302
Annexe B : Op´ erateurs en coordonn´ ees curvilignes orthogonales
Syst` eme de coordonn´ ees cylindriques
Divergence de la vitesse
∇ · V =
1
r
∂
∂ r
(r V r ) +
1
r
∂V θ
∂ θ
+
∂V z
∂ z
Gradient de la vitesse
∂V r
∂ r
1
r
∂V r
∂ θ −
V θ
r
∂V r
∂ z
∂V θ
∂ r
1
r
∂V θ
∂ θ +
V r
r
∂V θ
∂ z
∂V z
∂ r
1
r
∂V z
∂ θ
∂V z
∂ z
Rotationnel
∇ × V =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1
r
∂V z
∂ θ
−
∂V θ
∂ z
∂V r
∂ z
−
∂V z
∂ r
1
r
∂
∂ r
(r V θ ) −
1
r
∂V r
∂ θ
Laplacien de la vitesse
∇ × V =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∂
∂ r
� 1
r
∂
∂ r
(rV r )
�
+
1
r 2
∂ 2 V r
∂ θ 2 −
2
r 2
∂V θ
∂ θ
+
∂ 2 V r
∂ z 2
∂
∂ r
� 1
r
∂
∂ r
(rV θ )
�
+
1
r 2
∂ 2 V θ
∂ θ 2 +
2
r 2
∂V r
∂ θ
+
∂ 2 V θ
∂ z 2
1
r
∂
∂ r
�
r
∂V z
∂ r
�
+
1
r 2
∂ 2 V z
∂ θ 2 +
∂ 2 V z
∂ z 2
Tenseur des taux de d´ eformations D
d rr =
∂V r
∂ r
; d θθ =
1
r
∂V θ
∂ θ
+
V r
r
; d zz =
∂V z
∂ z
d rθ = d θ r =
1
2
�
r
∂
∂ r
� V θ
r
�
+
1
r
∂V r
∂ θ
