Full text of "Elementary Fluid Mechanics"

00 a:

OU 160141 >m

ELEMENTARY
FLUID MECHANICS

JOHN K. VENNARD

Assistant Professor of Fluid Mechanics
New York University

NEW YORK

JOHN WILEY & SONS, INC.

LONDON: CHAPMAN & HALL, LIMITED

1940

This book or any part thereof must not
be reproduced in any form without
the written permission of the publisher.

PRINTED IN U. S. A.

PRESS OP
BRAONWORTH A CO.. INC.

BRIDGEPORT. CONN.

ELEMENTARY
FLUID MECHANICS

PREFA^R

Fluid mechanics is the study of all ttuYds under all possible condi-
tions of rest and motion. Its approaches analytical, rational, and
mathematical rather than empirical ; it concerns itself with those basic
principles which lead to the solution of numerous diversified problems,
and it seeks results which are widely applicable to similar fluid sit-
uations and not limited to isolated special cases. Fluid mechanics
recognizes no arbitrary boundaries between fields of engineering knowl-
edge but attempts to solve all fluid problems, irrespective of their
occurrence or of the characteristics of the fluids involved.

This textbook is intended primarily for the beginner who knows
the principles of mathematics and mechanics but has had no previous
experience with fluid phenomena. The abilities of the average
beginner and the tremendous scope of fluid mechanics appear to be in
conflict, and the former obviously determine limits beyond which it is
not feasible to go; these practical limits represent the boundaries of
the subject which I have chosen to call elementary fluid mechanics.
The apparent conflict between scope of subject and beginner f s ability
is only along mathematical lines, however, and the physical ideas of
fluid mechanics are well within the reach of the beginner in the field.
Holding to the belief that physical concepts are the sine qua non of
mechanics, I have sacrificed mathematical rigor and detail in develop-
ing physical pictures and in many cases have stated general laws only
(without numerous exceptions and limitations) in order to convey basic
ideas; such oversimplification is necessary in introducing a new subject
to the beginner.

Like other courses in mechanics, fluid mechanics must include
disciplinary features as well as factual information the beginner must
follow theoretical developments, develop imagination in visualizing
physical phenomena, and be forced to think his way through problems
of theory and application. The text attempts to attain these objec-
tives in the following ways: omission of subsidiary conclusions is
designed to encourage the student to come to some conclusions by
himself; application of bare principles to specific problems should
develop ingenuity; illustrative problems are included to assist in
overcoming numerical difficulties; and many numerical problems for

vi PREFACE

the student to solve are intended not only to develop ingenuity but to
show practical applications as well.

Presentation of the subject begins with a discussion of funda-
mentals, physical properties and fluid statics. Frictionless flow is
then discussed to bring out the applications of the principles of con-
servation of mass and energy, and of impulse-momentum law, to fluid
motion. The principles of similarity and dimensional analysis are
next taken up so that these principles may be used as tools in later
developments. Frictional processes are discussed in a semi-quanti-
tative fashion, and the text proceeds to pipe and open-channel flow.
A chapter is devoted to the principles and apparatus for fluid measure-
ments, and the text ends with an elementary treatment of flow about
immersed objects. Throughout the text, the foot-pound-second sys-
tem of dimensions has been used, and problems of conversion from the
metric system, which so frequently divert the beginner's attention
from the physical ideas, have been avoided; justifications for experi-
mental results and empirical formulas have been presented except
at points where the student should discover them for himself;
bibliographies have been included to guide the inquiring reader to
more exhaustive treatments of the subject.

For criticism of my Notes on Elementary Fluid Mechanics which
have been expanded into the present text, I wish to extend my appre-
ciation to many of my colleagues at New York University, Professor
Boris A. Bakhmeteff of Columbia University, and Professor William
Allan of the College of the City of New York.

I am deeply indebted to Mr. William H. Peters of the Curtiss-
Wright Corporation for carefully reviewing the first eight chapters of
the manuscript, and to Professor Frederick K. Teichmann of New
York University for critical comments on the last chapter. I also
wish to thank Mr. J. Charles Morgan for general comments and
assistance in reading proof and Miss Katherine Williams for her care
and patience in typing the manuscript.

JOHN K. VENNARD

NEW YORK, N.Y.
May, 1040

CONTENTS

CHAPTER FACE

I . FUNDAMENTALS 1

Art. 1 Development of Fluid Mechanics. 2. Physical Character-
istics of the Fluid State^. 3. Density, Specific Weight, Specific
Volume, ,and Specific Gravity. 4. Compressibility, Elasticity.
5. Viscosity. 6. Surface Tension, Capillarity. 7. Vapor Pressure.

II. FLUID STATICS 22

Art.^-Pressure-Density-Height Relationships. 9. Absolute and
Gage Pressures. 10. Manometry. 11. Forces on Submerged Plane
Surfaces. 12. Forces on Submerged Curved Surfaces. 13. Buoy-
ancy and Flotation. 14. Stresses in Circular Pipes and Tanks.
15. Fluid Masses Subjected to Acceleration.

III. THE FLOW OF AN IDEAL FLUID 55

Art. 16 Definitions. 17. Equation of Continuity. 18. Euler's
Equation. 19. Bernoulli's Equation. 20. Energy Relationships.
21. Flow of an Incompressible Fluid. 22. Flow of a Compressible
Fluid. 23. Impulse-Momentum Relationships. 24. Flow Curva-
tures, Types of Vortices, Circulation.

IV. THE FLOW OF A REAL FLUID 106

Art. 25 Laminar and Turbulent Flow. 26. Fluid Flow Past Solid
Boundaries. 27. Fluid Flow between Parallel Plates. 28. Flow
about Immersed Objects. 29. Stability Secondary Flows.

V. SIMILARITY AND DIMENSIONAL ANALYSIS 125

Art. 30 Similarity and Models. 31. Dimensional Analysis.

