Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations
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Transcript of Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations
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Jump conditions across phase boundaries for the Navier-Stokes-
Korteweg equations
Dietmar Kröner, Freiburg
Paris, Nov.2, 2009
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Co-workers:
• D. Diehl
• A. Dressel
• K. Hermsdörfer
• C. Kraus
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Outline
• Introduction, numerical experiments for the NSK system• Jump conditions across the interface for NSK, static
case• The pressure jump for the incompressible Navier-Stokes
equations• Low Mach number limit for the compressible Navier-
Stokes system• Low Mach number limit for the NSK system• NSK system dynamical case• Phase field like scaling
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Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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ρψ(ρ)
Double well
p
β2
Pressure
β1 β2
β1 ρρ
= ~W(½)
p(½) :=½2Ã0(½)
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ρψ(ρ)
Double well
p
β2
Pressure
β1 β2
β1 ρρ
= ~W(½)
W(½)
p(½) :=½2Ã0(½)
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MinimizeZ
W(½)+"2jr ½j2dx
Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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Initialdata
Long timebehaviour of ½;v?
MinimizeZ
W(½)+"2jr ½j2dx
Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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Theoretical results
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Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.
Global existence result for initial data close to stableequilibrium, d=2,3;
local in timeexistence for ½0 ¸ c> 0:
Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:
J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.
H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.
H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.
R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. ( = <n , isotherm)
D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.
D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.
S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004
S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005
M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )
1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001
Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.
Global existence result for initial data close to stableequilibrium, d=2,3;
local in timeexistence for ½0 ¸ c> 0:
Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:
J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.
H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.
H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.
R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. ( = <n , isotherm)
D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.
D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.
S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004
S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005
M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )
1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001
Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.
Global existence result for initial data close to stableequilibrium, d=2,3;
local in timeexistence for ½0 ¸ c> 0:
Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:
J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.
H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.
H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.
R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. ( = <n , isotherm)
D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.
D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.
S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004
S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005
M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )
1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001
Danchin, Desjardin :Existence of solutions for compressible °uid models ofKorteweg type. Annales de l'IHP, Analysenon lineaire, 18,(2001), 97-133.
Global existence result for initial data close to stableequilibrium, d=2,3;
local in timeexistence for ½0 ¸ c> 0:
Bresch, Desjardin, Lin 1: Existenceof global weak solutions in energy spacesif ¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3:
J .E. Dunn, J . Serrin: On the thermodynamics of interstitial work. Arch.Rat. Mech. Anal. 88 (1985), 95{133.
H. Hattori, D. Li: Theexistenceof global solutions toa °uid dynamicmodelfor materials for Korteweg type. J . Partial Di®erential Equations 9 (4) (1996)323-342.
H. Hattori, D. Li: Global solutions of a high-dimensional system for Ko-rtewegmaterials. J . Math. Anal. Appl. 198, No. 1, (1996), 84-97.
R. Danchin, B. Desjardin, : Existence of solutions for compressible °uidmodels of Korteweg type. Annales de l'IHP, Analyse non lineaire, 18,(2001),97-133. ( = <n , isotherm)
D. Bresch, B. Desjardin, C.K.Lin: On some compressible °uid model: Ko-rteweg, lubrication and shallow water systems, 2001.
D. Bresch, B. Desjardins, C.-K. Lin. On some compressible °uid models:Korteweg, lubrication and shallow water systems. Commun. Partial Di®er.Equations, 28(3):843-868, Mars 2003.
S. Benzoni-Gavage, R.Danchin, S. Descombes: Well-posednessof one-dimensionalKortewegmodels, preprint 2004
S. Benzoni-Gavage, R. Danchin, S. Descombes: On the well-posedness fortheEuler-Kortewegmodel in several spacedimensions, preprint 2005
M. Kotschote: Strong well-posedness for a Korteweg-type model for thedynamicsof a compressiblenon-isothermal °uid. Preprint Leipzig2006. (Initialboundary value )
1D. Bresch, B. Desjardin, C.K .Lin: On some compressible °uid model:K orteweg, lubrica-tion and shallow water systems, 2001Danchin,Desjardin:Existenceofsolutionsforcompressiblefluid
modelsofKortewegtype.Annalesdel'IHP,Analysenonlineaire,18,(2001),97-133.
