Objectives
•
•
•
•
•
•
Define drag.
Explain the difference between laminar and turbulent flow.
Explain the difference between frictional drag and pressure drag.
Define viscosity and explain how it can be measured.
Explain why a freely falling object has a terminal speed.
Use Stokes’ law and Poiseuille’s law to solve problems involving fluid
resistance.
When one solid object slides against another, a force of friction opposes the
motion. The direction of the force is opposite the direction of the object’s
velocity. When a solid object moves through a fluid, there is also a force that
opposes the motion. This force is called drag. When a boat moves through
water or when an airplane moves through air, a drag force is exerted in the
direction opposite the velocity of the boat or airplane.
To find out more about
resistance in fluid systems,
follow the links at
www.learningincontext.com.
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C HAPTER 4
You can feel a drag force when you stand in a high wind or when you put
your hand out the window of a moving car. In the first case, you are
stationary and the fluid is moving past you; in the second case you are
moving past the fluid. Drag occurs only when there is relative movement
between an object and a fluid.
RESISTANCE
Laminar (Streamlined) and Turbulent Flow
The drag exerted on an object by a fluid depends on many factors. The most
important are the speed of the object (or fluid), the size and shape of the
object, and the physical properties of the fluid. These factors make it difficult
to calculate drag exactly, but you can make approximations.
The simplest approximation is to ignore drag forces when they are small. For
example, you can usually ignore drag for an object moving slowly in fluids
such as air or water. But even very slow speeds produce significant drag in
fluids such as molasses and motor oil.
When drag forces cannot be ignored, you can make two approximations
about the fluid flow—the flow can be laminar or turbulent. Laminar, or
streamlined flow is a slow, smooth flow over a surface, in which the paths
of individual particles do not cross. Each path is called a streamline, as
illustrated in Figure 4.10. The fluid speed at the surface is zero, and fluid
moves in theoretical layers, or laminates, with increasing speed away from
the surface. Drag is produced by the friction between successive layers of
fluid. This is called frictional drag.
(a) Laminar flow
(b) Turbulent flow
(c) Turbulent flow
Figure 4.10
Laminar and turbulent flows around obstacles.
A turbulent wake can be created by a different shape or a higher fluid speed.
Turbulent flow is irregular flow with eddies and whorls causing fluid to
move in different directions. Turbulence is produced by high speeds, by
shapes that are not streamlined, and by sharp bends in the path of a fluid.
Turbulence produces the visible wake behind a moving boat and an invisible
wake behind a moving airplane or car.
Changing the direction of the fluid into
eddies and whorls requires work. When
the fluid does work, the pressure drops.
(Remember, W = –VΔP.) Thus, the fluid
pressure in the wake is less than the
fluid pressure in the streamlined flow.
This pressure difference causes a force
to act on the object in the direction
opposite its relative velocity. This force
is called pressure drag.
S ECTION 4.2
Figure 4.11
Turbulence creates a region of low
pressure in the fluid. The higher
pressure over the front surface area
of the object causes a drag force.
RESISTANCE
IN
FLUID SYSTEMS
185
Frictional drag and pressure drag both increase as speed increases.
Figure 4.12 shows how the total drag force on an automobile increases as its
speed increases. (The axes do not have a scale because numerical values of
drag depend on the aerodynamic design of the automobile. But the linear and
nonlinear shape of the graph is similar for all designs.)
Figure 4.12
The drag force on a car increases
as the car’s speed increases.
At low speeds, the drag force on the car is frictional drag. The force
increases linearly with speed. This means that doubling the speed doubles
the frictional drag force.
At higher speeds, turbulence and pressure drag are more and more important.
This force increases as the square of the speed. Doubling the speed increases
the pressure drag by a factor of four.
Viscosity
Friction between two solid surfaces causes a resistance to movement
between the surfaces. On a microscopic level, the resistance is due to
electrical forces between atoms and molecules in the surfaces of the solids.