VI. FLUID FLOW IN PIPES 139

Art. 32 Energy Relationships. 33. General Mechanics of Fluid
Flow in Pipe Lines. 34. Laminar Flow. 35. Dimensional Analysis
of the Pipe-Friction Problem. 36. Results of Pipe-Friction Experi-
ments. 37. Velocity Distribution in Circular Pipes The Pipe
Coefficient. 38. Approximate Thickness of the Laminar Film.
39. Pipe Friction for Compressible Fluids. 40. Pipe Friction in
Non-Circular Pipes. 41. Pipe-Friction Calculations by the Hazen-
Williarns Method. 42. Minor Losses in Pipe Lines. 43. The
Pressure Grade Line and Its Use. 44. Branching Pipes.

vii

viii CONTENTS

CHAPTER PAGE

VII. FLUID FLOW IN OPEN CHANNELS 196

Art. 45 Fundamentals. 46. Uniform Flow The Chezy Equation.
47. The Chezy Coefficient. 48. Best Hydraulic Cross Section.
49. Variation of Velocity and Rate of Flow with Depth in Closed
Conduits. 50. Specific Energy. 51. Critical Depth Relationships.
52. Occurrence of Critical Depth. 53. Varied Flow. 54. The
Hydraulic Jump.

VIII. FLUID MEASUREMENTS 229

Art. 55 Measurement of Fluid Properties. 56. Measurement of
Static Pressure. 57. Measurement of Surface Elevation. 5&. Meas-
urement of Stagnation Pressure. 59. The Pitot (Pitot-Static)
Tube. y '60. The Venturi Tube. 6t^The Pitot-Venturi, 62. Ane-
mometers and Current Meters. 63. Total Quantity Methods.
&4. Venturi Meters. 65. Nozzles. &fl Orifices. 67. Flow Bends.
S&r- 1 Pitot -Tube Methods. 69. Dilution anj Thermal Methods.
70. Salt-Velocity Method. 71. Weirs. ^^ Current- Meter Meas-
urements. v ?3. Float Measurements.

IX. FLOW ABOUT IMMERSED OBJECTS 294

Art. 74 Fundamentals and Definitions. 75. Dimensional Analysis
of the Drag Problem. 76. Frictional Drag. 77. Profile Drag.
78. Drag at High Velocities. 79. Lift by Change of Momentum.
80. Circulation Theory of Lift. 81. Origin of Circulation. 82. Foils
of Finite Length. 83. Lift and Drag Diagrams.

APPENDICES 335

I. Description and Dimensions of Symbols. II. Specific Weight and
Density of Water. III. Velocity of a Pressure Wave through a
Fluid. IV. Viscosities of Liquids and Gases. V. Cavitation.

INDEX 347

ELEMENTARY FLUID MECHANICS

CHAPTER I
FUNDAMENTALS

1. Development of Fluid Mechanics. Man's desire for knowl-
edge of fluid phenomena began with his problems of water supply
and disposal and the use of water for obtaining power. With only a
rudimentary appreciation for the physics of fluid flow he dug wells,
operated crude water wheels and pumping devices, and, as his cities
increased in size, constructed ever larger aqueducts, which reached their
greatest size and grandeur in those of the City of Rome. However,
with the exception of the thoughts of Archimedes (250 B.C.) on the
principles of buoyancy little of the scant knowledge of the ancients
appears in modern fluid mechanics. After the fall of the Roman
Empire (A.D. 476) no progress was made in fluid mechanics until the
time of Leonardo Da Vinci (1452-1519). This great genius designed
and built the first chambered canal lock near Milan and ushered in a
new era in hydraulic engineering; he also studied the flight of birds and
developed some ideas on the origin of the forces which support them.

After the time of Da Vinci, the accumulation of hydraulic knowledge
rapidly gained momentum, the contributions of Galileo, Torricelli,
Newton, Pitot, D. Bernoulli, and D'Alembert to the fundamentals of
the science being outstanding. Although the theories proposed by
these scientists were in general confirmed by crude experiments, diver-
gences between theory and fact led D'Alembert to observe in 1744 that,
"The theory of fluids must necessarily be based upon experiment."
D'Alembert showed that there is no resistance to motion when a body
moves through an ideal (non-viscous) fluid, yet obviously this con-
clusion is not valid for bodies moving through real fluid. This dis-
crepancy between theory and practice is called the "D'Alembert
paradox" and serves to demonstrate the limitations of theory alone in
solving fluid problems.

Because of the conflict between theory and practice, two schools
of thought arose in the treatment of fluid problems, one dealing with

2 FUNDAMENTALS

the theoretical and the other with the practical aspects of fluid flow,
and in a sense these two schools of thought have persisted down to the
present day, resulting in the theoretical field of " hydrodynamics" and
the practical one of "hydraulics." Notable contributions to theo-
retical hydrodynamics have been made by Euler, La Grange, Helm-
hoi tz, Kirchhoff, Lord Rayleigh, Rankine, Lord Kelvin, and Lamb.
In a broad sense, experimental hydraulics became a study of the laws
of fluid resistance, mainly in pipes and open channels. Among the
many scientists who devoted their energies to this field were Brahms,
Bossut, Chezy, Dubuat, Fabre, Coulomb, Eytelwein, Belanger, Dupuit,
d'Aubisson, Hagen, and Poisseuille.

Toward the middle of the last century, Navier and Stokes succeeded
in modifying the general equations for ideal fluid motion to fit that
of a viscous fluid and in so doing showed the possibilities of adjusting
the differences between hydraulics and hydrodynamics. At about the
same time, theoretical and experimental work on vortex motion by
Helmholtz was aiding in explaining away many of the divergent results
of theory and practice.

Meanwhile, hydraulic research went on apace, and large quantities
of excellent data were collected or formulas proposed for fluid re-
sistance, notably by Darcy, Bazin, Weisbach, Fanning, Ganguillet,
Kutter, and Manning; among researchers on other hydraulic prob-
lems were Thomson, Fteley, Stearns, and H. Smith. Unfortunately,
researches led frequently to empirical formulas obtained by fitting
curves to experimental data or merely presenting the results in tabular
form, and in many cases the relationship between the physical facts
and the resulting formula was not apparent.

