ) 2 v j y j y i + b i = d v i dt | x=const = v i t | y k =const + 1 2 y i ( v k v k )+ ijk ω j v k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaaGymaaqaaiabeg 8aYbaadaWcaaqaaiabgkGi2kabec8aWnaaBaaaleaacaWGLbGaamyC aaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaaki abgUcaRmaalaaabaGaeq4TdGgabaGaeqyWdihaamaalaaabaGaeyOa Iy7aaWbaaSqabeaacaaIYaaaaOGaamODamaaBaaaleaacaWGPbaabe aaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaGccqGHciIT caWG5bWaaSbaaSqaaiaadQgaaeqaaaaakiabgUcaRmaabmaabaWaaS aaaeaacqaH6oWAaeaacqaHbpGCaaGaeyOeI0YaaSaaaeaacaaIYaGa eq4TdGgabaGaaG4maiabeg8aYbaaaiaawIcacaGLPaaadaWcaaqaai abgkGi2oaaCaaaleqabaGaaGOmaaaakiaadAhadaWgaaWcbaGaamOA aaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaey OaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGccqGHRaWkcaWGIbWa aSbaaSqaaiaadMgaaeqaaOGaaGPaVlabg2da9maaeiaabaWaaSaaae aacaWGKbGaamODamaaBaaaleaacaWGPbaabeaaaOqaaiaadsgacaWG 0baaaaGaayjcSdWaaSbaaSqaaiaahIhacqGH9aqpcaWGJbGaam4Bai aad6gacaWGZbGaamiDaaqabaGccqGH9aqpdaabcaqaamaalaaabaGa eyOaIyRaamODamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaads haaaaacaGLiWoadaWgaaWcbaGaamyEamaaBaaameaacaWGRbaabeaa liabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaakiabgU caRmaalaaabaGaaGymaaqaaiaaikdaaaWaaSaaaeaacqGHciITaeaa cqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiaacIcacaWG2b WaaSbaaSqaaiaadUgaaeqaaOGaamODamaaBaaaleaacaWGRbaabeaa kiaacMcacqGHRaWkcqGHiiIZdaWgaaWcbaGaamyAaiaadQgacaWGRb aabeaakiabeM8a3naaBaaaleaacaWGQbaabeaakiaadAhadaWgaaWc baGaam4Aaaqabaaaaa@9F43@

 

* The Vorticity transport equation governs evolution of vorticity in a fluid with density independent viscosity at constant temperature.  It has the form 

+ η ρ 2 ω i y j y j 1 ρ 2 ijk ρ y j { η 2 v k y l y l +( κ 2η 3 ) 2 v l y l y k }+ ijk x j ( b k )+ D ij ω j v k y k ω i = ω i t | x=const MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgUcaRmaalaaabaGaeq4TdGgabaGaeq yWdihaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyY dC3aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaale aacaWGQbaabeaakiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaa aOGaeyOeI0YaaSaaaeaacaaIXaaabaGaeqyWdi3aaWbaaSqabeaaca aIYaaaaaaakiabgIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqa aOWaaSaaaeaacqGHciITcqaHbpGCaeaacqGHciITcaWG5bWaaSbaaS qaaiaadQgaaeqaaaaakmaacmaabaGaeq4TdG2aaSaaaeaacqGHciIT daahaaWcbeqaaiaaikdaaaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaa GcbaGaeyOaIyRaamyEamaaBaaaleaacaWGSbaabeaakiabgkGi2kaa dMhadaWgaaWcbaGaamiBaaqabaaaaOGaey4kaSYaaeWaaeaacqaH6o WAcqGHsisldaWcaaqaaiaaikdacqaH3oaAaeaacaaIZaaaaaGaayjk aiaawMcaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaam ODamaaBaaaleaacaWGSbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWc baGaamiBaaqabaGccqGHciITcaWG5bWaaSbaaSqaaiaadUgaaeqaaa aaaOGaay5Eaiaaw2haaiabgUcaRiabgIGiopaaBaaaleaacaWGPbGa amOAaiaadUgaaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWG4b WaaSbaaSqaaiaadQgaaeqaaaaakiaacIcacaWGIbWaaSbaaSqaaiaa dUgaaeqaaOGaaiykaiaaykW7cqGHRaWkcaWGebWaaSbaaSqaaiaadM gacaWGQbaabeaakiabeM8a3naaBaaaleaacaWGQbaabeaakiabgkHi TmaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGRbaabeaaaOqaai abgkGi2kaadMhadaWgaaWcbaGaam4AaaqabaaaaOGaeqyYdC3aaSba aSqaaiaadMgaaeqaaOGaeyypa0ZaaqGaaeaadaWcaaqaaiabgkGi2k abeM8a3naaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaaa caGLiWoadaWgaaWcbaGaaCiEaiabg2da9iaadogacaWGVbGaamOBai aadohacaWG0baabeaaaaa@A59B@

This result is not especially useful for viscous, compressible flows, but simplifies considerably if the fluid is incompressible, or if the viscosity can be neglected.   The result can be derived as follows:  

(i)                 Recall first that π eq = ρ 2 ψ ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabec8aWnaaBaaaleaacaWGLbGaamyCaa qabaGccqGH9aqpcqaHbpGCdaahaaWcbeqaaiaaikdaaaGcdaWcaaqa aiabgkGi2kabeI8a5bqaaiabgkGi2kabeg8aYbaaaaa@3E5E@ .  It follows (by direct substitution, noting that temperature is constant) that

1 ρ π eq y i = y i ( ψ+ π eq ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGymaaqaaiabeg8aYbaada WcaaqaaiabgkGi2kabec8aWnaaBaaaleaacaWGLbGaamyCaaqabaaa keaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9m aalaaabaGaeyOaIylabaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaa beaaaaGcdaqadaqaaiabeI8a5jabgUcaRmaalaaabaGaeqiWda3aaS baaSqaaiaadwgacaWGXbaabeaaaOqaaiabeg8aYbaaaiaawIcacaGL Paaaaaa@4BA4@

 

(ii)               The Navier-Stokes equation can therefore be rearranged as

y i ( ψ+ π eq ρ )+ η ρ 2 v i y j y j +( κ 2η 3 ) 1 ρ 2 v j y j y i + b i = d v i dt | x=const MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaeyOaIylabaGaey OaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGcdaqadaqaaiabeI8a 5jabgUcaRmaalaaabaGaeqiWda3aaSbaaSqaaiaadwgacaWGXbaabe aaaOqaaiabeg8aYbaaaiaawIcacaGLPaaacqGHRaWkdaWcaaqaaiab eE7aObqaaiabeg8aYbaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaG OmaaaakiaadAhadaWgaaWcbaGaamyAaaqabaaakeaacqGHciITcaWG 5bWaaSbaaSqaaiaadQgaaeqaaOGaeyOaIyRaamyEamaaBaaaleaaca WGQbaabeaaaaGccqGHRaWkdaqadaqaaiabeQ7aRjabgkHiTmaalaaa baGaaGOmaiabeE7aObqaaiaaiodaaaaacaGLOaGaayzkaaWaaSaaae aacaaIXaaabaGaeqyWdihaamaalaaabaGaeyOaIy7aaWbaaSqabeaa caaIYaaaaOGaamODamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2k aadMhadaWgaaWcbaGaamOAaaqabaGccqGHciITcaWG5bWaaSbaaSqa aiaadMgaaeqaaaaakiabgUcaRiaadkgadaWgaaWcbaGaamyAaaqaba GccaaMc8Uaeyypa0ZaaqGaaeaadaWcaaqaaiaadsgacaWG2bWaaSba aSqaaiaadMgaaeqaaaGcbaGaamizaiaadshaaaaacaGLiWoadaWgaa WcbaGaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baa beaaaaa@798C@

(iii)             Taking the curl of both sides, recalling curl(grad(f))=0, and expressing the curl of the acceleration in terms of vorticity we then obtain

+ η ρ 2 ω i y j y j 1 ρ 2 ijk ρ y j { η 2 v k y l y l +( κ 2η 3 ) 2 v l y l y k }+ ijk y j ( b k )= ω i t | x=const D ij ω j + v k y k ω i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgUcaRmaalaaabaGaeq4TdGgabaGaeq yWdihaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyY dC3aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaale aacaWGQbaabeaakiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaa aOGaeyOeI0YaaSaaaeaacaaIXaaabaGaeqyWdi3aaWbaaSqabeaaca aIYaaaaaaakiabgIGiopaaBaaaleaacaWGPbGaamOAaiaadUgaaeqa aOWaaSaaaeaacqGHciITcqaHbpGCaeaacqGHciITcaWG5bWaaSbaaS qaaiaadQgaaeqaaaaakmaacmaabaGaeq4TdG2aaSaaaeaacqGHciIT daahaaWcbeqaaiaaikdaaaGccaWG2bWaaSbaaSqaaiaadUgaaeqaaa GcbaGaeyOaIyRaamyEamaaBaaaleaacaWGSbaabeaakiabgkGi2kaa dMhadaWgaaWcbaGaamiBaaqabaaaaOGaey4kaSYaaeWaaeaacqaH6o WAcqGHsisldaWcaaqaaiaaikdacqaH3oaAaeaacaaIZaaaaaGaayjk aiaawMcaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaam ODamaaBaaaleaacaWGSbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWc baGaamiBaaqabaGccqGHciITcaWG5bWaaSbaaSqaaiaadUgaaeqaaa aaaOGaay5Eaiaaw2haaiabgUcaRiabgIGiopaaBaaaleaacaWGPbGa amOAaiaadUgaaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWG5b WaaSbaaSqaaiaadQgaaeqaaaaakiaacIcacaWGIbWaaSbaaSqaaiaa dUgaaeqaaOGaaiykaiaaykW7cqGH9aqpdaabcaqaamaalaaabaGaey OaIyRaeqyYdC3aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiD aaaaaiaawIa7amaaBaaaleaacaWH4bGaeyypa0Jaam4yaiaad+gaca WGUbGaam4CaiaadshaaeqaaOGaeyOeI0IaamiramaaBaaaleaacaWG PbGaamOAaaqabaGccqaHjpWDdaWgaaWcbaGaamOAaaqabaGccqGHRa WkdaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaam4Aaaqabaaakeaa cqGHciITcaWG5bWaaSbaaSqaaiaadUgaaeqaaaaakiabeM8a3naaBa aaleaacaWGPbaabeaaaaa@A59C@

 

 

* The Entropy Equation is a useful result for analyzing gas flows.   It states that the rate of change of specific entropy at a point in a compressible viscous fluid can be computed from

ρθ s dt | x=const =κ(ρ,θ) D kk D jj +2η( D ij D ij D kk D jj /3) q i y i +q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaGaeqyWdiNaeqiUde3aaSaaae aacqGHciITcaWGZbaabaGaamizaiaadshaaaaacaGLiWoadaWgaaWc baGaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabe aakiabg2da9iabeQ7aRjaacIcacqaHbpGCcaGGSaGaeqiUdeNaaiyk aiaadseadaWgaaWcbaGaam4AaiaadUgaaeqaaOGaamiramaaBaaale aacaWGQbGaamOAaaqabaGccqGHRaWkcaaIYaGaeq4TdGMaaiikaiaa dseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamiramaaBaaaleaaca WGPbGaamOAaaqabaGccqGHsislcaWGebWaaSbaaSqaaiaadUgacaWG RbaabeaakiaadseadaWgaaWcbaGaamOAaiaadQgaaeqaaOGaai4lai aaiodacaGGPaGaeyOeI0YaaSaaaeaacqGHciITcaWGXbWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabe aaaaGccqGHRaWkcaWGXbaaaa@6B05@

 

To see this, start with the identity ψ=εθs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabg2da9iabew7aLjabgkHiTi abeI7aXjaadohaaaa@387E@ .   It follows that

ψ ˙ = ψ ρ ρ ˙ + ψ θ θ ˙ = ε ˙ θ s ˙ s θ ˙ θ s ˙ = ε ˙ π eq ρ 2 ρ ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqbeI8a5zaacaGaeyypa0ZaaSaaaeaacq GHciITcqaHipqEaeaacqGHciITcqaHbpGCaaGafqyWdiNbaiaacqGH RaWkdaWcaaqaaiabgkGi2kabeI8a5bqaaiabgkGi2kabeI7aXbaacu aH4oqCgaGaaiabg2da9iqbew7aLzaacaGaeyOeI0IaeqiUdeNabm4C ayaacaGaeyOeI0Iaam4CaiqbeI7aXzaacaGaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7cqGHshI3cqaH4oqCceWG ZbGbaiaacqGH9aqpcuaH1oqzgaGaaiabgkHiTmaalaaabaGaeqiWda 3aaSbaaSqaaiaadwgacaWGXbaabeaaaOqaaiabeg8aYnaaCaaaleqa baGaaGOmaaaaaaGccuaHbpGCgaGaaaaa@6BE5@

Next, recall the mass conservation and energy conservation equations

ρ t | x=const +ρ D kk =0ρ ε t | x=const = σ ij D ij q i y i +q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcqaHbp GCaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahIhacqGH 9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGHRaWkcq aHbpGCcaWGebWaaSbaaSqaaiaadUgacaWGRbaabeaakiabg2da9iaa icdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaeqyWdi3aaqGaaeaadaWcaaqaaiabgkGi2kabew7aLbqa aiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaaCiEaiabg2da9i aadogacaWGVbGaamOBaiaadohacaWG0baabeaakiabg2da9iabeo8a ZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWGebWaaSbaaSqaaiaadM gacaWGQbaabeaakiabgkHiTmaalaaabaGaeyOaIyRaamyCamaaBaaa leaacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaa qabaaaaOGaey4kaSIaamyCaaaa@8DC8@

The constitutive equations also show that

σ ij D ij =( π eq κ D kk ) D jj +2η( D ij D ij D kk D jj /3) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo8aZnaaBaaaleaacaWGPbGaamOAaa qabaGccaWGebWaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9iab gkHiTiaacIcacqaHapaCdaWgaaWcbaGaamyzaiaadghaaeqaaOGaey OeI0IaeqOUdSMaamiramaaBaaaleaacaWGRbGaam4AaaqabaGccaGG PaGaamiramaaBaaaleaacaWGQbGaamOAaaqabaGccqGHRaWkcaaIYa Gaeq4TdGMaaiikaiaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa amiramaaBaaaleaacaWGPbGaamOAaaqabaGccqGHsislcaWGebWaaS baaSqaaiaadUgacaWGRbaabeaakiaadseadaWgaaWcbaGaamOAaiaa dQgaaeqaaOGaai4laiaaiodacaGGPaaaaa@5908@

Thus

θ s ˙ =κ D kk D jj +2η( D ij D ij D kk D jj /3) q i y i +q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXjqadohagaGaaiabg2da9iabeQ 7aRjaadseadaWgaaWcbaGaam4AaiaadUgaaeqaaOGaamiramaaBaaa leaacaWGQbGaamOAaaqabaGccqGHRaWkcaaIYaGaeq4TdGMaaiikai aadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamiramaaBaaaleaa caWGPbGaamOAaaqabaGccqGHsislcaWGebWaaSbaaSqaaiaadUgaca WGRbaabeaakiaadseadaWgaaWcbaGaamOAaiaadQgaaeqaaOGaai4l aiaaiodacaGGPaGaeyOeI0YaaSaaaeaacqGHciITcaWGXbWaaSbaaS qaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaa beaaaaGccqGHRaWkcaWGXbaaaa@57E4@

 

 

7.5 Special forms of field equations for elastic fluids

 

For elastic fluids, we can show the following results

 

* Euler’s equations of motion are the special cases of the linear momentum balance equation for elastic fluids

