Munson B.R. Fundamentals of Fluid Mechanics

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Problems 87

frictionless horizontal shaft be located? (b) What is the magnitude

of the force on the gate when it opens?

2.73 A thin 4-ft-wide, right-angle gate with negligible mass is free

to pivot about a frictionless hinge at point O, as shown in Fig.

P2.73. The horizontal portion of the gate covers a 1-ft-diameter

drain pipe which contains air at atmospheric pressure. Determine

the minimum water depth, h, at which the gate will pivot to allow

water to flow into the pipe.

contains a spout that is closed by a 6-in.-diameter circular gate

that is hinged along one side as illustrated. The horizontal axis of

the hinge is located 10 ft below the water surface. Determine the

minimum torque that must be applied at the hinge to hold the

gate shut. Neglect the weight of the gate and friction at the hinge.

2.77 A 4-ft-tall, 8-in.-wide concrete (150 lbft

) retaining wall is

built as shown in Fig. P2.77. During a heavy rain, water fills the

space between the wall and the earth behind it to a depth h. Deter-

mine the maximum depth of water possible without the wall tipping

over. The wall simply rests on the ground without being anchored

to it.

2.79 (See Fluids in the News article titled “The Three Gorges

Dam,” Section 2.8.) (a) Determine the horizontal hydrostatic force

on the 2309-m-long Three Gorges Dam when the average depth of

the water against it is 175 m. (b) If all of the 6.4 billion people on

Earth were to push horizontally against the Three Gorges Dam,

could they generate enough force to hold it in place? Support your

answer with appropriate calculations.

Section 2.10 Hydrostatic Force on a Curved Surface

2.80 Obtain a photograph/image of a situation in which the

hydrostatic force on a curved surface is important. Print this

photo and write a brief paragraph that describes the situation

involved.

F I G U R E P2.73

Water

Hinge

Right-angle gate

Width = 4 ft

1-ft-diameter pipe

3 ft

F I G U R E P2.77

8 in.

4 ft

F I G U R E P2.76

Water

Air

10 psi

Axis

6-in.-diameter

gate

10 ft

F I G U R E P2.78

= h

Water

= 10 ft

80 ft

ᐉ

2.74 An open rectangular tank is 2 m wide and 4 m long. The

tank contains water to a depth of 2 m and oil on top of

the water to a depth of 1 m. Determine the magnitude and location

of the resultant fluid force acting on one end of the tank.

*2.75 An open rectangular settling tank contains a liquid suspen-

sion that at a given time has a specific weight that varies

approximately with depth according to the following data:

The depth corresponds to the free surface. Determine, by

means of numerical integration, the magnitude and location of the

resultant force that the liquid suspension exerts on a vertical wall of

the tank that is 6 m wide. The depth of fluid in the tank is 3.6 m.

2.76 The closed vessel of Fig. P2.76 contains water with an air

pressure of 10 psi at the water surface. One side of the vessel

h  0

1SG  0.82

h (m) ( )

0 10.0

0.4 10.1

0.8 10.2

1.2 10.6

1.6 11.3

2.0 12.3

2.4 12.7

2.8 12.9

3.2 13.0

3.6 13.1



*2.78 Water backs up behind a concrete dam as shown in

Fig. P2.78. Leakage under the foundation gives a pressure distribu-

tion under the dam as indicated. If the water depth, h, is too great,

the dam will topple over about its toe (point A). For the dimensions

given, determine the maximum water depth for the following widths

of the dam: Base your analysis on a

unit length of the dam. The specific weight of the concrete is

150 lb



/  20, 30, 40, 50, and 60 ft.

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88 Chapter 2 ■ Fluid Statics

F I G U R E P2.84

6 ft

Gate

Hinge

Water

2.85 The air pressure in the top of the 2-liter pop bottle shown in

Video V2.5 and Fig. P2.85 is 40 psi, and the pop depth is 10 in. The

bottom of the bottle has an irregular shape with a diameter of 4.3 in.