Toward the end of the last century, new industries arose which
demanded data on the flow of fluids other than water; this fact and
many significant advances in knowledge tended to arrest the increasing
empiricism of hydraulics. These advances were: (1) the theoretical
and experimental researches of Reynolds; (2) the development of di-
mensional analysis by Lord Rayleigh ; (3) the use of models by Froude,
Reynolds, Fargue, and Engels in the solution of fluid problems; and
(4) the rapid progress of theoretical and experimental aeronautics in
the work of Lanchester, Lilienthal, Kutta, Joukowski, and Prandtl.
These advances allowed new tools to be applied to the solution of
fluid problems and gave birth to modern fluid mechanics.

Since the beginning of the present century, empiricism has waned
and fluid problems have been solved by increasingly rational methods;
these methods have produced so many fruitful results and have aided

PHYSICAL CHARACTERISTICS OF THE FLUID STATE 3

so materially in increasing our knowledge of the details of fluid phenom-
ena that the trend appears likely to continue into the future. Among
the foremost contributors to modern fluid mechanics are Prandtl,
Blasius, Karman, Stanton, Nikuradse, Bakhmeteff, Koch, Bucking-
ham, Gibson, Rehbock, Durand, and Taylor.

2. Physical Characteristics of the Fluid State. Matter exists in
two states the solid and the fluid, the fluid state being commonly
divided into the liquid and gaseous states.

Solids differ from liquids and liquids from gases in the spacing and
latitude of motion of their molecules, these variables being large in a
gas, smaller in a liquid, and extremely small in a solid. It follows
that inter molecular cohesive forces are large in a solid, smaller in a
liquid, and extremely small in a gas^ These fundamental facts account
for the familiar compactness and rigidity of form possessed by solids,
the ability of liquid molecules to move freely within a liquid mass, and
the capacity of gases to fill com-
pletely the containers in which
they are placed.

A more fruitful and rigorous
mechanical definition of the solid
and fluid states may be made on
the basis of their actions under the
various types of stress. Applica- FIG. 1.

tion of tension7~c6rnpression , or

shear stresseVto a solid results first in elastic deformation, and later,
if these~stresses exceed the elastic limits, in permanent distortion
of the material. Fluids, however, possess elastic properties under
compression stress, but application of infinitesimal shear stress results
in continual and permanent distortion. This inability to resist shear
stress gives fluids their characteristic ability to " flow". Fluids
will support tension stress to the extent of the cohesive forces be-
tween their molecules. Since such forces are extremely small, it is
customary in engineering problems to assume that fluid can support
no tension stress.

Since shear stress applied to fluids always results in distortion or
"flow," it is evident that in fluids at rest no shear stresses can. exist
and compression stress, or "pressure/' becomes the only stress to be
considered.

Fluids being continuous media, it follows that pressures occurring
or imposed at a point in a fluid will ^transmitted undiminished to all
other points in the fluid (neglecting the weight of the fluid).

U UU H I i V

132 SIMILARITY AND DIMENSIONAL ANALYSIS

Before examining the methods of dimensional analysis, recall that
there are two different systems by which the dimensions of physical
quantities may be expressed. These systems are the force-length-time
system and the mass-length-time system. The former system, gener-
ally preferred by engineers, becomes the familiar "foot-pound-second"
system when expressed in English dimensions; the latter system in
English dimensions becomes the "foot-slug-second" system. The
latter system is generally preferred in dimensional analysis, and, since
the student is familiar with the former system, the use of the latter
will serve to develop versatility in the use of dimensions.

A summary of the fundamental quantities of fluid mechanics and
their dimensions in the various systems is given in Table VI, the con-
ventional system of capital letters being followed to indicate the dimen-
sions of quantities. The basic relation between the force-length-time
and mass-length-time systems of dimensions is given by the Newtonian
law, force or weight = (Mass) X (Acceleration) and, therefore,
dimensionally,

p = yr .

r m ^2

from which the dimensions of any quantity may be converted from
one system to the other.

To illustrate the mathematical steps in a simple dimensional prob-
lem, consider the familiar equation of fluid statics

p wh

but assume that the dimensions of w and h are known and those of p
unknown. The dimensions of p can be only some combination of
M, L, and T, and this combination may be discovered by writing the
equation dimensionally as

Unknown dimensions of p (Dimensions of w) X (Dimensions of h)
or

M*L b T* = (jj^ X (L)

in which a, b, and c are unknowns. The principle of dimensional
homogeneity being applied, the exponents of each of the quantities is
the same on each side of the equation, giving

a = 1, b =~ 2 + 1 =- 1, c =- 2

DIMENSIONAL ANALYSIS

133

TABLE VI
DIMENSIONS OF FUNDAMENTAL QUANTITIES USED IN FLUID MECHANICS

Quantity

Symbol

English Engi-
neering
Dimensions

Force-Length-
Time
Dimensions

Mass-Length-
Time
Dimensions

Acceleration

ft/sec 2

L/T 2

Acceleration due to gravity
Area

g
A

ft/sec 2
ft 2

L/T 2
L 2

Density . . . ...

Ib sec 2 /ft 4

FT 2 /L 4

lf/L 3

Force

ML/T 2

Kinematic viscosity . . .

ft 2 /sec

i: 2 /r 2

I 2 /T

Length

Mass

Ib sec 2 /ft

FT 2 /L

Power

ft Ib/sec

PL /T

ML 2 /T*

Pressure ....

lb/ft 2

F/L 2

MILT"

Rate of flow

ft 3 /scc

L*/T

Specific weight

lb/ft 3

F/L*

M/L 2 T 2

Time

sec

Velocity

ft /sec

L/T

Viscosity

fj,

Ib sec/ft 2

FT/L 2

M/LT

Weight

ML/T 2

Weight rate of flow

Ib/soc

F/T

ML/T*

whence

Dimensions of p ML 1 T 2

It is obvious, of course, that this result might have been obtained more
directly by cancellation of L on the right-hand side of the equation,
for this has been, and will continue to be, the usual method of obtain-
ing the unknown dimensions of a quantity, It is of utmost impor-
tance, however, to note the mathematical steps which lie unrevealed
in this hasty cancellation, if the basis of dimensional analysis is to be
understood.

