If the motion takes place purely in the radial direction we obtain the following
momentum equation,
∂u
∂t
þ u
∂u
∂r
¼ À
1
ρ
∂p
∂r
,
ð1:65Þ
where we have set u ¼ u r as the material velocity is assumed to be wholly in the
radial direction.
1.7.3 Equation of Energy Conservation
We have seen that the energy balance equation in plane geometry where the motion
takes place purely in the x-direction is
∂
∂t
ρ
u
2
2
þ e
!
þ
∂
∂x
ρu
u
2
2
þ e
þ pu
!
¼ 0:
Generalizing this equation for 3-dimensional motion, the following equation in
vector form is obtained,
∂
∂t
ρ e þ
V
2
2
!
þ ∇
! Á ρ e þ
V
2
2
V
! þ pV
!
!
¼ 0,
where V
!
is given by
V
! ¼ ub x þ vb y þ wb z,
and (u, v, w) are the velocity components in the x, y and z-directions, respectively,
and b x, b y,b z
ð
Þ are unit vectors in these directions. In spherical geometry, where the
motion takes place purely in the radial direction with velocity u, the 3-dimensional
form of the energy balance equation reduces to the following 1-dimensional form;
∂
∂t
ρ
1
2
u
2
þ e
h
i
þ
1
r 2
∂
∂r
r
2
ρu
1
2
u
2
þ e
þ pu
n
o
h
i
¼ 0
ð1:66aÞ
after noting that
∇
! Á ub r ¼
1
r 2
∂
∂r
r
2 u
À Á
,
where b r is a unit vector in the radial direction. We can also write this energy balance
equation as
32
1 Brief Outline of the Equations of Fluid Flow