• Aucun résultat trouvé

Macro-scale modeling of two-phase flows within structured packings

N/A
N/A
Protected

Academic year: 2021

Partager "Macro-scale modeling of two-phase flows within structured packings"

Copied!
36
0
0

Texte intégral

(1)

O

pen

A

rchive

T

OULOUSE

A

rchive

O

uverte (

OATAO

)

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.

This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 16118

To cite this version : Pasquier, Sylvain and Quintard, Michel and Davit,

Yohan Macro-scale modeling of two-phase Flows within structured

packings. (2016) In: Séminaire Fermat franco-allemand à Toulouse : «

Gas-Liquid Flows », 6 June 2016 - 8 June 2016 (Toulouse, France). (Unpublished)

Any correspondence concerning this service should be sent to the repository administrator: [email protected]

(2)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✴✷✷

▼❛❝#♦✲&❝❛❧❡ ♠♦❞❡❧✐♥❣ ♦❢ /✇♦✲♣❤❛&❡ ✢♦✇& ✇✐/❤✐♥

&/#✉❝/✉#❡❞ ♣❛❝❦✐♥❣&

❙②❧✈❛✐♥ '❛()✉✐❡,✱ ❨♦❤❛♥ ❉❛✈✐2✱ ▼✐❝❤❡❧ ◗✉✐♥2❛,❞

■♥"#✐#✉# ❞❡ ▼)❝❛♥✐,✉❡ ❞❡" ❋❧✉✐❞❡" ❞❡ ❚♦✉❧♦✉"❡ ✭■▼❋❚✮ ✲ ❯♥✐✈❡6"✐#) ❞❡ ❚♦✉❧♦✉"❡✱ ❈◆❘❙✲■◆<❚✲❯<❙✱ ❚♦✉❧♦✉"❡ ❋❘❆◆❈❊

(3)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✷✴✷✷

❙!"✉❝!✉"❡❞ ♣❛❝❦✐♥❣- ❛- ❛ -!"✉❝!✉"❡❞ ♣♦"♦✉- ♠❡❞✐✉♠

❆ ♠✉❧!✐✲-❝❛❧❡

♣"♦❝❡--❆ ❝♦♠♣❧❡① ✐♥!❡"❛❝!✐♦♥ ❜❡!✇❡❡♥ ❣❛- ❛♥❞ ❧✐6✉✐❞

(4)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✷✴✷✷

❙!"✉❝!✉"❡❞ ♣❛❝❦✐♥❣- ❛- ❛ -!"✉❝!✉"❡❞ ♣♦"♦✉- ♠❡❞✐✉♠

❆ ♠✉❧!✐✲-❝❛❧❡

♣"♦❝❡--❆ ❝♦♠♣❧❡① ✐♥!❡"❛❝!✐♦♥ ❜❡!✇❡❡♥ ❣❛- ❛♥❞ ❧✐6✉✐❞

(5)
(6)
(7)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✸✴✷✷

❯♣✲#❝❛❧✐♥❣ ♠❡,❤♦❞

❱♦❧✉♠❡ ❛✈❡3❛❣✐♥❣ ♠❡,❤♦❞

⋆ βγ

h

i h

i

h

i h

i

❊①❛♠♣❧❡✿ ❉❛3❝② ❡9✉❛,✐♦♥

ε

!

+ ∇ · h

i =

h

i = −

µ

·



∇ h

i

− ρ



(8)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✸✴✷✷

❯♣✲#❝❛❧✐♥❣ ♠❡,❤♦❞

❱♦❧✉♠❡ ❛✈❡3❛❣✐♥❣ ♠❡,❤♦❞

⋆ βγ

h

i h

i

h

i h

i

❊①❛♠♣❧❡✿ ❉❛3❝② ❡9✉❛,✐♦♥

ε

!