π eq y i +ρ b i =ρ v i t | y k =const + 1 2 ρ y i ( v k v k )+ρ ijk ω j v k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaeyOaIyRaeqiWda 3aaSbaaSqaaiaadwgacaWGXbaabeaaaOqaaiabgkGi2kaadMhadaWg aaWcbaGaamyAaaqabaaaaOGaey4kaSIaeqyWdiNaamOyamaaBaaale aacaWGPbaabeaakiaaykW7cqGH9aqpcqaHbpGCdaabcaqaamaalaaa baGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2k aadshaaaaacaGLiWoadaWgaaWcbaGaamyEamaaBaaameaacaWGRbaa beaaliabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaaki abgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaeqyWdi3aaSaaaeaa cqGHciITaeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaaki aacIcacaWG2bWaaSbaaSqaaiaadUgaaeqaaOGaamODamaaBaaaleaa caWGRbaabeaakiaacMcacqGHRaWkcqaHbpGCcqGHiiIZdaWgaaWcba GaamyAaiaadQgacaWGRbaabeaakiabeM8a3naaBaaaleaacaWGQbaa beaakiaadAhadaWgaaWcbaGaam4Aaaqabaaaaa@6DC3@

(the acceleration can be written in many other forms, of course)

 

* The Vorticity transport equation for an elastic fluid reduces to 

ijk x j ( b k )+ D ij ω j v k y k ω i = ω i t | x=const MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgIGiopaaBaaaleaacaWGPbGaamOAai aadUgaaeqaaOWaaSaaaeaacqGHciITaeaacqGHciITcaWG4bWaaSba aSqaaiaadQgaaeqaaaaakiaacIcacaWGIbWaaSbaaSqaaiaadUgaae qaaOGaaiykaiaaykW7cqGHRaWkcaWGebWaaSbaaSqaaiaadMgacaWG QbaabeaakiabeM8a3naaBaaaleaacaWGQbaabeaakiabgkHiTmaala aabaGaeyOaIyRaamODamaaBaaaleaacaWGRbaabeaaaOqaaiabgkGi 2kaadMhadaWgaaWcbaGaam4AaaqabaaaaOGaeqyYdC3aaSbaaSqaai aadMgaaeqaaOGaeyypa0ZaaqGaaeaadaWcaaqaaiabgkGi2kabeM8a 3naaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiW oadaWgaaWcbaGaaCiEaiabg2da9iaadogacaWGVbGaamOBaiaadoha caWG0baabeaaaaa@60E2@

An important conclusion from this result is that if the body force can be derived from a potential (so its curl vanishes), and the flow is irrotational at time t=0, it remains irrotational.

 

* Bernoulli’s equation is a special result governing steady flow of an elastic fluid that is subjected to conservative body forces.  Conservative body forces can be described by a potential energy, so that b i =Φ/ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkgadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcqGHsislcqGHciITcqqHMoGrcaGGVaGaeyOaIyRaamyEamaa BaaaleaacaWGPbaabeaaaaa@3B77@ .  Then, the Bernoulli equation states that, along a streamline

H=ψ+ π eq ρ + 1 2 v i v i +Φ=constant MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIeacqGH9aqpcqaHipqEcqGHRaWkda Wcaaqaaiabec8aWnaaBaaaleaacaWGLbGaamyCaaqabaaakeaacqaH bpGCaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaacaWG2bWaaS baaSqaaiaadMgaaeqaaOGaamODamaaBaaaleaacaWGPbaabeaakiab gUcaRiabfA6agjabg2da9iaabogacaqGVbGaaeOBaiaabohacaqG0b Gaaeyyaiaab6gacaqG0baaaa@4C18@

Recall that a streamline is a curve that is everywhere tangent to the velocity vector (for steady flow, the streamlines are coincident with the trajectories traced by material particles).  For the particular case of irrotational flow, this quantity is constant and independent of position. 

 

This result can be derived as follows.   First, recall that

1 ρ π eq y i = y i ( ψ+ π eq ρ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaaGymaaqaaiabeg8aYbaada WcaaqaaiabgkGi2kabec8aWnaaBaaaleaacaWGLbGaamyCaaqabaaa keaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9m aalaaabaGaeyOaIylabaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaa beaaaaGcdaqadaqaaiabeI8a5jabgUcaRmaalaaabaGaeqiWda3aaS baaSqaaiaadwgacaWGXbaabeaaaOqaaiabeg8aYbaaaiaawIcacaGL Paaaaaa@4BA4@

Noting that steady flow implies that v i /t| y=const =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaGaeyOaIyRaamODamaaBaaale aacaWGPbaabeaakiaac+cacqGHciITcaWG0baacaGLiWoadaWgaaWc baGaamyEaiabg2da9iaadogacaWGVbGaamOBaiaadohacaWG0baabe aakiabg2da9iaaicdaaaa@414F@ , the momentum balance equation therefore reads

y i ( ψ+ π eq ρ +Φ )= 1 2 y i ( v k v k )+ ijk ω j v k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaeyOaIylabaGaey OaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGcdaqadaqaaiabeI8a 5jabgUcaRmaalaaabaGaeqiWda3aaSbaaSqaaiaadwgacaWGXbaabe aaaOqaaiabeg8aYbaacqGHRaWkcqqHMoGraiaawIcacaGLPaaacaaM c8Uaeyypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaWcaaqaaiabgk Gi2cqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaaiik aiaadAhadaWgaaWcbaGaam4AaaqabaGccaWG2bWaaSbaaSqaaiaadU gaaeqaaOGaaiykaiabgUcaRiabgIGiopaaBaaaleaacaWGPbGaamOA aiaadUgaaeqaaOGaeqyYdC3aaSbaaSqaaiaadQgaaeqaaOGaamODam aaBaaaleaacaWGRbaabeaaaaa@5BAE@

For irrotational flow ω j =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3naaBaaaleaacaWGQbaabeaaki abg2da9iaaicdaaaa@351A@ , and this equation can then be integrated to obtain the Bernoulli equation.  Similarly, taking the dot product of both sides of this equation with the velocity vector and rearranging shows that

v i y i ( ψ+ π eq ρ + 1 2 v k v k +Φ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaadAhadaWgaaWcbaGaamyAaa qabaGcdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadMhadaWgaaWcbaGa amyAaaqabaaaaOWaaeWaaeaacqaHipqEcqGHRaWkdaWcaaqaaiabec 8aWnaaBaaaleaacaWGLbGaamyCaaqabaaakeaacqaHbpGCaaGaey4k aSYaaSaaaeaacaaIXaaabaGaaGOmaaaacaWG2bWaaSbaaSqaaiaadU gaaeqaaOGaamODamaaBaaaleaacaWGRbaabeaakiabgUcaRiabfA6a gbGaayjkaiaawMcaaiaaykW7cqGH9aqpcaaIWaaaaa@4EA0@

and therefore ψ+ π eq ρ + 1 2 v i v i +Φ=constant MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI8a5jabgUcaRmaalaaabaGaeqiWda 3aaSbaaSqaaiaadwgacaWGXbaabeaaaOqaaiabeg8aYbaacqGHRaWk daWcaaqaaiaaigdaaeaacaaIYaaaaiaadAhadaWgaaWcbaGaamyAaa qabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaeuOPdyKa eyypa0Jaae4yaiaab+gacaqGUbGaae4CaiaabshacaqGHbGaaeOBai aabshaaaa@4A45@  along streamlines. 

 

 

 

 

7.6 Incompressible flow

 

In many cases fluids can be approximated as incompressible, which simplifies the governing equations further.  

 

* The mass balance equation reduces to

v i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamODamaaBaaale aacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqa baaaaOGaeyypa0JaaGimaaaa@3945@

* The Incompressible Navier-Stokes Equation can be simplified to

1 ρ p y i + η ρ 2 v i y j v j + b i = d v i dt | x=const = v i t | y k =const + 1 2 y i ( v k v k )+ ijk ω j v k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaaGymaaqaaiabeg 8aYbaadaWcaaqaaiabgkGi2kaadchaaeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgUcaRmaalaaabaGaeq4TdGgabaGaeq yWdihaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamOD amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcba GaamOAaaqabaGccqGHciITcaWG2bWaaSbaaSqaaiaadQgaaeqaaaaa kiabgUcaRiaadkgadaWgaaWcbaGaamyAaaqabaGccqGH9aqpdaabca qaamaalaaabaGaamizaiaadAhadaWgaaWcbaGaamyAaaqabaaakeaa caWGKbGaamiDaaaaaiaawIa7amaaBaaaleaacaWH4bGaeyypa0Jaam 4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGaeyypa0ZaaqGaaeaa daWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaakeaacq GHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaadMhadaWgaaadbaGa am4AaaqabaWccqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaa qabaGccqGHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaamaalaaabaGa eyOaIylabaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGcca GGOaGaamODamaaBaaaleaacaWGRbaabeaakiaadAhadaWgaaWcbaGa am4AaaqabaGccaGGPaGaey4kaSIaeyicI48aaSbaaSqaaiaadMgaca WGQbGaam4AaaqabaGccqaHjpWDdaWgaaWcbaGaamOAaaqabaGccaWG 2bWaaSbaaSqaaiaadUgaaeqaaaaa@836E@

The quantity ν=η/ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabe27aUjabg2da9iabeE7aOjaac+cacq aHbpGCaaa@3745@  is known as the kinematic viscosity.

 

* The Vorticity transport equation reduces to 

+ η ρ 2 ω i y j y j + ijk x j ( b k )+ D ij ω j = ω i t | x=const MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgUcaRmaalaaabaGaeq4TdGgabaGaeq yWdihaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaeqyY dC3aaSbaaSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaale aacaWGQbaabeaakiabgkGi2kaadMhadaWgaaWcbaGaamOAaaqabaaa aOGaey4kaSIaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4Aaaqaba GcdaWcaaqaaiabgkGi2cqaaiabgkGi2kaadIhadaWgaaWcbaGaamOA aaqabaaaaOGaaiikaiaadkgadaWgaaWcbaGaam4AaaqabaGccaGGPa GaaGPaVlabgUcaRiaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa eqyYdC3aaSbaaSqaaiaadQgaaeqaaOGaeyypa0ZaaqGaaeaadaWcaa qaaiabgkGi2kabeM8a3naaBaaaleaacaWGPbaabeaaaOqaaiabgkGi 2kaadshaaaaacaGLiWoadaWgaaWcbaGaaCiEaiabg2da9iaadogaca WGVbGaamOBaiaadohacaWG0baabeaaaaa@678F@

 

 

 

7.7 Incompressible, inviscid, irrotational flow (potential flow)

 

If a flow field is irrotational, the velocity field can be derived from a scalar potential

v i = ϕ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiabgkGi2kabew9aMbqaaiabgkGi2kaadMhadaWg aaWcbaGaamyAaaqabaaaaaaa@3A49@

(it is easy to show that v i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaaaaa@327D@  of this form is irrotational.  The converse can be shown for fluids occupying a simply connected region MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  the potential may be multiple valued for multiply connected regions). 

 

 If a fluid is incompressible its velocity field (assuming irrotational flow) can be computed by solving for ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@3230@ .   In particular

  Mass Conservation yields the governing equation for ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@3230@ : v i y i = 2 ϕ y i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamODamaaBaaale aacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqa baaaaOGaeyypa0ZaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaa GccqaHvpGzaeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaOGa eyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGccqGH9aqpcaaIWa aaaa@458B@

* If the fluid is inviscid, then Bernoulli’s equation (generalized to time dependent flow) yields the governing equation for the pressure

p ρ + 1 2 ϕ y i ϕ y i +Φ+ ϕ t =f(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamiCaaqaaiabeg8aYbaacq GHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaamaalaaabaGaeyOaIyRa eqy1dygabaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGcda WcaaqaaiabgkGi2kabew9aMbqaaiabgkGi2kaadMhadaWgaaWcbaGa amyAaaqabaaaaOGaey4kaSIaeuOPdyKaey4kaSYaaSaaaeaacqGHci ITcqaHvpGzaeaacqGHciITcaWG0baaaiabg2da9iaadAgacaGGOaGa amiDaiaacMcaaaa@503F@

where Φ/ y i = b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiabgkGi2kabfA6agjaac+cacq GHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamOyamaa BaaaleaacaWGPbaabeaaaaa@3B77@   is the potential for body force per unit mass, and f(t) is an arbitrary function of time.   Since the flow is irrotational, this expression holds everywhere (not just along streamlines).

 

Together with appropriate boundary conditions, these equations determine both the velocity and pressure associated with the flow.

 

 

7.8 Stokes Flow

 

The general Navier-Stokes equation is nonlinear, because of the terms involving the square of the velocity in the acceleration.   If these terms can be neglected (as is the case for flows with low values of Reynolds number Re=ρVL/η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiGackfacaGGLbGaeyypa0JaeqyWdiNaam OvaiaadYeacaGGVaGaeq4TdGgaaa@38FA@ , where V and L are a representative velocity and length), the Navier-Stokes equation can be approximated as

1 ρ p y i + η ρ 2 v i y j v j + b i v i t | y k =const MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaaGymaaqaaiabeg 8aYbaadaWcaaqaaiabgkGi2kaadchaaeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgUcaRmaalaaabaGaeq4TdGgabaGaeq yWdihaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamOD amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcba GaamOAaaqabaGccqGHciITcaWG2bWaaSbaaSqaaiaadQgaaeqaaaaa kiabgUcaRiaadkgadaWgaaWcbaGaamyAaaqabaGccqGHijYUdaabca qaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaamyEamaaBaaame aacaWGRbaabeaaliabg2da9iaadogacaWGVbGaamOBaiaadohacaWG 0baabeaaaaa@5E08@

Velocity fields that satisfy this equation are known as Stokes Flows.  For steady flows, or for particularly slow flows, the equation can be simplified further by setting v i /t=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaa qabaGccaGGVaGaeyOaIyRaamiDaiabg2da9iaaicdaaaa@38BF@ .

 

 

7.9 Boundary conditions for fluid flow problems

 

There are three common types of fluid mechanics problem:

1.     A rigid or elastic solid which moves through a stationary fluid or gas;

2.     A bubble of gas or fluid of a second phase moves through a surrounding fluid;

3.     A fluid or gas flows through rigid or elastic walls of some sort.   

 

The following boundary conditions are most commonly used in these problems:

  1. Where a viscous fluid meets a solid boundary, the fluid is usually assumed to have the same velocity as the solid surface (both normal and tangential components of velocity are equal).  This is the no slip boundary condition.  Normal and tangential components of traction must also be continuous across the interface.

 

  1. Where an inviscid fluid meets a solid boundary, the normal component of velocity must be equal in both solid and fluid where the two meet.   The tangential component of velocity may be discontinuous.   The normal component of traction is continuous across the boundary, and the shear traction vanishes.

 

 

  1. A bubble inside a fluid is usually assumed to be inviscid, and tractions are required to be continuous across the fluid/gas interface.   For small bubbles, it may be necessary to account for the effects of surface tension as well.   In this case, the solid/fluid interface must be modeled as a separate phase, with its own constitutive behavior.  It is above our pay-grade to discuss this in detail here, but the basic concepts we develop here for solid and fluid phases should be sufficient for you to be able to follow texts and papers that work through the thermodynamics of interfaces.

 

  1. In some high-speed compressible viscous flow problems; or in microfluidics applications, the no-slip boundary condition is found to be inaccurate.  In this case the tangential component of velocity may be discontinuous across the boundary.  An additional constitutive law must specify a relationship between the tractions acting on the interface and the velocity discontinuity.

 

 

7.10 Control surface method for fluid flow problems

 

The control surface method is a helpful approach to computing average quantities such as resultant forces exerted by a fluid flow on a surface.   

