fluid_mechanics

5.2 Newton’s Second Law—The Linear Momentum and Moment-of-Momentum Equations 201
F D
F E
Coincident
control volume
F F
F G
F C
F B
System
F A
F I G U R E 5.2
coincident control volume.
External forces acting on system and
V5.5 Marine
propulsion
Flow out
or
time rate of change
of the linear
momentum of the
system
time rate of change
of the linear
momentum of the
contents of the
control volume
Equation 5.21 states that the time rate of change of system linear momentum is expressed
as the sum of the two control volume quantities: the time rate of change of the linear momentum
of the contents of the control volume, and the net rate of linear momentum flow through the control
surface. As particles of mass move into or out of a control volume through the control surface,
they carry linear momentum in or out. Thus, linear momentum flow should seem no more
unusual than mass flow.
For a control volume that is fixed 1and thus inertial2 and nondeforming, Eqs. 5.19, 5.20, and 5.21
provide an appropriate mathematical statement of Newton’s second law of motion as
net rate of flow
of linear momentum
through the
control surface
0
0t cv
Vr dV cs
VrV nˆ dA a F contents of the
control volume
(5.22)
Flow in
F fluid out
F fluid in
Control volume
F wall
We call Eq. 5.22 the linear momentum equation.
In our application of the linear momentum equation, we initially confine ourselves to fixed,
nondeforming control volumes for simplicity. Subsequently, we discuss the use of a moving but
inertial, nondeforming control volume. We do not consider deforming control volumes and accelerating
1noninertial2 control volumes. If a control volume is noninertial, the acceleration components
involved 1for example, translation acceleration, Coriolis acceleration, and centrifugal acceleration2
require consideration.
The forces involved in Eq. 5.22 are body and surface forces that act on what is contained in
the control volume as shown in the sketch in the margin. The only body force we consider in this
chapter is the one associated with the action of gravity. We experience this body force as weight, w.
The surface forces are basically exerted on the contents of the control volume by material just outside
the control volume in contact with material just inside the control volume. For example, a wall
in contact with fluid can exert a reaction surface force on the fluid it bounds. Similarly, fluid just
outside the control volume can push on fluid just inside the control volume at a common interface,
usually an opening in the control surface through which fluid flow occurs. An immersed object
can resist fluid motion with surface forces.
The linear momentum terms in the momentum equation deserve careful explanation. We clarify
their physical significance in the following sections.
V5.6 Force due to a
water jet
5.2.2 Application of the Linear Momentum Equation
The linear momentum equation for an inertial control volume is a vector equation 1Eq. 5.222. In
engineering applications, components of this vector equation resolved along orthogonal coordinates,
for example, x, y, and z 1rectangular coordinate system2 or r, u, and x 1cylindrical coordinate
system2, will normally be used. A simple example involving steady, incompressible flow is considered
first.