12
2 Equations de conservation
J
�
=
D
x 0
1 , x 0
2 , x 0
3
D (x 1 , x 2 , x 3 )
=
1
J
Calcul de 1/J · dJ/dt
D´ eriv´ ee d’un d´ eterminant
dJ
dt
=
𨐿
∂
dx 1
dt
∂ x
0
1
∂
dx 1
dt
∂ x
0
2
∂
dx 1
dt
∂ x
0
3
∂ x 2
∂ x
0
1
∂ x 2
∂ x
0
2
∂ x 2
∂ x
0
3
∂ x 3
∂ x
0
1
∂ x 3
∂ x
0
2
∂ x 3
∂ x
0
3
+
∂ x 1
∂ x
0
1
∂ x 1
∂ x
0
2
∂ x 1
∂ x
0
3
∂
dx 2
dt
∂ x
0
1
∂
dx 2
dt
∂ x
0
2
∂
dx 2
dt
∂ x
0
3
∂ x 3
∂ x
0
1
∂ x 3
∂ x
0
2
∂ x 3
∂ x
0
3
+
∂ x 1
∂ x
0
1
∂ x 1
∂ x
0
2
∂ x 1
∂ x
0
3
∂ x 2
∂ x
0
1
∂ x 2
∂ x
0
2
∂ x 2
∂ x
0
3
∂
dx 3
dt
∂ x
0
1
∂
dx 3
dt
∂ x
0
2
∂
dx 3
dt
∂ x
0
3
dJ
dt
=
D (V 1 , x 2 , x 3 )
D
x 0
1 , x 0
2 , x 0
3
+
D (x 1 ,V 2 , x 3 )
D
x 0
1 , x 0
2 , x 0
3
+
D (x 1 , x 2 ,V 3 )
D
x 0
1 , x 0
2 , x 0
3
1
J
dJ
dt
=
D (V 1 , x 2 , x 3 )
D
x 0
1 , x 0
2 , x 0
3
D
x 0
1 , x 0
2 , x 0
3
D (x 1 , x 2 , x 3 )
+ ...
1
J
dJ
dt
=
D (V 1 , x 2 , x 3 )
D (x 1 , x 2 , x 3 )
+
D (x 1 ,V 2 , x 3 )
D (x 1 , x 2 , x 3 )
+
D (x 1 , x 2 ,V 3 )
D (x 1 , x 2 , x 3 )
Compte tenu de ∂ x i /∂ x j = δ i j :
1
J
dJ
dt
=
∂V 1
∂ x 1
+
∂V 2
∂ x 2
+
∂V 3
∂ x 3
= ∇ · V
o` u ∇ · V est le taux de dilatation cubique du fluide
Soit
d
dt
Ω
A(M,t) dv =
Ω
dA
dt
+ A∇ · V
dv
=
Ω
∂ A
∂t
+ ∇ · (AV)
dv
=
Ω
∂ A
∂t
dv +
Σ
A V · n ds