 

The method relies on the conservation laws for a control volume

 

* Mass Conservation: d dt R ρdV + B ρvndA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaamizaiaadAfaaSqaaiaadkfaaeqaniab gUIiYdGccqGHRaWkdaWdrbqaaiabeg8aYjaahAhacqGHflY1caWHUb GaamizaiaadgeaaSqaaiaadkeaaeqaniabgUIiYdGccqGH9aqpcaaI Waaaaa@475E@

* Linear Momentum Balance B nσdA + R ρbdV = d dt R ρv dV+ B (ρv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaCOBaiabgwSixlaaho8aca WGKbGaamyqaaWcbaGaamOqaaqab0Gaey4kIipakiabgUcaRmaapefa baGaeqyWdiNaaCOyaiaadsgacaWGwbaaleaacaWGsbaabeqdcqGHRi I8aOGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizaiaadshaaaWaa8qu aeaacqaHbpGCcaWH2baaleaacaWGsbaabeqdcqGHRiI8aOGaamizai aadAfacqGHRaWkdaWdrbqaaiaacIcacqaHbpGCcaWH2bGaaiykaiaa hAhacqGHflY1caWHUbGaamizaiaadgeaaSqaaiaadkeaaeqaniabgU IiYdaaaa@5BC7@

* Angular Momentum Balance B y×(nσ)dA + R y×(ρb)dA = d dt R y×ρvdV + B (y×ρv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaCyEaiabgEna0kaacIcaca WHUbGaeyyXICTaaC4WdiaacMcacaWGKbGaamyqaaWcbaGaamOqaaqa b0Gaey4kIipakiabgUcaRmaapefabaGaaCyEaiabgEna0kaacIcacq aHbpGCcaWHIbGaaiykaiaadsgacaWGbbaaleaacaWGsbaabeqdcqGH RiI8aOGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizaiaadshaaaWaa8 quaeaacaWH5bGaey41aqRaeqyWdiNaaCODaiaadsgacaWGwbaaleaa caWGsbaabeqdcqGHRiI8aOGaey4kaSYaa8quaeaacaGGOaGaaCyEai abgEna0kabeg8aYjaahAhacaGGPaGaaCODaiabgwSixlaah6gacaWG KbGaamyqaaWcbaGaamOqaaqab0Gaey4kIipaaaa@6AC8@

* Mechanical Power Balance

B (nσ)vdA + R ρbv dV= R σ:D dV+ d dt R 1 2 ρ(vv)dV + B 1 2 ρ(vv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaiikaiaah6gacqGHflY1ca WHdpGaaiykaiabgwSixlaahAhacaWGKbGaamyqaaWcbaGaamOqaaqa b0Gaey4kIipakiabgUcaRmaapefabaGaeqyWdiNaaCOyaiabgwSixl aahAhaaSqaaiaadkfaaeqaniabgUIiYdGccaWGKbGaamOvaiabg2da 9maapefabaGaaC4WdiaacQdacaWHebaaleaacaWGsbaabeqdcqGHRi I8aOGaamizaiaadAfacqGHRaWkdaWcaaqaaiaadsgaaeaacaWGKbGa amiDaaaadaWdrbqaamaalaaabaGaaGymaaqaaiaaikdaaaGaeqyWdi NaaiikaiaahAhacqGHflY1caWH2bGaaiykaiaadsgacaWGwbaaleaa caWGsbaabeqdcqGHRiI8aOGaey4kaSYaa8quaeaadaWcaaqaaiaaig daaeaacaaIYaaaaiabeg8aYjaacIcacaWH2bGaeyyXICTaaCODaiaa cMcacaWH2bGaeyyXICTaaCOBaiaadsgacaWGbbaaleaacaWGcbaabe qdcqGHRiI8aaaa@7756@

* First law of thermodynamics

B (nσ)v dA+ R ρbv dV B qndA + V qdV = d dt R ρ( ε+ 1 2 vv )dV + B ρ( ε+ 1 2 vv )vn dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaiikaiaah6gacqGHflY1ca WHdpGaaiykaiabgwSixlaahAhaaSqaaiaadkeaaeqaniabgUIiYdGc caWGKbGaamyqaiabgUcaRmaapefabaGaeqyWdiNaaCOyaiabgwSixl aahAhaaSqaaiaadkfaaeqaniabgUIiYdGccaWGKbGaamOvaiabgkHi TmaapefabaGaaCyCaiabgwSixlaah6gacaWGKbGaamyqaaWcbaGaam Oqaaqab0Gaey4kIipakiabgUcaRmaapefabaGaamyCaiaadsgacaWG wbaaleaacaWGwbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaWGKb aabaGaamizaiaadshaaaWaa8quaeaacqaHbpGCdaqadaqaaiabew7a LjabgUcaRmaalaaabaGaaGymaaqaaiaaikdaaaGaaCODaiabgwSixl aahAhaaiaawIcacaGLPaaacaWGKbGaamOvaaWcbaGaamOuaaqab0Ga ey4kIipakiabgUcaRmaapefabaGaeqyWdi3aaeWaaeaacqaH1oqzcq GHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiaahAhacqGHflY1caWH 2baacaGLOaGaayzkaaGaaCODaiabgwSixlaah6gaaSqaaiaadkeaae qaniabgUIiYdGccaWGKbGaamyqaaaa@84E4@

* Second law of thermodynamics d dt R ρsdV + B ρs(vn)dA + B qn θ dA R q θ dV 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaam4CaiaadsgacaWGwbaaleaacaWGsbaa beqdcqGHRiI8aOGaey4kaSYaa8quaeaacqaHbpGCcaWGZbGaaiikai aahAhacqGHflY1caWHUbGaaiykaiaadsgacaWGbbaaleaacaWGcbaa beqdcqGHRiI8aOGaey4kaSYaa8quaeaadaWcaaqaaiaahghacqGHfl Y1caWHUbaabaGaeqiUdehaaiaadsgacaWGbbaaleaacaWGcbaabeqd cqGHRiI8aOGaeyOeI0Yaa8quaeaadaWcaaqaaiaadghaaeaacqaH4o qCaaGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHLjYS caaIWaaaaa@5FAC@

 

 

 

In addition, the Bernoulli equation can be applied to inviscid, incompressible flows along a streamline .  In this case it has the form

p ρ + 1 2 v i v i +Φ=constant MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamiCaaqaaiabeg8aYbaacq GHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiaadAhadaWgaaWcbaGa amyAaaqabaGccaWG2bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaeu OPdyKaeyypa0Jaae4yaiaab+gacaqGUbGaae4CaiaabshacaqGHbGa aeOBaiaabshaaaa@44B7@

where Φ/ y i = b i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiabgkGi2kabfA6agjaac+cacq GHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamOyamaa BaaaleaacaWGPbaabeaaaaa@3B77@  is the body force per unit mass.

 

 

Example 1: A jet of incompressible, inviscid fluid with mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@330E@ , cross sectional area A 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadgeadaWgaaWcbaGaaGimaaqabaaaaa@3214@ , and speed v 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGimaaqabaaaaa@3249@  is incident on an inclined wall, as shown in the figure. The surrounding atmostpheric pressure is p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGimaaqabaaaaa@3243@  and gravity may be neglected. Calculate the force acting on the wall.

 

 

 

  1. Consider a control volume as shown in the figure, with interior R and boundary B.
  2. Note that B p 0 ndA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaamiCamaaBaaaleaacaaIWa aabeaakiaah6gacaWGKbGaamyqaaWcbaGaamOqaaqab0Gaey4kIipa kiabg2da9iaaicdaaaa@39CE@
  3. Note that the fluid is inviscid, and therefore the shear stress acting on the wall vanishes.  Momentum balance in the j direction thus shows that

A ( p 0 njdA )j A 0 ρ 0 v 0 2 sinαj+ A 3 (p p 0 )dAj =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaiikaiabgkHiTiaadchada WgaaWcbaGaaGimaaqabaGccaWHUbGaeyyXICTaaCOAaiaadsgacaWG bbaaleaacaWGbbaabeqdcqGHRiI8aOGaaiykaiaahQgacqGHsislca WGbbWaaSbaaSqaaiaaicdaaeqaaOGaeqyWdi3aaSbaaSqaaiaaicda aeqaaOGaamODamaaDaaaleaacaaIWaaabaGaaGOmaaaakiGacohaca GGPbGaaiOBaiabeg7aHjaahQgacqGHRaWkdaWdrbqaaiaacIcacaWG WbGaeyOeI0IaamiCamaaBaaaleaacaaIWaaabeaakiaacMcacaWGKb GaamyqaiaahQgaaSqaaiaadgeadaWgaaadbaGaaG4maaqabaaaleqa niabgUIiYdGccqGH9aqpcaaIWaaaaa@5A5B@

  1. We recognize the last integral as the force exerted by the wall on the fluid.  The force exerted by the fluid on the wall must be equal and opposite, and so F= A 0 ρ 0 v 0 2 sinαj MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAeacqGH9aqpcqGHsislcaWGbbWaaS baaSqaaiaaicdaaeqaaOGaeqyWdi3aaSbaaSqaaiaaicdaaeqaaOGa amODamaaDaaaleaacaaIWaaabaGaaGOmaaaakiGacohacaGGPbGaai OBaiabeg7aHjaahQgaaaa@3FA2@

 

 

Example 2: The figure shows an idealized centrifugal pump.   Fluid enters the rotating blades with radial and tangential velocity ( v r0, v t0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG2bWaaSbaaSqaaiaadkhaca aIWaGaaiilaaqabaGccaWG2bWaaSbaaSqaaiaadshacaaIWaaabeaa kiaacMcaaaa@3837@  and exits with velocity ( v r1, v t1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG2bWaaSbaaSqaaiaadkhaca aIXaGaaiilaaqabaGccaWG2bWaaSbaaSqaaiaadshacaaIXaaabeaa kiaacMcaaaa@3839@ .   The tangential velocity of the fluid is equal to that of the blades at both entry and exit.   The blades rotate with angular speed ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeM8a3baa@3235@  and the mass flow rate through the pump is m ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiqad2gagaGaaaaa@3163@ .   Calculate the torque required to rotate the pump.

 

  1. Consider a control volume consisting of the annular region occupied by the blades, but excluding the blades themselves (only one piece of the CV is shown in the figure)
  2. Assume axisymmetric flow MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  so fluid velocity is independent of θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXbaa@321E@
  3. Mass conservation gives 2π r 0 z v r0 ρ=2π r 1 z v r1 ρ= m ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacqaHapaCcaWGYbWaaSbaaSqaai aaicdaaeqaaOGaamOEaiaadAhadaWgaaWcbaGaamOCaiaaicdaaeqa aOGaeqyWdiNaeyypa0JaaGOmaiabec8aWjaadkhadaWgaaWcbaGaaG ymaaqabaGccaWG6bGaamODamaaBaaaleaacaWGYbGaaGymaaqabaGc cqaHbpGCcqGH9aqpceWGTbGbaiaaaaa@4773@ , where z is the out of plane depth of the blades.
  4. The angular momentum equation gives

B y×(nσ)dA = B (y×ρv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaCyEaiabgEna0kaacIcaca WHUbGaeyyXICTaaC4WdiaacMcacaWGKbGaamyqaaWcbaGaamOqaaqa b0Gaey4kIipakiabg2da9maapefabaGaaiikaiaahMhacqGHxdaTcq aHbpGCcaWH2bGaaiykaiaahAhacqGHflY1caWHUbGaamizaiaadgea aSqaaiaadkeaaeqaniabgUIiYdaaaa@4F6B@

The integral on the left represents the net moment exerted by the blades on the fluid.  The contributions to the integral on the right from the surfaces adjacent to the blades cancel, leaving only the contributions from the inner and outer surfaces of the annulus.   Evaluating the integrals gives

M z =2π r 1 (z r 1 v t1 ) v n1 ρ2π r 0 (z r 0 v t0 ) v n0 ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaamOEaaqabaGccq GH9aqpcaaIYaGaeqiWdaNaamOCamaaBaaaleaacaaIXaaabeaakiaa cIcacaWG6bGaamOCamaaBaaaleaacaaIXaaabeaakiaadAhadaWgaa WcbaGaamiDaiaaigdaaeqaaOGaaiykaiaadAhadaWgaaWcbaGaamOB aiaaigdaaeqaaOGaeqyWdiNaeyOeI0IaaGOmaiabec8aWjaadkhada WgaaWcbaGaaGimaaqabaGccaGGOaGaamOEaiaadkhadaWgaaWcbaGa aGimaaqabaGccaWG2bWaaSbaaSqaaiaadshacaaIWaaabeaakiaacM cacaWG2bWaaSbaaSqaaiaad6gacaaIWaaabeaakiabeg8aYbaa@54A8@

  1. Note that v t1 = r 1 ω v t0 = r 0 ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamiDaiaaigdaae qaaOGaeyypa0JaamOCamaaBaaaleaacaaIXaaabeaakiabeM8a3jaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaamODamaaBaaaleaacaWG0b GaaGimaaqabaGccqGH9aqpcaWGYbWaaSbaaSqaaiaaicdaaeqaaOGa eqyYdChaaa@475D@ , which gives M z = m ˙ ω( r 1 2 r 0 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad2eadaWgaaWcbaGaamOEaaqabaGccq GH9aqpceWGTbGbaiaacqaHjpWDcaGGOaGaamOCamaaDaaaleaacaaI XaaabaGaaGOmaaaakiabgkHiTiaadkhadaqhaaWcbaGaaGimaaqaai aaikdaaaGccaGGPaaaaa@3DCC@ .

 

 

Example 3: For our third example, we work through Von Karman’s classic estimate of the drag force on a stationary plate inside a steadily flowing stream.   The figure shows the problem to be solved.   Ahead of the plate, the fluid has a uniform horizontal velocity V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfaaaa@3143@  with direction parallel to the edge of the plate.   A boundary layer in which the fluid flow is slowed develops adjacent to the plate, and as a result the horizontal velocity has profile v 1 ( y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamyEamaaBaaaleaacaaIYaaabeaakiaacMcaaaa@359D@  at the trailing edge of the plate.    The velocity field is assumed to be uniform outside the boundary layer. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOaa@3067@  We assume that the width of the boundary layer δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKbaa@320D@  at the trailing edge of the plate can be determined somehow.   Our goal is then to determine the drag force on the plate in terms of v 1 ( y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamyEamaaBaaaleaacaaIYaaabeaakiaacMcaaaa@359D@ , L  and the mass density of the fluid ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@ .  

 

We choose a control volume that is bounded by the plate, and at the top follows a streamline outside the boundary layer which terminates at the trailing edge of the boundary layer.   The pressure outside this control volume is uniform.  No fluid crosses the upper or lower surfaces of the control volume, and we assume steady-state conditions.