+ ∇ · h

i =

h

i = −

µ

·



∇ h

i

− ρ



(9)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✹✴✷✷

❉❛"❝②✬& ❧❛✇& ❢♦" ❣❛&✴❧✐.✉✐❞ ✢♦✇& ✐♥ ♣❛❝❦✐♥❣&❄

❉❛"❝②✲❣❡♥❡"❛❧✐+❡❞ ❡-✉❛/✐♦♥ ✭+✐♥❣❧❡ ♣❤❛+❡ ❛♣♣"♦①✐♠❛/✐♦♥✮ ❙♦✉❧❛✐♥❡ ❡/ ❛❧✳✱ :❛+-✉✐❡" ❡/✳ ❛❧✳ h✉❣i = −❑✵ µ · (∇ h♣❣i ❣ − ρ ❣❣) − ❋ · h✉❣i 1 2 3 0 5 10 F ∇ hpgi g ❊①♣❡"✐♠❡♥/❛❧ ❞❛/❛ ❡①/"❛❝/❡❞ ❢"♦♠ ❇"✉♥❛③③✐ ❡/ ❛❧✳ ✭✷✵✵✷✮ 0 1 2 3 8 10 12 14 F hl (%) ❙/"♦♥❣ ✐♥/❡"❛❝/✐♦♥ ❜❡/✇❡❡♥ ❧✐-✉✐❞ ❛♥❞ ❣❛+ ✐♥ /❤❡ ❧♦❛❞✐♥❣ "❡❣✐♠❡

♣"❡++✉"❡ ❞"♦♣ "✐+❡ ∇ h♣❣i❣

❧✐-✉✐❞ "❡/❡♥/✐♦♥ ❤❧ ✐♥❝"❡❛+❡+ ❘❡-✉✐"❡+ ❛ ❝♦✉♣❧❡❞ ♠♦❞❡❧ ❛/ /❤❡ ♠❛❝"♦✲+❝❛❧❡

(10)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✹✴✷✷

❉❛"❝②✬& ❧❛✇& ❢♦" ❣❛&✴❧✐.✉✐❞ ✢♦✇& ✐♥ ♣❛❝❦✐♥❣&❄

❉❛"❝②✲❣❡♥❡"❛❧✐+❡❞ ❡-✉❛/✐♦♥ ✭+✐♥❣❧❡ ♣❤❛+❡ ❛♣♣"♦①✐♠❛/✐♦♥✮ ❙♦✉❧❛✐♥❡ ❡/ ❛❧✳✱ :❛+-✉✐❡" ❡/✳ ❛❧✳ h✉❣i = −❑✵ µ · (∇ h♣❣i ❣ − ρ ❣❣) − ❋ · h✉❣i 1 2 3 0 5 10 F ∇ hpgi g ❊①♣❡"✐♠❡♥/❛❧ ❞❛/❛ ❡①/"❛❝/❡❞ ❢"♦♠ ❇"✉♥❛③③✐ ❡/ ❛❧✳ ✭✷✵✵✷✮ 0 1 2 3 8 10 12 14 F hl (%) ❙/"♦♥❣ ✐♥/❡"❛❝/✐♦♥ ❜❡/✇❡❡♥ ❧✐-✉✐❞ ❛♥❞ ❣❛+ ✐♥ /❤❡ ❧♦❛❞✐♥❣ "❡❣✐♠❡

