fluid_mechanics

5.2 Newton’s Second Law—The Linear Momentum and Moment-of-Momentum Equations 215
and involve some type of constraint such as a vane, channel, or conduit to guide the flowing fluid.
A flowing fluid can cause a vane, channel or conduit to move. When this happens, power is produced.
The selection of a control volume is an important matter. For determining anchoring forces,
consider including fluid and its constraint in the control volume. For determining force between a
fluid and its constraint, consider including only the fluid in the control volume.
The angular momentum
equation is
derived from Newton’s
second law.
5.2.3 Derivation of the Moment-of-Momentum Equation 2
In many engineering problems, the moment of a force with respect to an axis, namely, torque, is important.
Newton’s second law of motion has already led to a useful relationship between forces and
linear momentum flow. The linear momentum equation can also be used to solve problems involving
torques. However, by forming the moment of the linear momentum and the resultant force associated
with each particle of fluid with respect to a point in an inertial coordinate system, we will develop a
moment-of-momentum equation that relates torques and angular momentum flow for the contents of
a control volume. When torques are important, the moment-of-momentum equation is often more convenient
to use than the linear momentum equation.
Application of Newton’s second law of motion to a particle of fluid yields
D
Dt 1Vr dV2 dF particle
(5.30)
where V is the particle velocity measured in an inertial reference system, r is the particle density,
dV is the infinitesimally small particle volume, and dF particle is the resultant external force acting
on the particle. If we form the moment of each side of Eq. 5.30 with respect to the origin of an
inertial coordinate system, we obtain
r D Dt 1Vr dV2 r dF particle
(5.31)
where r is the position vector from the origin of the inertial coordinate system to the fluid particle
1Fig. 5.32. We note that
and
Thus, since
D
Dr
D1Vr dV2
31r V2r dV4 Vr dV r
Dt Dt Dt
Dr
Dt V
V V 0
by combining Eqs. 5.31, 5.32, 5.33, and 5.34, we obtain the expression
D
Dt 31r V2r dV4 r dF particle
(5.32)
(5.33)
(5.34)
(5.35)
z
V
r
dF particle
y
x
F I G U R E 5.3
Inertial coordinate system.
2 This section may be omitted, along with Sections 5.2.4 and 5.3.5, without loss of continuity in the text material. However, these sections
are recommended for those interested in Chapter 12.