 

  Mass conservation then implies that

d dt R ρdV + B ρvndA =0ρVh+ 0 δ ρ v 1 ( y 2 )d y 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaamizaiaadAfaaSqaaiaadkfaaeqaniab gUIiYdGccqGHRaWkdaWdrbqaaiabeg8aYjaahAhacqGHflY1caWHUb GaamizaiaadgeaaSqaaiaadkeaaeqaniabgUIiYdGccqGH9aqpcaaI WaGaeyO0H4TaeyOeI0IaeqyWdiNaamOvaiaadIgacqGHRaWkdaWdXb qaaiabeg8aYjaadAhadaWgaaWcbaGaaGymaaqabaGccaGGOaGaamyE amaaBaaaleaacaaIYaaabeaakiaacMcacaWGKbGaamyEamaaBaaale aacaaIYaaabeaakiabg2da9iaaicdaaSqaaiaaicdaaeaacqaH0oaz a0Gaey4kIipaaaa@5F68@

 

Linear momentum balance shows that

B nσdA + R ρbdV = d dt R ρv dV+ B (ρv)vndA B nσdA =ρ V 2 h+ 0 δ ρ [ v 1 ( y 2 ) ] 2 d y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaWaa8quaeaacaWHUbGaeyyXICTaaC 4WdiaadsgacaWGbbaaleaacaWGcbaabeqdcqGHRiI8aOGaey4kaSYa a8quaeaacqaHbpGCcaWHIbGaamizaiaadAfaaSqaaiaadkfaaeqani abgUIiYdGccqGH9aqpdaWcaaqaaiaadsgaaeaacaWGKbGaamiDaaaa daWdrbqaaiabeg8aYjaahAhaaSqaaiaadkfaaeqaniabgUIiYdGcca WGKbGaamOvaiabgUcaRmaapefabaGaaiikaiabeg8aYjaahAhacaGG PaGaaCODaiabgwSixlaah6gacaWGKbGaamyqaaWcbaGaamOqaaqab0 Gaey4kIipaaOqaaiabgkDiEpaapefabaGaaCOBaiabgwSixlaaho8a caWGKbGaamyqaaWcbaGaamOqaaqab0Gaey4kIipakiabg2da9iabgk HiTiabeg8aYjaadAfadaahaaWcbeqaaiaaikdaaaGccaWGObGaey4k aSYaa8qCaeaacqaHbpGCdaWadaqaaiaadAhadaWgaaWcbaGaaGymaa qabaGccaGGOaGaamyEamaaBaaaleaacaaIYaaabeaakiaacMcaaiaa wUfacaGLDbaadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamyEamaaBa aaleaacaaIYaaabeaaaeaacaaIWaaabaGaeqiTdqganiabgUIiYdaa aaa@8045@

Only the plate exerts a force on the control volume (the pressure is uniform, and there is no shear stress acting on the top surface of the control volume because the velocity is uniform), so we recognize the integral on the left as the resultant force exerted by the plate on the control volume.   The drag force exerted by the fluid on the plate must be equal and opposite.

 

Substituting for h from the first equation into the second, and noting that a boundary layer must form on both sides of the plate, gives the following expression for the total drag force

 

F D =2 0 δ ρ v 1 ( y 2 )[ V v 1 ( y 2 ) ]d y 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamiraaqabaGccq GH9aqpcaaIYaWaa8qCaeaacqaHbpGCcaWG2bWaaSbaaSqaaiaaigda aeqaaOGaaiikaiaadMhadaWgaaWcbaGaaGOmaaqabaGccaGGPaWaam WaaeaacaWGwbGaeyOeI0IaamODamaaBaaaleaacaaIXaaabeaakiaa cIcacaWG5bWaaSbaaSqaaiaaikdaaeqaaOGaaiykaaGaay5waiaaw2 faaiaadsgacaWG5bWaaSbaaSqaaiaaikdaaeqaaaqaaiaaicdaaeaa cqaH0oaza0Gaey4kIipaaaa@4B64@

The formula is only useful if the velocity profile at the trailing edge of the plate is known.  As a rough guess, we could take a parabolic velocity distribution which satisfies v=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhacqGH9aqpcaaIWaaaaa@3323@  at y=0 and v=V,dv/dy=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhacqGH9aqpcaWGwbGaaiilaiaads gacaWG2bGaai4laiaadsgacaWG5bGaeyypa0JaaGimaaaa@3A32@  at y=δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhacqGH9aqpcqaH0oazaaa@3411@ .   This then yields

v 1 ( y 2 )=V( 2 y 2 δ ( y 2 δ ) 2 ) F D = 4ρδ V 2 15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaaGymaaqabaGcca GGOaGaamyEamaaBaaaleaacaaIYaaabeaakiaacMcacqGH9aqpcaWG wbWaaeWaaeaacaaIYaWaaSaaaeaacaWG5bWaaSbaaSqaaiaaikdaae qaaaGcbaGaeqiTdqgaaiabgkHiTmaabmaabaWaaSaaaeaacaWG5bWa aSbaaSqaaiaaikdaaeqaaaGcbaGaeqiTdqgaaaGaayjkaiaawMcaam aaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaeyO0H4TaamOramaaBaaaleaacaWGebaabeaakiabg2da9maa laaabaGaaGinaiabeg8aYjabes7aKjaadAfadaahaaWcbeqaaiaaik daaaaakeaacaaIXaGaaGynaaaaaaa@6217@

To use this expression we still need to find the thickness of the boundary layer, but that’s beyond our pay-grade.   An asymptotic analysis gives δ=4.65L/(ρVL/η) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjabg2da9iaaisdacaGGUaGaaG OnaiaaiwdacaWGmbGaai4laiaacIcacqaHbpGCcaWGwbGaamitaiaa c+cacqaH3oaAcaGGPaaaaa@3EAA@ , where η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeE7aObaa@3214@  is the fluid viscosity.

 

 

 

7.11 Exact solutions to simple incompressible, inviscid, irrotational flow problems (potential flow)

 

The governing equations for potential flow can be reduced to:

* The solution can be generated from a harmonic potential satisfying 2 ϕ y i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaeqy1dygabaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaa beaakiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaeyypa0 JaaGimaaaa@3D68@

* The velocity components follow as v i = ϕ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiabgkGi2kabew9aMbqaaiabgkGi2kaadMhadaWg aaWcbaGaamyAaaqabaaaaaaa@3A49@

* The velocity must satisfy v i n i = V i n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGcca WGUbWaaSbaaSqaaiaadMgaaeqaaOGaeyypa0JaamOvamaaBaaaleaa caWGPbaabeaakiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@39B0@  at any point where it meets a surface with normal n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@3275@  moving with velocity V i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaa@325D@

* The pressure can be computed from the Bernoulli equation

p ρ + 1 2 ϕ y i ϕ y i +Φ+ ϕ t = p 0 ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamiCaaqaaiabeg8aYbaacq GHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaamaalaaabaGaeyOaIyRa eqy1dygabaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGcda WcaaqaaiabgkGi2kabew9aMbqaaiabgkGi2kaadMhadaWgaaWcbaGa amyAaaqabaaaaOGaey4kaSIaeuOPdyKaey4kaSYaaSaaaeaacqGHci ITcqaHvpGzaeaacqGHciITcaWG0baaaiabg2da9maalaaabaGaamiC amaaBaaaleaacaaIWaaabeaaaOqaaiabeg8aYbaaaaa@50B7@

where p 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGimaaqabaaaaa@3243@  is a reference pressure at some point where the velocity and body force potential are zero.

 

It is relatively straightforward to solve the Laplace equation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in 3D, many solutions have been found by guesswork; techniques such as Fourier transforms can be used to solve more complex problems.   In 2D, complex variable methods are very effective.   It turns out that both the real and imaginary parts of a differentiable function of a complex number (eg a polynomial function of a complex number z=x+iy) are solutions to Laplace’s equation.   Techniques such as conformal mapping and analytic continuation are also useful.  We don’t have time in this course to discuss these procedures in detail MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  instead, we just list a few important solutions.

 

 

 

 

 

 

7.11.1 Flow surrounding a moving sphere

 

The flow surrounding a rigid sphere with radius a that starts at the origin, and then moves without rotation with velocity V i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaa@325D@  can be computed from the following potential

ϕ= a 3 V i ( y i V i t) 2 r 3 r= ( y k V k t)( y k V k t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjabg2da9iabgkHiTmaalaaaba GaamyyamaaCaaaleqabaGaaG4maaaakiaadAfadaWgaaWcbaGaamyA aaqabaGccaGGOaGaamyEamaaBaaaleaacaWGPbaabeaakiabgkHiTi aadAfadaWgaaWcbaGaamyAaaqabaGccaWG0bGaaiykaaqaaiaaikda caWGYbWaaWbaaSqabeaacaaIZaaaaaaakiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGYbGaeyypa0ZaaOaaae aacaGGOaGaamyEamaaBaaaleaacaWGRbaabeaakiabgkHiTiaadAfa daWgaaWcbaGaam4AaaqabaGccaWG0bGaaiykaiaacIcacaWG5bWaaS baaSqaaiaadUgaaeqaaOGaeyOeI0IaamOvamaaBaaaleaacaWGRbaa beaakiaadshacaGGPaaaleqaaaaa@6BA3@

 

7.11.2 Flow surrounding a moving cylinder

 

The flow surrounding a rigid cylinder that moves without rotation with velocity V i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaa@325D@  can be computed from the following potential

ϕ= a 2 V α ( y α V α t) 2 r 2 r= ( y α V α t)( y α V α t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMjabg2da9iabgkHiTmaalaaaba GaamyyamaaCaaaleqabaGaaGOmaaaakiaadAfadaWgaaWcbaGaeqyS degabeaakiaacIcacaWG5bWaaSbaaSqaaiabeg7aHbqabaGccqGHsi slcaWGwbWaaSbaaSqaaiabeg7aHbqabaGccaWG0bGaaiykaaqaaiaa ikdacaWGYbWaaWbaaSqabeaacaaIYaaaaaaakiaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGYbGaeyypa0ZaaO aaaeaacaGGOaGaamyEamaaBaaaleaacqaHXoqyaeqaaOGaeyOeI0Ia amOvamaaBaaaleaacqaHXoqyaeqaaOGaamiDaiaacMcacaGGOaGaam yEamaaBaaaleaacqaHXoqyaeqaaOGaeyOeI0IaamOvamaaBaaaleaa cqaHXoqyaeqaaOGaamiDaiaacMcaaSqabaaaaa@7070@

 

As an exercise, you might like to calculate (i) The components of the velocity field; (ii) the pressure distribution acting on the sphere and cylinder; (iii) The drag force acting on the sphere and cylinder (the result is surprising at first, but note that there is no dissipation so at steady-state there can be no drag; (iv) the total kinetic energy in the fluid.

 

 

 

7.12 Solutions to simple viscous flow problems (Stokes flow)

 

The equations governing Stokes flow are also fairly easy to solve, because the equations are linear.  This means that Fourier transform techniques and complex variable methods both work.   We won’t try to cover these here, but list a few examples of solutions to Stokes flow problems

 

 

 

7.12.1 Laminar flow of an incompressible viscous fluid between plates

 

The problem to be solved is illustrated in the figure.  An incompressible fluid with mass density ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYbaa@3228@  and viscosity η MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeE7aObaa@3214@  is confined between parallel plates.  External boundary constraints at the ends of the plate induce a (constant) pressure gradient Δp/ΔL MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabfs5aejaadchacaGGVaGaeuiLdqKaam itaaaa@35AD@  in a direction parallel to the plates.  The top plate moves with speed V, and the bottom is stationary.

 

The velocity field in the fluid has the form

v=[ V y 2 h Δp 2ΔL y 2 (h y 2 ) ] e 1 σ= ηV h + Δp ΔL ( h 2 y 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaCODaiabg2da9maadmaabaGaam OvamaalaaabaGaamyEamaaBaaaleaacaaIYaaabeaaaOqaaiaadIga aaGaeyOeI0YaaSaaaeaacqqHuoarcaWGWbaabaGaaGOmaiabfs5aej aadYeaaaGaamyEamaaBaaaleaacaaIYaaabeaakiaacIcacaWGObGa eyOeI0IaamyEamaaBaaaleaacaaIYaaabeaakiaacMcaaiaawUfaca GLDbaacaWHLbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaaC4Wdiabg2da 9maalaaabaGaeq4TdGMaamOvaaqaaiaadIgaaaGaey4kaSYaaSaaae aacqqHuoarcaWGWbaabaGaeuiLdqKaamitaaaadaqadaqaamaalaaa baGaamiAaaqaaiaaikdaaaGaeyOeI0IaamyEamaaBaaaleaacaaIYa aabeaaaOGaayjkaiaawMcaaaaaaa@58E7@

These results can be derived as follows. 

1.     Boundary conditions at the top and bottom surfaces of the plate suggest that the flow velocity must be parallel to the plates, and independent of y 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaaGymaaqabaaaaa@324D@ .  We can assume that v=f( y 2 ) e 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahAhacqGH9aqpcaWGMbGaaiikaiaadM hadaWgaaWcbaGaaGOmaaqabaGccaGGPaGaaCyzamaaBaaaleaacaaI Xaaabeaaaaa@3876@

2.     The flow is steady, so the general Navier-Stokes equation reduces to

1 ρ p y i + η ρ 2 v i y j v j + b i v i t | y k =const Δp ΔL +η 2 f y 2 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaaGymaaqaaiabeg 8aYbaadaWcaaqaaiabgkGi2kaadchaaeaacqGHciITcaWG5bWaaSba aSqaaiaadMgaaeqaaaaakiabgUcaRmaalaaabaGaeq4TdGgabaGaeq yWdihaamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaamOD amaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaadMhadaWgaaWcba GaamOAaaqabaGccqGHciITcaWG2bWaaSbaaSqaaiaadQgaaeqaaaaa kiabgUcaRiaadkgadaWgaaWcbaGaamyAaaqabaGccqGHijYUdaabca qaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaamyEamaaBaaame aacaWGRbaabeaaliabg2da9iaadogacaWGVbGaamOBaiaadohacaWG 0baabeaakiabgkDiElabgkHiTmaalaaabaGaeuiLdqKaamiCaaqaai abfs5aejaadYeaaaGaey4kaSIaeq4TdG2aaSaaaeaacqGHciITdaah aaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamyEamaaDaaale aacaaIYaaabaGaaGOmaaaaaaGccqGH9aqpcaaIWaaaaa@71B3@

3.     We can easily integrate this equation to see that f=A y 2 +B+ Δp 2ΔL y 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacqGH9aqpcaWGbbGaamyEamaaBa aaleaacaaIYaaabeaakiabgUcaRiaadkeacqGHRaWkdaWcaaqaaiab fs5aejaadchaaeaacaaIYaGaeuiLdqKaamitaaaacaWG5bWaa0baaS qaaiaaikdaaeaacaaIYaaaaaaa@3F9B@

4.     Boundary conditions at  the plate surfaces imply that f(0)=0f(h)=V MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAgacaGGOaGaaGimaiaacMcacqGH9a qpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaamOzaiaacIcacaWGObGaaiykaiabg2da9iaadA faaaa@481B@ .  We can then solve for the unknown constants A and B, giving the answer stated.

