In terms of its components Eq. (1.43) becomes,
∂ρ
∂t
þ u
∂ρ
∂x
þ v
∂ρ
∂y
þ w
∂ρ
∂z
¼ Àρ
∂u
∂x
þ
∂v
∂y
þ
∂w
∂z
or
∂ρ
∂t
þ
∂ ρu
ð Þ
∂x
þ
∂ ρv
ð Þ
∂y
þ
∂ ρw
ð Þ
∂z
¼ 0:
By writing the derivative in Eq. (1.42) in terms of the specific volume υ ¼ 1/ρ
rather than the density ρ we have,
ρ
Dυ
Dt
¼
∂u
∂x
:
ð1:44Þ
1.4.2 Equation of Motion: The Momentum Equation
The law of the conservation of momentum implies that the rate of change in
momentum of the fluid in a volume element between x and x + dx is equal to the
rate at which momentum flows into the volume element at x minus the rate at which
momentum flows out at x + dx plus the net force acting on the volume element.
Expressing this mathematically, we have [8],
∂
∂t
ρu
ð ÞAdx ¼ ρ x, t
ð Þu
2 x, t
ð ÞA À ρ x þ dx, t
ð
Þ u
2 x þ dx, t
ð
Þ A þ p x, t
ð ÞA
À p x þ dx, t
ð
Þ A
hence,
∂
∂t
ρu
ð ÞAdx ¼ À
∂
∂x
ρ x, t
ð Þu
2 x, t
ð Þ
Â
Ã
Adx À
∂p x, t
ð Þ
∂x
Adx,
so that
∂
∂t
ρu
ð Þ þ
∂
∂x
ρu
2
À Á ¼ À
∂p
∂x
,
ð1:45Þ
which is the momentum equation. By using the continuity equation it is easy to show
that the latter equation can be written in final form as
1.4 Conservation Equations in Plane Geometry
15