ds
c V
¼ γ
dυ
υ
þ
dp
p
,
where γ ¼ c P /c V . Integrating the latter equation we have
Z s
s 0
ds
c V
¼ γ
Z υ
υ 0
dυ
υ
þ
Z p
p 0
dp
p
,
hence,
s À s 0
c V
¼ γ ln
υ
υ 0
þ ln
p
p 0
¼ ln
pυ
γ
p 0 υ 0
γ ,
so that
pυ
γ
p 0 υ 0
γ ¼ e
sÀs 0
c V :
Accordingly, constancy of the entropy as expressed by Eq. (1.58) implies that the
fluid element lies on the same adiabatic curve when it commenced its motion, hence,
∂
∂t
þ u
∂
∂x
pυ
γ
ð Þ ¼ 0,
ð1:59Þ
or expressed in terms of the density ρ as,
∂
∂t
þ u
∂
∂x
pρ
Àγ
ð
Þ¼0:
ð1:60Þ
1.7 Spherical Geometry
Let us now turn our attention to the conservation equations in spherical geometry as
we will be requiring these equations later on when spherical motion is discussed.
Initially, we will present the general form of these equations in spherical coordinates
before considering the case where the motion is confined to take place in the radial
direction and the equations reduce to their one-dimensional form.
24
1 Brief Outline of the Equations of Fluid Flow
c V
¼ γ
dυ
υ
þ
dp
p
,
where γ ¼ c P /c V . Integrating the latter equation we have
Z s
s 0
ds
c V
¼ γ
Z υ
υ 0
dυ
υ
þ
Z p
p 0
dp
p
,
hence,
s À s 0
c V
¼ γ ln
υ
υ 0
þ ln
p
p 0
¼ ln
pυ
γ
p 0 υ 0
γ ,
so that
pυ
γ
p 0 υ 0
γ ¼ e
sÀs 0
c V :
Accordingly, constancy of the entropy as expressed by Eq. (1.58) implies that the
fluid element lies on the same adiabatic curve when it commenced its motion, hence,
∂
∂t
þ u
∂
∂x
pυ
γ
ð Þ ¼ 0,
ð1:59Þ
or expressed in terms of the density ρ as,
∂
∂t
þ u
∂
∂x
pρ
Àγ
ð
Þ¼0:
ð1:60Þ
1.7 Spherical Geometry
Let us now turn our attention to the conservation equations in spherical geometry as
we will be requiring these equations later on when spherical motion is discussed.
Initially, we will present the general form of these equations in spherical coordinates
before considering the case where the motion is confined to take place in the radial
direction and the equations reduce to their one-dimensional form.
24
1 Brief Outline of the Equations of Fluid Flow