 

 

7.12.3 Rigid sphere moving at steady speed through a viscous fluid

 

The velocity field surrounding a rigid sphere that moves without rotation with instantaneous velocity V i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaa@325D@  through a viscous fluid is

v i = 3a V j 4 r 5 ( r 2 a 2 )( y j V j t)( y i V i t)+ a V i 4 r 3 (3 r 2 + a 2 ) r= ( y i V i t)( y i V i t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamODamaaBaaaleaacaWGPbaabe aakiabg2da9maalaaabaGaaG4maiaadggacaWGwbWaaSbaaSqaaiaa dQgaaeqaaaGcbaGaaGinaiaadkhadaahaaWcbeqaaiaaiwdaaaaaaO GaaiikaiaadkhadaahaaWcbeqaaiaaikdaaaGccqGHsislcaWGHbWa aWbaaSqabeaacaaIYaaaaOGaaiykaiaacIcacaWG5bWaaSbaaSqaai aadQgaaeqaaOGaeyOeI0IaamOvamaaBaaaleaacaWGQbaabeaakiaa dshacaGGPaGaaiikaiaadMhadaWgaaWcbaGaamyAaaqabaGccqGHsi slcaWGwbWaaSbaaSqaaiaadMgaaeqaaOGaamiDaiaacMcacqGHRaWk daWcaaqaaiaadggacaWGwbWaaSbaaSqaaiaadMgaaeqaaaGcbaGaaG inaiaadkhadaahaaWcbeqaaiaaiodaaaaaaOGaaiikaiaaiodacaWG YbWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIaamyyamaaCaaaleqaba GaaGOmaaaakiaacMcaaeaacaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caWGYbGaeyypa0ZaaOaaaeaaca GGOaGaamyEamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadAfadaWg aaWcbaGaamyAaaqabaGccaWG0bGaaiykaiaacIcacaWG5bWaaSbaaS qaaiaadMgaaeqaaOGaeyOeI0IaamOvamaaBaaaleaacaWGPbaabeaa kiaadshacaGGPaaaleqaaaaaaa@7AB3@

For r>>a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhacqGH+aGpcqGH+aGpcaWGHbaaaa@3455@  the velocity field can be approximated as

v i = 3a V j 4 r 3 ( y j V j t)( y i V i t)+ 3a V i 4r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaaiodacaWGHbGaamOvamaaBaaaleaacaWGQbaa beaaaOqaaiaaisdacaWGYbWaaWbaaSqabeaacaaIZaaaaaaakiaacI cacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyOeI0IaamOvamaaBaaa leaacaWGQbaabeaakiaadshacaGGPaGaaiikaiaadMhadaWgaaWcba GaamyAaaqabaGccqGHsislcaWGwbWaaSbaaSqaaiaadMgaaeqaaOGa amiDaiaacMcacqGHRaWkdaWcaaqaaiaaiodacaWGHbGaamOvamaaBa aaleaacaWGPbaabeaaaOqaaiaaisdacaWGYbaaaaaa@4EF4@

This solution is called a ‘Stokeslet.’  The Stokeslet solution can also be expressed in terms of the force exerted by the sphere on the fluid MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  for a sphere at the origin we have

v i = P j 8ηr ( y i y j r 2 + δ ij ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaadcfadaWgaaWcbaGaamOAaaqabaaakeaacaaI 4aGaeq4TdGMaamOCaaaadaqadaqaamaalaaabaGaamyEamaaBaaale aacaWGPbaabeaakiaadMhadaWgaaWcbaGaamOAaaqabaaakeaacaWG YbWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRiabes7aKnaaBaaale aacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaaaaa@455E@

 

 

7.12.3 Spherical bubble moving at steady speed through a viscous fluid

 

The velocity field surrounding a spherical bubble that moves with instantaneous velocity V i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAfadaWgaaWcbaGaamyAaaqabaaaaa@325D@  through a viscous fluid is

v i = a V j 2r ( δ ij + ( y i V i t)( y j V j t) r 2 )r= ( y i V i t)( y i V i t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaadggacaWGwbWaaSbaaSqaaiaadQgaaeqaaaGc baGaaGOmaiaadkhaaaWaaeWaaeaacqaH0oazdaWgaaWcbaGaamyAai aadQgaaeqaaOGaey4kaSYaaSaaaeaacaGGOaGaamyEamaaBaaaleaa caWGPbaabeaakiabgkHiTiaadAfadaWgaaWcbaGaamyAaaqabaGcca WG0bGaaiykaiaacIcacaWG5bWaaSbaaSqaaiaadQgaaeqaaOGaeyOe I0IaamOvamaaBaaaleaacaWGQbaabeaakiaadshacaGGPaaabaGaam OCamaaCaaaleqabaGaaGOmaaaaaaaakiaawIcacaGLPaaacaaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaamOCaiabg2da9m aakaaabaGaaiikaiaadMhadaWgaaWcbaGaamyAaaqabaGccqGHsisl caWGwbWaaSbaaSqaaiaadMgaaeqaaOGaamiDaiaacMcacaGGOaGaam yEamaaBaaaleaacaWGPbaabeaakiabgkHiTiaadAfadaWgaaWcbaGa amyAaaqabaGccaWG0bGaaiykaaWcbeaaaaa@6ABA@

 

 

 

7.13 Solutions to simple compressible flow problems

 

Many compressible flow problems are concerned with flows in air, which can often be approximated as an ideal gas.  The governing equations for an ideal gas reduce to

 

* Mass conservation

ρ t | x=const +ρ v i y i =0or    ρ t | y=const + ρ v i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcqaHbp GCaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahIhacqGH 9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGHRaWkcq aHbpGCdaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyAaaqabaaa keaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaakiabg2da9i aaicdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaab+gacaqGYbGa aeiiaiaabccacaqGGaWaaqGaaeaadaWcaaqaaiabgkGi2kabeg8aYb qaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaaCyEaiabg2da 9iaadogacaWGVbGaamOBaiaadohacaWG0baabeaakiabgUcaRmaala aabaGaeyOaIyRaeqyWdiNaamODamaaBaaaleaacaWGPbaabeaaaOqa aiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaeyypa0JaaG imaaaa@6F3D@

 * Navier Stokes

p y i +ρ b i =ρ d v i dt | x=const =ρ v i t | y k =const + 1 2 ρ y i ( v k v k )+ρ ijk ω j v k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaeyOaIyRaamiCaa qaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaey4kaSIa eqyWdiNaamOyamaaBaaaleaacaWGPbaabeaakiaaykW7cqGH9aqpcq aHbpGCdaabcaqaamaalaaabaGaamizaiaadAhadaWgaaWcbaGaamyA aaqabaaakeaacaWGKbGaamiDaaaaaiaawIa7amaaBaaaleaacaWH4b Gaeyypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGaeyyp a0JaeqyWdi3aaqGaaeaadaWcaaqaaiabgkGi2kaadAhadaWgaaWcba GaamyAaaqabaaakeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqa aiaadMhadaWgaaadbaGaam4AaaqabaWccqGH9aqpcaWGJbGaam4Bai aad6gacaWGZbGaamiDaaqabaGccqGHRaWkdaWcaaqaaiaaigdaaeaa caaIYaaaaiabeg8aYnaalaaabaGaeyOaIylabaGaeyOaIyRaamyEam aaBaaaleaacaWGPbaabeaaaaGccaGGOaGaamODamaaBaaaleaacaWG RbaabeaakiaadAhadaWgaaWcbaGaam4AaaqabaGccaGGPaGaey4kaS IaeqyWdiNaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGc cqaHjpWDdaWgaaWcbaGaamOAaaqabaGccaWG2bWaaSbaaSqaaiaadU gaaeqaaaaa@7B38@

* Energy conservation

ρ ε t | x=const =p v i y i q i y i +q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaeiaabaWaaSaaaeaacqGHci ITcqaH1oqzaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaa hIhacqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccq GH9aqpcqGHsislcaWGWbWaaSaaaeaacqGHciITcaWG2bWaaSbaaSqa aiaadMgaaeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabe aaaaGccqGHsisldaWcaaqaaiabgkGi2kaadghadaWgaaWcbaGaamyA aaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaaki abgUcaRiaadghaaaa@5419@

* Entropy equation

θ s t | x=const = q i y i +q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeI7aXnaaeiaabaWaaSaaaeaacqGHci ITcaWGZbaabaGaeyOaIyRaamiDaaaaaiaawIa7amaaBaaaleaacaWH 4bGaeyypa0Jaam4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGaey ypa0JaeyOeI0YaaSaaaeaacqGHciITcaWGXbWaaSbaaSqaaiaadMga aeqaaaGcbaGaeyOaIyRaamyEamaaBaaaleaacaWGPbaabeaaaaGccq GHRaWkcaWGXbaaaa@4A61@

* Constitutive law: internal energy, free energy and stress response function

ε= c v θψ= c v θθ( c v logθRlogρ+ s 0 )p=ρRθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLjabg2da9iaadogadaWgaaWcba GaamODaaqabaGccqaH4oqCcaaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaeqiYdKNaeyypa0Jaam4yamaaBaaaleaacaWG2baabeaakiabeI7a XjabgkHiTiabeI7aXnaabmaabaGaam4yamaaBaaaleaacaWG2baabe aakiGacYgacaGGVbGaai4zaiabeI7aXjabgkHiTiaadkfaciGGSbGa ai4BaiaacEgacqaHbpGCcqGHRaWkcaWGZbWaaSbaaSqaaiaaicdaae qaaaGccaGLOaGaayzkaaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaamiCaiabg2da9iabeg8aYjaadk facqaH4oqCaaa@79AA@

where R is the gas constant, c v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamODaaqabaaaaa@3277@  is the specific heat capacity (a constant), and s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohadaWgaaWcbaGaaGimaaqabaaaaa@3246@  is an arbitrary constant.  Ideal gases are also characterized by the specific heat at constant pressure c p = c v +R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaamiCaaqabaGccq GH9aqpcaWGJbWaaSbaaSqaaiaadAhaaeqaaOGaey4kaSIaamOuaaaa @3753@  and the ratio γ= c p / c v MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeo7aNjabg2da9iaadogadaWgaaWcba GaamiCaaqabaGccaGGVaGaam4yamaaBaaaleaacaWG2baabeaaaaa@37EA@ .

 

 

7.13.1 Sound waves in an ideal gas

 

Small amplitude sound waves can be approximated as a small variation in density and pressure superimposed on a static state.    The velocity field for a small amplitude acoustic wave in still air is irrotational, and the velocity field and the change in density in the gas satisfy the equations

2 v i t 2 c s 2 2 v j y i y j =0 2 δρ t 2 c s 2 δρ y i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaamODamaaBaaaleaacaWGPbaabeaaaOqaaiabgkGi2kaa dshadaahaaWcbeqaaiaaikdaaaaaaOGaeyOeI0Iaam4yamaaDaaale aacaWGZbaabaGaaGOmaaaakmaalaaabaGaeyOaIy7aaWbaaSqabeaa caaIYaaaaOGaamODamaaBaaaleaacaWGQbaabeaaaOqaaiabgkGi2k aadMhadaWgaaWcbaGaamyAaaqabaGccqGHciITcaWG5bWaaSbaaSqa aiaadQgaaeqaaaaakiabg2da9iaaicdacaaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8+aaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccq aH0oazcqaHbpGCaeaacqGHciITcaWG0bWaaWbaaSqabeaacaaIYaaa aaaakiabgkHiTiaadogadaqhaaWcbaGaam4CaaqaaiaaikdaaaGcda WcaaqaaiabgkGi2kabes7aKjabeg8aYbqaaiabgkGi2kaadMhadaWg aaWcbaGaamyAaaqabaGccqGHciITcaWG5bWaaSbaaSqaaiaadMgaae qaaaaakiabg2da9iaaicdaaaa@77F6@

where

c s = p ρ | s=const MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaaqabaGccq GH9aqpdaGcaaqaamaaeiaabaWaaSaaaeaacqGHciITcaWGWbaabaGa eyOaIyRaeqyWdihaaaGaayjcSdWaaSbaaSqaaiaadohacqGH9aqpca WGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaaabeaaaaa@41A5@

is the speed of sound. In actual calculations it is often convenient to introduce a flow potential ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew9aMbaa@3230@  that satisfies

v i = ϕ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiabgkGi2kabew9aMbqaaiabgkGi2kaadMhadaWg aaWcbaGaamyAaaqabaaaaaaa@3A49@

The flow potential satisfies the wave equation

  2 ϕ t 2 c s 2 2 ϕ y i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIy7aaWbaaSqabeaaca aIYaaaaOGaeqy1dygabaGaeyOaIyRaamiDamaaCaaaleqabaGaaGOm aaaaaaGccqGHsislcaWGJbWaa0baaSqaaiaadohaaeaacaaIYaaaaO WaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaa cqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaeyOaIyRaamyEam aaBaaaleaacaWGPbaabeaaaaGccqGH9aqpcaaIWaGaaGPaVdaa@4A37@

 

These results are derived as follows. 

  • It is usually assumed that heat flow can be neglected on the time-scales associated with acoustic vibrations.   This means that the entropy of the gas is constant (see the entropy equation).
  • For small perturbations, the change in pressure of the gas is linearly related to its density.  We can write

δp= p ρ | s=const δρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjaadchacqGH9aqpdaabcaqaam aalaaabaGaeyOaIyRaamiCaaqaaiabgkGi2kabeg8aYbaaaiaawIa7 amaaBaaaleaacaWGZbGaeyypa0Jaam4yaiaad+gacaWGUbGaam4Cai aadshaaeqaaOGaeqiTdqMaeqyWdihaaa@4588@

 

For an ideal gas, it is possible to find a formula for the sound speed. Recall that

s= ψ θ s= s 0 +log( θ c v ρ R )θ= ρ R/ c v exp((s+ s 0 )/ c v ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadohacqGH9aqpcqGHsisldaWcaaqaai abgkGi2kabeI8a5bqaaiabgkGi2kabeI7aXbaacaaMc8UaaGPaVlaa ykW7caaMc8UaeyO0H4Taam4Caiabg2da9iaadohadaWgaaWcbaGaaG imaaqabaGccqGHRaWkciGGSbGaai4BaiaacEgacaGGOaGaeqiUde3a aWbaaSqabeaacaWGJbWaaSbaaWqaaiaadAhaaeqaaaaakiabeg8aYn aaCaaaleqabaGaeyOeI0IaamOuaaaakiaacMcacqGHshI3cqaH4oqC cqGH9aqpcqaHbpGCdaahaaWcbeqaaiaadkfacaGGVaGaam4yamaaBa aameaacaWG2baabeaaaaGcciGGLbGaaiiEaiaacchacaGGOaGaaiik aiaadohacqGHRaWkcaWGZbWaaSbaaSqaaiaaicdaaeqaaOGaaiykai aac+cacaWGJbWaaSbaaSqaaiaadAhaaeqaaOGaaiykaaaa@69AA@

and hence, since s=const , R/ c v =γ1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkfacaGGVaGaam4yamaaBaaaleaaca WG2baabeaakiabg2da9iabeo7aNjabgkHiTiaaigdaaaa@3860@  and p=ρRθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacqGH9aqpcqaHbpGCcaWGsbGaeq iUdehaaa@36B0@  the pressure is related to density by p=k ρ γ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchacqGH9aqpcaWGRbGaeqyWdi3aaW baaSqabeaacqaHZoWzaaaaaa@36E7@  where k is a constant.  It follows that

c s = kγ ρ γ1 = γ p ρ = γRθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaaqabaGccq GH9aqpdaGcaaqaaiaadUgacqaHZoWzcqaHbpGCdaahaaWcbeqaaiab eo7aNjabgkHiTiaaigdaaaaabeaakiabg2da9maakaaabaGaeq4SdC 2aaSaaaeaacaWGWbaabaGaeqyWdihaaaWcbeaakiabg2da9maakaaa baGaeq4SdCMaamOuaiabeI7aXbWcbeaaaaa@465D@ .

·         For small amplitude perturbations, we approximate the Navier-Stokes equation as

p y i +ρ b i ρ v i t | y k =const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaeyOaIyRaamiCaa qaaiabgkGi2kaadMhadaWgaaWcbaGaamyAaaqabaaaaOGaey4kaSIa eqyWdiNaamOyamaaBaaaleaacaWGPbaabeaakiaaykW7cqGHijYUcq aHbpGCdaabcaqaamaalaaabaGaeyOaIyRaamODamaaBaaaleaacaWG PbaabeaaaOqaaiabgkGi2kaadshaaaaacaGLiWoadaWgaaWcbaGaam yEamaaBaaameaacaWGRbaabeaaliabg2da9iaadogacaWGVbGaamOB aiaadohacaWG0baabeaaaaa@5092@

Taking the time derivative of this expression, and assuming body forces to be independent of time then gives

2 p y i t ρ 2 v i t 2 | y k =const MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTmaalaaabaGaeyOaIy7aaWbaaS qabeaacaaIYaaaaOGaamiCaaqaaiabgkGi2kaadMhadaWgaaWcbaGa amyAaaqabaGccqGHciITcaWG0baaaiabgIKi7kabeg8aYnaaeiaaba WaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG2bWaaSba aSqaaiaadMgaaeqaaaGcbaGaeyOaIyRaamiDamaaCaaaleqabaGaaG OmaaaaaaaakiaawIa7amaaBaaaleaacaWG5bWaaSbaaWqaaiaadUga aeqaaSGaeyypa0Jaam4yaiaad+gacaWGUbGaam4Caiaadshaaeqaaa aa@4F92@

·         Note that

p t = c s 2 δρ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamiCaaqaaiabgk Gi2kaadshaaaGaeyypa0Jaam4yamaaDaaaleaacaWGZbaabaGaaGOm aaaakmaalaaabaGaeyOaIyRaeqiTdqMaeqyWdihabaGaeyOaIyRaam iDaaaaaaa@4045@

and linearizing the mass conservation equation shows that

δρ t | x=const +ρ v i y i =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcqaH0o azcqaHbpGCaeaacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaa hIhacqGH9aqpcaWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccq GHRaWkcqaHbpGCdaWcaaqaaiabgkGi2kaadAhadaWgaaWcbaGaamyA aaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadMgaaeqaaaaaki abg2da9iaaicdacaaMc8oaaa@4D3E@

Combining these results gives the wave equations.