Globalexistenceresultforinitialdataclosetostableequilibrium,d=2,3;localintimeexistencefor
Bresch,Desjardin,Lin:ExistenceofglobalweaksolutionsinenergyspacesifPreprint2002(?)
H.Hattori,D.Li:TheexistenceofglobalsolutionstoafluiddynamicmodelformaterialsforKortewegtype.J.PartialDifferentialEquations9(4)(1996)323-342.
H. Hattori, D. Li: Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198, No. 1, (1996), 84-97.
R. Danchin, B. Desjardin, : Existence of solutions for compressible fluid models of Korteweg type. Annales de l'IHP, Analyse nonlineaire, 18,(2001), 97-133.
½0 ¸ c> 0:
¹ ¢ u is replaced by ¹ div(½D(u));d= 2;3
( = <n ; isotherm)
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D.Bresch,B.Desjardins,C.-K.Lin.Onsomecompressiblefluidmodels:Korteweg,lubricationandshallowwatersystems.CommunPartialDiffer.Equations,28(3):843-868,2003.
S.Benzoni-Gavage,R.Danchin,S.Descombes:Well-posednessofone-dimensionalKortewegmodels,preprint2004
S.Benzoni-Gavage,R.Danchin,S.Descombes:Onthewell-posednessfortheEuler-Kortewegmodelinseveralspacedimensions,preprint2005
M.Kotschote:Strongwell-posednessforaKorteweg-typemodelforthedynamicsofacompressiblenon-isothermalfluid.PreprintLeipzig2006.(Initialboundaryvalue)
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Numerical results PhD Thesis Dennis Diehl
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Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
Stabilization:
@t½+r ¢(½v) = ±¢½
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
Stabilization:
@t½+r ¢(½v) = ±¢½
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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MinimizeZ
W(½)+"2jr ½j2dx
Euler Lagrange equation static case: ¡ W0(½) +°"2¢½= const:
· = ¡ W0(½) +°"2¢½= const:
Stabilization:
@t½+r ¢(½v) = ±¢½
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
@t½+r ¢(½v) = ±¢ ·
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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Jump conditions across the interface (static case)
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pl ¡ pv = c· "
Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
(Luckhaus,Modica,Dreyer,Kraus)
Stationary case:
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Navier Stokes K orteweg system(dynamical case):
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Jumpconditions:??????
liquidvapor
Navier Stokes K orteweg system (dynamical case):
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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A ssumptions For a sequence " ! 0weassume that
(1) ½" and v" are smooth solutions of theNSK in T := £ ]0;T[.
(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ; and inL1( ); j½"(x;t)j ·c for all (x;t) 2 T and c does not depend on ".
(3) ½0 has only two values: ¯1;¯2. De neE0(t) := fx 2 j½0(x;t) = ¯1g:
(4) v" ! v0 a.e. in T and v0 is su±ciently smooth in E0 and nE0.
(5) lim"1"
R
¡"2jr ½" j2+W(½")
¢dx exists uniformly in t.
Navier Stokes K orteweg system (dynamical case):
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ®¢ v+°"2½r ¢½
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Navier Stokes K orteweg system:
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
Multiply by a smooth testfunction ψ
Integration by partsZ T
0
Z
(½"@tà +(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢ Ã +°"2½"r ¢½"Ã
¢dxdt:
Z T
0
Z
(@t½+r ¢(½v))Ãdxdt = 0
Z T
0
Z
¡@t(½v) + r ¢(½vvt +p(½)I )
¢Ãdxdt =
Z T
0
Z
¡¹ ¢ v+°"2½r ¢½
¢Ãdxdt:
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Z T
0
Z
(½"@tÃ+(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢Ã +°"2½"r ¢½"Ã
¢dxdt:
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Z T
0
Z
(½"@tÃ+(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢Ã +°"2½"r ¢½"Ã
¢dxdt:
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
![Page 45: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/45.jpg)
Z T
0
Z
(½"@tÃ+(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢Ã +°"2½"r ¢½"Ã
¢dxdt:
¡RT0
R ½0v0@tÃ+(½0v0vt0+p(½0)I )) r Ãdxdt
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
![Page 46: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/46.jpg)
Z T
0
Z
(½"@tÃ+(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢Ã +°"2½"r ¢½"Ã
¢dxdt:
¡RT0
R ½0v0@tÃ+(½0v0vt0+p(½0)I )) r Ãdxdt
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
RT0
R ¹ v0¢ Ãdxdt
![Page 47: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/47.jpg)
Z T
0
Z
(½"@tÃ+(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢Ã +°"2½"r ¢½"Ã
¢dxdt:
¡RT0
R ½0v0@tÃ+(½0v0vt0+p(½0)I )) r Ãdxdt
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
?