Electrical forces also exist between atoms and molecules of a fluid. These
forces create internal friction in the fluid, which causes a resistance to
movement. Viscosity is the property of a fluid that describes this internal
friction. We use the Greek letter η (eta) to represent viscosity. Molasses and
bubble gum have high resistance to internal movement, and high viscosities.
Air and water have much lower viscosities.
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C HAPTER 4
RESISTANCE
Viscosity for fluid resistance is similar to the coefficients of friction for
mechanical resistance, but viscosity is not a simple coefficient. For example,
Figure 4.13 shows a layer of fluid of thickness Δy between two plates. The
bottom plate is held in place, and the top plate (of area A) is pulled to the
right at a constant speed v. The fluid in contact with the top plate moves with
the plate at speed v, and the fluid in contact with the bottom plate remains
motionless. The speed of the fluid between the top and bottom varies
linearly. Electrical forces between layers of fluid resist this variation in
motion between layers. (The top plate “drags” layers of fluid with it. This is
the source of the word that describes the resistance.) The force F is required
to overcome the resistance and keep the plate moving at constant speed.
Figure 4.13
The viscosity η of a fluid can be measured by pulling a plate
at constant speed across a layer of the fluid.
η=
FΔ y
Av
When the plate moves to the right at constant speed, no net force is acting on
the plate. Therefore, the fluid exerts a force of friction, or drag force Fdrag on
the plate to the left, opposing motion. The magnitude of the drag force
equals F.
As long as the plate speed v is not so large that turbulence occurs, the fluid
flow between the plates is laminar. The force F required to maintain a
constant speed for most fluids in laminar flow is found to be:
• proportional to A and v, and
• inversely proportional to the thickness of the fluid layer, Δy.
The proportionality constant is the viscosity of the fluid.
Fdrag = F = η
Av
Δy
Viscosity has units of (pressure) (time). The SI units for viscosity are
or Pa • s. The English units are
lb
• s or psi • s.
ft 2
S ECTION 4.2
RESISTANCE
IN
N •
s
m2
FLUID SYSTEMS
187
Different fluids resist motion differently, and therefore have different
viscosities. Table 4.2 lists values of viscosity for several fluids.
Table 4.2 Viscosities of Common Fluids
Fluid
Gases
Air
Water Vapor
Liquids
Water
Blood
Cooking Oil
Motor Oil
Corn Syrup
Molten Lava
Temperature
°C
0
20
100
100
0
20
100
37
20
20
20
950
Viscosity
Pa • s
1.7 × 10–5
1.9 × 10–5
2.2 × 10–5
1.3 × 10–5
0.0018
0.0010
0.00028
∼ 0.005
∼ 0.01
1
8
1000
The viscosity of most liquids decreases as the temperature increases. For
example, cold honey is very “thick” and difficult to pour (high viscosity).
But if you heat honey in a microwave oven it becomes “watery” (lower
viscosity). As the temperature increases, the molecules in the honey become
less and less tightly bound to each other. Thus, the force required to separate
the molecules also becomes less.
On the other hand, the viscosity of most gases increases with temperature.
Forces between gas molecules are exerted only during collisions, and there
are more collisions per second with higher temperature.
Motor oil is rated by viscosity by the Society of Automotive Engineers
(SAE). Oil with an SAE10 rating has lower viscosity than SAE40 oil (at the
same temperature). Oil rated as 10W40 has a viscosity at low engine
temperatures equivalent to SAE10 (at low temperatures) and a viscosity at
higher engine temperatures equivalent to SAE40 (at higher temperatures).
Stokes’ Law
In 1845, the Irish mathematician and physicist George Stokes used viscosity
and the equations of fluid flow to predict the drag force on a sphere moving
through a fluid. The result is called Stokes’ law. It applies to objects moving
at low enough speeds that the flow of fluid around the objects is streamlined,
or laminar. In these cases, there is no turbulence and the only drag force on
the objects is due to frictional drag.
