Etude mathématique et analyse asymptotique de quelques problèmes de lubrification par des fluides incompressibles essentiellement non-Newtoniens avec des conditions de non adhérence aux bords.

HAL Id: tel-00011638

https://tel.archives-ouvertes.fr/tel-00011638

Submitted on 17 Feb 2006

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Etude mathématique et analyse asymptotique de

quelques problèmes de lubrification par des fluides

incompressibles essentiellement non-Newtoniens avec

des conditions de non adhérence aux bords.

Rachid El Mir

To cite this version:

^F2= @C= J857@CBC23= 6C TF9?@C J>F9==C74 C234C 6C7L =74TF8C= =5@96C= 52 F>>458RC @C= JO7F3952= 6C 35 C= 57 6C FG9C4 35 C= >F4 72C JO7F3952 693C 6C ICN25@6= 69G

(h

3

_{∇p) =}

69G

(S h),

5

h

C=3 @KJ>F9==C74 6C @KJ857@CBC23 C3

S

72 GC83C74 6522J 4C>4J=C23F23 @C 89=F9@@CBC23 6C @K72C 6C= 6C7L =74TF8C= =5@96C=0

57= 257=923J4C==52= 6528  6C= J857@CBC23=34969BC2=9522C@==3F39522F94C= FGC8 S@9= =CBC23 6C= 796C= 252 C3529C2= =F7T C2 3459=9ABC 8RF>934C 5 @C 796C C=3 =7>>5=J C3529C20 57= 4F>>C@52= 6K F?546 O7K72 796C C3529C2 C=3 72 B9@9C7 8523927 6523 @C 3C2=C74 6C= 85234F923C=

σ

C=3 6522J >F4

σ(u, p) = −pI + µD(u),

5

u

C=3 @F G93C==C

p

C=3 @F >4C==952

I

C=3 @C 3C2=C74 96C2393J

µ

C=3 @F G9=85=93J

D(u) = (d

ij

(u))

1≤i,j≤3

Z

ω

h

3

12µ

∇p

?

−

h

₂

s

?

+ ˜

F

_∇φdx

0

_{= −}

Z

∂ω

φh

_{G.n ∀φ ∈ H}

˜ˆ

1

(ω),

V0 V0/P FGC8

˜ˆ

G =

1

h(x

0

₎

Z

h(x

0

₎

0

ˆ

G(x

0

, z)dz.

^F2=  ,
$  257= 852=96J452= 72 796C 6C 92SRFB 6F2= @C BBC 65BF92C

Ω

ε

6522J F7 8RF>934C /FGC8 6C=852693952=252 @92JF94C==74@C ?546057=852=C4 G52= @C= BBC= 852693952= F7L @9B93C= =74 @F G93C==C =7>>5=JC= C2 >4CB9C4 8RF>934C =F7T =74

Γ

ε

1

 5 257= =7>>5=52= O7C

σ

_T

ε

(u

ε

) + l

ε

u

ε

= 0

=74

Γ

ε

₁

,

V0 V0/H

u

ε

.n = 0

=74

Γ

ε

₁

,

V0 V0/] FGC8

l

ε

_{> 0}

C=3 72C 6522JC 67 >45?@ABC0

ε

LJ 257=B523452=@KCL9=3C28C C3@K729893J6C==5@73952=TF9?@C=6C8C >45?@ABC0<79= 257= 739@9=52= 72 4J=7@3F3 8528C42F23 @K92JSF@93J 6C 542 5?3C27 6F2= P >574 3457GC4 @C= C=39BF3952= F >49549 =74 @F G93C==C C3 @F >4C==9520 U2=793C 257= J376952= @K F2F@N=C F=NB>3539O7C 67 >45?@ABC 6C @F BBC BF29A4C O7C 6F2=@C=8RF>934C= >4J8J6C23=054=O7C

