fluid_mechanics

3.9 Chapter Summary and Study Guide 131
Since V 1 V 2 V 0 and z 1 z 2 0, it follows that p 1 p 2 p 0
and the Bernoulli constant for this streamline, C 12 , is given by
C 12 1 2rV 2 0 p 0
Along the streamline from 132 to 142 we note that V 3 V 4 V 0
and z 3 z 4 h. As was shown in Example 3.5, application of
F ma across the streamline 1Eq. 3.122 gives p 3 p 1 gh because
the streamlines are straight and horizontal. The above facts
combined with the Bernoulli equation applied between 132 and 142
show that p 3 p 4 and that the Bernoulli constant along this
streamline is the same as that along the streamline between
112 and 122. That is, C 34 C 12 , or
p 3 1 2rV 2 3 gz 3 p 4 1 2rV 2 4 gz 4 C 34 C 12
Similar reasoning shows that the Bernoulli constant is the same
for any streamline in Fig. E3.18. Hence,
p 1 2rV 2 gz constant throughout the flow
Again from Example 3.5 we recall that
p 4 p 5 gH gH
If we apply the Bernoulli equation across streamlines from 142 to
152, we obtain the incorrect result “H p 4g V 2 42g.” The correct
result is H p 4g.
From the above we see that we can apply the Bernoulli equation
across streamlines 112–122 and 132–1421i.e., C 12 C 34 2 but not across
streamlines from 142 to 152. The reason for this is that while the flow
in the channel is “irrotational,” it is “rotational” between the flowing
fluid in the channel and the stationary fluid in the piezometer tube.
Because of the uniform velocity profile across the channel, it is seen
that the fluid particles do not rotate or “spin” as they move. The flow
is “irrotational.” However, as seen in Fig. E3.18b, there is a very thin
shear layer between 142 and 152 in which adjacent fluid particles interact
and rotate or “spin.” This produces a “rotational” flow. A more
complete analysis would show that the Bernoulli equation cannot be
applied across streamlines if the flow is “rotational” 1see Chapter 62.
V3.12 Flow over a
cavity
The Bernoulli
equation is not
valid for flows that
involve pumps or
turbines.
As is suggested by Example 3.18, if the flow is “irrotational” 1i.e., the fluid particles do not
“spin” as they move2, it is appropriate to use the Bernoulli equation across streamlines. However,
if the flow is “rotational” 1fluid particles “spin”2, use of the Bernoulli equation is restricted to flow
along a streamline. The distinction between irrotational and rotational flow is often a very subtle
and confusing one. These topics are discussed in more detail in Chapter 6. A thorough discussion
can be found in more advanced texts 1Ref. 32.
3.8.4 Other Restrictions
Another restriction on the Bernoulli equation is that the flow is inviscid. As is discussed in Section
3.4, the Bernoulli equation is actually a first integral of Newton’s second law along a streamline. This
general integration was possible because, in the absence of viscous effects, the fluid system considered
was a conservative system. The total energy of the system remains constant. If viscous effects
are important the system is nonconservative 1dissipative2 and energy losses occur. A more detailed
analysis is needed for these cases. Such material is presented in Chapter 5.
The final basic restriction on use of the Bernoulli equation is that there are no mechanical
devices 1pumps or turbines2 in the system between the two points along the streamline for which
the equation is applied. These devices represent sources or sinks of energy. Since the Bernoulli
equation is actually one form of the energy equation, it must be altered to include pumps or turbines,
if these are present. The inclusion of pumps and turbines is covered in Chapters 5 and 12.
In this chapter we have spent considerable time investigating fluid dynamic situations governed
by a relatively simple analysis for steady, inviscid, incompressible flows. Many flows can
be adequately analyzed by use of these ideas. However, because of the rather severe restrictions
imposed, many others cannot. An understanding of these basic ideas will provide a firm foundation
for the remainder of the topics in this book.
3.9 Chapter Summary and Study Guide
In this chapter, several aspects of the steady flow of an inviscid, incompressible fluid are discussed.
Newton’s second law, F ma, is applied to flows for which the only important forces are those
due to pressure and gravity (weight)—viscous effects are assumed negligible. The result is the oftenused
Bernoulli equation, which provides a simple relationship among pressure, elevation, and velocity
variations along a streamline. A similar but less often used equation is also obtained to describe
the variations in these parameters normal to a streamline.
The concept of a stagnation point and the corresponding stagnation pressure is introduced
as are the concepts of static, dynamic, and total pressure and their related heads.