♣"❡++✉"❡ ❞"♦♣ "✐+❡ ∇ h♣❣i❣

❧✐-✉✐❞ "❡/❡♥/✐♦♥ ❤❧ ✐♥❝"❡❛+❡+ ❘❡-✉✐"❡+ ❛ ❝♦✉♣❧❡❞ ♠♦❞❡❧ ❛/ /❤❡ ♠❛❝"♦✲+❝❛❧❡

(11)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✹✴✷✷

❉❛"❝②✬& ❧❛✇& ❢♦" ❣❛&✴❧✐.✉✐❞ ✢♦✇& ✐♥ ♣❛❝❦✐♥❣&❄

❉❛"❝②✲❣❡♥❡"❛❧✐+❡❞ ❡-✉❛/✐♦♥ ✭+✐♥❣❧❡ ♣❤❛+❡ ❛♣♣"♦①✐♠❛/✐♦♥✮ ❙♦✉❧❛✐♥❡ ❡/ ❛❧✳✱ :❛+-✉✐❡" ❡/✳ ❛❧✳ h✉❣i = −❑✵ µ · (∇ h♣❣i ❣ − ρ ❣❣) − ❋ · h✉❣i 1 2 3 0 5 10 F ∇ hpgi g ❊①♣❡"✐♠❡♥/❛❧ ❞❛/❛ ❡①/"❛❝/❡❞ ❢"♦♠ ❇"✉♥❛③③✐ ❡/ ❛❧✳ ✭✷✵✵✷✮ 0 1 2 3 8 10 12 14 F hl (%) ❙/"♦♥❣ ✐♥/❡"❛❝/✐♦♥ ❜❡/✇❡❡♥ ❧✐-✉✐❞ ❛♥❞ ❣❛+ ✐♥ /❤❡ ❧♦❛❞✐♥❣ "❡❣✐♠❡

♣"❡++✉"❡ ❞"♦♣ "✐+❡ ∇ h♣❣i❣

❧✐-✉✐❞ "❡/❡♥/✐♦♥ ❤❧ ✐♥❝"❡❛+❡+ ❘❡-✉✐"❡+ ❛ ❝♦✉♣❧❡❞ ♠♦❞❡❧ ❛/ /❤❡ ♠❛❝"♦✲+❝❛❧❡

(12)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✺✴✷✷

❯♣✲#❝❛❧✐♥❣ ✲ ❈♦✉♣❧❡❞ ♠♦❞❡❧

❱✐"❝♦✉" &❡❣✐♠❡ ✭❲❤✐-❛❦❡&✱ ▲❛""❡✉① ❡- ❛❧✳✮   h✉i h✉❣i   = −     ❑⋆ ❧❧ µ ❑⋆ ❧❣ µ ❑⋆ ❣❧ µ ❑⋆ ❣❣ µ       ∇ h♣i❧ ∇ h♣❣i❣   ❆♣♣❧✐❝❛-✐♦♥✿ ❚✇♦✲♣❤❛"❡ ✢♦✇ ✐♥ ♣❛&-✐❝❧❡ ❜❡❞" ❈❛❧✐❞❡ ❡①♣❡&✐♠❡♥- ✲ ■❘❙◆ ✲ "❡❡ ❈❤✐❦❤✐ ❡-❛❧✳ ✷✵✶✻ ✶❉ &❡"♦❧✉-✐♦♥ ♦❢ -❤❡ ❝♦✉♣❧❡❞ "②"-❡♠ ✭❖♣❡♥❋♦❛♠✮ ❈❧♦"✉&❡" ♦❢ ❑⋆ ❧❧ , ❑⋆❣❣ , ❑⋆❧❣ , ❑⋆❣❧ ❢&♦♠ ❡①♣❡&✐♠❡♥-❛❧ &❡"✉❧-"

(13)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✻✴✷✷

◆✉♠❡$✐❝❛❧ $❡)♦❧✉+✐♦♥ ✿ ■▼0❊❙ ▼❡+❤♦❞

■▼♣❧✐❝✐+ 0$❡))✉$❡ ❊①♣❧✐❝✐+ ❙❛+✉$❛+✐♦♥

❙❡7✉❡♥+✐❛❧ $❡)♦❧✉+✐♦♥ ♦❢ ❛♥ ❡7✉❛+✐♦♥ ♦♥ ♣$❡))✉$❡ ❛♥❞ ♦♥ )❛+✉$❛+✐♦♥

∂ ∂   ❙ ❙❣  + ∇ ·   h✉i h✉❣i   = ✵   h✉i h✉❣i   = −     ❑⋆ ❧❧ µ ❑⋆ ❧❣ µ ❑⋆ ❣❧ µ ❑⋆ ❣❣ µ       ∇ h♣i❧ ∇ h♣❣i❣  