5 3R457SR F 3R92 =@F?0

M

2

_AN

 G5@ ;]25 / : ;; /]]W0 /D 0M0FF45GM=NB>35398 =5@73952 5T3RC FG9C4 35 C=>45?@CB 52 3RC 5 5TF 3R92 @FNC4 5T 7960 9?C49F2 F3RCBF398F@ 5742F@ :/ 205 ; ;]Y :VP /]]V0 /W I0 <93 0 C4GC3 0 JSC4 9=C C2 JG96C28C 694C83C 6KJ857@CBC23=  @F >F459  69GC4=C= 923C4TF8C= RCLF6J8F2C =5@96C0 ICG7C 6C J3F@@74S9C Q1 89C28C C3 J29C 6C= F3J49F7L>>0 /Y] /PD EJG49C4 ;VV/0 /Y E0 F969 74 O7C@O7C= >45?@ABC= 6C @7?49 8F3952 >F4 6C= 796C= 2C3529C2= 252 9=53RC4BC= C3 9285B>4C==9?@C= FGC8 6C= 852693952= F7L ?546= 252 @92JF94C=0 X376C BF3RJBF39O7C=C3F=NB>3539O7C0RA=C29GC4=93J CF2 522C3 F923 U39C22C;VVD0 /P 00 9@ 92=52 52 C3529F2 796= E@796 BC8RF298= B9L92S F26 RCF3 34F2=TC40 123C42F3952F@ C49C= 5T 525S4F>R=52 QRCB98F@ U2S92CC492SG5@0/0<C4SFB52 <4C==







457GC4

u

ε

_{∈ K}

ε

div

3C@ O7C

(A(u

ε

_{), φ − u}

ε

_{) + j}

ε

_{(φ) − j}

ε

_(u

ε

_{) + δ}

K

ε

div

(φ) − δ

K

div

ε

(u

ε

_{) ≥ (f}

ε

_{, φ − u}

ε

_),

∀φ ∈ (W

Γ

1,r

ε

1

(Ω

ε

₎₎

3

_.

MGC8

δ

K

ε

div

6J=9S2C @F T5283952 92698F3498C 67 852GCLC

K

ε

div

6J 29C >F4

δ

K

ε

div

=



0

=9

v ∈ K

ε

div

+∞

=9252 57=4F>>C@52=O7C @F T5283952 92698F3498C 6K72 852GCLC TC4BJC=3852GCLC C3=CB9 8523927C 92TJ49C74CBC230

       593

V

72 C=>F8C 6C F2F8R C3

V

0

=52 67F@0 2 6J=9S2C >F4   @C >456793 6C 67F@93J 6F2=

V

0   * !  ! 

A

   

V → V

0



•

        
       

(λ

n

)

n

   

λ

 

A(u + λ

n

v), w

→ A(u + λv), w

∀(u, v, w) ∈ V

3

.

•

          
 

A(u) − A(v), u − v) ≥ 0 ∀(u, v) ∈ V

2

.

•



         
 

A(u) − A(v), u − v) > 0 ∀(u, v) ∈ V

2

3C@ O7C

λ −→| D(u(x) + λv(x)) |

r−2

D(u(x)) : D(w(x)),

C=3 8523927C 6528

| D(u(x) + λ

n

v(x)) |

r−2

D(u(x)) : D(w(x)) −→| D(u(x) + λv(x)) |

r−2

D(u(x)) : D(w(x))

) 59C23

Ω

1

= ω×]0, h

?

[

72 57GC43 ?542J 6C

R

3

926J>C26F23 6C

ε

3C@@C O7C

h

?

_{> h}

max

 C3

ϕ

72C T5283952 6C

K

ε

0 ICBF4O752= 6K F?546 O7C

Ω

ε

_{⊂ Ω}

1

057= 6J 29==52=

¯

ϕ

6F2=

Ω

1

>F4

¯

ϕ =



ϕ

6F2=

Ω

ε

_,

0

6F2=

Ω

1

/Ω

ε

.

@K92JSF@93J 6C 542 =74 @C 65BF92C LC

Ω

1

9B>@9O7C @KCL9=3C28C 6K72C 852=3F23C

C > 0

926J>C26F23C 6C

ε

3C@@C O7C

k ∇ϕ k

L

r

_(Ω

1

)

≤ C k D(ϕ) k

L

r

(Ω

1

)

.

4

k ∇ ¯

ϕ k

L

r

_(Ω

₁

₎

=k ∇ϕ k

_L

r

_(Ω

ε

₎

,

k D( ¯

ϕ) k

L

r

_(Ω

₁

₎

=k D(ϕ) k

_L

r

_(Ω

ε

₎

,

6K5 /0D0 P 6J857@C0



$, ! Z !   
    