RT0
R ¹ v0¢ Ãdxdt
![Page 48: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/48.jpg)
Z T
0
Z
(½"@tÃ+(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢Ã +°"2½"r ¢½"Ã
¢dxdt:
¡RT0
R ½0v0@tÃ+(½0v0vt0+p(½0)I )) r Ãdxdt
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
=:R
RT0
R ¹ v0¢ Ãdxdt
![Page 49: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/49.jpg)
R : = "2Z T
0
Z
(½"Ã)r ¢½"dxdt = ¡ "2
Z T
0
Z
r ¢(½"Ã)¢½"dxdt
= ¡ "2Z T
0
Z
(@j½"Ãj +½"@j Ãj )¢½"dxdt
= ¡ "2X
k
Z T
0
Z
(@j½"Ãj +½"@j Ãj )@2k½"dxdt
= "2X
k
Z T
0
Z
(@k@j½"Ãj +@j½"@kÃj +@k½"@j Ãj +½"@j@kÃj )@k½"dxdt
= "2X
k
Z T
0
Z
(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj +½"@k½"@j@kÃj )dxdt
![Page 50: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/50.jpg)
R : = "2X
k
Z T
0
Z
(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj +½"@k½"@j@kÃj )dxdt
![Page 51: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/51.jpg)
R : = "2X
k
Z T
0
Z
(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj +½"@k½"@j@kÃj )dxdt
= "2X
k
Z T
0
Z
(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj +
12@k½2"@j@kÃj )dxdt
= "2X
k
Z T
0
Z
(@k½"@k@j½"Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj ¡
12½2"@j@
2kÃj )dxdt
= "2X
k
Z T
0
Z
(¡
12(@k½")2@j Ãj +@k½"@j½"@kÃj +(@k½")2@j Ãj ¡
12½2"@j@
2kÃj )dxdt
= "2X
k
Z T
0
Z
(@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32(@k½")2@j Ãj ¡
12½2"@j@
2kÃj )dxdt
= "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
![Page 52: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/52.jpg)
= "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
![Page 53: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/53.jpg)
= "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
Remember the assumptions:
A ssumptions For a sequence " ! 0weassume that
(1) ½" and v" are smooth solutions of theNSK in T := £ ]0;T[.
(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ; and inL1( ); j½"(x;t)j ·c for all (x;t) 2 T and c does not depend on ".
(3) De neE0(t) := fx 2 j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ¡ E0(t).
(4) v" ! v0 a.e. in T and v0 is su±ciently smooth in E0 and nE0.
(5) lim"1"
R
¡"2jr ½" j2+W(½")
¢dx exists uniformly in t.
![Page 54: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/54.jpg)
= "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
Remember the assumptions:
A ssumptions For a sequence " ! 0weassume that
(1) ½" and v" are smooth solutions of theNSK in T := £ ]0;T[.
(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ; and inL1( ); j½"(x;t)j ·c for all (x;t) 2 T and c does not depend on ".
(3) De neE0(t) := fx 2 j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ¡ E0(t).
(4) v" ! v0 a.e. in T and v0 is su±ciently smooth in E0 and nE0.
(5) lim"1"
R
¡"2jr ½" j2+W(½")
¢dx exists uniformly in t.
![Page 55: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/55.jpg)
= "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
Remember the assumptions:
A ssumptions For a sequence " ! 0weassume that
(1) ½" and v" are smooth solutions of theNSK in T := £ ]0;T[.
(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ; and inL1( ); j½"(x;t)j ·c for all (x;t) 2 T and c does not depend on ".
(3) De neE0(t) := fx 2 j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ¡ E0(t).