❋✐♥✐+❡ ✈♦❧✉♠❡ ♠❡+❤♦❞

❙❝❛❧❛$ ♦$ +❡♥)♦$✐❛❧ ❡✛❡❝+✐✈❡ ♣❛$❛♠❡+❡$) ❑

(14)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✼✴✷✷

❯♣✲#❝❛❧✐♥❣ ✲ ❈♦✉♣❧❡❞ ♠♦❞❡❧

❉✐♠❡♥%✐♦♥❧❡%% ♣)❡%%✉)❡ ❞)♦♣ ✐♥ ,❤❡ ✈✐%❝♦✉% )❡❣✐♠❡ −|∇ h♣❣i ❣| ρ❣ ❊①♣❡$✐♠❡♥(❛❧ $❡+✉❧(+ ❈❧❛✈✐❡$ ❡( ❛❧✳ ✭✷✵✶✺✮ 0 20 40 60 80 0 0.5 1 Reg −|∇ hpgi g | ρlg Rel= 0 Rel= 64 Rel= 128 ✇✐"❤♦✉" ❝'♦(( "❡'♠( ◆✉♠❡$✐❝❛❧ ♠♦❞❡❧✐♥❣ ♦❢ (❤❡ ❝♦✉♣❧❡❞ +②+(❡♠ ✭❖♣❡♥❋♦❛♠✮

(15)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✽✴✷✷

❯♣✲#❝❛❧✐♥❣ ✲ ❈♦✉♣❧❡❞ ♠♦❞❡❧

■♥❡#$✐❛❧ #❡❣✐♠❡ ✭▲❛,,❡✉① ❡$ ❛❧✳✱ ✷✵✵✽✮   h✉❧i h✉❣i   = −     ❑⋆ ❧❧ µ ❑⋆ ❧❣ µ ❑⋆ ❣❧ µ ❑⋆ ❣❣ µ       ∇ h♣❧i❧ ∇ h♣❣i❣  −   ❋⋆ ❧❧ ❋⋆❧❣ ❋⋆ ❣❧ ❋⋆❣❣     h✉❧i h✉❣i   −|∇ h♣❣i ❣| ρ❣ 0 20 40 60 80 0 0 .5 1 1 .5 Reg −|∇ hpgi g | ρlg Stokes Rel = 0 Inertia Rel = 0 Stokes Rel = 128 Inertia Rel = 128

(16)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✾✴✷✷

❆♣♣❧✐❝❛&✐♦♥ &♦ )&*✉❝&✉*❡❞ ♣❛❝❦✐♥❣)

✶❉ *❡)♦❧✉&✐♦♥ ♦❢ ❛ ❛✐* ✴ ✇❛&❡* )②)&❡♠

❈♦♠♣❛*✐)♦♥ ♦❢ ❛ ❉❛*❝② ♠♦❞❡❧ ❛♥❞ ❝♦✉♣❧❡❞ ♠♦❞❡❧) ✭✈✐)❝♦✉) ❛♥❞ ✐♥❡*&✐❛❧✮

❈❧♦)✉*❡)✿

❱✐)❝♦✉)✿ ❛♥❛❧♦❣② &♦ ❛ ❧✐=✉✐❞

✜❧♠ ✐♥ ❛ ❝②❧✐♥❞❡* &✉❜❡

■♥❡*&✐❛❧✿ ❜❛)❡❞ ♦♥ &❤❡

❝❧♦)✉*❡) ❢*♦♠ &❤❡ ❈❛❧✐❞❡

❡①♣❡*✐♠❡♥&

❣❣

=

(

✶ − ❙

)

;

❧❧

=

❧✸

❧❣

=

µ

µ

✷ ❧

(

✶ − ❙

)

;

❣❧

=

(

✶ − ❙

)

(17)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✾✴✷✷

❆♣♣❧✐❝❛&✐♦♥ &♦ )&*✉❝&✉*❡❞ ♣❛❝❦✐♥❣)