  
  
           

 

r = 2

   
 
   
    
   

1 < r < ∞

 


         


 
  

[



,



,



,



,



]

   

r = 1

 

     

         -
$ ! Z ! 
       
  

   

C

  

ε



 

X

1≤i,j≤2

k ε

∂ ˆ

u

ε

i

∂x

j

k

r

L

r

_(Ω)

+ k ε

∂ ˆ

u

ε

3

∂z

k

r

L

r

_(Ω)

+

+

2

X

i=1



k

∂ ˆ

u

ε

i

∂z

k

r

L

r

_(Ω)

+ k ε

2

∂ ˆ

u

ε

3

∂x

i

k

r

L

r

_(Ω)



≤ C,

/0D0 H

k

∂ ˆ

p

ε

∂x

i

k

W

−1,r0

_(Ω)

≤ C i = 1, 2,

/0D0 ]

k

∂ ˆ

p

ε

∂z

k

W

−1,r0

(Ω)

≤ C.ε.

/0D0/V ) 593

u

ε

@F =5@73952 67 >45?@ABC

P b(K

ε

div

)

0^528

$$ ! [ ! 
          
 
 


u

?

_{, p}

?

  
 
   

p

?

(x

0

, z) = p

?

(x

0

) presque partout dans Ω,

/0 W0/

<4C2F23

v = (1 − λ)u + λw, λ ∈]0, 1[

 C2 4CB>@FF23 6F2= /0W0Y 52 5?39C23

(F

0

_{(1 − λ)u + λw), λ(w − u)) + G((1 − λ)u + λw) − G(u) ≥ 0,}

FGC8 @F 852GCL93J 6C

G

52 F

λ(F

0

_{(1 − λ)u + λw), (w − u)) + λ(G(w) − G(u)) ≥ 0}

2 5?39C23 @KC=39BF3952  ;0 :0 ] 85BBC 6F2= @C 3RJ54ABC /0D0/ 67 >4CB9C4 8RF >934C0 593

(θ

n

)

n

72C =793C O79 852GCSC T543CBC23 GC4=

θ

6F2=

W

_Γ

1,q

ε

1

∪Γ

ε

L

(Ω

ε

₎

0 ^C ;0:0] 52 6J6793 O7K9@ CL9=3C

u

ε

_{∈ K}

ε

3C@@C O7C

u

ε

θ

n

* u

ε

 B523452= 6K F?546 O7C

u

ε

_{= u}

ε

θ

.

U2 CC3 52 F

a(θ

n

; u

ε

θ

n

, φ − u

ε

θ

n

) + j

ε

(φ) − j

ε

(u

ε

_θ

_n

_{) ≥ (f}

ε

_{, φ − u}

ε

_θ

_n

_{) ∀φ ∈ K}

_div

ε

,

6KF>4A= @C @CBBC 6C 923N 8C33C 92JSF@93J C=3 JO79GF@C23C 

a(θ

n

; φ, φ − u

ε

θ

n

) + j

ε

(φ) − (f

ε

, φ − u

ε

θ

n) ≥ j

ε

(u

ε

θ

n) ∀φ ∈ K

div

ε

,

C3 85BBC

θ

n

−→ θ

T543CBC23 6F2=

W

_Γ

1,q

ε

1

∪Γ

ε

L

(Ω

ε

₎

C3

µ

ε

_{∈ C}

0

_(Ω

ε

₎

 F@54= 52 >C73 CL34F94C 72C =57= =793C

θ

n

3C@@C O7C

µ

ε

_(θ

n

) −→ µ

ε

(θ)

>4C=O7C >F435730 U2 >F==F23  @F @9B93C 6F2= @K92JO7F3952 >4J8J6C23C 9@ G9C23 O7C

a(θ; φ, φ − u

ε

) + j

ε

_{(φ) − (f}

ε

_{, φ − u}

ε

_{) ≥ lim inf}

F@54= 9@ CL9=3C 72C 852=3F23C

C

ε

1

O79 2C 6J>C26 29 6C

δ

29 6C

θ

29 6C

u

ε

θ

3C@@C O7C

k m

δ

(θ, u

ε

θ

) k

L

1

_(Ω

ε

₎

≤ C

₁

ε

.