The above methods may now be used in quite another and more
important way. To illustrate by another familiar example, suppose
that it is known that the power P, which can be derived from a hy-
draulic motor, is dependent upon the rate of flow through the motor
Q, the specific weight of the fluid flowing w, and the unit energy E
which is given up by every pound of fluid as it passes through the

4 FUNDAMENTALS

Since pressure ^cornyBr^sioruatress, the equilibrium of a mass of
fluid at rest occurs from pressure acting inward upon its boundary
surface as shown in Fig. 1. If this mass of fluid is reduced to infini-
tesimal size, it becomes evident that at a point in a fluid the pressure
is the same in all directions. Since fluids are unable to support tan-
gential (shear) stresses, no component of force can exist along the solid
boundary of Fig. 1, and thus pressure must be transmitted from a
fluid to a solid boundary normal to the boundary at every point.

PHYSICAL PROPERTIES OF FLUIDS

3. Density, Specific Weight, Specific Volume, and Specific Gravity.

Density * is the mass of fluid contained in a unit of volume ; specific
weight, 1 the weight of fluid contained in a unit of volume. Both these
terms are fundamentally measures of the number of molecules per
unit of volume. Since molecular activity and spacing increase with
temperature fewer molecules will exist in a given unit volume as tem-
perature rises, thus causing density and specific weight to decrease
with increasing temperature. 2 Since a larger number of molecules can
be forced into a given volume by application of pressure, it will be
found that density and specific weight will increase with increasing
pressure.

Density, p (rho), will be expressed in the mass-length-time system
of dimensions and will have the dimensions of mass units (slugs) per
cubic foot (slugs/ft 3 ).

Specific weight, w, will be expressed in the force-length-time system
of dimensions and will have the dimensions of pounds per cubic
foot (lb/ft 8 ).

Since a mass, M, is related to its weight, W, by the equation

M W

M =

in which g is the acceleration due to gravity, density and specific
weight (the mass and weight of a unit volume of fluid) will be related
by a similar equation

p = or w = pg
g

1 In American engineering practice, specific weight is frequently termed " density r '
and density " mass density/'

2 A variation in temperature from 32Fto212F will decrease the specific weight
of water 4 per cent (Appendix II) and will decrease the density of gases 37 per cent
(assuming no pressure variation).

PHYSICAL PROPERTIES OF FLUIDS 5

Using the fact that physical equations are dimensionally homogeneous,
the foot-pound-second dimensions of p (which are equivalent to slugs
per cubic foot) may be calculated as follows :

_ . . f Dimensions of w ft 3 Ib sec 2

Dimensions of p = : = - = 4

Dimensions of g ft ft

sec 2

This algebraic use of the dimensions of quantities in the equation ex-
pressing physical relationship will be employed extensively and will
prove to be an invaluable check on engineering calculations. 3

The specific volume, v, defined as volume per unit of weight, will
have dimensions of cubic feet per pound (ft 3 /lb). This definition
identifies specific volume as the reciprocal of specific weight and intro-
duces the equations

1 1

i) = or w = -

W V

Specific gravity, 5, is the ratio of specific weight or density of a sub-
stance to the specific weight or density of pure water. Since all these
items vary with temperature, temperatures must be quoted when spe-
cific gravity is used in precise calculations of specific weight or density.
Specific gravities of a few common liquids at 68 F. (except as noted),
are presented in Table I, from which the specific weights of liquids

TABLE I *

SPECIFIC GRAVITIES, S, OF VARIOUS LIQUIDS AT 68 F f
(Referred to water at 39.2 F)

Ethyl alcohol 0. 789

Turpentine (d-pinene) . 862

Benzene 0.888

Linseed oil 0.934 (59.9 F)

Castor oil 0.960

Water 0.998

Glycerine 1 .264 (57 F)

Carbon tetrachloride 1 .594

Mercury 13 . 546

* Smithsonian Physical Tables, Eighth Ed., 1933, Smithsonian Institution.
t Except as noted.

3 A summary of quantities and their dimensions is given in Appendix I.

6 FUNDAMENTALS

may be readily calculated by

w = 5 X 62.45 lb/ft 3

The specific weight of gases may be calculated by means of Boyle's
law and Charles' law. Using the specific volume of a gas, Boyle's
law may be stated as 4

pv = Constant

which expresses the law of compression or expansion of a gas at con-
stant temperature. Charles' law, expressing the variation of pressure
with temperature in a constant volume of gas, is 4

= Constant

Obviously the only combination of variables which will satisfy both
Boyle's and Charles' laws simultaneously is

which is called the "equation of state" of the gas in which the constant,
JR, is called the "gas constant" and has dimensions of feet/degree
Fahrenheit absolute. Since w = l/v 9 the above equation may be
transformed into

, = --

W RT

from which specific weights of gases may be readily calculated.

Application of Avogadro's law, that "all gases at the same pressures
and temperatures have the same number of molecules per unit of
volume," allows the calculation of a "universal gas constant." Con-
sider two gases having constants R\ and R%, specific weights w\ and w^
and existing at the same pressure and temperature, p and T. Dividing
their equations of state

P _
w\T

4 p is the absolute pressure in pounds per square foot, T is the temperature in
degrees F absolute (degrees F -}- 459.6), and the above "constants" are constant if the
gas is " perfect." Common gases in the ordinary engineering range of pressures and
temperatures may be considered to be "perfect" for most engineering calculations.

PHYSICAL PROPERTIES OF FLUIDS
results in

' but, according to Avogadro's principle, the specific weight of a gas must
be proportional to its molecular weight, giving w<2/Wi = w 2 /%i, in
which mi and ra 2 are the respective molecular weights of the gases.
Combining this equation with the preceding one gives m^/mi

In other words, the product of molecular weight and gas constant is
the same 5 for all gases. This product mR is called the "universal
gas constant" and is preferred for general use by many engineers.
Values of these gas constants are given in Table II.