✶❉ *❡)♦❧✉&✐♦♥ ♦❢ ❛ ❛✐* ✴ ✇❛&❡* )②)&❡♠

❈♦♠♣❛*✐)♦♥ ♦❢ ❛ ❉❛*❝② ♠♦❞❡❧ ❛♥❞ ❝♦✉♣❧❡❞ ♠♦❞❡❧) ✭✈✐)❝♦✉) ❛♥❞ ✐♥❡*&✐❛❧✮

❈❧♦)✉*❡)✿

❱✐)❝♦✉)✿ ❛♥❛❧♦❣② &♦ ❛ ❧✐=✉✐❞

✜❧♠ ✐♥ ❛ ❝②❧✐♥❞❡* &✉❜❡

■♥❡*&✐❛❧✿ ❜❛)❡❞ ♦♥ &❤❡

❝❧♦)✉*❡) ❢*♦♠ &❤❡ ❈❛❧✐❞❡

❡①♣❡*✐♠❡♥&

❣❣

=

(

✶ − ❙

)

;

❧❧

=

❧✸

❧❣

=

µ

µ

✷ ❧

(

✶ − ❙

)

;

❣❧

=

(

✶ − ❙

)

(18)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✾✴✷✷

❆♣♣❧✐❝❛&✐♦♥ &♦ )&*✉❝&✉*❡❞ ♣❛❝❦✐♥❣)

✶❉ *❡)♦❧✉&✐♦♥ ♦❢ ❛ ❛✐* ✴ ✇❛&❡* )②)&❡♠

❈♦♠♣❛*✐)♦♥ ♦❢ ❛ ❉❛*❝② ♠♦❞❡❧ ❛♥❞ ❝♦✉♣❧❡❞ ♠♦❞❡❧) ✭✈✐)❝♦✉) ❛♥❞ ✐♥❡*&✐❛❧✮

❈❧♦)✉*❡)✿

❱✐)❝♦✉)✿ ❛♥❛❧♦❣② &♦ ❛ ❧✐=✉✐❞

✜❧♠ ✐♥ ❛ ❝②❧✐♥❞❡* &✉❜❡

■♥❡*&✐❛❧✿ ❜❛)❡❞ ♦♥ &❤❡

❝❧♦)✉*❡) ❢*♦♠ &❤❡ ❈❛❧✐❞❡

❡①♣❡*✐♠❡♥&

❣❣

=

(

✶ − ❙

)

;

❧❧

=

❧✸

❧❣

=

µ

µ

✷ ❧

(

✶ − ❙

)

;

❣❧

=

(

✶ − ❙

)

(19)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✵✴✷✷

❆♣♣❧✐❝❛&✐♦♥ &♦ )&*✉❝&✉*❡❞ ♣❛❝❦✐♥❣)

❆♥❛❧②%✐% ♦❢ )❤❡ ♣-❡%%✉-❡ ❞-♦♣ ∇ h♣❣i❣ ❛♥❞ -❡)❡♥)✐♦♥ ❤❧ 100 101 102 103 104 10−3 10−1 101 103 Reg |∇ hpg i g − ρg g | Darcy Viscous coupling Inertial coupling 100 101 102 103 104 0.02 0.03 0.04 0.05 Reg hl(%) Darcy Viscous coupling Inertial coupling ❚❤❡ ✐♥❡-)✐❛❧ ❝♦✉♣❧❡❞ ♠♦❞❡❧ -❡✢❡❝)% )❤❡ ✐♥❝-❡❛%❡ ✐♥ -❡)❡♥)✐♦♥ ❛♥❞ ♣-❡%%✉-❡ ❞-♦♣ ❈♦♠♣❛-✐%♦♥ ❢-♦♠ ❙✉❡%% ❛♥❞ ❙♣✐❡❣❡❧ ✭✶✾✾✶✮ ❢♦- ✐❧❧✉%)-❛)✐♦♥