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C HAPTER 4
RESISTANCE
Figure 4.14
The drag force on a sphere
moving through a fluid
opposes the sphere’s velocity.
The drag force acts in the direction opposite the object’s velocity (it opposes
motion). According to Stokes’ law, the drag force equals the product of a
constant (6π for a sphere), the radius r of the object, the speed v of the object
(or the relative speed between the object and fluid), and the fluid’s viscosity
η:
Fdrag = 6πrvη
Terminal Speed
When an object moves through a fluid, the drag force on the object increases
as the speed increases. If you drop a baseball from a high tower, it has a very
low speed and very small drag at first. The force of gravity (weight) acting
downward is greater than the drag force acting upward. Therefore, a net
force acts downward on the baseball and it accelerates downward. As the
speed increases the drag increases, until at some point the upward drag
equals the weight. At this point the forces acting on the baseball are balanced
and it no longer accelerates. The speed becomes constant. The terminal
speed of a falling object is the constant speed that occurs when the drag
force equals the gravitational force.
The terminal speed of a baseball is about 40 m/s, but the terminal speed of a
basketball is only about 20 m/s. Which ball has a greater drag force at any
given speed?
Example 4.5 Terminal Speed of a Dust Particle
When volcanoes erupt, rocks, dust, and ash are thrown into the
atmosphere. These particles fall back to Earth at terminal speeds that
depend on their size. The terminal speed of a particle in a volcanic dust
cloud can be estimated using a sphere to model the particle.
Estimate the terminal speed of a 50-μm-diameter particle if the density of
the volcanic material is 2500 kg/m3.
S ECTION 4.2
RESISTANCE
IN
FLUID SYSTEMS
189
Solution:
When the particle moves at terminal speed, no net
force is acting on it. The force of gravity equals
the drag force. Use Stokes’ law to calculate the
drag force.
The radius r of the particle is one-half the
diameter:
r=
50 μm
−5
= 25 μm or 2.5 × 10 m
2
The mass m of the particle is the product of the density ρ and
4
volume V. The volume of a spherical particle is
πr3:
3
m = ρV
kg ⎞ ⎡ 4
⎛
−5
3⎤
= ⎜ 2500 3 ⎟ ⎢ π 2.5 × 10 m ⎥
m ⎠ ⎣3
⎝
⎦
(
= 1.64 × 10
−10
)
kg
The weight of the particle is Fg:
Fg = mg = (1.64 × 10−10 kg) (9.80 m/s 2 )
−9
2
−9
= 1.61 × 10 kg • m/s or 1.61 × 10 N
Use Stokes’ law, and equate the drag to the weight. Solve for
the speed v. Use the value of viscosity from Table 4.2 for air at
20°C:
Fdrag = Fg
6πrv η = 1.61 × 10−9 N
v=
=
1.61 × 10−9 N
6π r η
1.61 × 10−9 N
N ⎞
⎛
6π 2.5 × 10−5 m ⎜1.9 × 10−5 2 • s ⎟
m
⎠
⎝
(
)
= 0.18 m/s or 18 cm/s
The terminal speed of a 50-μm particle of volcanic dust is about
18 centimeters per second.
190
C HAPTER 4
RESISTANCE
Poiseuille’s Law
Poiseuille’s law gives the volume flow rate of a fluid flowing through a tube
or pipe. Like Stokes’ law, Poiseuille’s law applies to laminar flow.
Figure 4.15 illustrates laminar flow in a pipe. The fluid layer at the center
moves the fastest, and layers nearer the wall move more slowly. (Fluid in
contact with the wall does not move.)
Figure 4.15
Cross section through the
center of a pipe, showing
laminar flow. The fluid at
the center of the pipe has
the highest speed. The
speed decreases closer
to the wall of the pipe.
Jean Louis Poiseuille was a physician who was also trained as a physicist
and mathematician. In the mid-1840s, he experimented with water flowing
through glass capillary tubes as a simulation of blood flowing through small
blood vessels. Poiseuille learned that the rate at which fluid flows through a
tube increases proportionately to the pressure applied and to the fourth
power of the radius of the tube.