TABLE II
GAS CONSTANTS FOR COMMON GASES *

R, ft/F abs mR

Sulphur dioxide ..................... 23.6 1512

Carbon dioxide ........... .......... 34.9 1536

Oxygen ............................ 48.3 1546

Air ................................ 53.3 1545

Nitrogen ........................... 55.1 1543

Ammonia .......................... 89.5 1516

Hydrogen .......................... 767.0 1546

* O. W. Eshbach, Handbook of Engineering Fundamentals, p. 7-16, John Wiley & Sons, 1936.

ILLUSTRATIVE PROBLEM

Calculate the density of carbon dioxide at a temperature of 80 F and absolute
pressure of 100 lb/in. 2

, x 1536

m = 12 + 2(16) =44, R = - = 34.9

p 100 X 144 Ib

w __ _ __ - __ n 703 -

RT 34.9 X (80 + 460) ' ft 3

0.793 / ,

P - - 0.0246 slugs/ft 3

6 The constancy of mR is particularly true for the monatomic and diatomic gases.
Gases having more than two atoms per molecule tend to deviate from the law
mR Constant. See Table II.

8 FUNDAMENTALS

4. Compressibility, Elasticity. All fluids may be compressed with
consequent increase in density, the process of compression taking
place at the expense of the space between molecules. Therefore, as
fluids are compressed the molecular spacing is diminished and the
fluids become increasingly difficult to compress further. Fluids also
become more difficult to compress as temperature increases because
of increased molecular activity reducing the molecular spacing available
for compression. Owing to these facts, it is obviously an approxima-
tion to express elastic compression of fluids by Hooke's law

Stress

*-' == 7^ ~.

Strain

because E is not a constant but increases with increased temperature
or pressure. Such an approximation, however, is justified for ordinary
engineering calculations since the range of pressure encountered in
engineering is comparatively small and the change in E over this pres-
sure range is usually negligible. 6 The above equation then becomes

the strain ( AF/F) being the decrease in volume (AF) per original
volume (F) obtained by an increment of pressure (A/>).

Compression of gases may take place according to various laws of
thermodynamics. The isothermal compression of a volume of perfect
gas, FI, existing at an absolute pressure, pi, to a volume Fg at a
pressure p2 will be accomplished according to Boyle's law.

P\V\ = p2^2 or pV = Constant
Using the specific volume, v, in this equation there results

pv = Constant or = Constant
w

Frequently expansion of gases occurs so rapidly that there is
no opportunity for flow of heat during the process. Such an expansion

follows the adiabatic law

PiVf = p2V 2 k or pV k = Constant
8 E for water is commonly taken as a constant, 300,000 lb/in. 2

COMPRESSIBILITY, ELASTICITY 9

which, written in terms of specific volume and specific weight, becomes

t P

pv K = Constant or ^ = Constant

in which k, called the "adiabatic constant/' is the ratio of the two
specific heats of the gas, that at constant pressure, c p , to that at con-
stant volume, c v . Values of k for common gases are given in Table III.

TABLE III
ADIABATIC CONSTANTS, k t FOR COMMON GASES *

Sulphur dioxide 1.26

Carbon dioxide 1 . 30

Oxygen 1 . 40

Air 1.40

Nitrogen 1 . 40

Ammonia 1.32

Hydrogen 1 . 40

* O. W. Eshbach, Handbook of Engineering Fundamentals, p. 7-17, John Wiley & Sons, 1936.

Values of modulus of elasticity of gases, E, may be derived for
isothermal and adiabatic processes for use in subsequent developments.
Writing the law of elastic compression in differential form

V
For isothermal compression

pV = Constant
Differentiating this equation in respect to V results in

whence

dp_ __ p_

dV~ V
and substituting this in the first expression

E-p

for an isothermal process.

10 FUNDAMENTALS

By a similar analysis E for an adiabatic process may be shown to be
given by

E = kp

Imposed pressures or pressure disturbances are not transmitted
instantaneously from point to point in a fluid, but move in waves at
finite velocity. The velocity or celerity of propagation of such waves is
dependent upon the elastic properties of the fluid ; fluids which are more
easily compressible (having low values of E) transmit pressures with
smaller velocity than those which are difficult to compress. A pressure
disturbance is transmitted in a fluid with a celerity, c, which is given by
the equation 7

in which c is frequently termed the "sonic" or "acoustic" velocity since
it is the velocity with which sound, a pressure disturbance, travels. In
a gas, sound moves by a series of adiabatic compressions and rarefac-
tions. Thus the sonic velocity in a gas may be calculated from

P
an equation which is accurately confirmed by experiment.

ILLUSTRATIVE PROBLEMS

Ten cubic feet of water exist at atmospheric pressure. When a pressure of
2000 lb/in. 2 is applied, what reduction in volume results?

A 2000

E = - 77P ^ 300,000 = - -

A V A V

~v To""

- AF = 0.0667 ft 8
Calculate the velocity with which sound travels in water.

300,000 X 144
-457 - 4720 ft/sec

32.2
7 For derivation see Appendix III.

VISCOSITY

T-

~*~

-*-j.

-- T-

~^~

3J Viscosity. The property of viscosity, which is exhibited by all
fltFWns, is due fundamentally to the existence of cohesion and inter-
action between fluid molecules. As fluids flow these cohesions and
interactions result in tangential or
shear stresses between the moving
fluid layers. Consider the thin viscous
fluid layers shown in relative motion
in Fig. 2. Let them have a thickness,
dy, and areas of contact, A, the lower
layer moving with velocity, v, the
upper one with velocity v + dv. To maintain this velocity difference
a force F must be continually exerted on the upper layer as indicated,
which results in a shear or friction stress, r (tau), between the layers,
given by

f TTTf T
p

FIG. 2.

For viscous fluid motion the shear stress, r, has been found to be por-
portional to the rate of change of velocity along y or

r oc

dv
~dy

and the "coefficient of viscosity," M (mu), 8 is defined as the constant
of proportionality in the above equation, or

Indicating

r, and thus the coemaenj^ofjdscosit^ independent 9 of pressure.