(20)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✶✴✷✷

▼♦❞❡❧✐♥❣ ♦❢ )❤❡ ❞✐+♣❡-+✐♦♥ ♦❢ ❧✐.✉✐❞❄ ✭✶✮

▼❛❤# ❛♥❞ ▼❡✇❡( ✭✶✾✾✾✮

❆ )✇♦✲❡.✉❛)✐♦♥ ♠♦❞❡❧ ❢♦- )❤❡ ❧✐.✉✐❞

♣❤❛+❡

❚♦♠♦❣-❛♣❤② ✈✐+✉❛❧✐+❛)✐♦♥

❋♦✉#❛0✐ ❡0 ❛❧✳ ✭✷✵✶✷✮ ❙❝❤✉❣ ❡0 ❛❧✳ ✭✷✵✶✺✮

(21)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✶✴✷✷

▼♦❞❡❧✐♥❣ ♦❢ )❤❡ ❞✐+♣❡-+✐♦♥ ♦❢ ❧✐.✉✐❞❄ ✭✶✮

▼❛❤# ❛♥❞ ▼❡✇❡( ✭✶✾✾✾✮

❆ )✇♦✲❡.✉❛)✐♦♥ ♠♦❞❡❧ ❢♦- )❤❡ ❧✐.✉✐❞

♣❤❛+❡

❚♦♠♦❣-❛♣❤② ✈✐+✉❛❧✐+❛)✐♦♥

❋♦✉#❛0✐ ❡0 ❛❧✳ ✭✷✵✶✷✮ ❙❝❤✉❣ ❡0 ❛❧✳ ✭✷✵✶✺✮

(22)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✶✴✷✷

▼♦❞❡❧✐♥❣ ♦❢ )❤❡ ❞✐+♣❡-+✐♦♥ ♦❢ ❧✐.✉✐❞❄ ✭✶✮

▼❛❤# ❛♥❞ ▼❡✇❡( ✭✶✾✾✾✮

❆ )✇♦✲❡.✉❛)✐♦♥ ♠♦❞❡❧ ❢♦- )❤❡ ❧✐.✉✐❞

♣❤❛+❡

❚♦♠♦❣-❛♣❤② ✈✐+✉❛❧✐+❛)✐♦♥

❋♦✉#❛0✐ ❡0 ❛❧✳ ✭✷✵✶✷✮ ❙❝❤✉❣ ❡0 ❛❧✳ ✭✷✵✶✺✮

(23)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✷✴✷✷

▼♦❞❡❧✐♥❣ ♦❢ )❤❡ ❞✐+♣❡-+✐♦♥ ♦❢ ❧✐.✉✐❞❄ ✭✷✮

❚✇♦✲❡%✉❛(✐♦♥ ♠♦❞❡❧ ❢♦/ (❤❡ ❧✐%✉✐❞ ♣❤❛2❡ ✭▼❛❤/ ❛♥❞ ▼❡✇❡2✱ ❙♦✉❧❛✐♥❡ ❡( ❛❧✳✮ ✉❧✶ = − ❑⋆ ❧❧✶ µ ·  ∇♣ ❧✶ − ρ❣ ; ✉ = −❑ ⋆ ❧❧✷ µ ·  ∇♣ ❧✷ − ρ❣ ❑⋆ ❧❧ = ❑❧❧⋆      ✵ ✵ ✵ ✵ ❝♦$✷θ$✐♥θ±❝♦$θ$✐♥θ⋆ ✵ ±❝♦$θ⋆$✐♥θ❝♦$θ$✐♥θ⋆      ✐ = ✶, ✷

θ

▲✐.✉✐❞ ❡①❝❤❛♥❣❡ ❛) )❤❡ ❝♦♥)❛❝) ♣♦✐♥)+ ❄

∂❙ ∂( + ∇ · ✉ = ˚♠ ; ∂❙❧✷ ∂( + ∇ · ✉ = −˚♠

(24)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✷✴✷✷

▼♦❞❡❧✐♥❣ ♦❢ )❤❡ ❞✐+♣❡-+✐♦♥ ♦❢ ❧✐.✉✐❞❄ ✭✷✮

❚✇♦✲❡%✉❛(✐♦♥ ♠♦❞❡❧ ❢♦/ (❤❡ ❧✐%✉✐❞ ♣❤❛2❡ ✭▼❛❤/ ❛♥❞ ▼❡✇❡2✱ ❙♦✉❧❛✐♥❡ ❡( ❛❧✳✮ ✉❧✶ = − ❑⋆ ❧❧✶ µ ·  ∇♣ ❧✶ − ρ❣ ; ✉ = −❑ ⋆ ❧❧✷ µ ·  ∇♣ ❧✷ − ρ❣ ❑⋆ ❧❧ = ❑❧❧⋆      ✵ ✵ ✵ ✵ ❝♦$✷θ$✐♥θ±❝♦$θ$✐♥θ⋆ ✵ ±❝♦$θ⋆$✐♥θ❝♦$θ$✐♥θ⋆      ✐ = ✶, ✷

