Chapter 2
Momentum Equations
2.1
Introduction
When a fluid flows over/on a solid body, it exert force on it. For example, force exerted
on a solid surface by a jet of fluid impinging on it, aerodynamic forces ( lift and drag)
on aircraft wing, the force on a pipebend by fluid flowing within it, etc. These forces
are hydrodynamic forces (due to a moving fluid) and are associated with a change in
momentum.
The magnitude of the hydrodynamic forces on the body due to a moving fluid is
determined by Newton’s second law of motion ” the net force acting on a body in any
direction is equal to the rate of increase of momentum of the body in that direction”.i.e
(equation 2.1)
F
=
ma
(2.1)
The law usually need to be expressed in a form particularly suited to steady flow of a
fluid.
2.2
Linear Momentum Equations for Steady Flows
Consider a steady flow which is nonuniform flowing in a control volume (stream tube)
as shown in figure 2.1. Where
A
is cross sectional area,
u
is velocity and
ρ
is density
and subscript 1 and 2 represent conditions at entry and exit respectively. In a short
interval
δt
, a volume of the fluid moves from the inlet a distance
uδt
.
The mass of fluid entering the control volume in time
δt
is given by equation 2.2
Mass entering control volume = volume
×
density =
ρ
1
A
1
u
1
δt
(2.2)
16
Figure 2.1: Nonuniform linear flow stream tube
Hence momentum entering the control volume is given by equation 2.3
Momentum entering control volume = mass
×
velocity =
ρ
1
A
1
u
1
u
1
δt
(2.3)
Similarly equation 2.4 expresses the momentum leaving the stream tube
Momentum leaving control volume = mass
×
velocity =
ρ
2
A
2
u
2
u
2
δt
(2.4)
The force exerted by the fluid is calculated using Newton’s 2
nd
law, equation 2.5
Force, F = Rate of change of momentum
=
ρ
2
A
2
u
2
u
2
δt

ρ
1
A
1
u
1
u
1
δt
δt
(2.5)
For a steady flow, continuity requires that
Q
=
A
1
u
1
=
A
2
u
2
(2.6)
and for a constant density,
ρ
1
=
ρ
2
=
ρ
, then equation 2.5 reduces to equation 2.7
F
=
Qρ
(
u
2

u
1
)
= ˙
m
(
u
2

u
1
)
(2.7)
2.3
Angular Momentum Equations for Steady Flows
Consider a steady flow which is nonuniform flowing in a control volume (stream tube)
as shown in figure 2.2. The inlet velocity vector,
u
1
makes an angle
θ
1
with the xaxis,
while at the outlet velocity
u
2
make an angle
θ
2
to the xaxis. Therefore, the forces are
resolved in the direction of the coordinate axes as;
17
Figure 2.2: Nonuniform anhular flow stream tube
Force in
x
direction,
F
x
is given by equation 2.8
F
x
= Rate of change of momentum in xdirection
= ˙
m
(
u
2
x

u
1
x
)
=
Qρ
(
u
2
cosθ
2

u
1
cosθ
1
)
(2.8)
Similarly, equation 2.9 gives the force in
y
direction as
F
y
= Rate of change of momentum in ydirection
= ˙
m
(
u
2
y

u
1
y
)
=
Qρ
(
u
2
sinθ
2

u
1
sinθ
1
)
(2.9)
The resultant force,
F
resultant
is given by equation 2.10 (figure 2.3)
F
resultant
=
F
2
x
+
F
2
y
(2.10)
and the angle,
φ
at which this force act is given as
φ
=
Tan

1
(
F
y
F
x
)
(2.11)
Note
: The force exerted by the fluid on the solid body touching the control volume
is opposite to
F
. So the reaction force,
R
is given as
R
=

F
R
.
F
is the total force