(a) If the bottle cap has a diameter of 1 in. what is the magnitude of

the axial force required to hold the cap in place? (b) Determine the

force needed to secure the bottom 2 in. of the bottle to its cylindri-

cal sides. For this calculation assume the effect of the weight of the

pop is negligible. (c) By how much does the weight of the pop in-

crease the pressure 2 in. above the bottom? Assume the pop has the

same specific weight as that of water.

2.86 Hoover Dam (see Video 2.4) is the highest arch-gravity type

of dam in the United States. A cross section of the dam is shown in

Fig. P2.86(a). The walls of the canyon in which the dam is located

are sloped, and just upstream of the dam the vertical plane shown in

Figure P2.86(b) approximately represents the cross section of the

water acting on the dam. Use this vertical cross section to estimate

the resultant horizontal force of the water on the dam, and show

where this force acts.

2.87 A plug in the bottom of a pressurized tank is conical in shape,

as shown in Fig. P2.87. The air pressure is 40 kPa and the liquid in

F I G U R E P2.83

Sphere diameter = 3 ft

Vent

Cable

12 in.

10 in.

1-in. diameter

4.3-in. diameter

air

= 40 psi

F I G U R E P2.85

45 ft

727 ft

660 ft

715 ft.

(b)(a)

880 ft

290 ft

F I G U R E P2.86

Air

Liquid

40 kPa

3 m

1 m

60°

F I G U R E P2.87

F I G U R E P2.82

Area = A Area = A

2.81 A 2-ft-diameter hemispherical plexiglass “bubble” is to be

used as a special window on the side of an above-ground swimming

pool. The window is to be bolted onto the vertical wall of the pool

and faces outward, covering a 2-ft-diameter opening in the wall.

The center of the opening is 4 ft below the surface. Determine the

horizontal and vertical components of the force of the water on the

hemisphere.

2.82 Two round, open tanks containing the same type of fluid rest

on a table top as shown in Fig. P2.82. They have the same bottom

area, A, but different shapes. When the depth, h, of the liquid in

the two tanks is the same, the pressure force of the liquids on the

bottom of the two tanks is the same. However, the force that the

table exerts on the two tanks is different because the weight in each

of the tanks is different. How do you account for this apparent

paradox?

2.83 Two hemispherical shells are bolted together as shown in Fig.

P2.83. The resulting spherical container, which weighs 300 lb, is

filled with mercury and supported by a cable as shown. The

container is vented at the top. If eight bolts are symmetrically located

around the circumference, what is the vertical force that each bolt

must carry?

2.84 The 18-ft-long gate of Fig. P2.84 is a quarter circle and is

hinged at H. Determine the horizontal force, P, required to hold

the gate in place. Neglect friction at the hinge and the weight of

the gate.

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Problems 89

the tank has a specific weight of Determine the magni-

tude, direction, and line of action of the force exerted on the curved

surface of the cone within the tank due to the 40-kPa pressure and

the liquid.

2.88 The homogeneous gate shown in Fig. P2.88 consists of one

quarter of a circular cylinder and is used to maintain a water depth

of 4 m. That is, when the water depth exceeds 4 m, the gate opens

slightly and lets the water flow under it. Determine the weight of

the gate per meter of length.

27 kN



2.89 The concrete seawall of Fig.

P2.89 has a curved surface and restrains seawater at a depth of 24 ft.

The trace of the surface is a parabola as illustrated. Determine the

moment of the fluid force (per unit length) with respect to an axis

through the toe (point A).

1specific weight



150 lb



2.90 A cylindrical tank with its axis horizontal has a diameter of

2.0 m and a length of 4.0 m. The ends of the tank are vertical planes.

A vertical, 0.1-m-diameter pipe is connected to the top of the tank.

The tank and the pipe are filled with ethyl alcohol to a level of 1.5 m

above the top of the tank. Determine the resultant force of the

alcohol on one end of the tank and show where it acts.

