viscosite

R1R2> R1hR2
Ω
ρ
η
R2−R1R1∂v
∂z =∂v ∂θ = 0
v(r)−→ uθ
v(r) = Ar +B
r
A B
−→
A=f(r)−→ uθ∆−→ A=∂2f
∂r2+1 r
∂f
∂r −f r2−→
uθ.
v(r)−→ uθdiv−→ v= 0
NS ρ −→ v .−−→ grad−→ v=η∆−→ v−−→ ∇p= 0 −→ ∇p.−→ uθ= 0
vθ=Ar +B
r∂2vθ ∂r2+1 r
∂vθ
∂r −vθ r2=A0 + 1 r−r r2+B2 r3−1 r
1
r2−1/r r2= 0
AR1+B
R1= 0 ⇔B=−AR2
1AR2+B
R2=R2Ω⇔A=R2
2Ω
R2
2−R2 1
B=−R2
1R2 2Ω
R2
2−R2 1
R1d−→ F=η∂v
∂r dS−→
uθ=ηA−B
R2
1dS−→
uθ= 2ηAdS−→ uθ
2ηAdSR1−→ uz2ηR2
2Ω
R2
2−R2 1
2πR2
1h−4πηhR2
1R2 2Ω
R2
2−R2 1
R L
ρ η. ∆p
α
ρ η h
v(x, z)−→ ux
L

l
L h
div−→ v= 0 ∂xv= 0 v(z)−→ ux. ∂tv= 0
−→
v .−−→ grad−→ v= 0 η∆−→ v−−→ ∇p+ρ−→ g= 0 (η∂2v
∂z2−∂p ∂x +ρgsinα = 0
−∂p
∂z +ρgcosα = 0 p(x, z) =
ρgcosα +f(x)p(x, h) = p0∀x p(x, z) = p0+ρgcosα(h−z)∂zv=−ρg
ηsinαz +K
∂zv=−ρg
ηsinα(h−z)v=ρgsinα 2η(2hz −z2) + K0v(0) = 0 K0= 0
˜ρ−→
v .−→ dS =ρgsinα
2ηρL(hh2−h3 3) = ρ2gsinα 3ηLh3
dxdydz dP =−→ dF .−→ v=η∆vvdτ =−ηρg
ηsinα ρgsinα 2η(2hz −z2)dτ =ρ2g2sin2α 2η(z2−
2hz)dxdydz P =ρ2g2sin2α
2η(h3 3−hh2)Ll =−Qglsinα
div−→ A=1
r
∂
∂r (rAr) + 1 r
∂Aθ
∂θ +∂Az ∂z
∂vz
∂z (r, z)=0
−−→
grad =∂
∂r
−→
ur+1
r
∂
∂θ
−→
uθ+∂
∂z
−→
uz−→ v .−−→ grad−→ v=vz∂
∂z vz−→
uz= 0
ρ∂−→
v
∂t +−→
v .−−→ grad−→ v=−−−→ gradP +η∆−→ v∆vz−→ uz=1
r
∂
∂r r∂vz ∂r +1 r2
∂2vz
∂θ2+∂2vz ∂z2−→
uz
0 = −∂P
∂r
0 = −1
r
∂P
∂θ
0 = −∂P
∂z +η1 r
∂
∂r r∂vz ∂r 
P(r, θ, z) = P(z)
∂P
∂z =η1 r
∂
∂r r∂vz ∂r =κ
P(z) = P0+κz ∆P=P(L)−P(0)
P(z) = P(0) + ∆P
Lz
η1
r
∂
∂r r∂vz ∂r =∆P L⇔∂ ∂r r∂vz ∂r =∆P ηL r
⇔r∂vz
∂r =∆P
ηL
r2
2+A
⇔∂vz
∂r =∆P
ηL
r
2+A r
⇔vz(r) = ∆P
ηL
r2
4+A ln r r0+B

|vz(0)|<∞
|vz(R)|= 0
vz(r) = −R2∆P
4ηL 1−r2 R2