•
According to Poiseuille’s law, the volume flow rate V (m3/s) of a fluid of
viscosity η through a tube or pipe of radius r and length L is:
•
V =−
π r 4Δ P
8 ηL
In this equation, ΔP is the change in pressure of the fluid as it flows the
length L. The internal friction of the fluid causes the pressure to decrease as
the fluid flows. Therefore, ΔP = P2 – P1 is negative because P2 < P1. With a
•
negative ΔP, V is positive.
Figure 4.16
A fluid’s viscosity,
or internal friction,
causes the pressure
to drop along the
direction of flow.
S ECTION 4.2
RESISTANCE
IN
FLUID SYSTEMS
191
Example 4.6 Blood Flow Rate
The flow rate of blood through a vein is 3.2 cm3/s. The diameter of the
vein is 3.6 mm. If the pressure drops by 1100 Pa between two points in
the vein, estimate the length of the vein between the points.
Solution:
•
Convert units of V and r to SI:
•
V = 3.2
cm3 1 m3
3
−6
•
= 3.2 × 10 m /s
s 106 cm3
r = d /2 = (3.6 mm)/2 = 1.8 mm = 1.8 × 10−3 m
Solve the equation for Poiseuille’s law for the length L. Use
the value of viscosity from Table 4.2:
V =−
π r 4Δ P
8 ηL
L=−
π r 4Δ P
•
8 ηV
•
=−
π
8
(1.8 × 10−3 m) 4 (−1100 Pa)
3⎞
⎛
(0.005 Pa • s) ⎜ 3.2 × 10−6 m ⎟
s ⎠
⎝
= 0.28 m or 28 cm
The vein is about 28 cm long.
Factors Affecting Flow Through a Pipe
•
Resistance decreases the flow rate V of fluid through a pipe. Poiseuille’s law
shows how this resistance depends on three factors: (1) the radius (or crosssectional area) of the pipe, (2) the length of the pipe, and (3) the viscosity of
the fluid. These dependencies can be illustrated using graphs of volume flow
rate versus pressure drop. For each graph, you can define a “fluid resistance”
R as the ratio of the prime mover to the volume flow rate. The prime mover in
fluid systems is pressure change, or in this case pressure drop. Pressure drop
is –ΔP. (ΔP is negative, so pressure drop and fluid resistance are positive.)
R=
−Δ P
pressure drop
= •
volume flow rate
V
Therefore, R is the slope of each line graphed on the following pages.
A high-resistance pipe has a large slope, and a low-resistance pipe has a
small slope.
192
C HAPTER 4
RESISTANCE
(1) Dependence on Radius
Compare the flow of a fluid through two pipes of the same length, but one
with a small radius and the other with a large radius. The larger pipe has a
greater cross-sectional area and can move a greater volume of fluid per
second. This pipe also has a lower resistance to flow than the smaller pipe.
According to Poiseuille’s law, the volume flow rate increases as the fourth
power of the radius. If the radius of the large pipe is twice the radius of the
small pipe, the volume flow rate is sixteen times higher (if r → 2r, then
r 4 → 16r 4).
Figure 4.17 shows graphically the effect of increasing the pipe radius. The
volume flow rates and pressure drops are shown for water flowing through a
1-m length of pipe for three pipe radii. Which graph has the highest slope?
The highest resistance? For a pressure drop of 25 Pa, if you increase the
radius of the pipe from 1.0 cm to 1.5 cm, you increase the cross-sectional
area, decrease the flow resistance, and increase the volume flow rate from
98.2 cm3/s to 497 cm3/s.
Figure 4.17
Fluid resistance decreases
as pipe radius
and cross-sectional area
increase.
(2) Dependence on Length
Figure 4.18 shows how fluid resistance changes when the pipe length
changes. Longer pipes have higher fluid resistance. If you double the length
of a pipe, you double the resistance and the volume flow rate is halved.