(4) v" ! v0 a.e. in T and v0 is su±ciently smooth in E0 and nE0.
(5) lim"1"
R
¡"2jr ½" j2+W(½")
¢dx exists uniformly in t.
![Page 56: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/56.jpg)
= "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
Remember the assumptions:
A ssumptions For a sequence " ! 0weassume that
(1) ½" and v" are smooth solutions of theNSK in T := £ ]0;T[.
(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ; and inL1( ); j½"(x;t)j ·c for all (x;t) 2 T and c does not depend on ".
(3) De neE0(t) := fx 2 j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ¡ E0(t).
(4) v" ! v0 a.e. in T and v0 is su±ciently smooth in E0 and nE0.
(5) lim"1"
R
¡"2jr ½" j2+W(½")
¢dx exists uniformly in t.
![Page 57: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/57.jpg)
= "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
Remember the assumptions:
A ssumptions For a sequence " ! 0weassume that
(1) ½" and v" are smooth solutions of theNSK in T := £ ]0;T[.
(2) ½"(¢;t) ! ½0(¢;t) for all t 2 [0;T] a.e. in ; and inL1( ); j½"(x;t)j ·c for all (x;t) 2 T and c does not depend on ".
(3) De neE0(t) := fx 2 j½0(x;t) = ¯1g. Thenwehave½0(¢;t) = ¯1 in E0(t)und ½0(¢;t) = ¯2 in ¡ E0(t).
(4) v" ! v0 a.e. in T and v0 is su±ciently smooth in E0 and nE0.
(5) lim"1"
R
¡"2jr ½" j2+W(½")
¢dx exists uniformly in t.
![Page 58: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/58.jpg)
Lemma (Luckhaus, M odica):
WehaveR ! 0 if " ! 0 and
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt = "
Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o("):
(1)
R = "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
![Page 59: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/59.jpg)
Lemma (Luckhaus, M odica):
WehaveR ! 0 if " ! 0 and
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt = "
Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o("):
(1)
R = "2Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2+
1"2W(½")
¶@j Ãj
¡32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt
Lemma (Luckhaus, M odica):
For " ! 0wehave
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt
= "Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")
and thereforeR ! 0 for " ! 0:
Lemma (Luckhaus, M odica):
For " ! 0wehave
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt
= "Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")
and thereforeR ! 0 for " ! 0:
![Page 60: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/60.jpg)
Z T
0
Z
(½"@tÃ+(½"v" )r Ã)dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p(½")I )
¢r Ãdxdt:
=Z T
0
Z
¡¹ v"¢Ã +°"2½"r ¢½"Ã
¢dxdt:
" ! 0
Z T
0
Z
(½0@tÃ+(½0v0)r Ã) dxdt = 0:
¡Z T
0
Z
½0v0@tÃ+
¡½v0vt0+p(½0)I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v0¢Ã) dxdt:
![Page 61: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/61.jpg)
Z T
0
Z
(½0@tÃ+(½0v0)r Ã) dxdt = 0:
¡Z T
0
Z
½0v0@tÃ+
¡½v0vt0+p(½0)I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v0¢Ã) dxdt:
liquidvapor
![Page 62: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/62.jpg)
Z T
0
Z
(½0@tÃ+(½0v0)r Ã) dxdt = 0:
¡Z T
0
Z
½0v0@tÃ+
¡½v0vt0+p(½0)I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v0¢Ã) dxdt:
liquidvapor
@t½0+r ¢(½0v0) = 0 in E0(t) and in nE0(t)
localize
½0@tv0+r ¢¡½0v0vt0+p(½0)I )
¢= ¹ ¢ v0 in E0(t) and in nE0(t):
![Page 63: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/63.jpg)
Z T
0
Z
(½0@tÃ+(½0v0)r Ã) dxdt = 0:
¡Z T
0
Z
½0v0@tÃ+
¡½v0vt0+p(½0)I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v0¢Ã) dxdt:
liquidvapor
@t½0+r ¢(½0v0) = 0 in E0(t) and in nE0(t)
½0@tv0+r ¢¡½0v0vt0+p(½0)I )
¢= ¹ ¢ v0 in E0(t) and in nE0(t):
(1)
localize
Jump conditions ????