2.91 If the tank ends in Problem 2.90 are hemispherical, what is

the magnitude of the resultant horizontal force of the alcohol on

one of the curved ends?

2.92 An open tank containing water has a bulge in its vertical side

that is semicircular in shape as shown in Fig. P2.92. Determine the

horizontal and vertical components of the force that the water ex-

erts on the bulge. Base your analysis on a 1-ft length of the bulge.

2.93 A closed tank is filled with water and has a 4-ft-diameter hemi-

spherical dome as shown in Fig. P2.93. A U-tube manometer is con-

nected to the tank. Determine the vertical force of the water on the

dome if the differential manometer reading is 7 ft and the air pressure

at the upper end of the manometer is 12.6 psi.

2.94 A 3-m-diameter open cylindrical tank contains water and has a

hemispherical bottom as shown in Fig. P2.94. Determine the magni-

tude, line of action, and direction of the force of the water on the

curved bottom.

2.95 Three gates of negligible weight are used to hold back water

in a channel of width b as shown in Fig. P2.95 on the next page. The

force of the gate against the block for gate (b) is R. Determine (in

terms of R) the force against the blocks for the other two gates.

Section 2.11 Buoyancy, Flotation, and Stability

2.96 Obtain a photograph/image of a situation in which

Archimedes’ principle is important. Print this photo and write a

brief paragraph that describes the situation involved.

2.97 A freshly cut log floats with one fourth of its volume pro-

truding above the water surface. Determine the specific weight of

the log.

F I G U R E P2.88

4 m

1 m

Pivot

F I G U R E P2.92

6 ft

3 ft

Water

Bulge

F I G U R E P2.93

20 psi

4-ft diameter

Air

Gage

fluid

(

SG = 3.0)

Water

2 ft

5 ft

Seawater

24 ft

y = 0.2x

15 ft

F I G U R E P2.89

Water

8 m

3 m

F I G U R E P2.94

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90 Chapter 2 ■ Fluid Statics

2.100 When the Tucurui Dam was constructed in northern

Brazil, the lake that was created covered a large forest of valuable

hardwood trees. It was found that even after 15 years underwater

the trees were perfectly preserved and underwater logging was

started. During the logging process a tree is selected, trimmed,

and anchored with ropes to prevent it from shooting to the surface

like a missile when cut. Assume that a typical large tree can be ap-

proximated as a truncated cone with a base diameter of 8 ft, a top

diameter of 2 ft, and a height of 100 ft. Determine the resultant

vertical force that the ropes must resist when the completely sub-

merged tree is cut. The specific gravity of the wood is approxi-

mately 0.6.

†2.101 Estimate the minimum water depth needed to float a canoe

carrying two people and their camping gear. List all assumptions

and show all calculations.

F I G U R E P2.99

Δh

(a)(b)

F I G U R E P2.104

4 m

1 m

diameter

Water

(a)(b)

Hinge

Block

(c)

F I G U R E P2.95

Air

Plastic bottle

Water

Test tube

F I G U R E P2.102

F I G U R E P2.105

Fluid

surface

Hydrometer

2.98 A river barge, whose cross section is approximately rectan-

gular, carries a load of grain. The barge is 28 ft wide and

90 ft long. When unloaded its draft (depth of submergence)

is 5 ft, and with the load of grain the draft is 7 ft. Determine:

(a) the unloaded weight of the barge, and (b) the weight of the

grain.

2.99 A tank of cross-sectional area A is filled with a liquid of

specific weight as shown in Fig. P2.99a. Show that when a

cylinder of specific weight and volume V

–

is floated in the liq-

uid (see Fig. P2.99b), the liquid level rises by an amount

¢h  1g



2 V



2.102 An inverted test tube partially filled with air floats in a plas-

tic water-filled soft drink bottle as shown in Video V2.7 and Fig.