 ;0 :0/: Q52=96J452= F@54= @C >45?@ABC 923C4BJ69F94C =79GF23 457GC4

T

_δ

ε

_{∈ W}

_Γ

1,q

ε

1

∪Γ

ε

L

(Ω

ε

₎

3C@@C O7C

:

Z

Ω

ε

K

ε

_∇T

_δ

ε

_{∇ψ +}

Z

Ω

ε

r

ε

T

_δ

ε

ψ =

Z

Ω

ε

m

δ

(θ, u

ε

θ

)ψ,

∀ψ ∈ H

Γ

1

ε

1

∪Γ

ε

L

(Ω

ε

_).

 ;0 :0/D

U2 F>>@9O7F23 @C @CBBC 6C FL 9@S4FB 8C >45?@ABC F6BC3 72C =5@73952 729O7C0

2 F

ε

2

∂ ˆ

u

ε

3

∂x

i

* 0 (1 ≤ i ≤ 2) faiblement dans L

r

_(Ω),

 ;0 W0D

ε ˆ

u

ε

3

* 0 f aiblement dans L

r

(Ω),

 ;0 W0W

ˆ

p

ε

_{* p}

?

_{f aiblement dans L}

r

0

(Ω),

 ;0 W0 Y

ˆ

T

ε

_{* T}

?

_{f aiblement dans V}

q

z

,

 ;0 W0P

ε

∂ ˆ

T

ε

∂x

i

* 0 f aiblement dans L

q

(Ω), i = 1, 2.

 ;0 W0 H ) ^KF>4A=  ;0D0W  ;0D0P 257= 5?3C252= ;0 W0/ ;0 W0Y 85BBC 6F2= @C 3RJ54ABC /0D0/ 67 >4CB9C4 8RF>934C ;0W0 P ;0W0H =K5?39C22C23  >F4394 6C ;0D0/W ;0D0/Y0



-
$ \ [ \ 
    
          

( ˆ

K)

0

_∈

L

∞

_(R)


   


u

?

_{, p}

?

_{, T}

?



∀r > 1

u

ˆ

ε

i

−→ u

?

i

f ortement dans V

z

,

 ;0 W0 ]

p

?

(x

1

, x

2

, z) = p

?

(x

1

, x

2

) presque partout dans Ω,

a : V

ε

_{× V}

ε

_{→ R}

(u, v) → a(u, v) =

Z

Ω

ε

d

ij

(u)d

ij

(v)dx

0

dx

n

b : V

ε

_{× V}

ε

_{× V}

ε

_{→ R}

(u, v, w) → b(u, v, w) =

Z

Ω

ε

u

i

v

j,i

w

j

dx

0

dx

n

j

ε

: V

ε

_{→ R}

+

v → j

ε

(v) =

Z

ω

k

ε

_{| v − ε}

β

_{s | dσ.}

M T54BF@ F>>@98F3952 5T :0 ;0/ :0 ;0/V @CF6= 35 3RC T5@@592S GF49F3952F@ >45?@CB F= 92 P $ ! 

u

ε

_{∈ V}

ε

div

, p

ε

∈ L

2

0

(Ω

ε

),

=78R 3RF3

2µa(u

ε

_{, φ − u}

ε

) + ε

γ

b(u

ε

, u

ε

_{, φ − u}

ε

_{) − (p}

ε

,

69G

(φ)) + j

ε

_{(φ) − j}

ε

(u

ε

)

≥ (f

ε

, φ − u

ε

) ∀φ ∈ V

ε

,

:0:0 : C 4CBF4 3RF3

a



b

F26

j

ε

=F39=TN 3RC T5@@592S >45>C439C= 9

a

9= F ?9@92CF4 T54B

•

852392757=

| a(u, v) |≤k u k

(H

1

_(Ω

ε

₎₎

n

k v k

_(H

1

_(Ω

ε

₎₎

n

.