Volume flow rate is inversely proportional to length.
Figure 4.18
Fluid resistance increases
as pipe length increases.
S ECTION 4.2
RESISTANCE
IN
FLUID SYSTEMS
193
(3) Dependence on Viscosity
Figure 4.19 shows how resistance varies with the viscosity of the fluid.
Which graph has the highest slope? The highest resistance? Volume flow rate
is inversely proportional to viscosity. If you use a fluid with half the
viscosity, you double the volume flow rate.
Figure 4.19
Fluid resistance increases
as viscosity increases.
If the flow becomes turbulent, resistance increases rapidly. As illustrated in
Figure 4.20, bends and Ts in a pipe or air duct cause turbulence. When it is
important to maintain laminar flow and reduce resistance, designers use
curves with radii as large as possible rather than abrupt changes in the path
of a fluid.
Figure 4.20
Abrupt changes in the direction of fluid flow
can cause turbulence and increase resistance.
Obstructions or restrictions also cause turbulence. For example, the grill of a
car is an obstruction that causes turbulence, affecting the aerodynamic drag
of an automobile. Filters in air ducts are restrictions. Figure 4.21 shows the
pressure drops along an air duct containing a filter. There is a small pressure
drop from P1 to P2 (exaggerated in the graph) along the length of the duct,
and there is a much larger drop from P2 to P3 because of the filter. If the
filter is dirty, the pressure drop is even larger. In fact, a clogged filter can
almost stop airflow.
194
C HAPTER 4
RESISTANCE
(a) Pressure vs position for a clean filter
(b) Pressure vs position for a dirty filter. The volume flow rate for this situation is lower
than that for (a).
Figure 4.21
Pressure drop along the flow path for an air
duct and filter. Note the pressure drop from P1 to P2 is
exaggerated, as is the drop from P3 to the right.
Summary
• Drag is the force that opposes the motion of an object moving through a
fluid or the force a moving fluid exerts on a stationary object.
• Laminar flow is slow, smooth flow over a surface, where particles
follow streamlines. The streamlines define theoretical layers of fluid
that do not mix. The friction between successive layers of fluid is called
frictional drag.
S ECTION 4.2
RESISTANCE
IN
FLUID SYSTEMS
195
• Turbulent flow is irregular flow with eddies and whorls that mix the
fluid. Turbulence causes a wake behind a moving object. The pressure
difference between the fluid outside the wake and the fluid inside the
wake causes pressure drag.
• Drag increases with speed. When turbulence is created, pressure drag
increases more rapidly than friction drag.
• Viscosity is the property of a fluid that describes its internal friction.
The SI units of viscosity are Pa • s.
• Stokes’ law can be used to calculate the drag force on a sphere moving
at constant speed in a viscous fluid.
• When the drag equals the gravitational force acting on a falling body,
the body falls at a constant speed, called the terminal speed of the body.
• Poiseuille’s law can be used to calculate the volume flow rate or
pressure drop of a viscous fluid flowing through a tube or pipe.
Exercises
1. Drag on an object moving through a stationary fluid ____________
(increases or decreases) as the object’s speed increases.
2. If a fluid flows past a stationary object, the drag on the object
____________ (increases or decreases) as the fluid speed increases.
3. Fluid resistance increases as flow becomes ____________ (laminar or
turbulent).
4. Why do downhill ski racers and bicycle racers wear specially formed
clothing and helmets, and form their bodies in the shape of an egg?
5. The forces acting on four objects moving through a fluid are shown
below. The force vectors are drawn to scale. Which object(s) moves at
constant velocity?
(a)
196
C HAPTER 4
RESISTANCE
(b)
(c)
(d)
6. Two metal plates are separated by a 1.5-mm thickness of motor oil, as
shown in Figure 4.13. The top plate measures 12 cm by 15 cm.
(a) What force is required to move the top plate at a constant speed of
0.4 cm/s? Use the value of viscosity in Table 4.2.
