Fluid Mechanics with teacher's notes

The expression for the conservation of energy in fluids is called Bernoulli’s
equation, and it can be expressed as follows:
BERNOULLI’S EQUATION
P + ⎯ 1
2 ⎯rv 2 + rgh = constant
pressure + kinetic energy per unit volume +
gravitational potential energy per unit volume =
constant along a given streamline
Note that Bernoulli’s equation differs slightly from the law of conservation of
energy given in Chapter 5. First of all, two of the terms on the left side of the
equation look like the terms for kinetic energy and gravitational potential
energy, but they contain density, r, instead of mass, m. That is because the
conserved quantity in Bernoulli’s equation is energy per unit volume—not
just energy—and density is equivalent to mass per unit volume. This statement
of the conservation of energy in fluids also includes an additional term:
pressure, P. Note that the units of pressure are equivalent to the units for energy
per unit volume.
If you wish to compare the energy in a given volume of fluid at two different
points, Bernoulli’s equation takes the following equivalent form:
P 1 + ⎯ 1
2 ⎯rv 1 2 + rgh1 = P 2 + ⎯ 1
2 ⎯rv 2 2 + rgh2
Bernoulli’s principle is a special case of Bernoulli’s equation
Two special cases of Bernoulli’s equation are worth mentioning here. First, if
the fluid is not moving, then both speeds are zero. This case is a static situation,
such as a column of water in a cylinder. If the height at the top of the
column, h 1, is defined as zero, and h 2 is the depth, then Bernoulli’s equation
reduces to the equation for pressure as a function of depth, introduced in
Section 9-2:
P 1 = P 2 + rgh 2 (static fluid)
Second, imagine again a fluid flowing through a horizontal pipe with a constriction.
Because the height of the fluid is constant, the gravitational potential
energy does not change. Bernoulli’s equation then reduces to the following:
P 1 + ⎯ 1
2 ⎯rv 1 2 = P2 + ⎯ 1
2 ⎯rv 2 2 (horizontal pipe)
This equation suggests that if v 1 is greater than v 2 at two different points
in the flow, then P 1 must be less than P 2. In other words, the pressure
decreases as speed increases. This is Bernoulli’s principle again, which we
now can see as a special case of Bernoulli’s equation. The conditions required
for this case tell us that Bernoulli’s principle is strictly true only when elevation
is constant.
Copyright © by Holt, Rinehart and Winston. All rights reserved.
P1A1
y1
∆x 1
v1
Bernoulli’s Principle
MATERIALS LIST
✔ single sheet of paper
You can see the effects of
Bernoulli’s principle on a sheet of
paper by holding the edge of the
sheet horizontally and blowing
across the top surface. The sheet
should rise as a result of the lower
air pressure above the sheet.
NSTA
TOPIC: Bernoulli’s principle
GO TO: www.scilinks.org
sciLINKS CODE: HF2094
y2
∆x2
v2
P 2A 2
Figure 9-15
As a fluid flows through this pipe, it
may change velocity, pressure, and
elevation.
SECTION 9-3
Teaching Tip
The equivalence of units in
Bernoulli’s equation can be seen
more clearly if you multiply the
units of pressure, N/m 2 , by a factor
equivalent to unity, m/m:
N
⎯
m
2⎯ × ⎯ m
m ⎯ = = ⎯ N •m J
⎯m 3
m
3⎯
TEACHER’S NOTES
For this lab to be effective, it is
best to hold a piece of very light
paper or soft plastic with both
hands and blow over it through
a straw.
Demonstration 9
Atomizer
Purpose Demonstrate an application
of Bernoulli’s principle.
Materials glass jar, water, plastic
straw, bicycle-horn bulb attached
to thin hollow tube
Procedure Place the straw in the
water. Use the bulb and tube to
blow air across the top of the
straw. Water will come up the
straw and spray out. The increased
air flow above the straw
results in a pressure decrease
inside it.
Fluid Mechanics 335
335