![Page 64: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/64.jpg)
liquidvapor
Z T
0
Z
(½0@tÃ+(½0v0)r Ã) dxdt = 0:
¡Z T
0
Z
½0v0@tÃ+
¡½v0vt0+p(½0)I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v0¢Ã) dxdt:
![Page 65: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/65.jpg)
[½0]nt +[½0v0]nx =0 on ¡ :
liquidvapor
Z T
0
Z
(½0@tÃ+(½0v0)r Ã) dxdt = 0:
¡Z T
0
Z
½0v0@tÃ+
¡½v0vt0+p(½0)I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v0¢Ã) dxdt:
![Page 66: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/66.jpg)
[½0]nt +[½0v0]nx =0 on ¡ :
liquidvapor
Z T
0
Z
(½0@tÃ+(½0v0)r Ã) dxdt = 0:
¡Z T
0
Z
½0v0@tÃ+
¡½v0vt0+p(½0)I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v0¢Ã) dxdt:
Z
¡
¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxÃ
¢d¾
=Z
¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾:
![Page 67: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/67.jpg)
[½0]nt +[½0v0]nx =0 on ¡ :Z
¡
¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxÃ
¢d¾
=Z
¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾:
![Page 68: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/68.jpg)
[½0]nt +[½0v0]nx =0 on ¡ :Z
¡
¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxÃ
¢d¾
=Z
¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾:
[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡
Use any testfunction Á on ¡ and extend it to a function à on such thatnxr à = 0 in a small layer around ¡ :
![Page 69: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/69.jpg)
[v0]= 0 on ¡ :
[½0]nt +[½0v0]nx =0 on ¡ :Z
¡
¡[½0v0]ntÃ+[½0v0vt0+p(½0)]nxÃ
¢d¾
=Z
¡(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾:
[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡
Use any testfunction Á on ¡ and extend it to a function à on such thatnxr à = 0 in a small layer around ¡ :
![Page 70: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/70.jpg)
Summary
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 71: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/71.jpg)
Summary
Jump conditions: liquid
vapor
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 72: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/72.jpg)
[v0]= 0 on ¡ :
[½0]nt +[½0v0]nx =0 on ¡ :
[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡
Summary
Jump conditions: liquid
vapor
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 73: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/73.jpg)
[v0]= 0 on ¡ :
[½0]nt +[½0v0]nx =0 on ¡ :
[½0v0]nt +[½0v0vt0+p(½0)]nx = ¡ ¹ nx[r v0] on ¡
Summary
Jump conditions: liquid
vapor
No curvature term !
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 74: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/74.jpg)
Different scaling to get the curvature term
Phase field like scaling
![Page 75: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/75.jpg)
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 76: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/76.jpg)
In thestatic caseNSK reduces to
r ¢(p(½)I ) = °"2½r ¢½:
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 77: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/77.jpg)
In the static caseNSK reduces to
r ¢(p(½)I ) = °"2½r ¢½:
This is equivalent to
~W0(½) = °"2¢½+¸" :
Herewehaveused ~W(½) =½Ã(½) and p(½) =½2Ã0(½) .