P2.102. The amount of air in the tube has been adjusted so that it

just floats. The bottle cap is securely fastened. A slight squeezing of

the plastic bottle will cause the test tube to sink to the bottom of the

bottle. Explain this phenomenon.

2.103 An irregularly shaped piece of a solid material weighs 8.05 lb

in air and 5.26 lb when completely submerged in water. Determine

the density of the material.

2.104 A 1-m-diameter cylindrical mass, M, is connected to a 2-

m-wide rectangular gate as shown in Fig. P2.104. The gate is to

open when the water level, h, drops below 2.5 m. Determine the

required value for M. Neglect friction at the gate hinge and the

pulley.

2.105 When a hydrometer (see Fig. P2.105 and Video V2.8) hav-

ing a stem diameter of 0.30 in. is placed in water, the stem pro-

trudes 3.15 in. above the water surface. If the water is replaced with

a liquid having a specific gravity of 1.10, how much of the stem

would protrude above the liquid surface? The hydrometer weighs

0.042 lb.

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Problems 91

2.106 A 2-ft-thick block constructed of wood (SG  0.6) is sub-

merged in oil (SG  0.8), and has a 2-ft-thick aluminum (specific

weight  168 lbft

) plate attached to the bottom as indicated in Fig.

P2.106. Determine completely the force required to hold the block

in the position shown. Locate the force with respect to point A.

2.107 (See Fluids in the News article titled “Concrete canoe,”

Section 2.11.1.) How much extra water does a 147-lb concrete ca-

noe displace compared to an ultralightweight 38-lb Kevlar canoe of

the same size carrying the same load?

2.108 An iceberg (specific gravity 0.917) floats in the ocean (spe-

cific gravity 1.025). What percent of the volume of the iceberg is

under water?

Section 2.12 Pressure Variation in a Fluid

with Rigid-Body Motion

2.109 Obtain a photograph/image of a situation in which the pres-

sure variation in a fluid with rigid-body motion is involved. Print

this photo and write a brief paragraph that describes the situation

involved.

2.110 It is noted that while stopping, the water surface in a glass of

water sitting in the cup holder of a car is slanted at an angle of 15º

relative to the horizontal street. Determine the rate at which the car

is decelerating.

2.111 An open container of oil rests on the flatbed of a truck that

is traveling along a horizontal road at As the truck slows

uniformly to a complete stop in 5 s, what will be the slope of the oil

surface during the period of constant deceleration?

2.112 A 5-gal, cylindrical open container with a bottom area of

is filled with glycerin and rests on the floor of an elevator.

(a) Determine the fluid pressure at the bottom of the container

when the elevator has an upward acceleration of (b) What

resultant force does the container exert on the floor of the elevator

during this acceleration? The weight of the container is negligible.

(Note: )

2.113 An open rectangular tank 1 m wide and 2 m long contains

gasoline to a depth of 1 m. If the height of the tank sides is 1.5 m,

what is the maximum horizontal acceleration (along the long axis of

the tank) that can develop before the gasoline would begin to spill?

2.114 If the tank of Problem 2.113 slides down a frictionless plane

that is inclined at with the horizontal, determine the angle the

free surface makes with the horizontal.

2.115 A closed cylindrical tank that is 8 ft in diameter and 24 ft

long is completely filled with gasoline. The tank, with its long axis

horizontal, is pulled by a truck along a horizontal surface. Deter-

mine the pressure difference between the ends (along the long axis

of the tank) when the truck undergoes an acceleration of

5 ft



30°

1 gal  231 in.

3 ft



120 in.

55 mi



hr.

2.116 The open U-tube of Fig. P2.116 is partially filled with a liq-

uid. When this device is accelerated with a horizontal acceleration

a, a differential reading h develops between the manometer legs

which are spaced a distance apart. Determine the relationship be-

tween a, and h./,

F I G U R E P2.106

Oil

Aluminum

4 ft

0.5 ft

6 ft

10 ft

F I G U R E P2.116

ᐉ

F I G U R E P2.121

Receiver

Light rays

6 ft

Δh

= 7 rpm

Mercury

■ Lab Problems

2.122 This problem involves the force needed to open a gate that

covers an opening in the side of a water-filled tank. To proceed with

this problem, go to Appendix H which is located on the book’s web

site, www.wiley.com/college/munson.