θ

▲✐.✉✐❞ ❡①❝❤❛♥❣❡ ❛) )❤❡ ❝♦♥)❛❝) ♣♦✐♥)+ ❄

∂❙ ∂ + ∇ · ✉ = ˚♠ ; ∂❙❧✷ ∂ + ∇ · ✉ = −˚♠

(25)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✸✴✷✷

▼♦❞❡❧✐♥❣ ♦❢ )❤❡ ❞✐+♣❡-+✐♦♥ ♦❢ ❧✐.✉✐❞❄ ✭✸✮

❈❧♦+✉-❡ ♦❢ )❤❡ ❧✐.✉✐❞ )-❛♥+❢❡- ❛) ❝♦♥)❛❝) ♣♦✐♥)+❄

˚

♠ = ❤



h

i

❧✷

− h

i

❧✶



❆++✉♠✐♥❣ ❝❛♣✐❧❧❛-② ♣-❡++✉-❡ ❡✛❡❝)+

❛) ❝♦♥)❛❝) ♣♦✐♥)+

#

❝✐

(

) = h

i

− h

❧✐

i

❧✐

˚

♠ = ❤

(

#

❝✷

(

❧✷

) −

#

❝✶

(

❧✶

))

❇-♦♦❦+ ❛♥❞ ❈♦-❡② -❡❧❛)✐♦♥

0 0 .5 1 0 0 .5 1 Sli Pc i Pci= Pc0S 0.5 li

(26)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✹✴✷✷

❚❡"# ❝❛"❡ ✭✶✮

❱❛*✐❛#✐♦♥ ♦❢ #❤❡ ❧✐1✉✐❞ ❡①❝❤❛♥❣❡ ❝♦❡✣❝✐❡♥#

✵✳✶ ✶ ✶✵

✾ ✐♥❥❡❝#✐♦♥ ♣♦✐♥#"

❑⋆ ❧❧✐ =❑❧❧✐⋆     ✵ ✵ ✵ ✵ ❝♦#✷θ⋆#✐♥θ⋆ ±❝♦#θ⋆#✐♥θ⋆ ✵ ±❝♦#θ⋆#✐♥θ❝♦#θ#✐♥θ⋆     ✐ = ✶, ✷ ❑⋆ ❧❧✐ =❑❧❧✐⋆     ❝♦#✷θ⋆#✐♥θ ±❝♦#θ#✐♥θ⋆ ✵ ✵ ✵ ±❝♦#θ⋆#✐♥θ ❝♦#θ#✐♥θ⋆     ✐ = ✶, ✷