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 78: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/78.jpg)
In thestatic caseNSK reduces to
r ¢(p(½)I ) = °"2½r ¢½:
This is equivalent to
~W0(½) = °"2¢½+¸" :
Here we have used ~W(½) = ½Ã(½) and p(½) = ½2Ã0(½) . This is just the EulerLagrangeequationwith theLagrangemultiplier ¸ " of thefollowingminimizationproblem: Minimize
~J "(½) :=Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
under the constraintR ½dx =M0 (conservation of mass):
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 79: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/79.jpg)
In thestatic caseNSK reduces to
r ¢(p(½)I ) = °"2½r ¢½:
This is equivalent to
~W0(½) = °"2¢½+¸" :
Here we have used ~W(½) = ½Ã(½) and p(½) = ½2Ã0(½) . This is just the EulerLagrangeequationwith theLagrangemultiplier ¸ " of thefollowingminimizationproblem: Minimize
~J "(½) :=Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
under the constraintR ½dx =M0 (conservation of mass):
d‘Alambert variation principle
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
![Page 80: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/80.jpg)
Minimize
~J "(½) :=Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
under the constraintR ½dx =M0 (conservation of mass):
New scaling: Instead of
![Page 81: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/81.jpg)
Minimize
~J "(½) :=Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
under the constraintR ½dx =M0 (conservation of mass):
New scaling: Instead of
consider
Minimize
J "(½) :=1"
Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
under the constraintR ½dx =M0 (conservation of mass):
![Page 82: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/82.jpg)
~J "(½) :=Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
d‘Alambert variation principle
![Page 83: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/83.jpg)
~J "(½) :=Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
J "(½) :=1"
Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
d‘Alambert variation principle
d‘Alambert variation principle
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
NSK : @t(½v) + r ¢(½vvt +p"(½)I ) = ¹ ¢ v+°"½r ¢½
![Page 84: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/84.jpg)
~J "(½) :=Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
J "(½) :=1"
Z
µ~W(½) +°"2
jr ½j2
2
¶dx (total energy)
d‘Alambert variation principle
d‘Alambert variation principle
p"(½) := 1"p(½) +
1¡ "" d1
NSK : @t(½v) + r ¢(½vvt +p(½)I ) = ¹ ¢ v+°"2½r ¢½
NSK : @t(½v) + r ¢(½vvt +p"(½)I ) = ¹ ¢ v+°"½r ¢½
![Page 85: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/85.jpg)
@t½+r ¢(½v) = 0
@t(½v) + r ¢(½vvt +p"(½)I ) = ¹ ¢ v+°"½r ¢½:
Variational formulation:Z T
0
Z
(@t½+r ¢(½v))Ãdxdt = 0
Z T
0
Z
¡@t(½v) + r ¢(½vvt +p"(½)I )
¢Ãdxdt =
Z T
0
Z
(¹ ¢ v+°"½r ¢½)Ãdxdt:
Now consider for " ! 0 :
Z T
0
Z
(½"@tà +(½"v")r Ã) dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p"(½")I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v"¢ Ã +°"½"r ¢½"Ã)dxdt:
Integration by parts:
![Page 86: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/86.jpg)
Z T
0
Z
(½"@tà +(½"v")r Ã) dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p"(½")I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v"¢ Ã +°"½"r ¢½"Ã)dxdt:
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
![Page 87: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/87.jpg)
Z T
0
Z
(½"@tà +(½"v")r Ã) dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p"(½")I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v"¢ Ã +°"½"r ¢½"Ã)dxdt:
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
![Page 88: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/88.jpg)
Z T
0
Z
(½"@tà +(½"v")r Ã) dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p"(½")I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v"¢ Ã +°"½"r ¢½"Ã)dxdt:
RT0
R (½0@tà +(½0v0)r Ã) dxdt = 0;
?
?
![Page 89: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/89.jpg)
Z T
0
Z
(½"@tà +(½"v")r Ã) dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p"(½")I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v"¢ Ã +°"½"r ¢½"Ã)dxdt:
1"R : =
Z T
0
Z
°"½"r ¢½"Ã dxdt
= "Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2 ¡
1"2W(½")
¶@j Ãj
+32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt:
As before:
![Page 90: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/90.jpg)
1"R : =
Z T
0
Z
°"½"r ¢½"Ã dxdt
= "Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2 ¡
1"2W(½")
¶@j Ãj
+32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt:
![Page 91: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/91.jpg)
Lemma (Luckhaus, M odica):
For " ! 0wehave
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt
= "Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")
and thereforeR ! 0:
1"R : =
Z T
0
Z
°"½"r ¢½"Ã dxdt
= "Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2 ¡
1"2W(½")
¶@j Ãj
+32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt:
![Page 92: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/92.jpg)
Lemma (Luckhaus, M odica):
For " ! 0wehave
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt
= "Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")
and thereforeR ! 0:
1"R : =
Z T
0
Z
°"½"r ¢½"Ã dxdt
= "Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2 ¡
1"2W(½")
¶@j Ãj
+32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt:
o(1)
Lemma Luckhaus Modica
![Page 93: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/93.jpg)
Lemma (Luckhaus, M odica):
For " ! 0wehave
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt
= "Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")
and thereforeR ! 0:
1"R : =
Z T
0
Z
°"½"r ¢½"Ã dxdt
= "Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2 ¡
1"2W(½")
¶@j Ãj
+32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt:
o(1)
Lemma Luckhaus Modica
o(1)
![Page 94: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/94.jpg)
Lemma (Luckhaus, M odica):
For " ! 0wehave
"2Z T
0
Z
@k½"@j½"@kÃj (¢;t) ¡ (@k½")2@j Ãj (¢;t)dxdt
= "Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+o(")
(and thereforeR ! 0:)
1"R : =
Z T
0
Z
°"½"r ¢½"Ã dxdt
= "Z T
0
Z
@k½"@j½"@kÃj ¡ (@k½")2@j Ãj +
32
µ(@k½")2 ¡
1"2W(½")
¶@j Ãj
+32"2
W(½")@j Ãj ¡12½2"@j@
2kÃj dxdt:
o(1)
Lemma Luckhaus Modica
O(ε)
?