·         If the fluid is at rest at time t=0 its vorticity vanishes.   The vorticity transport equation then shows that the vorticity must vanish at all times, i.e. the flow is irrotational.   The velocity field therefore satisfies curl(v)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacaWG1bGaamOCaiaadYgacaGGOa GaaCODaiaacMcacqGH9aqpcaWHWaaaaa@3849@  and hence we can set v i =ϕ/ y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcqGHciITcqaHvpGzcaGGVaGaeyOaIyRaamyEamaaBaaaleaa caWGPbaabeaaaaa@3AEC@ .

 

 

The wave equations can be solved easily for an infinite region.   Two solutions are:

 

Plane waves.  A plane wave can be visualized as a propagating ‘wall’ of sound.  Let v i = n i f( y k n k c s t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWGUbWaaSbaaSqaaiaadMgaaeqaaOGaamOzaiaacIcacaWG 5bWaaSbaaSqaaiaadUgaaeqaaOGaamOBamaaBaaaleaacaWGRbaabe aakiabgkHiTiaadogadaWgaaWcbaGaam4CaaqabaGccaWG0bGaaiyk aaaa@4021@ , where n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@3275@  is a unit vector, and f  is an arbitrary function.   This corresponds to a plane wave travelling in a direction parallel to the unit vector n i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaamyAaaqabaaaaa@3275@ .  It is easy to show that this velocity field satisfies the wave equation for any f(x).   The density change follows as δρ=(ρ/ c s )f( y k n k c s t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes7aKjabeg8aYjabg2da9iaacIcacq aHbpGCcaGGVaGaam4yamaaBaaaleaacaWGZbaabeaakiaacMcacaWG MbGaaiikaiaadMhadaWgaaWcbaGaam4AaaqabaGccaWGUbWaaSbaaS qaaiaadUgaaeqaaOGaeyOeI0Iaam4yamaaBaaaleaacaWGZbaabeaa kiaadshacaGGPaaaaa@4532@ , and of course the pressure is proportional to the density change.

 

Monopole.    A monopole is a point source of sound.   It would be produced, e.g. by a vanishingly small radially vibrating sphere.  The velocity field has the form

v i = x i 4π r 2 f(r c s t)r= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaaakeaacaaI 0aGaeqiWdaNaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaWGMbGaai ikaiaadkhacqGHsislcaWGJbWaaSbaaSqaaiaadohaaeqaaOGaamiD aiaacMcacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGYbGa eyypa0ZaaOaaaeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEam aaBaaaleaacaWGRbaabeaaaeqaaaaa@75ED@

where f(x) is again an arbitrary function. 

 

 

Dipole.  A dipole is two nearby monopoles emitting equal and opposite signals MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  it would represent, eg, the sound produced by a sphere moving backwards and forwards along a line.   A loudspeaker that is removed from its box behaves somewhat like a dipole.    The displacement field has the form

v i = ( r 2 δ ij x i x j ) n j 4π r 4 f(r c s t)r= x k x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpdaWcaaqaaiaacIcacaWGYbWaaWbaaSqabeaacaaIYaaaaOGa eqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHiTiaadIhada WgaaWcbaGaamyAaaqabaGccaWG4bWaaSbaaSqaaiaadQgaaeqaaOGa aiykaiaad6gadaWgaaWcbaGaamOAaaqabaaakeaacaaI0aGaeqiWda NaamOCamaaCaaaleqabaGaaGinaaaaaaGccaWGMbGaaiikaiaadkha cqGHsislcaWGJbWaaSbaaSqaaiaadohaaeqaaOGaamiDaiaacMcaca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWGYbGaeyypa0ZaaO aaaeaacaWG4bWaaSbaaSqaaiaadUgaaeqaaOGaamiEamaaBaaaleaa caWGRbaabeaaaeqaaaaa@8211@

where n j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaad6gadaWgaaWcbaGaamOAaaqabaaaaa@3276@  is a unit vector parallel to the direction of the dipole (eg the direction of the line along which the sphere moves).

 

7.13.2 Flow behind an accelerating piston

 

This is a classic problem that is often used to illustrate the method of characteristics.   The figure shows a piston in a tube that is filled with an ideal gas with mass density ρ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIWaaabeaaaa a@330E@  and wave speed c 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaaGimaaqabaaaaa@3236@  (it is more convenient to specify the wave speed than the pressure, but of course the two are related).   At time t=0 the gas and piston are at rest.   The piston then starts to move to the left with constant acceleration a.

 

The velocity field in the gas for 2 c 0 /a(γ1)>t>0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaWGJbWaaSbaaSqaaiaaicdaae qaaOGaai4laiaadggacaGGOaGaeq4SdCMaeyOeI0IaaGymaiaacMca cqGH+aGpcaWG0bGaeyOpa4JaaGimaaaa@3D00@  is given by

v={ 1 γ [ at(γ+1) 2 + ( at(γ+1) 2 ) 2 +2yaγ ]y< c 0 t 0y c 0 t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhacqGH9aqpdaGabaqaauaabeqace aaaeaacqGHsisldaWcaaqaaiaaigdaaeaacqaHZoWzaaWaamWaaeaa daWcaaqaaiaadggacaWG0bGaaiikaiabeo7aNjabgUcaRiaaigdaca GGPaaabaGaaGOmaaaacqGHRaWkdaGcaaqaamaabmaabaWaaSaaaeaa caWGHbGaamiDaiaacIcacqaHZoWzcqGHRaWkcaaIXaGaaiykaaqaai aaikdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaaGOmaiaadMhacaWGHbGaeq4SdCgaleqaaaGccaGLBbGaayzxaa GaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caWG5bGaeyipaWJaam4y amaaBaaaleaacaaIWaaabeaakiaadshaaeaacaaIWaGaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaamyEaiabgwMiZkaadogadaWgaaWcbaGaaGimaa qabaGccaWG0baaaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7aiaawUhaaaaa@01E2@

The wave speed can be calculated from the condition

c= c 0 +( γ1 )v/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacqGH9aqpcaWGJbWaaSbaaSqaai aaicdaaeqaaOGaey4kaSYaaeWaaeaacqaHZoWzcqGHsislcaaIXaaa caGLOaGaayzkaaGaamODaiaac+cacaaIYaaaaa@3C52@

 

These results can be derived as follows. For one-dimensional flow in the y direction, the Navier-Stokes and mass conservation equations reduce to

v t | y=const +v v y + c s 2 ρ ρ y =0 ρ t | y=const +v ρ y +ρ v y =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaeiaabaWaaSaaaeaacqGHciITcaWG2b aabaGaeyOaIyRaamiDaaaaaiaawIa7amaaBaaaleaacaWH5bGaeyyp a0Jaam4yaiaad+gacaWGUbGaam4CaiaadshaaeqaaOGaey4kaSIaam ODamaalaaabaGaeyOaIyRaamODaaqaaiabgkGi2kaadMhaaaGaey4k aSYaaSaaaeaacaWGJbWaa0baaSqaaiaadohaaeaacaaIYaaaaaGcba GaeqyWdihaamaalaaabaGaeyOaIyRaeqyWdihabaGaeyOaIyRaamyE aaaacqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVpaaeiaabaWaaSaaaeaacqGHciITcqaHbpGCae aacqGHciITcaWG0baaaaGaayjcSdWaaSbaaSqaaiaahMhacqGH9aqp caWGJbGaam4Baiaad6gacaWGZbGaamiDaaqabaGccqGHRaWkcaWG2b WaaSaaaeaacqGHciITcqaHbpGCaeaacqGHciITcaWG5baaaiabgUca Riabeg8aYnaalaaabaGaeyOaIyRaamODaaqaaiabgkGi2kaadMhaaa Gaeyypa0JaaGimaaaa@88E4@

where c s = k ρ γ1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaam4CaaqabaGccq GH9aqpdaGcaaqaaiaadUgacqaHbpGCdaahaaWcbeqaaiabeo7aNjab gkHiTiaaigdaaaaabeaaaaa@39C0@  is the sound speed.  Solving this equation is a bit above our pay-grade in this course, but you are probably covering this material in math, so we’ll work through it anyway. The first step is to write the equations in matrix form

q i t + A ij q j y =0q=[ v ρ ]A=[ v c s 2 /ρ ρ v ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaeyOaIyRaamyCamaaBaaale aacaWGPbaabeaaaOqaaiabgkGi2kaadshaaaGaey4kaSIaamyqamaa BaaaleaacaWGPbGaamOAaaqabaGcdaWcaaqaaiabgkGi2kaadghada WgaaWcbaGaamOAaaqabaaakeaacqGHciITcaWG5baaaiabg2da9iaa icdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaCyCaiabg2da9maadmaabaqbaeqabiqaaaqa aiaadAhaaeaacqaHbpGCaaaacaGLBbGaayzxaaGaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWHbbGaeyypa0ZaamWaaeaafaqabeGacaaabaGaamOD aaqaaiaadogadaqhaaWcbaGaam4CaaqaaiaaikdaaaGccaGGVaGaeq yWdihabaGaeqyWdihabaGaamODaaaaaiaawUfacaGLDbaaaaa@8399@

Now, let λ, r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabgYcaSiaadkhadaWgaaWcba GaamyAaaqabaaaaa@3511@  be a left eigenvalue/eigenvector pair of A.  The eigenvalues/vectors are easily shown to be (v+ c s ),[ρ, c s ];(v c s ),[ρ, c s ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG2bGaey4kaSIaam4yamaaBa aaleaacaWGZbaabeaakiaacMcacaGGSaGaai4waiabeg8aYjaacYca caWGJbWaaSbaaSqaaiaadohaaeqaaOGaaiyxaiaaykW7caGG7aGaai ikaiaadAhacqGHsislcaWGJbWaaSbaaSqaaiaadohaaeqaaOGaaiyk aiaacYcacaGGBbGaeqyWdiNaaiilaiabgkHiTiaadogadaWgaaWcba Gaam4CaaqabaGccaGGDbaaaa@4C2E@ . It follows that

r i ( q i t +λ q j y )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhadaWgaaWcbaGaamyAaaqabaGcda qadaqaamaalaaabaGaeyOaIyRaamyCamaaBaaaleaacaWGPbaabeaa aOqaaiabgkGi2kaadshaaaGaey4kaSIaeq4UdW2aaSaaaeaacqGHci ITcaWGXbWaaSbaaSqaaiaadQgaaeqaaaGcbaGaeyOaIyRaamyEaaaa aiaawIcacaGLPaaacqGH9aqpcaaIWaGaaGPaVdaa@45D1@

If we set λ=y/t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSjabg2da9iabgkGi2kaadMhaca GGVaGaeyOaIyRaamiDaaaa@3898@  we see further that

r i q i t =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhadaWgaaWcbaGaamyAaaqabaGcda WcaaqaaiabgkGi2kaadghadaWgaaWcbaGaamyAaaqabaaakeaacqGH ciITcaWG0baaaiabg2da9iaaicdaaaa@3A32@

along characteristic lines such that y/t=v± c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadMhacaGGVaGaeyOaIyRaam iDaiabg2da9iaadAhacqGHXcqScaWGJbWaaSbaaSqaaiaadohaaeqa aaaa@3BD9@ .  Substituting for r i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadkhadaWgaaWcbaGaamyAaaqabaaaaa@3279@ , q i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadghadaWgaaWcbaGaamyAaaqabaaaaa@3278@ , we see that

ρ v t ± c s ρ t =0v± c s ρ dρ=constantv± 2 γ1 c s =constant MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaalaaabaGaeyOaIyRaamODaa qaaiabgkGi2kaadshaaaGaeyySaeRaam4yamaaBaaaleaacaWGZbaa beaakmaalaaabaGaeyOaIyRaeqyWdihabaGaeyOaIyRaamiDaaaacq GH9aqpcaaIWaGaeyO0H4TaamODaiabgglaXoaapeaabaWaaSaaaeaa caWGJbWaaSbaaSqaaiaadohaaeqaaaGcbaGaeqyWdihaaiaadsgacq aHbpGCcqGH9aqpcaqGJbGaae4Baiaab6gacaqGZbGaaeiDaiaabgga caqGUbGaaeiDaiabgkDiElaadAhacqGHXcqSdaWcaaqaaiaaikdaae aacqaHZoWzcqGHsislcaaIXaaaaiaadogadaWgaaWcbaGaam4Caaqa baaabeqab0Gaey4kIipakiabg2da9iaabogacaqGVbGaaeOBaiaabo hacaqG0bGaaeyyaiaab6gacaqG0baaaa@6C9D@

These conditions hold along lines satisfying y/t=v± c s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadMhacaGGVaGaeyOaIyRaam iDaiabg2da9iaadAhacqGHXcqScaWGJbWaaSbaaSqaaiaadohaaeqa aaaa@3BD9@

 

We can now apply this result to the piston problem. 

