ρ
Dv
Dt
¼ ρ
∂v
∂t
þ ρV
! Á ∇
!
v ¼ À
∂p
∂y
ð1:48bÞ
ρ
Dw
Dt
¼ ρ
∂w
∂t
þ ρV
! Á ∇
!
w ¼ À
∂p
∂z
;
ð1:48cÞ
leading to the following momentum equations in the x, y and z-directions,
respectively.
ρ
∂u
∂t
þ u
∂u
∂x
þ v
∂u
∂y
þ w
∂u
∂z
¼ À
∂p
∂x
ð1:49aÞ
ρ
∂v
∂t
þ u
∂v
∂x
þ v
∂v
∂y
þ w
∂v
∂z
¼ À
∂p
∂y
ð1:49bÞ
ρ
∂w
∂t
þ u
∂w
∂x
þ v
∂w
∂y
þ w
∂w
∂z
¼ À
∂p
∂z
ð1:49cÞ
1.4.3 Energy Balance Equation
Let us now consider how the energy within the small volume element Adx changes
with time, this energy is comprised of two parts; kinetic energy due to gas motion
and internal energy due to molecular motion. The rate of change of this energy with
time is
∂
∂t
ρ
u
2
2
þ e
!
Adx
where e is the internal energy per unit mass (for an ideal gas e ¼ p/(γ À 1)ρ). This
rate is equal to the rate of flow of energy into this volume element at x minus the rate
of flow of energy out of the volume element at x + dx, plus the rate at which pressure
forces does work at x minus the rate at which pressure forces does work at x + dx.
Expressing this mathematically, we have [8],
∂
∂t
ρ
u
2
2
þ e
!
Adx ¼
1
2
ρ x, t
ð Þu
2 x, t
ð Þ þ ρ x, t
ð Þe x, t
ð Þ
h
i
u x, t
ð ÞAÀ
1.4 Conservation Equations in Plane Geometry
17