Viscosity varies ^widely with temperature, but temperature varia-
tion has an opposite effect upon the viscosities of liquids and gases due
to their fundamentally different intermolecular characteristics. In
gases, where intermolecular cohesion is negligible, the shear stress, r,

8 The dimensions of /* may be obtained by writing

Dimensions of r lb/ft 2

Dimensions of

Dimensions of
dy

dv ft/se

= Ib sec/ft 2

9 Viscosity actually increases slightly with pressure, but this variation is negligible
in most engineering problems. Oils manifest the greatest increase of viscosity with
pressure.

FUNDAMENTALS

between moving layers of fluid results from an exchange of momentum
between these layers brought about by molecular agitation normal to

.060

.050

.040

,030

.020

100

200 300 400

Temperature in Degrees T.

FIG. 3. Viscosities of Gases. 10

500

600

the general direction of motion. Since this molecular activity is known
to increase with temperature, the shear stress, and thus the viscosity

10 Data on viscosities from Smithsonian Physical Tables, Eighth Edition, 1933,
Smithsonian Institution. (For original data see Appendix IV.) Data on viscos-
ity of steam from Fluid Meters, Their Theory and Application, Fourth Edition,
1937, A.S.M.E.

VISCOSITY

10,000

5,000

1,000

500

100

1.0

0.5

0.1

Gycer

I enzene

.Ca

stor oil

x-0 ive oil

Mercury

Carbon tetr chloride

Linsee oil

50 100 150

Temperature in Degrees F.

FIG. 4. Viscosities of Liquids. 10

200

Water

Turpentin

250

10 Data on viscosities from Smithsonian Physical Tables, Eighth Edition, 1933,
Smithsonian Institution. (For original data, see Appendix IV.) Data on viscos-
ity of steam from Fluid Meters, Their Theory and Application, Fourth Edition,
1937, A.S.M.E.

14 FUNDAMENTALS

of gases, will increase with temperature (Fig. 3). This reasoning is
borne out by tests and by considerations of the kinetic theory of gases
which indicate that gas viscosities vary directly with the square root

In a viscous liquid, momentum exchange due to molecular agitation
is small compared to the cohesive forces between the molecules, and
thus shear stress, r, and viscosity, jo,, are primarily dependent on the
magnitude of these cohesive forces. Since these forces decrease rapidly
with increases of temperature, liquid viscosities decrease as tempera-
ture increases (Fig. 4).

jj,

Owing to the continual appearance of the ratio - in subsequent

developments, this term has been defined by

z/(nu) = -
P

in which v is called the " kinematic viscosity. " The kinematic viscosity
embraces both the viscosity and density properties of a fluid. Dimen-
sional consideration of the above equation shows the dimensions of v
to be square feet per second, a combination of kinematic terms, which
explains the name "kinematic" viscosity.

ILLUSTRATIVE PROBLEM

Calculate the kinematic viscosity of glycerine at 80 F.
From Fig. 4, ju = 0.0103 Ib sec/ft. 2
From Table I, S = 1.26

JLI 0.0103

v = - = - - - = 0.00423 ft 2 /sec
p 62.4 X 1.26

32.2

6. Surface Tension, Capillarity. The apparent tension effects,
which occur on the free surfaces of liquids, where the surfaces are in
contact with another fluid or a solid, depend fundamentally upon the
relative sizes of intermolecular cohesive and adhesive forces. On a
free liquid surface in contact with the atmosphere, surface tension
manifests itself as an apparent "skin" over the surface which will
support small loads. 11 The magnitude of surface tension, T, is the
force in the surface and normal to a line of unit length drawn in the

11 A small needle placed gently upon a water surface will not sink but will be
supported by the tension in the liquid surface.

SURFACE TENSION, CAPILLARITY

liquid surface ; thus it will have dimensions of pounds per foot. Since
surface tension is directly dependent upon intermolecular cohesive
forces, its magnitude will decrease as temperature increases. 12 Surface
tension is also dependent upon the gas in contact with the liquid surface,
thus surface tensions are usually quoted "in contact with air" as indi-
cated in Table IV.

TABLE IV

SURFACE TENSION, T, OF COMMON LIQUIDS *
(At 68 F in contact with air)

Ib/ft

Ethyl alcohol 0.001527

Carbon tetrachloride 001832

Turpentine 0.001857

Benzene .001980

Olive oil 002295

Water 0.004985

Mercury 0.03562

* International Critical Tables, First Edition, 1926-1933, McGraw-Hill Book Company.

Surface tension in the surface of a droplet of liquid causes the pres-
sure inside of the droplet to be greater than that outside. The rela-
tion of this excess pressure to the surface tension can be found by a
simple mechanical analysis as follows. Consider the droplet of diam-

Tension in surface
V= T Ib. per ft.

FIG. 5.

eter d, indicated in Fig. 5. If the droplet is halved the forces on one
half are seen to be (1) the force due to surface tension, T, existing
around the circumference of the droplet and acting to the right, and

12 For example, the surface tension of water decreases from 0.00498 Ib/ft at 68 F
to 0,00421 Ib/ft at 212 F,

16 FUNDAMENTALS

(2) the force due to the excess pressure, p, acting to the left. These
forces are in equilibrium resulting in

Surface tension force Pressure force
or

vdT - p =
4

giving

thus relating excess pressure to surface tension and indicating that
these pressures increase as the size of droplet decreases.

The angle of contact made by a liquid on a horizontal surface illus-
trates another surface-tension phenomenon of more complex nature.
Consider the mercury and water on a glass surface illustrated by Fig. 6.

Glass T a t * A-

surta c,v * y-rxX"

/S x g /\ } C _s?S y

Water Mercury

FIG. 6.

The familiar large angle of contact assumed by the mercury indicates
a comparatively large affinity of mercury molecules for each other
(cohesion) and small affinity of these molecules for those of the glass
(adhesion). The opposite effect is exhibited by the water. The
water is said to "wet" the glass since its angle of contact is less than
90 degrees. The stability of these liquids on a solid surface may be
characterized by the equilibrium of assumed surface tensions 18 at
their points of contact. Thus the following equation may be written

Tis + T a i cos a = T a8

Frequently equilibrium^does not exist between these tensions, and the
following inequality results

T a8 > Ti 8 + T a i cos a

18 T a i = surface tension between liquid and air.
Tis = surface tension between liquid arid solid.
T a8 = surface tension between air and solid.

