fluid_mechanics

4.4 The Reynolds Transport Theorem 177
V A = Absolute velocity of A
V B
V CV
W B = Velocity of
W A = Velocity of
B relative
V CV
A relative
to control F I G U R E 4.22
to control
volume Relationship between absolute and relative
volume
velocities.
The Reynolds transport
theorem for a
moving control volume
involves the
relative velocity.
velocity, V, and the control volume velocity, V cv , will not be in the same direction so that the
relative and absolute velocities will have different directions 1see Fig. 4.222.
The Reynolds transport theorem for a moving, nondeforming control volume can be derived
in the same manner that it was obtained for a fixed control volume. As is indicated in Fig. 4.23, the
only difference that needs be considered is the fact that relative to the moving control volume the
fluid velocity observed is the relative velocity, not the absolute velocity. An observer fixed to
the moving control volume may or may not even know that he or she is moving relative to some
fixed coordinate system. If we follow the derivation that led to Eq. 4.19 1the Reynolds transport
theorem for a fixed control volume2, we note that the corresponding result for a moving control
volume can be obtained by simply replacing the absolute velocity, V, in that equation by the relative
velocity, W. Thus, the Reynolds transport theorem for a control volume moving with constant
velocity is given by
DB sys
Dt
where the relative velocity is given by Eq. 4.22.
0 0t
cv
rb dV cs
rb W nˆ dA
(4.23)
4.4.7 Selection of a Control Volume
Any volume in space can be considered as a control volume. It may be of finite size or it may be
infinitesimal in size, depending on the type of analysis to be carried out. In most of our cases,
the control volume will be a fixed, nondeforming volume. In some situations we will consider
control volumes that move with constant velocity. In either case it is important that considerable
thought go into the selection of the specific control volume to be used.
The selection of an appropriate control volume in fluid mechanics is very similar to the selection
of an appropriate free-body diagram in dynamics or statics. In dynamics, we select the body in which
we are interested, represent the object in a free-body diagram, and then apply the appropriate governing
laws to that body. The ease of solving a given dynamics problem is often very dependent on the
specific object that we select for use in our free-body diagram. Similarly, the ease of solving a given
fluid mechanics problem is often very dependent on the choice of the control volume used. Only by
practice can we develop skill at selecting the “best” control volume. None are “wrong,” but some are
“much better” than others.
Solution of a typical problem will involve determining parameters such as velocity, pressure,
and force at some point in the flow field. It is usually best to ensure that this point is located on
the control surface, not “buried” within the control volume. The unknown will then appear in the
convective term 1the surface integral2 of the Reynolds transport theorem. If possible, the control
Control volume
and system at time t
System at time
t + δt
Pathlines as
seen from the
moving control
volume
W = V – V CV
Flow as seen by an
observer moving with
velocity V CV
F I G U R E 4.23
Control volume and system as seen
by an observer moving with the
control volume.