2.117 An open 1-m-diameter tank contains water at a depth of 0.7

m when at rest. As the tank is rotated about its vertical axis the cen-

ter of the fluid surface is depressed. At what angular velocity will

the bottom of the tank first be exposed? No water is spilled from the

tank.

2.118 An open, 2-ft-diameter tank contains water to a depth of 3 ft

when at rest. If the tank is rotated about its vertical axis with an an-

gular velocity of 180 revmin, what is the minimum height of the

tank walls to prevent water from spilling over the sides?

2.119 A child riding in a car holds a string attached to a floating,

helium-filled balloon. As the car decelerates to a stop, the balloon

tilts backwards. As the car makes a right-hand turn, the balloon

tilts to the right. On the other hand, the child tends to be forced

forward as the car decelerates and to the left as the car makes a

right-hand turn. Explain these observed effects on the balloon and

child.

2.120 A closed, 0.4-m-diameter cylindrical tank is completely

filled with oil and rotates about its vertical longitudinal

axis with an angular velocity of Determine the difference

in pressure just under the vessel cover between a point on the cir-

cumference and a point on the axis.

2.121 (See Fluids in the News article titled “Rotating mercury

mirror telescope,” Section 2.12.2.) The largest liquid mirror tele-

scope uses a 6-ft-diameter tank of mercury rotating at 7 rpm to pro-

duce its parabolic-shaped mirror as shown in Fig. P2.121. Deter-

mine the difference in elevation of the mercury, , between the

edge and the center of the mirror.

¢h

40 rad



1SG  0.92

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92 Chapter 2 ■ Fluid Statics

2.123 This problem involves the use of a cleverly designed appa-

ratus to investigate the hydrostatic pressure force on a submerged

rectangle. To proceed with this problem, go to Appendix H which is

located on the book’s web site, www.wiley.com/college/munson.

2.124 This problem involves determining the weight needed to hold

down an open-bottom box that has slanted sides when the box is filled

with water. To proceed with this problem, go to Appendix H which is

located on the book’s web site, www.wiley.com/college/munson.

2.125 This problem involves the use of a pressurized air pad to

provide the vertical force to support a given load. To proceed with

this problem, go to Appendix H which is located on the book’s web

site, www.wiley.com/college/munson.

■ Life Long Learning Problems

2.126 Although it is relatively easy to calculate the net hydrostatic

pressure force on a dam, it is not necessarily easy to design and

construct an appropriate, long-lasting, inexpensive dam. In fact, in-

spection of older dams has revealed that many of them are in peril

of collapse unless corrective action is soon taken. Obtain informa-

tion about the severity of the poor conditions of older dams

throughout the country. Summarize your findings in a brief report.

2.127 Over the years the demand for high-quality, first-growth

timber has increased dramatically. Unfortunately, most of the trees

that supply such lumber have already been harvested. Recently,

however, several companies have started to reclaim the numerous

high-quality logs that sank in lakes and oceans during the logging

boom times many years ago. Many of these logs are still in excel-

lent condition. Obtain information, particularly that associated with

the use of fluid mechanics concepts, about harvesting sunken logs.

Summarize your findings in a brief report.

2.128 Liquid-filled manometers and Bourdon tube pressure gages

have been the mainstay for measuring pressure for many, many

years. However, for many modern applications, these tried-and-true

devices are not sufficient. For example, many new uses need small,

accurate, inexpensive pressure transducers with digital outputs.

Obtain information about some of the new concepts used for pres-

sure measurement. Summarize your findings in a brief report.

■ FE Exam Problems

Sample FE (Fundamentals of Engineering) exam question for fluid

mechanics are provided on the book’s web site, www.wiley.com/

college/munson.