![Page 95: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/95.jpg)
Consider:Z T
0
Z
32"W(½")@j Ãj dxdt
We need several steps:
² Á(s) =Rs0
qW (r )2 dr, w" = Á±½" and w0 =Á±½0:
² lim"! 0R
1"W(½"(x;t))Ãdx = lim"! 0
R
"2jr ½"(x;t)j
2Ãdx
= lim"! 0R jr w" jÃdx (HT)
² lim"! 0R Ãr w"dx = ¡ lim"! 0
R r Ãw"dx = ¡
R w0r Ãdx
= c0R¡ Ã(x)º0dHn¡ 1
² lim"! 0R Ã(x)jr w"(x)jdx = lim"! 0
R F (x;r w"(x))dx =
R F (x;w0(x))djDw0j
=R Ã(x)jw0(x)jdjDw0j = c0
R¡ tÃ(x)jnjdHn¡ 1 = c0
R¡ tÃ(x)dHn¡ 1: (LM)
² lim"! 0R
32"W(½")@j Ãj dxdt = 3
2c0R¡ t@j Ãj (x)dHn¡ 1:
![Page 96: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/96.jpg)
lim"! 0
Z T
0
Z
°"½"r ¢½"Ã dxdt =
Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt+
32c0
Z T
0
Z
¡ t@j Ãj (x)dHn¡ 1:
SummaryZ T
0
Z
(½"@tà +(½"v")r Ã) dxdt = 0:
¡Z T
0
Z
½"v"@tÃ+
¡½v"vt" +p"(½")I )
¢r Ãdxdt:
=Z T
0
Z
(¹ v"¢ Ã +°"½"r ¢½"Ã)dxdt:
By an additional assumption wehave
Z T
0
Z
p"(½")r Ãdxdt !
Z T
0
Z
p(x;t)r Ãdxdt
for somep which is smooth in each phase.
Thereforewecan go to the limit " ! 0:
![Page 97: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/97.jpg)
Z T
0
Z
(½0@tà +(½0v0)r Ã)dxdt = 0
¡Z T
0
Z
½0v0@tà +
¡½v0vt0+p(x;t)I )
¢r Ãdxdt =
Z T
0
Z
¹ v0¢Ãdxdt+
=Z T
0co
Z
¡ t
· tÃ(¢;t)nxdHn¡ 1dt+32c0
Z T
0
Z
¡ t
@j Ãj (x)dHn¡ 1:
liquidvapor
@t½0+r ¢(½0v0) = 0 in E0(t) and in nE0(t)
localize
½0@tv0+r ¢¡½0v0vt0+p(½0)I )
¢= ¹ ¢ v0 in E0(t) and in nE0(t):
[½0]nt +[½0v0]nx = 0 on ¡ :
![Page 98: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/98.jpg)
Z T
0
Z
(½0@tà +(½0v0)r Ã)dxdt = 0
¡Z T
0
Z
½0v0@tà +
¡½v0vt0+p(x;t)I )
¢r Ãdxdt =
Z T
0
Z
¹ v0¢Ãdxdt+
=Z T
0co
Z
¡ t
· tÃ(¢;t)nxdHn¡ 1dt+32c0
Z T
0
Z
¡ t
@j Ãj (x)dHn¡ 1:
liquidvapor
localize
Z T
0
Z
¡ t
¡[½0v0]ntÃ+[½0v0vt0+p(x;t)]nxÃ
¢d¾
=Z T
0
Z
¡ t(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾+
Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt
+32c0
Z T
0
Z
¡ t
@j Ãj (x)dHn¡ 1:
![Page 99: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/99.jpg)
Z T
0
Z
¡ t
¡[½0v0]ntÃ+[½0v0vt0+p(x;t)]nxÃ
¢d¾
=Z T
0
Z
¡ t(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾+
Z T
0co
Z
¡ t· tÃ(¢;t)nxdHn¡ 1dt
+32c0
Z T
0
Z
¡ t
@j Ãj (x)dHn¡ 1:
Sincer à = (nxr Ã)nx +(¿xr Ã)¿x
wealso have@iÃk = nxj@j Ãknxi +¿xj@j Ãk¿xi
and therefore
r ¢Ã = @iÃi =nxj@j Ãinxi +¿xj@j Ãi¿xi = nxr Ãinxi +(¿r Ãi )¿xi= nxr Ãinxi +r ¡ t ¢Ã
wherer ¡ t ¢Ã := (¿r Ãi )¿xi denotes thetangential divergenceto thesurface¡ t.