•

85C489GC ?N 542K= F26 <5928F4JK= 92CO7F@939C=

∃α > 0

=78R 3RF3

a(v, v) ≥ α k v k

2

(H

1

_(Ω

ε

₎₎

n

∀v ∈ (H

1

(Ω

ε

))

n

=78R 3RF3

v = 0

52

Γ

ε

1

99

b

9= F 349@92CF4 T54B

•

852392757=

∃K

1

> 0

=78R 3RF3

| b(u, v, w) |≤ K

1

k u k

(H

1

_(Ω

ε

₎₎

n

k v k

_(H

1

_(Ω

ε

₎₎

n

k w k

_(H

1

_(Ω

ε

₎₎

n

,

•

T45B :0;0 Y

b

9= F239=NBBC3498 =5

b(u, v, w) + b(u, w, v) = 0, ∀(u, v, w) ∈ (V

ε

)

3

.

k

∂ ˆ

p

ε

∂x

i

k

H

−1

_(Ω)

≤ C, i = 1, .., n − 1,

:0D0;P

k

∂ ˆ

p

ε

∂z

k

H

−1

(Ω)

≤ Cε.

:0D0;H + E45B  :0D0// 93R 3RC 8R598C

φ = G

ε

F26 F=

b(u

ε

_{, u}

ε

_{, u}

ε

_{) = 0}

F26

j

ε

_(G

ε

_{) = 0}

 C SC3

K(Γ

ε

1

) ≤ 2 max

τ ∈Γ

ε

1

|n

i,k

(τ )|.

Q5BBC

Γ

ε

₁

_{= {(x}

1

, x

2

, x

3

) ∈ R

3

: x

3

− h

ε

(x

1

, x

2

) = 0},

6528 @C GC83C74 254BF@ 7293F94C CL3J49C74 

Γ

ε

1

=KJ8493

n(η) =

−

∂h

ε

∂x

1

(x

1

, x

2

), −

∂h

ε

∂x

2

(x

1

, x

2

), 1

p

1 + |∇h

ε

_(x

0

_)|

2

= n(x

1

, x

2

).

<574

i = 1, 2

 52 F

n

3,i

(x

1

, x

2

) =

∂n

3

∂x

i

(x

1

, x

2

) = −(1 + |∇h

ε

|

2

)

−

3

2

∂h

ε

∂x

1

∂

2

h

ε

∂x

i

∂x

1

+

∂h

ε

∂x

2

∂

2

h

ε

∂x

i

∂x

2

6528

|n

3,i

(x

1

, x

2

)| ≤ 2|D

1

h

ε

|.|D

2

h

ε

| ≤ |D

2

h

ε

|(1 + |D

1

h

ε

|

2

).

<F4 72 8F@87@ =9B9@F94C 52 F >574

i = 1, .., 3

C3

j = 1, 2

|n

j,i

(x

1

, x

2

)| ≤ 2|D

1

h

ε

|.|D

2

h

ε

| ≤ |D

2

h

ε

|(1 + |D

1

h

ε

|

2

).

^K5

K(Γ

ε

1

) ≤ C(Γ

ε

1

)

 5 @F 852=3F23C

C(Γ

ε

1

)

C=3 6J 6J 29C 6F2= 

4.3.28

0 U2 739@9=F23 D0 :0 ;] C3 D0:0:; 52 5?39C23 D0 :0 ;P0



$$ Z  Z  

Z

Ω

ε

|∇u

ε

|

2

dx ≤

4

µ

a(u

ε

_{, u}

ε

_{) + 10}

Z

Ω

ε

|∇G

ε

| +

+ 4C(Γ

ε

₁

)

(Z

Γ

ε

1

|u

ε

|

2

dτ +

Z

Γ

ε

1

|G

ε

|

2

dτ

)

.

D0 :0 :: ) ^KF>4A= D0:0;P 52 F

Z

Ω

ε

|∇u

ε

|

2

dx ≤ 2

Z

Ω

ε

|∇(u

ε

− G

ε

)|

2

dx + 2

Z

Ω

ε

|∇G

ε

|

2

dx

≤

2

_µ

a(u

ε

_{− G}

ε

, u

ε

_{− G}

ε

) + 2C(Γ

ε

1

)

Z

Γ

ε

1

|u

ε

− G

ε

|

2

dτ + 2

Z

Ω

ε

|∇G

ε

|

2

dx.

D0 :0 :D 4