(b) If the force on the top plate is halved, what is the plate’s speed?
7. (a) The plate in Exercise 6 moves to the right at a constant speed of
0.4 cm/s. What are the magnitude and direction of the drag force on
the plate?
(b) The oil between the plates is replaced with water at 20°C. What is the
drag force on the top plate when it moves to the right at a constant
0.4 cm/s?
(c) The water is replaced with air at 20°C. What is the drag force for the
same motion of the top plate?
(d) Suppose you are approximating the drag force on a plate moving
through a fluid at 0.4 cm/s. For which fluid—oil, water, or air—are
you most accurate when you say the drag force is approximately
zero?
8. A sky diver falls at a terminal speed of 14 ft/s with her parachute open.
Her weight plus the weight of her clothing and gear is 168 lb. What are
the magnitude and direction of the drag force on the sky diver?
9. Metal particles produced in a milling machine are collected in an oil tank.
The particles can be modeled as spheres. One particle is 1.5 mm in
diameter and weighs 7.4 × 10–5 N. What is the terminal speed of the
particle in the oil if the oil’s viscosity is 2.8 Pa • s?
10. The volume flow rates and pressure drops for two pipes are graphed
below.
(a) Which pipe has the higher resistance?
(b) If the pipes are the same length, which one has the larger diameter?
(c) If the pipes have the same diameter, which one is longer?
S ECTION 4.2
RESISTANCE
IN
FLUID SYSTEMS
197
11. (a) Solve the Poiseuille’s law equation for fluid resistance.
R=
– ΔP
•
V
=?
What are the SI units of this quantity?
(b) A 40-cm length of copper tubing has an internal diameter of 0.32 cm.
Calculate the resistance of the tubing when water at 20°C flows
through it.
(c) What is the water flow rate through the tubing when there is a
pressure drop of 1.5 kPa between the tubing inlet and its outlet?
(d) A second, identical 40-cm length of tubing is connected to the outlet
of the first length. What is the water flow rate through the combined
length if the pressure drop is 1.5 kPa?
12. A 12-inch-diameter pipeline transports crude oil 54.4 miles. The
maximum pressure in the pipeline is 950 psi. The pressure drop is 850 psi.
The viscosity of the oil is 1.9 × 10– 4 psi • s.
(a) Find the volume flow rate (in in3/s) of crude oil through the pipeline.
(b) Calculate the amount of work (in ft • lb) that must be done by pumps
to operate the pipeline for one second.
(c) Suppose the pipeline is replaced with one whose diameter is
14 inches. What is the pressure drop required to produce the same
flow rate as in (a)?
13. The two water pipes shown below have the same diameter and length.
The water flow rate through the pipes is the same, but the pressure drop is
not the same. What could cause the difference in pressure drop? Explain.
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RESISTANCE
14. The piston in an air compressor has a diameter of 12.5 cm and a height of
15.0 cm. There is a 0.0127-cm space between the piston and cylinder wall
filled with lubricating oil. The oil has a viscosity of 0.35 Pa • s. If the
average speed of the piston is 43.0 m/s, what is the average drag on the
piston caused by the lubricating oil?
15. The speed of a sailboat is limited by the drag of water on the hull. At low
speeds, you can use a laminar flow model to estimate drag. The water
touching, or wetting the hull moves with the boat. Successive layers of
water slide against each other, with each layer moving at a lower speed
than previous layers. At a distance of approximately 0.3 mm, water does
not slide in layers. You can use this distance as the thickness of the fluid
layer in the viscosity equation.
(a) When moving at constant velocity, a sailboat has a wetted hull surface
area of 21.5 m2. The sails apply a force of 223 N in the direction the
boat is moving. What is the speed of the boat? (Assume the water’s
temperature is 20°C.)
(b) If the sailboat has a winged keel that reduces the wetted surface by
15%, what is the speed of the boat?
S ECTION 4.2
RESISTANCE
IN
FLUID SYSTEMS
199