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CHAPTER OPENING PHOTO: Flow past a blunt body: On any object placed in a moving fluid there is a stag-

nation point on the front of the object where the velocity is zero. This location has a relatively large pres-

sure and divides the flow field into two portions—one flowing to the left, and one flowing to the right of

the body. 1Dye in water.21Photograph by B. R. Munson.2

Learning Objectives

After completing this chapter, you should be able to:

■ discuss the application of Newton’s second law to fluid flows.

■ explain the development, uses, and limitations of the Bernoulli equation.

■ use the Bernoulli equation (stand-alone or in combination with the continuity

equation) to solve simple flow problems.

■ apply the concepts of static, stagnation, dynamic, and total pressures.

■ calculate various flow properties using the energy and hydraulic grade lines.

In this chapter we investigate some typical fluid motions (fluid dynamics) in an elementary way.

We will discuss in some detail the use of Newton’s second law (F  ma) as it is applied to fluid

particle motion that is “ideal” in some sense. We will obtain the celebrated Bernoulli equation

and apply it to various flows. Although this equation is one of the oldest in fluid mechanics and

the assumptions involved in its derivation are numerous, it can be used effectively to predict and

analyze a variety of flow situations. However, if the equation is applied without proper respect

for its restrictions, serious errors can arise. Indeed, the Bernoulli equation is appropriately called

“the most used and the most abused equation in fluid mechanics.”

A thorough understanding of the elementary approach to fluid dynamics involved in this chap-

ter will be useful on its own. It also provides a good foundation for the material in the following

chapters where some of the present restrictions are removed and “more nearly exact” results are

presented.

lementary Fluid

Dynamics—The

Bernoulli Equation

lementary Fluid

Dynamics—The

Bernoulli Equation

The Bernoulli equa-

tion may be the most

used and abused

equation in fluid

mechanics.

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94 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation

As a fluid particle moves from one location to another, it usually experiences an acceleration or de-

celeration. According to Newton’s second law of motion, the net force acting on the fluid particle

under consideration must equal its mass times its acceleration,

In this chapter we consider the motion of inviscid fluids. That is, the fluid is assumed to have zero

viscosity. If the viscosity is zero, then the thermal conductivity of the fluid is also zero and there

can be no heat transfer 1except by radiation2.

In practice there are no inviscid fluids, since every fluid supports shear stresses when it is

subjected to a rate of strain displacement. For many flow situations the viscous effects are rela-

tively small compared with other effects. As a first approximation for such cases it is often possi-

ble to ignore viscous effects. For example, often the viscous forces developed in flowing water

may be several orders of magnitude smaller than forces due to other influences, such as gravity or

pressure differences. For other water flow situations, however, the viscous effects may be the dom-

inant ones. Similarly, the viscous effects associated with the flow of a gas are often negligible, al-

though in some circumstances they are very important.

We assume that the fluid motion is governed by pressure and gravity forces only and exam-

ine Newton’s second law as it applies to a fluid particle in the form:

The results of the interaction between the pressure, gravity, and acceleration provide numerous use-

ful applications in fluid mechanics.

To apply Newton’s second law to a fluid 1or any other object2, we must define an appropri-

ate coordinate system in which to describe the motion. In general the motion will be three-

dimensional and unsteady so that three space coordinates and time are needed to describe it. There

are numerous coordinate systems available, including the most often used rectangular and

cylindrical systems shown by the figure in the margin. Usually the specific flow geometry

dictates which system would be most appropriate.

In this chapter we will be concerned with two-dimensional motion like that confined to the

x–z plane as is shown in Fig. 3.1a. Clearly we could choose to describe the flow in terms of the

components of acceleration and forces in the x and z coordinate directions. The resulting equations

are frequently referred to as a two-dimensional form of the Euler equations of motion in rectan-

gular Cartesian coordinates. This approach will be discussed in Chapter 6.