SURFACE TENSION, CAPILLARITY

causing the liquid to spread over the surface on which it is placed.
Such a condition exists when certain types of oil are placed on a water
surface.

Surface-tension effects like the above, existing when surfaces of
liquids come in contact with vertical solid surfaces, result in the
phenomenon known as "capillarity." Water and mercury in contact
with a vertical clean glass plate are illustrated in Fig. 7. Here again
are demonstrated the results
of attractions and repulsions,
cohesions and adhesions, be-
tween the molecules of liquid
and solid.

When a vertical tube

Glass
surface -

Water

Glass
surface -

Mercury

by*

FIG. 7.

FIG. 8.

placed. in a liquid as in Fig. 8, these surface phenomena form a "menis-
cus/' or curved surface, in the tube, and the liquid in the tube
stands above or below that outside, depending upon the size of the
angle of contact. The "capillary rise," h, in such a tube may be cal-
culated approximately by considering the equilibrium of the vertical
forces on the mass of fluid A BCD. Neglecting the fluid above the low
point of the meniscus the weight of A BCD is given by

which acts downward. The vertical component, FT, of the force due
to surface tension is given by

FT = irdT cos ft
which acts upward and is in equilibrium with the downward force, thus

,^ 2 JT

wh = irdl cos ft

18 FUNDAMENTALS

giving

42" cos ft

h = ~

allowing the capillary rise to be calculated approximately and confirm-
ing the familiar fact that capillary rise becomes greater as tube diam-
eter is decreased.

Similarly it may be shown that the capillary rise between vertical
parallel plates is given by

_ 2Tcos/3
wd

where d is the distance between the plates.

ILLUSTRATIVE PROBLEM

Of what diameter must a droplet of water be to have the pressure within it
0.1 lb/in. 2 greater than that outside?
From Table IV, T = 0.004985 Ib/ft.

4T 4 X 0.004985

p t d = = 0.001385 ft = 0.0166 in.

a 0.1 X 144

7. Vapor Pressure. All liquids possess a tendency to vaporize,
i.e., to change from the liquid state into the gaseous state. Such
vaporization occurs because molecules are continually projected
through the free liquid surface and lost from the body of liquid. Such
molecules, being gaseous, are capable of exerting a partial pressure, the
" vapor pressure" of the liquid, and since this pressure is dependent
primarily upon molecular activity it will increase with increasing tem-
perature. The variation of the vapor pressure of water with tem-
perature is indicated in Fig. 9.

For boiling to occur a liquid's temperature must be raised sufficiently
for the vapor pressure to become equal to the pressure imposed on the
liquid. This means that the boiling point of a liquid is depehdent
upon its pressure as well as its temperature. 14

Table V offers a comparison of the vapor pressures of a few common
liquids at the same temperature. The low vapor pressure of mercury
along with its high density makes this liquid well suited for use in
barometers and other pressure-measuring devices.

14 For instance, water boils at 212 F when exposed to an atmospheric pressure of
14.7 Ib/sq in., but will boil at 200 F if the imposed pressure is reduced to 1 1.4 Ib/sq in.

VAPOR PRESSURE

Vapor Pressure in Pounds per Square Inch
t^co ;- rolo'^cn *-* ro u> *> in o u

/
'

50 100 150 200

Temperature in Degrees F.
FIG. 9. Vapor Pressure of Water.

TABLE V*
VAPOR PRESSURE, p v , OF COMMON LIQUIDS AT 68 F

lb/ft 2 lb/in. 2

Ether 1231. 8.55

Carbon tetrachloride 250 . 1 . 738

Benzene 208 . 1 .448

Ethyl alcohol 122 .4 .850

Water 48.9 0.339

Turpentine 1 . 115 .00773

Mercury .00362 .0000251

* Smithsonian Physical Tables, Eighth Edition, 1933, Smithsonian Institution.

20 FUNDAMENTALS

BIBLIOGRAPHY
HISTORICAL

W. F. DURAND, "The Development of Our Knowledge of the Laws of Fluid Me-
chanics," Science, Vol. 78, No. 2025, p. 343, October 20, 1933.

R. GIACOMELLI and E. PISTOLESI. Historical Sketch, Aerodynamic Theory, Vol. I,
p. 305, Julius Springer, Berlin, 1934.

C. E. BARDSLEY, "Historical Resume of the Development of the Science of Hydrau-
lics," Pub. 39, Engineering Experiment Station of Oklahoma Agricultural and
Mechanical College, April, 1939.

VISCOSITY

E. C. BINGHAM, Fluidity and Plasticity, McGraw-Hill Book Co., 1922.
E. HATSCHEK, The Viscosity of Liquids, Van Nostrand, 1928.

PROBLEMS

1. The two pistons A and B have respectively cross-sectional areas of 2 in. 2 and
50 in. 2 What force, F, must be applied to piston A to support a weight of 100 Ib
on B1

2. If 186 ft 3 of a certain oil weigh 9860 Ib, calculate the
specific weight, density, and specific gravity of this oil.

3. Calculate the specific weight and density of mercury
at 68 F.

4. Calculate the specific weight and density of glycerine
at57F.

5. The density of alcohol is 1.53 slugs/ft 3 . Calculate its specific weight, specific
gravity, and specific volume.

6. A cubic foot of air at 14,7 lb/in. 2 and 59 F weighs 0.0765 lb/ft 3 . What is
its specific volume?

7. Calculate the specific weight, specific volume, and density of air at 40 F and
50 lb/in. 2 absolute.

8. Calculate the density, specific weight, and specific volume of carbon dioxide
at 100 lb/in. 2 absolute and 200 F.

9. Calculate the density, specific weight, and specific volume of chlorine gas at
50 lb/in. 2 absolute and 100 F.

10. Calculate the density of carbon monoxide at 20 lb/in. 2 absolute and 50 F.

11. The specific volume of a certain perfect gas at 30 lb/in. 2 absolute and 100 F
is 10 ft, 3 Calculate its gas constant and molecular weight.

