1.7.2 Equation of Motion
Recalling Eq. (1.48) and writing the components of the equation of motion in the
form,
∂u
∂t
þ V
! Á ∇
!
u ¼ À
1
ρ
∂p
∂x
∂v
∂t
þ V
! Á ∇
!
v ¼ À
1
ρ
∂p
∂y
∂w
∂t
þ V
! Á ∇
!
w ¼ À
1
ρ
∂p
∂z
:
Multiplying the first of these equations above by b x, the second equation by b y and
the third equation by b z, yielding,
∂ub x
∂t
þ V
! Á ∇
!
ub x ¼ À
1
ρ
b x
∂p
∂x
,
∂vb y
∂t
þ V
! Á ∇
!
vb y ¼ À
1
ρ
b y
∂p
∂y
∂wb z
∂t
þ V
! Á ∇
!
wb z ¼ À
1
ρ
b z
∂p
∂z
,
by adding all three components of the above equation we can write it as a single
equation in the form,
∂
∂t
ub x þ vb y þ wb z
ð
Þ þ V
! Á ∇
!
ub x þ vb y þ wb z
ð
޼1
ρ
b x
∂
∂x
þ b y
∂
∂y
þ b z
∂
∂z
p,
which gives the following momentum equation for inviscid flow in vector notation,
∂V
!
∂t
þ V
! Á ∇
!
V
! ¼ À
1
ρ
∇
!
p,
ð1:64Þ
where V
! ¼ ub x þ vb y þ wb z and ∇
! ¼ b x ∂=∂x
ð
Þþb y ∂=∂y
ð
Þþb z ∂=∂z
ð
Þ.
30
1 Brief Outline of the Equations of Fluid Flow