Annexe B : Op´ erateurs en coordonn´ ees curvilignes orthogonales
303
d θ z = d zθ =
1
2
∂V θ
∂ z
+
1
r
∂V z
∂ θ
훀
d zr = d rz =
1
2
∂V z
∂ r
+
∂V r
∂ z
Equation de Navier-Stokes en formulation incompressible `
a viscosit´ e constante
ρ
∂V r
∂t
+V r
∂V r
∂ r
+
V θ
r
∂V r
∂ θ
−
V 2
θ
r
+V z
∂V r
∂ z
= −
∂ p
∂ r
+ ρ g r
+μ
∂
∂ 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
ρ
∂V θ
∂t
+V r
∂V θ
∂ r
+
V θ
r
∂V θ
∂ θ
+
V r V θ
r
+V z
∂V θ
∂ z
= −
1
r
∂ p
∂ θ
+ ρ g θ
+μ
∂
∂ r
1
r
∂
∂ r
(rV θ )
𨐾
+
1
r 2
∂ 2 V θ
∂ θ 2 +
2
r 2
∂V r
∂ θ
+
∂ 2 V θ
∂ z 2
ρ
∂V z
∂t
+V r
∂V z
∂ r
+
V θ
r
∂V z
∂ θ
+V z
∂V z
∂ z
= −
∂ p
∂ z
+ ρ g z
+μ
1
r
∂
∂ r
r
∂V z
∂ r
+
1
r 2
∂ 2 V z
∂ θ 2 +
∂ 2 V z
∂ z 2
Equation de l’Energie en formulation incompressible et propri´ et´ es constantes
ρ c p
∂ T
∂t
+V r
∂ T
∂ x
+
V θ
r
∂ T
∂ θ
+V z
∂ T
∂ z
횾
= λ
1
r
∂
∂ r
횾
r
∂ T
∂ r
횾
+
1
r 2
∂ 2 T
∂ θ 2 +
∂ 2 T
∂ z 2
+ q + Φ