Relationsusuelles•
287
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equaTions de navier-sTokes
Pour un écoulemenT incomPressible
Coordonnées cartésiennes
−
∂
∂
+





 +
=
−
∂
∂
+
∂
∂
+
∂
∂
p
x
F
t
u
x
u
y
u
z
u
x
x
x
x
x
μ
ρ
2
2
2
2
2
2
d
d
∂ ∂
∂
+





 +
=
−
∂
∂
+
∂
∂
+
∂
∂
p
y
F
t
u
x
u
y
u
z
u
y
y
y
y
y
μ
ρ
2
2
2
2
2
2
d
d
∂ ∂
∂
+





 +
=
∂
∂
+
∂
∂
+
∂
∂

p
z
F
t
u
x
u
y
u
z
u
z
z
z
z
z
μ
ρ
2
2
2
2
2
2
d
d
 


 





Coordonnées cylindriques
−
∂
∂
+
∂
∂
∂ ( )
∂





 +
∂
∂
−
∂
∂
+
p
r
r r
r u
r
r
u
r
u
r
r
μ
θ
θ
θ
1
1
2
2
2
2
2
∂ ∂
∂
∂
∂





 +
=
−






−
∂
∂
+
2
2
2
1
u
z
u
u
r
r
F
t
r
p
r
r
r
ρ
θ
μ
θ
d
d
1 1
1
2
2
2
2
2
2
2
r
r u
r
r
u
r
u
u
z
r
∂ ( )
∂





 +
∂
∂
+
∂
∂
+
∂
∂


θ
θ
θ
θ
θ
 


 +
=
−
(
)
−
∂
∂
+
∂
∂
∂ ( )
∂




F
t
r
p
z
u
u u
r r
r u
r
r
z
θ
θ
θ
ρ
μ
d
d
1
 
 +
∂
∂
+
∂
∂














 +
=
1
2
2
2
2
2
r
u
u
z
u
F
t
z
z
z
z
θ
ρ
d
d
Coordonnées sphériques
−
∂
∂
+
−
−
∂
∂
−
−
∂
p
r
u
u
r
r
u
r
u
r
u
r
r
μ
θ
θ
θ
θ
θ
ϕ
Δ
2
2
2
2
2
2
2
2
cotg
sin ∂ ∂
+
+





 +
=
−






−
∂
∂
+
ϕ
ρ
θ
μ
θ
ϕ
θ
F
t
r
r
p
u
u
u
u
r
r
d
d
2
2
1
Δ
2 2
2
2
2
2
2
2
r
u
u
r
r
u
F
r
∂
∂
−
−
∂
∂





 +
=
θ
θ
θ
ϕ
θ
θ
ϕ
θ
sin
cos
sin
ρ ρ
θ ϕ
μ
θ
θ
θ
ϕ
ϕ
ϕ
d
d
cotg
u
u u
u
u
u
t
r
r
r
p
r
+
−






−
∂
∂
+
−
1
sin
Δ
r r
r
u
r
u
r
2
2
2
2
2
2
2
2
sin θ
ϕ
θ
ϕ
θ
θ
θ
+
∂
∂
+
∂
∂





 +
sin
sin
cos
F F ϕ =
                                                                                                   ρ
θ
ϕ
ϕ
θ ϕ
d
d
cotg
u
u u
u u
t
r
r
r
+
−

















 

relaTions usuelles
grad
grad
Δ
Δ
f
f
( ) = (
)
∇
∇
∇
( ) =
+
fg
f g g f
 
 
∇( ) = ∇ + ⊗ ∇
f u
f u u
f
∇ ⋅
∇
( ) ⋅ ∇
( ) ⋅
( ) =
+
u v
u v
v u
t
t
9782100549337-Livre.indb 287
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