![Page 100: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/100.jpg)
32c0
Z T
0
Z
¡ t@j Ãj (x)dHn¡ 1:
r ¢Ã = @iÃi =nxj@j Ãinxi +¿xj@j Ãi¿xi = nxr Ãinxi +(¿r Ãi )¿xi= nxr Ãinxi +r ¡ t ¢Ã
wherer ¡ t ¢Ã := (¿r Ãi )¿xi denotes thetangential divergenceto thesurface¡ t.Due to thespecial choiceof the testfunction à wehavenxr Ãi = 0and
@iÃi = r ¡ t ¢Ã:
Using theGauss theoremon surfaces weobtainZ
¡ t@j Ãj (x)dHn¡ 1 =
Z
¡ tr ¡ t ¢ÃdHn¡ 1 =
Z
¡ tr ¡ t ¢nxÃnxdHn¡ 1
=Z
¡ t· tÃnxdHn¡ 1:
![Page 101: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/101.jpg)
Z T
0
Z
¡ t
¡[½0v0]ntà +[½0v0vt0+p(x;t)]nxÃ
¢d¾
=Z T
0
Z
¡ t
(¹ [v0]nxr à ¡ ¹ nx[r v0]Ã) d¾+Z T
0co
Z
¡ t
· tÃ(¢;t)nxdHn¡ 1dt
+32c0
Z T
0
Z
¡ t
@j Ãj (x)dHn¡ 1:
Z
¡ t@j Ãj (x)dHn¡ 1 =
Z
¡ tr ¡ t ¢ÃdHn¡ 1 =
Z
¡ tr ¡ t ¢nxÃnxdHn¡ 1
=Z
¡ t· tÃnxdHn¡ 1:
Z T
0
Z
¡ t
¡[½0v0]ntÃ+[½0v0vt0+p(x;t)]nxÃ
¢d¾
= ¡Z T
0
Z
¡ t¹ nx[r v0]Ãd¾+
52c0
Z T
0
Z
¡ t· tÃnxdHn¡ 1
![Page 102: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/102.jpg)
Z T
0
Z
¡ t
¡[½0v0]ntÃ+[½0v0vt0+p(x;t)]nxÃ
¢d¾
= ¡Z T
0
Z
¡ t¹ nx[r v0]Ãd¾+
52c0
Z T
0
Z
¡ t· tÃnxdHn¡ 1
[½0v0]nt +[½0v0vt0+p(x;t)]nx = ¡ ¹ nx[r v0]+52co· tnx
![Page 103: Jump conditions across phase boundaries for the Navier-Stokes-Korteweg equations](https://reader036.fdocuments.us/reader036/viewer/2022062718/56812b94550346895d8fb6bf/html5/thumbnails/103.jpg)
Conclusion
• Introduction, numerical experiments for the NSK system
• Jump conditions across the interface for NSK, static case: no curvature term
• The pressure jump for the incompressible Navier-Stokes equations: curvature term
• Low Mach number limit for the NSK system: curvature term
• NSK system dynamical case: no curvature term• Phase field like scaling: curvature term