·         The (t,y) characteristic diagram is shown in the figure.  The red characteristic lines have slope y/t=vc MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadMhacaGGVaGaeyOaIyRaam iDaiabg2da9iaadAhacqGHsislcaWGJbaaaa@39B4@ ; the green ones have slope  y/t=v+c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadMhacaGGVaGaeyOaIyRaam iDaiabg2da9iaadAhacqGHRaWkcaWGJbaaaa@39A9@  (we drop the subscript s for clarity)

·         Note that the red lines terminate in a region where the gas is stationary, and the wave speed is c 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaWgaaWcbaGaaGimaaqabaaaaa@3236@ .  Therefore v2c/(γ1)=2 c 0 /(γ1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhacqGHsislcaaIYaGaam4yaiaac+ cacaGGOaGaeq4SdCMaeyOeI0IaaGymaiaacMcacqGH9aqpcqGHsisl caaIYaGaam4yamaaBaaaleaacaaIWaaabeaakiaac+cacaGGOaGaeq 4SdCMaeyOeI0IaaGymaiaacMcaaaa@4331@  along these lines.  It follows that the wave speed in the moving region of the gas is related to its velocity by c= c 0 +(γ1)v/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacqGH9aqpcaWGJbWaaSbaaSqaai aaicdaaeqaaOGaey4kaSIaaiikaiabeo7aNjabgkHiTiaaigdacaGG PaGaamODaiaac+cacaaIYaaaaa@3C22@

·         For the green lines, we know v+2c/(γ1)=B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhacqGHRaWkcaaIYaGaam4yaiaac+ cacaGGOaGaeq4SdCMaeyOeI0IaaGymaiaacMcacqGH9aqpcaWGcbaa aa@3B11@ , with B some constant.   Since each green lines intersects a red one, we already know the wave speed in terms of v, and substituting for c shows that 2v+2 c 0 /(γ1)=B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaaikdacaWG2bGaey4kaSIaaGOmaiaado gadaWgaaWcbaGaaGimaaqabaGccaGGVaGaaiikaiabeo7aNjabgkHi TiaaigdacaGGPaGaeyypa0JaamOqaaaa@3CBD@ .   It follows that v must be constant on the green lines, and therefore they have constant slope y/t=v+c=v(γ+1)/2+ c 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadMhacaGGVaGaeyOaIyRaam iDaiabg2da9iaadAhacqGHRaWkcaWGJbGaeyypa0JaamODaiaacIca cqaHZoWzcqGHRaWkcaaIXaGaaiykaiaac+cacaaIYaGaey4kaSIaam 4yamaaBaaaleaacaaIWaaabeaaaaa@4466@ .  We can apply this at the point where the green line intersects the piston (the gas velocity there is equal to that of the piston) to conclude that y/t=a t 0 (γ+1)/2+ c 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkGi2kaadMhacaGGVaGaeyOaIyRaam iDaiabg2da9iabgkHiTiaadggacaWG0bWaaSbaaSqaaiaaicdaaeqa aOGaaiikaiabeo7aNjabgUcaRiaaigdacaGGPaGaai4laiaaikdacq GHRaWkcaWGJbWaaSbaaSqaaiaaicdaaeqaaaaa@435C@

·         Integrating, we get y=a t 0 (γ+1)t/2+ c 0 t+D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhacqGH9aqpcqGHsislcaWGHbGaam iDamaaBaaaleaacaaIWaaabeaakiaacIcacqaHZoWzcqGHRaWkcaaI XaGaaiykaiaadshacaGGVaGaaGOmaiabgUcaRiaadogadaWgaaWcba GaaGimaaqabaGccaWG0bGaey4kaSIaamiraaaa@428B@  where D is a  constant of integration MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  and since we know y=a t 0 2 /2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhacqGH9aqpcqGHsislcaWGHbGaam iDamaaDaaaleaacaaIWaaabaGaaGOmaaaakiaac+cacaaIYaaaaa@3854@  at time t= t 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaWG0bWaaSbaaSqaai aaicdaaeqaaaaa@3446@  we can solve for D and find that y=aγ t 0 2 /2a t 0 (γ+1)t/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhacqGH9aqpcaWGHbGaeq4SdCMaam iDamaaDaaaleaacaaIWaaabaGaaGOmaaaakiaac+cacaaIYaGaeyOe I0IaamyyaiaadshadaWgaaWcbaGaaGimaaqabaGccaGGOaGaeq4SdC Maey4kaSIaaGymaiaacMcacaWG0bGaai4laiaaikdaaaa@43CF@

·         Finally, we can use this result to solve for the value of t 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaWgaaWcbaGaaGimaaqabaaaaa@3247@  corresponding to a general point y at time t in the tube.  The solution is

t 0 = 1 aγ [ at(γ+1) 2 + ( at(γ+1) 2 ) 2 +2yaγ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshadaWgaaWcbaGaaGimaaqabaGccq GH9aqpdaWcaaqaaiaaigdaaeaacaWGHbGaeq4SdCgaamaadmaabaWa aSaaaeaacaWGHbGaamiDaiaacIcacqaHZoWzcqGHRaWkcaaIXaGaai ykaaqaaiaaikdaaaGaey4kaSYaaOaaaeaadaqadaqaamaalaaabaGa amyyaiaadshacaGGOaGaeq4SdCMaey4kaSIaaGymaiaacMcaaeaaca aIYaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgUca RiaaikdacaWG5bGaamyyaiabeo7aNbWcbeaaaOGaay5waiaaw2faaa aa@4FDD@

Finally, the velocity at time t of the point y in the tube follows as v=a t 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAhacqGH9aqpcqGHsislcaWGHbGaam iDamaaBaaaleaacaaIWaaabeaaaaa@361B@  giving the solution stated.

·         The wave speed follows from c= c 0 +(γ1)v/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogacqGH9aqpcaWGJbWaaSbaaSqaai aaicdaaeqaaOGaey4kaSIaaiikaiabeo7aNjabgkHiTiaaigdacaGG PaGaamODaiaac+cacaaIYaaaaa@3C22@ .   Note that at time t=2 c 0 /a(γ1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadshacqGH9aqpcaaIYaGaam4yamaaBa aaleaacaaIWaaabeaakiaac+cacaWGHbGaaiikaiabeo7aNjabgkHi TiaaigdacaGGPaaaaa@3B3C@  the wave speed drops to zero at the piston.  The wave speed can’t be negative, so after this time the piston separates from the gas and a vacuum develops between the piston and the advancing gas.  After this time the gas/vacuum interface moves with speed 2 c 0 /(γ1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgkHiTiaaikdacaWGJbWaaSbaaSqaai aaicdaaeqaaOGaai4laiaacIcacqaHZoWzcqGHsislcaaIXaGaaiyk aaaa@3944@ .

 

 

7.13.3 Stationary Normal Shock in an ideal gas

 

Shock waves are surfaces in a compressible fluid or gas across which fluid properties and state experience a sharp discontinuity.   They occur between regions where flow is supersonic and subsonic.   The discontinuities across a shock are not arbitrary MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  they are related by the usual conservation laws and constitutive equations for a fluid.  

 

Consider a stationary shock that separates regions 1 and 2 of a flow.  For simplicity, assume that fluid flows in a direction perpendicular to the plane of the shock. Let ( v 1 , p 1 , q 1 , ρ 1 , ε 1 , s 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG2bWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaadchadaWgaaWcbaGaaGymaaqabaGccaGGSaGaamyC amaaBaaaleaacaaIXaaabeaakiaacYcacqaHbpGCdaWgaaWcbaGaaG ymaaqabaGccaGGSaGaeqyTdu2aaSbaaSqaaiaaigdaaeqaaOGaaiil aiaadohadaWgaaWcbaGaaGymaaqabaGccaGGPaaaaa@421C@ , ( v 2 , p 2 , q 2 , ρ 2 , ε 2 , s 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWG2bWaaSbaaSqaaiaaikdaae qaaOGaaiilaiaadchadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyC amaaBaaaleaacaaIYaaabeaakiaacYcacqaHbpGCdaWgaaWcbaGaaG OmaaqabaGccaGGSaGaeqyTdu2aaSbaaSqaaiaaikdaaeqaaOGaaiil aiaadohadaWgaaWcbaGaaGOmaaqabaGccaGGPaaaaa@4222@  denote the (uniform) velocity, pressure, heat flux, density , specific internal energy and specific entropy just adjacent to the two sides of the shock.

 

Conservation and thermodynamic laws relating these quantities have the following form:

 

* Mass Balance ρ 1 v 1 = ρ 2 v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaki aadAhadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqaHbpGCdaWgaaWc baGaaGOmaaqabaGccaWG2bWaaSbaaSqaaiaaikdaaeqaaaaa@3AA0@

* Linear Momentum Balance p 1 + ρ 1 v 1 2 = p 2 + ρ 2 v 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaGccq GHRaWkcqaHbpGCdaWgaaWcbaGaaGymaaqabaGccaWG2bWaa0baaSqa aiaaigdaaeaacaaIYaaaaOGaeyypa0JaamiCamaaBaaaleaacaaIYa aabeaakiabgUcaRiabeg8aYnaaBaaaleaacaaIYaaabeaakiaadAha daqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaa@41AB@

* Energy Conservation (1st law)  p 1 v 1 + ρ 1 v 1 ( ε 1 + 1 2 v 1 2 )+ q 1 = p 2 v 2 + ρ 2 v 2 ( ε 2 + 1 2 v 2 2 )+ q 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaGcca WG2bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqyWdi3aaSbaaSqa aiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakmaabmaaba GaeqyTdu2aaSbaaSqaaiaaigdaaeqaaOGaey4kaSYaaSaaaeaacaaI XaaabaGaaGOmaaaacaWG2bWaa0baaSqaaiaaigdaaeaacaaIYaaaaa GccaGLOaGaayzkaaGaey4kaSIaamyCamaaBaaaleaacaaIXaaabeaa kiabg2da9iaadchadaWgaaWcbaGaaGOmaaqabaGccaWG2bWaaSbaaS qaaiaaikdaaeqaaOGaey4kaSIaeqyWdi3aaSbaaSqaaiaaikdaaeqa aOGaamODamaaBaaaleaacaaIYaaabeaakmaabmaabaGaeqyTdu2aaS baaSqaaiaaikdaaeqaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGOm aaaacaWG2bWaa0baaSqaaiaaikdaaeaacaaIYaaaaaGccaGLOaGaay zkaaGaey4kaSIaamyCamaaBaaaleaacaaIYaaabeaaaaa@5C05@

* Entropy Inequality (2nd  law)  ( ρ 2 s 2 v 2 + q 2 θ 1 )( ρ 1 s 1 v 1 + q 1 θ 1 )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaeqyWdi3aaSbaaSqaaiaaik daaeqaaOGaam4CamaaBaaaleaacaaIYaaabeaakiaadAhadaWgaaWc baGaaGOmaaqabaGccqGHRaWkdaWcaaqaaiaadghadaWgaaWcbaGaaG OmaaqabaaakeaacqaH4oqCdaWgaaWcbaGaaGymaaqabaaaaaGccaGL OaGaayzkaaGaeyOeI0YaaeWaaeaacqaHbpGCdaWgaaWcbaGaaGymaa qabaGccaWGZbWaaSbaaSqaaiaaigdaaeqaaOGaamODamaaBaaaleaa caaIXaaabeaakiabgUcaRmaalaaabaGaamyCamaaBaaaleaacaaIXa aabeaaaOqaaiabeI7aXnaaBaaaleaacaaIXaaabeaaaaaakiaawIca caGLPaaacqGHLjYScaaIWaaaaa@4EF7@

 

 

For an ideal gas, with negligible heat flow, these can be combined to yield several additional relations across a normal shock.

 

* Rankine Hugoniot Relations

ε 1 ε 2 = 1 2 ( 1 ρ 2 1 ρ 1 )( p 1 + p 2 ) λ 2 ( p 1 ρ 1 p 2 ρ 2 ) p 1 ρ 2 + p 2 ρ 1 =0 λ 2 = γ1 γ+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIXaaabeaaki abgkHiTiabew7aLnaaBaaaleaacaaIYaaabeaakiabg2da9maalaaa baGaaGymaaqaaiaaikdaaaWaaeWaaeaadaWcaaqaaiaaigdaaeaacq aHbpGCdaWgaaWcbaGaaGOmaaqabaaaaOGaeyOeI0YaaSaaaeaacaaI XaaabaGaeqyWdi3aaSbaaSqaaiaaigdaaeqaaaaaaOGaayjkaiaawM caaiaacIcacaWGWbWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaamiC amaaBaaaleaacaaIYaaabeaakiaacMcacqGHshI3cqaH7oaBdaahaa WcbeqaaiaaikdaaaGcdaqadaqaamaalaaabaGaamiCamaaBaaaleaa caaIXaaabeaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaaaGccq GHsisldaWcaaqaaiaadchadaWgaaWcbaGaaGOmaaqabaaakeaacqaH bpGCdaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGaeyOeI0 YaaSaaaeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeqyWdi3a aSbaaSqaaiaaikdaaeqaaaaakiabgUcaRmaalaaabaGaamiCamaaBa aaleaacaaIYaaabeaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaa aaGccqGH9aqpcaaIWaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlabeU7aSnaaCaaaleqabaGaaGOmaaaakiabg2da9maa laaabaGaeq4SdCMaeyOeI0IaaGymaaqaaiabeo7aNjabgUcaRiaaig daaaaaaa@7B50@

 

* Prandtl’s Relation

v 1 v 2 = c * 2 c * 2 =(1 λ 2 ) c 1 2 + λ 2 v 1 2 =(1 λ 2 ) c 2 2 + λ 2 u 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamODamaaBaaaleaacaaIXaaabe aakiaadAhadaWgaaWcbaGaaGOmaaqabaGccqGH9aqpcaWGJbWaa0ba aSqaaiaacQcaaeaacaaIYaaaaaGcbaGaam4yamaaDaaaleaacaGGQa aabaGaaGOmaaaakiabg2da9iaacIcacaaIXaGaeyOeI0Iaeq4UdW2a aWbaaSqabeaacaaIYaaaaOGaaiykaiaadogadaqhaaWcbaGaaGymaa qaaiaaikdaaaGccqGHRaWkcqaH7oaBdaahaaWcbeqaaiaaikdaaaGc caWG2bWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaeyypa0Jaaiikai aaigdacqGHsislcqaH7oaBdaahaaWcbeqaaiaaikdaaaGccaGGPaGa am4yamaaDaaaleaacaaIYaaabaGaaGOmaaaakiabgUcaRiabeU7aSn aaCaaaleqabaGaaGOmaaaakiaadwhadaqhaaWcbaGaaGOmaaqaaiaa ikdaaaaaaaa@5943@

Prandtl’s relation implies that flow must be supersonic on one side of the shock and subsonic on the other. In addition, to satisfy the entropy inequality, the supersonic flow must be upstream of the shock.

 

 

These results can be derived as follows:  Begin by recalling the conservation laws for a control volume

Mass Conservation: d dt R ρdV + B ρvndA =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaamizaiaadAfaaSqaaiaadkfaaeqaniab gUIiYdGccqGHRaWkdaWdrbqaaiabeg8aYjaahAhacqGHflY1caWHUb GaamizaiaadgeaaSqaaiaadkeaaeqaniabgUIiYdGccqGH9aqpcaaI Waaaaa@475E@

Linear Momentum Balance B nσdA + R ρbdV = d dt R ρv dV+ B (ρv)vndA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaCOBaiabgwSixlaaho8aca WGKbGaamyqaaWcbaGaamOqaaqab0Gaey4kIipakiabgUcaRmaapefa baGaeqyWdiNaaCOyaiaadsgacaWGwbaaleaacaWGsbaabeqdcqGHRi I8aOGaeyypa0ZaaSaaaeaacaWGKbaabaGaamizaiaadshaaaWaa8qu aeaacqaHbpGCcaWH2baaleaacaWGsbaabeqdcqGHRiI8aOGaamizai aadAfacqGHRaWkdaWdrbqaaiaacIcacqaHbpGCcaWH2bGaaiykaiaa hAhacqGHflY1caWHUbGaamizaiaadgeaaSqaaiaadkeaaeqaniabgU IiYdaaaa@5BC7@

First law of thermodynamics

B (nσ)v dA+ R bv dV B qndA + V qdV = d dt R ρ( ε+ 1 2 vv )dV + B ρ( ε+ 1 2 vv )vn dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaapefabaGaaiikaiaah6gacqGHflY1ca WHdpGaaiykaiabgwSixlaahAhaaSqaaiaadkeaaeqaniabgUIiYdGc caWGKbGaamyqaiabgUcaRmaapefabaGaaCOyaiabgwSixlaahAhaaS qaaiaadkfaaeqaniabgUIiYdGccaWGKbGaamOvaiabgkHiTmaapefa baGaaCyCaiabgwSixlaah6gacaWGKbGaamyqaaWcbaGaamOqaaqab0 Gaey4kIipakiabgUcaRmaapefabaGaamyCaiaadsgacaWGwbaaleaa caWGwbaabeqdcqGHRiI8aOGaeyypa0ZaaSaaaeaacaWGKbaabaGaam izaiaadshaaaWaa8quaeaacqaHbpGCdaqadaqaaiabew7aLjabgUca RmaalaaabaGaaGymaaqaaiaaikdaaaGaaCODaiabgwSixlaahAhaai aawIcacaGLPaaacaWGKbGaamOvaaWcbaGaamOuaaqab0Gaey4kIipa kiabgUcaRmaapefabaGaeqyWdi3aaeWaaeaacqaH1oqzcqGHRaWkda WcaaqaaiaaigdaaeaacaaIYaaaaiaahAhacqGHflY1caWH2baacaGL OaGaayzkaaGaaCODaiabgwSixlaah6gaaSqaaiaadkeaaeqaniabgU IiYdGccaWGKbGaamyqaaaa@8324@

 

Second law of thermodynamics

d dt R ρsdV + B ρs(vn)dA + B qn θ dA R q θ dV 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaalaaabaGaamizaaqaaiaadsgacaWG0b aaamaapefabaGaeqyWdiNaam4CaiaadsgacaWGwbaaleaacaWGsbaa beqdcqGHRiI8aOGaey4kaSYaa8quaeaacqaHbpGCcaWGZbGaaiikai aahAhacqGHflY1caWHUbGaaiykaiaadsgacaWGbbaaleaacaWGcbaa beqdcqGHRiI8aOGaey4kaSYaa8quaeaadaWcaaqaaiaahghacqGHfl Y1caWHUbaabaGaeqiUdehaaiaadsgacaWGbbaaleaacaWGcbaabeqd cqGHRiI8aOGaeyOeI0Yaa8quaeaadaWcaaqaaiaadghaaeaacqaH4o qCaaGaamizaiaadAfaaSqaaiaadkfaaeqaniabgUIiYdGccqGHLjYS caaIWaaaaa@5FAC@

Then, introduce a control volume consisting of a narrow rectangular region enclosing the shock.   The thickness can be reduced to zero, so the volume integrals can be made arbitrarily small.  In addition, recall that the stress in an ideal gas is hydrostatic.  Finally, evaluating the separate integrals over the two external surfaces of the control volume and localizing the integrals leads to the conservation relations.