As is done in the study of dynamics 1Ref. 12, the motion of each fluid particle is described

in terms of its velocity vector, V, which is defined as the time rate of change of the position of the

particle. The particle’s velocity is a vector quantity with a magnitude 1the speed, 2and di-

rection. As the particle moves about, it follows a particular path, the shape of which is governed

by the velocity of the particle. The location of the particle along the path is a function of where

the particle started at the initial time and its velocity along the path. If it is steady flow 1i.e., noth-

ing changes with time at a given location in the flow field2, each successive particle that passes

through a given point [such as point 112in Fig. 3.1a] will follow the same path. For such cases the

V  0V0

1r, u, z2

1x, y, z2

1particle mass2 1particle acceleration2

1Net pressure force on a particle2 1net gravity force on particle2

F  ma

3.1 Newton’s Second Law

Inviscid fluid flow

is governed by

pressure and grav-

ity forces.

Rectangular

Cylindrical

(a)

(1)

(2)

Fluid particle

(b)



(s)

n = 0

n = n

Streamlines

F I G U R E 3.1 (a) Flow in the x–z plane. (b) Flow in terms of streamline and normal

coordinates.

JWCL068_ch03_093-146.qxd 8/19/08 10:20 PM Page 94

path is a fixed line in the x–z plane. Neighboring particles that pass on either side of point 112fol-

low their own paths, which may be of a different shape than the one passing through 112. The entire

x–z plane is filled with such paths.

For steady flows each particle slides along its path, and its velocity vector is everywhere

tangent to the path. The lines that are tangent to the velocity vectors throughout the flow field

are called streamlines. For many situations it is easiest to describe the flow in terms of the

“streamline” coordinates based on the streamlines as are illustrated in Fig. 3.1b. The particle

motion is described in terms of its distance, along the streamline from some convenient

origin and the local radius of curvature of the streamline, The distance along the

streamline is related to the particle’s speed by and the radius of curvature is related

to the shape of the streamline. In addition to the coordinate along the streamline, s, the coordi-

nate normal to the streamline, n, as is shown in Fig. 3.1b, will be of use.

To apply Newton’s second law to a particle flowing along its streamline, we must write the

particle acceleration in terms of the streamline coordinates. By definition, the acceleration is the

time rate of change of the velocity of the particle, For two-dimensional flow in the x–z

plane, the acceleration has two components—one along the streamline, the streamwise accel-

eration, and one normal to the streamline, the normal acceleration.

The streamwise acceleration results from the fact that the speed of the particle generally

varies along the streamline, For example, in Fig. 3.1a the speed may be at

point 112and at point 122. Thus, by use of the chain rule of differentiation, the s com-

ponent of the acceleration is given by We have used the

fact that speed is the time rate of change of distance, Note that the streamwise ac-

celeration is the product of the rate of change of speed with distance along the streamline,

and the speed, V. Since can be positive, negative, or zero, the streamwise acceleration

can, therefore, be positive (acceleration), negative (deceleration), or zero (constant speed).

The normal component of acceleration, the centrifugal acceleration, is given in terms of the

particle speed and the radius of curvature of its path. Thus, where both V and may

vary along the streamline. These equations for the acceleration should be familiar from the study

of particle motion in physics 1Ref. 22or dynamics 1Ref. 12. A more complete derivation and dis-

cussion of these topics can be found in Chapter 4.

Thus, the components of acceleration in the s and n directions, and are given by

(3.1)

where is the local radius of curvature of the streamline, and s is the distance measured along

the streamline from some arbitrary initial point. In general there is acceleration along the stream-

line 1because the particle speed changes along its path, 2and acceleration normal to the

streamline 1because the particle does not flow in a straight line, 2. Various flows and the ac-

celerations associated with them are shown in the figure in the margin. As discussed in Section

3.6.2, for incompressible flow the velocity is inversely proportional to the streamline spacing.

