Annexe B : Op´ erateurs en coordonn´ ees curvilignes orthogonales
305
Tenseur des taux de d´ eformations D
d rr =
∂V r
∂ r
; d θθ =
1
r
∂V θ
∂ θ
+
V r
r
; d ϕϕ =
1
r sin θ
∂V ϕ
∂ ϕ
+
V r
r
+
V θ cotgθ
r
d rϕ = d ϕr =
1
2
1
r sin θ
∂V r
∂ ϕ
+ r
∂
∂ r
횿 V ϕ
r
횿횿
d ϕθ = d θϕ =
1
2
sin θ
r
∂
∂ θ
V ϕ
sin θ
+
1
r sin θ
∂V θ
∂ ϕ
d ϕr = d rϕ =
1
2
1
r sin θ
∂V r
∂ ϕ
+ r
∂
∂ r
V ϕ
r
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 ϕ
r sin θ
∂V r
∂ ϕ
−
V 2
θ +V 2
ϕ
r 2
= −
∂ p
∂ r
+ ρ g r + μ∇
2
r V
ρ
∂V θ
∂t
+V r
∂V θ
∂ r
+
V θ
r
∂V θ
∂ θ
+
V r V θ
r
+
V ϕ
r sin θ
∂V θ
∂ ϕ
−
V 2
ϕ cotgθ
r
= −
1
r
∂ p
∂ θ
+ ρ g θ + μ∇
2
θ V
ρ
∂V ϕ
∂t
+V r
∂V ϕ
∂ r
+
V θ
r
∂V ϕ
∂ θ
+
V ϕ
r sin θ
∂V ϕ
∂ ϕ
+
V r V θ
r
+
V θ V ϕ cotgθ
r
= −
1
r sin θ
∂ p
∂ ϕ
+ ρ g ϕ + μ∇
2
ϕ V
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 ϕ
r sin θ
∂ T
∂ ϕ
=
λ
1
r 2
∂
∂ r
r
2 ∂ T
∂ r
+
1
r 2 sin θ
∂
∂ θ
sin θ
∂ T
∂ θ
+
1
r 2 sin
2
θ
∂ 2 T
∂ ϕ 2
+ q + Φ