(27)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✺✴✷✷

❚❡"# ❝❛"❡ ✭✷✮ ✲ ❘❡❣✐♠❡" ♦❢ ❞✐"♣❡2"✐♦♥

˚

♠ ր

(28)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✻✴✷✷

❡"#♣❡❝&✐✈❡#

❆❝❝✉"❛&❡ ❝❧♦#✉"❡ ♦❢ &❤❡ ❡✛❡❝&✐✈❡ ♣❛"❛♠❡&❡"#

◆✉♠❡"✐❝❛❧ ♠♦❞❡❧✐♥❣ ❛& &❤❡ ❧♦❝❛❧✲#❝❛❧❡ ✭❱❖❋✮

■♠♣❛❝& ♦❢ &❤❡ ❝♦✉♣❧❡❞ #②#&❡♠ ♦♥ &❤❡ ❞✐#♣❡"#✐♦♥❄

❈❧♦#✉"❡ ♦❢ &❤❡ ❧✐@✉✐❞ ❡①❝❤❛♥❣❡ &❡"♠ ˚

◆✉♠❡"✐❝❛❧ ♠♦❞❡❧✐♥❣ ❛& &❤❡ ♠✐❝"♦✲#❝❛❧❡ ✭@✉❛♥&✐✜❝❛&✐♦♥ ♦❢ &❤❡ ❧✐@✉✐❞

(29)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✻✴✷✷

❡"#♣❡❝&✐✈❡#

❆❝❝✉"❛&❡ ❝❧♦#✉"❡ ♦❢ &❤❡ ❡✛❡❝&✐✈❡ ♣❛"❛♠❡&❡"#

◆✉♠❡"✐❝❛❧ ♠♦❞❡❧✐♥❣ ❛& &❤❡ ❧♦❝❛❧✲#❝❛❧❡ ✭❱❖❋✮

■♠♣❛❝& ♦❢ &❤❡ ❝♦✉♣❧❡❞ #②#&❡♠ ♦♥ &❤❡ ❞✐#♣❡"#✐♦♥❄

❈❧♦#✉"❡ ♦❢ &❤❡ ❧✐@✉✐❞ ❡①❝❤❛♥❣❡ &❡"♠ ˚

◆✉♠❡"✐❝❛❧ ♠♦❞❡❧✐♥❣ ❛& &❤❡ ♠✐❝"♦✲#❝❛❧❡ ✭@✉❛♥&✐✜❝❛&✐♦♥ ♦❢ &❤❡ ❧✐@✉✐❞

(30)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✻✴✷✷

❡"#♣❡❝&✐✈❡#

❆❝❝✉"❛&❡ ❝❧♦#✉"❡ ♦❢ &❤❡ ❡✛❡❝&✐✈❡ ♣❛"❛♠❡&❡"#

◆✉♠❡"✐❝❛❧ ♠♦❞❡❧✐♥❣ ❛& &❤❡ ❧♦❝❛❧✲#❝❛❧❡ ✭❱❖❋✮

■♠♣❛❝& ♦❢ &❤❡ ❝♦✉♣❧❡❞ #②#&❡♠ ♦♥ &❤❡ ❞✐#♣❡"#✐♦♥❄

❈❧♦#✉"❡ ♦❢ &❤❡ ❧✐@✉✐❞ ❡①❝❤❛♥❣❡ &❡"♠ ˚

◆✉♠❡"✐❝❛❧ ♠♦❞❡❧✐♥❣ ❛& &❤❡ ♠✐❝"♦✲#❝❛❧❡ ✭@✉❛♥&✐✜❝❛&✐♦♥ ♦❢ &❤❡ ❧✐@✉✐❞

❡①❝❤❛♥❣❡✮

(31)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✼✴✷✷

(32)

❙②❧✈❛✐♥ '❛()✉✐❡, ✲ ●❊▼' ✶✽✴✷✷

❈❧♦2✉+❡2 ❢♦+ -❤❡ ✈✐2❝♦✉2 -❡+♠2 ✲ ❜❛2❡❞ ♦♥ ❡①♣❡+✐♠❡♥-❛❧ +❡2✉❧-2

❧❧

=

;

❧❣

=

✺.✺

µ

µ

✷ ❧

(

✶ − ❙

)

❣❣

=

(

✶ − ❙

)

;

❣❧

=

µ

µ

❣❣

❧❣

❧❧

❈❧♦2✉+❡2 ❢♦+ -❤❡ ✈✐2❝♦✉2 -❡+♠2 ✲ ❜❛2❡❞ ♦♥ ❡①♣❡+✐♠❡♥-❛❧ +❡2✉❧-2

❧❧

=

ρ

µ

η

h

i

;

❧❣

=

❧❣

(

✶ − ❙

)

(

✶ − ❙

)

+

❣❣

=

ρ

µ

η

h

i

;

❣❧

=

❧❣

(

✶ − ❙

)

(33)
(34)
(35)
(36)

Références

Documents relatifs