fluid_mechanics

3.2 F ma along a Streamline 99
V3.2 Balancing
ball
In general it is not possible to integrate the pressure term because the density may not be constant
and, therefore, cannot be removed from under the integral sign. To carry out this integration we
must know specifically how the density varies with pressure. This is not always easily determined.
For example, for a perfect gas the density, pressure, and temperature are related according to
r pRT, where R is the gas constant. To know how the density varies with pressure, we must also
know the temperature variation. For now we will assume that the density and specific weight are constant
1incompressible flow2. The justification for this assumption and the consequences of compressibility
will be considered further in Section 3.8.1 and more fully in Chapter 11.
With the additional assumption that the density remains constant 1a very good assumption
for liquids and also for gases if the speed is “not too high”2, Eq. 3.6 assumes the following simple
representation for steady, inviscid, incompressible flow.
p 1 2rV 2 gz constant along streamline
(3.7)
V3.3 Flow past a
biker
This is the celebrated Bernoulli equation—a very powerful tool in fluid mechanics. In 1738 Daniel
Bernoulli 11700–17822 published his Hydrodynamics in which an equivalent of this famous equation
first appeared. To use it correctly we must constantly remember the basic assumptions used
in its derivation: 112 viscous effects are assumed negligible, 122 the flow is assumed to be steady,
132 the flow is assumed to be incompressible, 142 the equation is applicable along a streamline. In
the derivation of Eq. 3.7, we assume that the flow takes place in a plane 1the x–z plane2. In general,
this equation is valid for both planar and nonplanar 1three-dimensional2 flows, provided it is
applied along the streamline.
We will provide many examples to illustrate the correct use of the Bernoulli equation and will
show how a violation of the basic assumptions used in the derivation of this equation can lead to
erroneous conclusions. The constant of integration in the Bernoulli equation can be evaluated if sufficient
information about the flow is known at one location along the streamline.
E XAMPLE 3.2
The Bernoulli Equation
GIVEN Consider the flow of air around a bicyclist moving
through still air with velocity V 0 , as is shown in Fig. E3.2.
V 2 = 0 V 1 = V 0
FIND Determine the difference in the pressure between points
(2)
(1)
112 and 122.
SOLUTION
In a coordinate fixed to the ground, the flow is unsteady as the bicyclist
rides by. However, in a coordinate system fixed to the bike,
it appears as though the air is flowing steadily toward the bicyclist
with speed V 0 . Since use of the Bernoulli equation is restricted to F I G U R E E3.2
steady flows, we select the coordinate system fixed to the bike. If
the assumptions of Bernoulli’s equation are valid 1steady, incompressible,
inviscid flow2, Eq. 3.7 can be applied as follows along
the velocity distribution along the streamline, V1s2, was known.
the streamline that passes through 112 and 122
The Bernoulli equation is a general integration of F ma. To
p 1 1 2rV 2 1 gz 1 p 2 1 2rV 2 2 gz 2
determine p 2 p 1 , knowledge of the detailed velocity distribution
is not needed—only the “boundary conditions” at 112 and
We consider 112 to be in the free stream so that V 1 V 0 and 122 to 122 are required. Of course, knowledge of the value of V along
be at the tip of the bicyclist’s nose and assume that z 1 z 2 and the streamline is needed to determine the pressure at points
V 2 0 1both of which, as is discussed in Section 3.4, are reasonable
assumptions2. It follows that the pressure at 122 is greater than termine the speed, V 0 . As discussed in Section 3.5, this is the
between 112 and 122. Note that if we measure p 2 p 1 we can de-
that at 112 by an amount
principle upon which many velocity measuring devices are
p 2 p 1 1 (Ans) based.
2rV 2 1 1 2
2rV 0
If the bicyclist were accelerating or decelerating, the flow
COMMENTS A similar result was obtained in Example 3.1 would be unsteady 1i.e., V 0 constant2 and the above analysis
by integrating the pressure gradient, which was known because would be incorrect since Eq. 3.7 is restricted to steady flow.