Fluid Mechanics with teacher's notes

transformed into internal energy because of the internal friction. Ideal fluids are
considered nonviscous, so they lose no kinetic energy due to friction as they flow.
Ideal fluids are also characterized by a steady flow. In other words, the
velocity, density, and pressure at each point in the fluid are constant. The flow
of an ideal fluid is also nonturbulent, which means that there are no eddy currents
in the moving fluid.
Although no real fluid has all the properties of an ideal fluid, the ideal fluid
model does help explain many properties of real fluids, so the model is a useful
tool for analysis. Unless otherwise stated, the fluids in the rest of our discussion
of fluid flow will be treated as ideal fluids.
PRINCIPLES OF FLUID FLOW
Fluid behavior is often very complex. A detailed analysis of the forces acting
on a fluid may be so complicated that even a supercomputer cannot create an
accurate model. However, several general principles describing the flow of fluids
can be derived relatively easily from basic physical laws.
The continuity equation results from mass conservation
Imagine that an ideal fluid flows into one end of a pipe and out the other end,
as shown in Figure 9-11. The diameter of the pipe is different on each end.
How does the speed of fluid flow change as the fluid passes through the pipe?
Because mass is conserved and because the fluid is incompressible, we
know that the mass flowing into the bottom of the pipe, m 1, must equal the
mass flowing out of the top of the pipe, m 2, during any given time interval:
m1 = m2 This simple equation can be expanded by recalling that m = rV and by
using the formula for the volume of a cylinder, V = A∆x.
r1V1 = r2V2 r1A1∆x1 = r2A2∆x2 The length of the cylinder, ∆x, is also the distance the fluid travels, which is
equal to the speed of flow multiplied by the time interval (∆x = v∆t).
r 1A 1v 1∆t = r 2A 2v 2∆t
For an ideal fluid, both the time interval and the density are the same on each
side of the equation, so they cancel each other out. The resulting equation is
called the continuity equation:
CONTINUITY EQUATION
A 1v 1 = A 2v 2
area speed in region 1 area speed in region 2
Copyright © by Holt, Rinehart and Winston. All rights reserved.
Motor-oil labels include a viscosity
index based on standards set by the
Society of Automotive Engineers
(SAE). High-viscosity oils (high SAE
numbers) are meant for use in hot
climates and in high-speed driving.
Low-viscosity oils (low SAE numbers)
are meant for use in severe
winter climates, when the engine is
cold for much of the time.
A1
v1
∆x1
Fluid Mechanics
v2
∆x2
A2
Figure 9-11
The mass flowing into the pipe must
equal the mass flowing out of the
pipe in the same time interval.
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SECTION 9-3
Teaching Tip
The viscosity index on motor oil
cans consists of an SAE number
(typically from 5 to 50) followed
by the letter W. The higher the
SAE number, the more viscous
the oil is. Low-viscosity oils are
meant for use in severe winter climates
because low-viscosity oils
flow more easily in cold temperatures.
For hot or high-speed driving
conditions, a high-viscosity
oil can be used because the excessive
heat effectively thins the oil.
Visual Strategy
Figure 9-11
Point out that the volumes of the
two shaded sections of the pipe
must be equal in order for the
continuity equation to apply.
Assume A1 is smaller than
Q pictured in Figure 9-11.
Would ∆x1 need to be longer or
shorter for the mass of liquid in
each section to still be equal?
A
longer
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