 

The Rankine-Hugoniot relations can be derived from the energy conservation equation

p 1 v 1 + ρ 1 v 1 ( ε 1 + 1 2 v 1 2 )= p 2 v 2 + ρ 2 v 2 ( ε 2 + 1 2 v 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadchadaWgaaWcbaGaaGymaaqabaGcca WG2bWaaSbaaSqaaiaaigdaaeqaaOGaey4kaSIaeqyWdi3aaSbaaSqa aiaaigdaaeqaaOGaamODamaaBaaaleaacaaIXaaabeaakmaabmaaba GaeqyTdu2aaSbaaSqaaiaaigdaaeqaaOGaey4kaSYaaSaaaeaacaaI XaaabaGaaGOmaaaacaWG2bWaa0baaSqaaiaaigdaaeaacaaIYaaaaa GccaGLOaGaayzkaaGaeyypa0JaamiCamaaBaaaleaacaaIYaaabeaa kiaadAhadaWgaaWcbaGaaGOmaaqabaGccqGHRaWkcqaHbpGCdaWgaa WcbaGaaGOmaaqabaGccaWG2bWaaSbaaSqaaiaaikdaaeqaaOWaaeWa aeaacqaH1oqzdaWgaaWcbaGaaGOmaaqabaGccqGHRaWkdaWcaaqaai aaigdaaeaacaaIYaaaaiaadAhadaqhaaWcbaGaaGOmaaqaaiaaikda aaaakiaawIcacaGLPaaaaaa@567C@

Recall that ρ 1 v 1 = ρ 2 v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaki aadAhadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqaHbpGCdaWgaaWc baGaaGOmaaqabaGccaWG2bWaaSbaaSqaaiaaikdaaeqaaaaa@3AA0@ , so this equation can be rearranged as

ε 2 ε 1 = p 1 ρ 1 p 2 ρ 2 + 1 2 v 1 2 1 2 v 2 2 = p 1 ρ 1 p 2 ρ 2 + 1 2 ρ 1 2 v 1 2 ( 1 ρ 1 2 1 ρ 2 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIYaaabeaaki abgkHiTiabew7aLnaaBaaaleaacaaIXaaabeaakiabg2da9maalaaa baGaamiCamaaBaaaleaacaaIXaaabeaaaOqaaiabeg8aYnaaBaaale aacaaIXaaabeaaaaGccqGHsisldaWcaaqaaiaadchadaWgaaWcbaGa aGOmaaqabaaakeaacqaHbpGCdaWgaaWcbaGaaGOmaaqabaaaaOGaey 4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaacaWG2bWaa0baaSqaaiaa igdaaeaacaaIYaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaGOmaa aacaWG2bWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaeyypa0ZaaSaa aeaacaWGWbWaaSbaaSqaaiaaigdaaeqaaaGcbaGaeqyWdi3aaSbaaS qaaiaaigdaaeqaaaaakiabgkHiTmaalaaabaGaamiCamaaBaaaleaa caaIYaaabeaaaOqaaiabeg8aYnaaBaaaleaacaaIYaaabeaaaaGccq GHRaWkdaWcaaqaaiaaigdaaeaacaaIYaaaaiabeg8aYnaaDaaaleaa caaIXaaabaGaaGOmaaaakiaadAhadaqhaaWcbaGaaGymaaqaaiaaik daaaGcdaqadaqaamaalaaabaGaaGymaaqaaiabeg8aYnaaDaaaleaa caaIXaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaaeaacq aHbpGCdaqhaaWcbaGaaGOmaaqaaiaaikdaaaaaaaGccaGLOaGaayzk aaaaaa@6AA5@

The momentum conservation equation and mass conservation can also be combined to see that

ρ 1 2 v 1 2 ( 1 ρ 1 1 ρ 2 )= p 2 p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaDaaaleaacaaIXaaabaGaaG OmaaaakiaadAhadaqhaaWcbaGaaGymaaqaaiaaikdaaaGcdaqadaqa amaalaaabaGaaGymaaqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaaa GccqGHsisldaWcaaqaaiaaigdaaeaacqaHbpGCdaWgaaWcbaGaaGOm aaqabaaaaaGccaGLOaGaayzkaaGaeyypa0JaamiCamaaBaaaleaaca aIYaaabeaakiabgkHiTiaadchadaWgaaWcbaGaaGymaaqabaaaaa@45A4@

and hence eliminating the velocity term from the energy equation

ε 2 ε 1 = p 1 ρ 1 p 2 ρ 2 + 1 2 ( p 2 p 1 )( 1 ρ 1 + 1 ρ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIYaaabeaaki abgkHiTiabew7aLnaaBaaaleaacaaIXaaabeaakiabg2da9maalaaa baGaamiCamaaBaaaleaacaaIXaaabeaaaOqaaiabeg8aYnaaBaaale aacaaIXaaabeaaaaGccqGHsisldaWcaaqaaiaadchadaWgaaWcbaGa aGOmaaqabaaakeaacqaHbpGCdaWgaaWcbaGaaGOmaaqabaaaaOGaey 4kaSYaaSaaaeaacaaIXaaabaGaaGOmaaaacaGGOaGaamiCamaaBaaa leaacaaIYaaabeaakiabgkHiTiaadchadaWgaaWcbaGaaGymaaqaba GccaGGPaWaaeWaaeaadaWcaaqaaiaaigdaaeaacqaHbpGCdaWgaaWc baGaaGymaaqabaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeqyWdi 3aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaaa@53A9@

which can be rearranged to give the Rankine Hugoniot relation.   The second identity follows by substituting ε= p (γ1)ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLjabg2da9maalaaabaGaamiCaa qaaiaacIcacqaHZoWzcqGHsislcaaIXaGaaiykaiabeg8aYbaacaaM c8UaaGPaVdaa@3D98@  for an ideal gas.

 

To show Prandtl’s relation recall that

c 2 =γ p ρ ε+ p ρ = 1 λ 2 2 λ 2 c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaGccq GH9aqpcqaHZoWzdaWcaaqaaiaadchaaeaacqaHbpGCaaGaeyO0H4Ta eqyTduMaey4kaSYaaSaaaeaacaWGWbaabaGaeqyWdihaaiabg2da9m aalaaabaGaaGymaiabgkHiTiabeU7aSnaaCaaaleqabaGaaGOmaaaa aOqaaiaaikdacqaH7oaBdaahaaWcbeqaaiaaikdaaaaaaOGaam4yam aaCaaaleqabaGaaGOmaaaaaaa@49F9@

where λ 2 = γ1 γ+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeU7aSnaaCaaaleqabaGaaGOmaaaaki abg2da9maalaaabaGaeq4SdCMaeyOeI0IaaGymaaqaaiabeo7aNjab gUcaRiaaigdaaaaaaa@3AB8@ .   The energy equation can then be re-written as

ε 2 + p 2 ρ 2 + 1 2 v 2 2 = ε 1 + p 1 ρ 1 + 1 2 v 1 2 1 λ 2 2 λ 2 c 2 2 + 1 2 v 2 2 = 1 λ 2 2 λ 2 c 1 2 + 1 2 v 1 2 = c * 2 2 λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabew7aLnaaBaaaleaacaaIYaaabeaaki abgUcaRmaalaaabaGaamiCamaaBaaaleaacaaIYaaabeaaaOqaaiab eg8aYnaaBaaaleaacaaIYaaabeaaaaGccqGHRaWkdaWcaaqaaiaaig daaeaacaaIYaaaaiaadAhadaqhaaWcbaGaaGOmaaqaaiaaikdaaaGc cqGH9aqpcqaH1oqzdaWgaaWcbaGaaGymaaqabaGccqGHRaWkdaWcaa qaaiaadchadaWgaaWcbaGaaGymaaqabaaakeaacqaHbpGCdaWgaaWc baGaaGymaaqabaaaaOGaeyOeI0Iaey4kaSYaaSaaaeaacaaIXaaaba GaaGOmaaaacaWG2bWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaeyO0 H49aaSaaaeaacaaIXaGaeyOeI0Iaeq4UdW2aaWbaaSqabeaacaaIYa aaaaGcbaGaaGOmaiabeU7aSnaaCaaaleqabaGaaGOmaaaaaaGccaWG JbWaa0baaSqaaiaaikdaaeaacaaIYaaaaOGaey4kaSYaaSaaaeaaca aIXaaabaGaaGOmaaaacaWG2bWaa0baaSqaaiaaikdaaeaacaaIYaaa aOGaeyypa0ZaaSaaaeaacaaIXaGaeyOeI0Iaeq4UdW2aaWbaaSqabe aacaaIYaaaaaGcbaGaaGOmaiabeU7aSnaaCaaaleqabaGaaGOmaaaa aaGccaWGJbWaa0baaSqaaiaaigdaaeaacaaIYaaaaOGaey4kaSYaaS aaaeaacaaIXaaabaGaaGOmaaaacaWG2bWaa0baaSqaaiaaigdaaeaa caaIYaaaaOGaeyypa0ZaaSaaaeaacaWGJbWaa0baaSqaaiaacQcaae aacaaIYaaaaaGcbaGaaGOmaiabeU7aSnaaCaaaleqabaGaaGOmaaaa aaaaaa@75F6@

Which then gives the first relation.   To show the second note that

c 2 = (1+ λ 2 ) (1 λ 2 ) p ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadogadaahaaWcbeqaaiaaikdaaaGccq GH9aqpdaWcaaqaaiaacIcacaaIXaGaey4kaSIaeq4UdW2aaWbaaSqa beaacaaIYaaaaOGaaiykaaqaaiaacIcacaaIXaGaeyOeI0Iaeq4UdW 2aaWbaaSqabeaacaaIYaaaaOGaaiykaaaadaWcaaqaaiaadchaaeaa cqaHbpGCaaaaaa@4163@

We can then use the first Prandtl relation to see that

(1+ λ 2 ) p 2 + λ 2 ρ 2 v 2 2 = ρ 2 c * 2 (1+ λ 2 ) p 1 + λ 2 ρ 1 v 1 2 = ρ 1 c * 2 (1+ λ 2 )( p 2 p 1 )+ λ 2 ρ 1 v 1 ρ 2 v 2 ( 1 ρ 2 1 ρ 1 )= c * 2 ( ρ 2 ρ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaaiikaiaaigdacqGHRaWkcqaH7o aBdaahaaWcbeqaaiaaikdaaaGccaGGPaGaamiCamaaBaaaleaacaaI YaaabeaakiabgUcaRiabeU7aSnaaCaaaleqabaGaaGOmaaaakiabeg 8aYnaaBaaaleaacaaIYaaabeaakiaadAhadaqhaaWcbaGaaGOmaaqa aiaaikdaaaGccqGH9aqpcqaHbpGCdaWgaaWcbaGaaGOmaaqabaGcca WGJbWaa0baaSqaaiaacQcaaeaacaaIYaaaaaGcbaGaaiikaiaaigda cqGHRaWkcqaH7oaBdaahaaWcbeqaaiaaikdaaaGccaGGPaGaamiCam aaBaaaleaacaaIXaaabeaakiabgUcaRiabeU7aSnaaCaaaleqabaGa aGOmaaaakiabeg8aYnaaBaaaleaacaaIXaaabeaakiaadAhadaqhaa WcbaGaaGymaaqaaiaaikdaaaGccqGH9aqpcqaHbpGCdaWgaaWcbaGa aGymaaqabaGccaWGJbWaa0baaSqaaiaacQcaaeaacaaIYaaaaaGcba GaeyO0H4TaaiikaiaaigdacqGHRaWkcqaH7oaBdaahaaWcbeqaaiaa ikdaaaGccaGGPaGaaiikaiaadchadaWgaaWcbaGaaGOmaaqabaGccq GHsislcaWGWbWaaSbaaSqaaiaaigdaaeqaaOGaaiykaiabgUcaRiab eU7aSnaaCaaaleqabaGaaGOmaaaakiabeg8aYnaaBaaaleaacaaIXa aabeaakiaadAhadaWgaaWcbaGaaGymaaqabaGccqaHbpGCdaWgaaWc baGaaGOmaaqabaGccaWG2bWaaSbaaSqaaiaaikdaaeqaaOWaaeWaae aadaWcaaqaaiaaigdaaeaacqaHbpGCdaWgaaWcbaGaaGOmaaqabaaa aOGaeyOeI0YaaSaaaeaacaaIXaaabaGaeqyWdi3aaSbaaSqaaiaaig daaeqaaaaaaOGaayjkaiaawMcaaiabg2da9iaadogadaqhaaWcbaGa aiOkaaqaaiaaikdaaaGccaGGOaGaeqyWdi3aaSbaaSqaaiaaikdaae qaaOGaeyOeI0IaeqyWdi3aaSbaaSqaaiaaigdaaeqaaOGaaiykaaaa aa@8D39@

where we have used mass conservation ρ 1 v 1 = ρ 2 v 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaki aadAhadaWgaaWcbaGaaGymaaqabaGccqGH9aqpcqaHbpGCdaWgaaWc baGaaGOmaaqabaGccaWG2bWaaSbaaSqaaiaaikdaaeqaaaaa@3AA0@ .  Finally, recall that

ρ 1 v 1 ρ 2 v 2 ( 1 ρ 1 1 ρ 2 )= p 2 p 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabeg8aYnaaBaaaleaacaaIXaaabeaaki aadAhadaWgaaWcbaGaaGymaaqabaGccqaHbpGCdaWgaaWcbaGaaGOm aaqabaGccaWG2bWaaSbaaSqaaiaaikdaaeqaaOWaaeWaaeaadaWcaa qaaiaaigdaaeaacqaHbpGCdaWgaaWcbaGaaGymaaqabaaaaOGaeyOe I0YaaSaaaeaacaaIXaaabaGaeqyWdi3aaSbaaSqaaiaaikdaaeqaaa aaaOGaayjkaiaawMcaaiabg2da9iaadchadaWgaaWcbaGaaGOmaaqa baGccqGHsislcaWGWbWaaSbaaSqaaiaaigdaaeqaaaaa@48C9@

Substituting this into the preceding result and simplifying gives the second Prandtl formula.