1.7.1 Continuity Equation
From Eq. (1.43) we have
∂ρ
∂t
þ u
∂ρ
∂x
þ v
∂ρ
∂y
þ w
∂ρ
∂z
þ ρ∇
! Á V
! ¼ 0
which can be written as
∂ρ
∂t
þ V
! Á ∇
! ρ þ ρ∇
! Á V
! ¼ 0
and, therefore, this latter equation becomes
∂ρ
∂t
þ ∇
! Á ρV
!
¼ 0:
This is the vector form of the continuity equation. Using Eq. (1.62) in this latter
equation we obtain the following continuity equation in spherical geometry;
∂ρ
∂t
þ
1
r 2
∂ ρr
2 u r
ð
Þ
∂r
þ
1
rSinθ
∂ ρu θ Sinθ
ð
Þ
∂θ
þ
1
rSinθ
∂ ρu ϕ
À Á
∂ϕ
¼ 0,
with V
! ¼ u r b r þ u θ b θ þ u ϕ b
ϕ, where u r , u θ and u ϕ are the components of the velocity
vector in the b r, b θ and b
ϕ directions, respectively. If the motion takes place purely in
the radial direction (u θ ¼ 0 and u ϕ ¼ 0), then the latter equation becomes
∂ρ
∂t
þ
1
r 2
∂ ρr
2 u r
ð
Þ
∂r
¼ 0,
and by carrying out the differentiation we obtain the following continuity equation,
∂ρ
∂t
þ u
∂ρ
∂r
þ ρ
∂u
∂r
þ
2u
r
¼ 0,
ð1:63Þ
where we have set u ¼ u r as the material velocity is assumed to be wholly in the
radial direction.
1.7 Spherical Geometry
29