Hence, converging streamlines produce positive streamwise acceleration. To produce this acceler-

ation there must be a net, nonzero force on the fluid particle.

To determine the forces necessary to produce a given flow 1or conversely, what flow results

from a given set of forces2, we consider the free-body diagram of a small fluid particle as is shown

in Fig. 3.2. The particle of interest is removed from its surroundings, and the reactions of the

r ⫽⬁

Ⲑ

0s ⫽ 0

⫽ V

⫽

⫽ V

Ⲑ

0s,

V ⫽ ds

Ⲑ

dt.

⫽ dV

Ⲑ

dt ⫽ 10V

Ⲑ

0s21ds

Ⲑ

dt2⫽ 10V

Ⲑ

0s2V.

50 ft

Ⲑ

100 ft

Ⲑ

sV ⫽ V1s2.

a ⫽ dV

Ⲑ

dt.

V ⫽ ds

Ⲑ

dt,

r ⫽ r1s2.

s ⫽ s1t2,

3.1 Newton’s Second Law 95

Fluid particles ac-

celerate normal to

and along stream-

lines.

> 0

= a

= 0

> 0

> 0, a

> 0

Streamline

Fluid particle

F I G U R E 3.2 Isolation of a small fluid particle in a flow field. (Photo courtesy

of Diana Sailplanes.)

V3.1 Streamlines

past an airfoil

JWCL068_ch03_093-146.qxd 10/17/08 4:18 PM Page 95

96 Chapter 3 ■ Elementary Fluid Dynamics—The Bernoulli Equation

Consider the small fluid particle of size by in the plane of the figure and normal to the

figure as shown in the free-body diagram of Fig. 3.3. Unit vectors along and normal to the stream-

line are denoted by and respectively. For steady flow, the component of Newton’s second law

along the streamline direction, s, can be written as

(3.2)

where represents the sum of the s components of all the forces acting on the particle, which

has mass and is the acceleration in the s direction. Here, is

the particle volume. Equation 3.2 is valid for both compressible and incompressible fluids. That

is, the density need not be constant throughout the flow field.

The gravity force 1weight2on the particle can be written as where is

the specific weight of the fluid Hence, the component of the weight force in the

direction of the streamline is

If the streamline is horizontal at the point of interest, then and there is no component of

particle weight along the streamline to contribute to its acceleration in that direction.

As is indicated in Chapter 2, the pressure is not constant throughout a stationary fluid

because of the fluid weight. Likewise, in a flowing fluid the pressure is usually not constant. In gen-

eral, for steady flow, If the pressure at the center of the particle shown in Fig. 3.3 is

denoted as p, then its average value on the two end faces that are perpendicular to the streamline are

and Since the particle is “small,” we can use a one-term Taylor series expansion

for the pressure field 1as was done in Chapter 2 for the pressure forces in static fluids2to obtain



p  dp

.p  dp

p  p1s, n2.

1§p  02

u  0,

dw sin u g dV sin u

1lb



or N



g  rgdw  g dV,

dVds dn dyV 0V



0sdm  r d V,

g dF

 dm a

 dm V

 r dV V

nˆ,sˆ

dydnds

3.2 along a Streamline

F ⴝ ma

Particle thickness = y

Along streamline

Normal to streamline



(p + p

) s y

δδδ

(p + p

) n y

δδδ

(p – p

) n y

δδδ

(p – p

) s y

δδδ

s y

δδτ

= 0

s y

δδτ

= 0



F I G U R E 3.3 Free-

body diagram of a fluid particle for

which the important forces are those

due to pressure and gravity.

surroundings on the particle are indicated by the appropriate forces present, and so forth.

For the present case, the important forces are assumed to be gravity and pressure. Other forces,

such as viscous forces and surface tension effects, are assumed negligible. The acceleration of grav-

ity, g, is assumed to be constant and acts vertically, in the negative z direction, at an angle rela-

tive to the normal to the streamline.

, F

In a flowing fluid

the pressure varies

from one location

to another.

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