We have already noted that the velocity vector V
!
in terms of the orthogonal
spherical velocity components, u r , u θ , u ϕ , is given by
V
! ¼ u r b r þ u θ b θ þ u ϕ b
ϕ
and we have shown that the vector differential operator ∇
!
for this spherical
coordinate system is
∇
! ¼ b r
∂
∂r
þ b θ
1
r
∂
∂θ
þ b
ϕ
1
rSinθ
∂
∂ϕ
,
hence V
! Á ∇
!
in Eq. (1.64) becomes,
V
! Á ∇
!
¼ u r
∂
∂r
þ
u θ
r
∂
∂θ
þ
u ϕ
rSinθ
∂
∂ϕ
and forming the product V
! Á ∇
!
V
!
we have
V
! Á ∇
!
V
! ¼ u r
∂
∂r
þ
u θ
r
∂
∂θ
þ
u ϕ
rSinθ
∂
∂ϕ
u r b r þ u θ b θ þ u ϕ b
ϕ
¼ u r
∂
∂r
u r b r
ð Þþu r
∂
∂r
u θ b θ
þ
∂
∂r
u ϕ b
ϕ
þ
u θ
r
∂
∂θ
u r b r
ð Þþ
u θ
r
∂
∂θ
u θ b θ
þ
u θ
r
∂
∂θ
u ϕ b
ϕ
þ
u ϕ
rSinθ
∂
∂ϕ
u r b r
ð Þþ
u ϕ
rSinθ
∂
∂ϕ
u θ b θ
þ
u ϕ
rSinθ
∂
∂ϕ
u ϕ b
ϕ
:
Carrying out the differentiation in this latter equation and using the previously
established results for the derivatives of the unit vectors, we can write the r-component, the θ-component and the ϕ-component of Eq.(1.64), in the following form,
∂u r
∂t
þ u r
∂u r
∂r
þ
u θ
r
∂u r
∂θ
þ
u ϕ
rSinθ
∂u r
∂ϕ
À
u
2
θ þ u
2
ϕ
r
¼ À
1
ρ
∂p
∂r
∂u θ
∂t
þ u r
∂u θ
∂r
þ
u θ
r
∂u θ
∂θ
þ
u ϕ
rSinθ
∂u θ
∂ϕ
þ
u r u θ
r
À
u
2
ϕ
r
Cotθ ¼ À
1
ρr
∂p
∂θ
∂u ϕ
∂t
þ u r
∂u ϕ
∂r
þ
u θ
r
∂u ϕ
∂θ
þ
u ϕ
rSinθ
∂u ϕ
∂ϕ
þ
u r u ϕ
r
þ
u θ u ϕ
r
Cotθ ¼ À
1
ρrSinθ
∂p
∂ϕ
:
1.7 Spherical Geometry
31