¼ ∇
! Ψ
r
dr þ ∇
! Ψ
θ
rdθ þ ∇
! Ψ
ϕ
rSinθdϕ,
after substituting for d r
! , hence,
∇
! Ψ
r
¼
∂Ψ
∂r
, ∇
! Ψ
θ
¼
1
r
∂Ψ
∂θ
and ∇
! Ψ
ϕ
¼
1
rSinθ
∂Ψ
∂ϕ
and, therefore, the vector differential operator in spherical coordinates becomes,
∇
! ¼ b r
∂
∂r
þ b θ
1
r
∂
∂θ
þ b
ϕ
1
rSinθ
∂
∂ϕ
:
ð1:61Þ
Divergence of a Vector A
→
in Spherical Coordinates
Performing the following dot product of ∇
!
with the vector A
!
we have
∇
! Á A
! ¼ b r
∂
∂r
þ b θ
1
r
∂
∂θ
þ b
ϕ
1
rSinθ
∂
∂ϕ
Á A r b r þ A θ b θ þ A ϕ b
ϕ
¼ b r Á
∂
∂r
A r b r
ð Þþb r Á
∂
∂r
A θ b θ
þ b r Á
∂
∂r
A ϕ b
ϕ
þ b θ Á
1
r
∂
∂θ
A r b r
ð Þþ b θ Á
1
r
∂
∂θ
A θ b θ
þ b θ Á
1
r
∂
∂θ
A ϕ b
ϕ
þ b
ϕ Á
1
rSinθ
∂
∂ϕ
A r b r
ð Þþ b
ϕ Á
1
rSinθ
∂
∂ϕ
A θ b θ
þ b
ϕ Á
1
rSinθ
∂
∂ϕ
A ϕ b
ϕ
Carrying out the differentiation of the various terms and by using the previously
developed derivatives of the unit vectors, we find that the latter equation reduces to,
∇
! Á A
! ¼
∂A r
∂r
þ
A r
r
þ
1
r
∂A θ
∂θ
þ
A r
r
þ
A θ Cosθ
rSinθ
þ
1
rSinθ
∂A ϕ
∂ϕ
,
and this is generally written in the following standard form,
∇
! Á A
! ¼
1
r 2
∂
∂r
r
2 A r
À
Á þ
1
rSinθ
∂
∂θ
A θ Sinθ
ð
Þþ
1
rSinθ
∂A ϕ
∂ϕ
:
ð1:62Þ
This completes our brief look at the various relationships that are required for the
subsequent discussion of the conservation equations in spherical geometry.
28
1 Brief Outline of the Equations of Fluid Flow