12. If h = . what are the dimensions of h?

13. If V v2gh, calculate the dimensions of V,

14. If F = QwV/g, what are the dimensions of Fl

15. Twelve cubic feet of water are placed under a pressure of 1000 lb/in. 2 . Calcu-
late the volume at this pressure.

16. If the volume of a liquid is reduced 0.035 per cent by application of a pressure
of 100 lb/in. 2 , what is its modulus of elasticity?

17. What pressure must be applied to water to reduce its volume 1 per cent?

18. Ten cubic feet of air at 100 F and 50 lb/in. 2 absolute are compressed isother-

PROBLEMS 21

mally to 2 cu ft. What is the pressure when the air is reduced to this volume? What
is the modulus of elasticity at the beginning and end of the compression?

19. If the air in the preceding problem is compressed adiabatically to 2 cu ft,
calculate the final pressure and temperature and the modulus of elasticity at beginning
and end of the compression.

20. Calculate the velocity of sound in air of standard conditions (32 F and
14.7 lb/in. 2 absolute).

21. Calculate the velocity of sound in fresh water.

22. Calculate the kinematic viscosity of turpentine at 68 F.

23. Calculate the kinematic viscosity of castor oil at 68 F.

24. Calculate the kinematic viscosity of nitrogen at 100 F and 80 lb/in. 2 absolute.

25. What is the ratio between the viscosities of air and water at 50 F? What is
the ratio between their kinematic viscosities at this temperature and standard baro-
metric pressure?

26. A space of 1-in. width between two large plane surfaces is filled with glycerine
at 68.5 F. What force is required to drag a very thin plate of 5 ft 2 area between the
surfaces at a speed of 0.5 ft/sec if this plate remains equidistant from the two sur-
faces? If it is at a distance of 0.25 in. from one of the surfaces?

27. Castor oil at 68 F fills the space between two concentric cylinders of 10-in.
height and 6-in. and 6.25-in. diameters. What torque is required to rotate the inner
cylinder at 12 rpm, the outer cylinder remaining stationary?

28. What force is necessary to overcome viscous action when removing the above
inner cylinder from the outer one at a speed of 1 ft/sec?

29. A circular disk of diameter d is rotated in a liquid of viscosity /x at a small
distance A/& from a fixed surface. Derive an expression for the torque T, necessary to
maintain an angular velocity co. Neglect centrifugal effects.

30. Calculate the excess pressure within a droplet of water at 68 F if the droplet
has a diameter of 0.01 in.

31. What excess pressure may be caused within a 0.20-in. -diameter cylindrical
jet of water by surface tension?

32. Calculate the capillary rise of water in a glass tube of 1-mm diameter at 68 F.

(ft - 0.)

33. Calculate the capillary rise of water (68 F) between two vertical, clean glass
plates spaced 1 mm apart. (|3 = 0.)

34. Develop the equation for theoretical capillary rise between parallel plates.

35. Calculate the capillary depression of mercury in a glass tube of 1-mm diameter
at 68 F. (ft = 140.)

36. A soap bubble 2 in. in diameter contains a pressure (in excess of atmospheric)
of 0.003 lb/in. 2 Calculate the surface tension of the soap film.

37. What force is necessary to lift a thin wire ring of 1-in. diameter from a water
surface at 68 F? Neglect weight of ring.

38. What is the minimum absolute pressure which may be maintained in the space
above the liquid, in a can of ether at 68 F?

39. To what value must the absolute pressure over carbon tetrachloride be re-
duced to make it boil at 68 F?

40. What reduction below standard atmospheric pressure must occur to cause
water to boil at 150 F?

41. A 6-in.-diameter cylinder containing a tight-fitting piston is completely filled
with water at 150 F. What force is necessary to withdraw the piston if atmospheric
pressure is 14.70 lb/in. 2 ?

CHAPTER II
FLUID STATICS

The subject of fluid statics involves fluid problems in which there
is no relative motion between fluid particles. If no relative motion
exists between particles of a fluid, viscosity can have no effect, and the
fluids involved may be treated as if they were completely devoid of
viscosity. With the effects of viscosity excluded from fluid statics
exact solutions of problems may be obtained by analytical methods
without the aid of experiment.

8. Pressure-Density-Height Relationships. The fundamental
equation of fluid statics is that relating pressure, density, and vertical
distance in a fluid. This equation may be derived readily by consider-
ing the vertical equilibrium of an element of fluid such as the small cube
of Fig. 10. Let this cube be differentially small and have dimensions
dx, dy, and dz, and assume that the density of the fluid in the cube is
uniform. If the pressure upward on the bottom face of this cube is p,
the force due to this pressure will be given by p dx dy. Assuming an
increase of pressure in the positive direction of z, the pressure down-

ward on the top face of the cube will be p + dz, and the force due

to this pressure will be ( p -\ -- dz) dx dy. The other vertical force
\ dz /

involved is the weight, dW, of the cube, given by

dW = w dxdy dz
The vertical equilibrium of the cube will be expressed by

/ dp \

[p + dzj

dxdy + wdxdydz pdxdy =

gvng

dz
22

PRESSURE-DENSITY-HEIGHT RELATIONSHIPS

the fundamental equation of fluid statics, which must be integrated for
the solution of engineering problems. Such integration may be
accomplished by transposing the terms w and dz, resulting in

FIG. 10.
which may be integrated as follows :

r pi d P r'

/ = - / dz = z 2 -

J w J*i

giving

FLUID STATICS

in which pi is the greater pressure existing at the lower point 1, p2
the lesser pressure existing at the upper point 2, and h the vertical
distance between these points. The integration of the left-hand side
of the equation cannot be carried out until w = f(p) is known. For
gases this relationship may be obtained from certain laws of thermo-
dynamics. For liquids the specific weight, w, is sensibly constant
allowing integration of the equation to

Pi -

= h or pi

permitting ready calculation of the increase in pressure in a liquid
as depth is gained. It should be noted that equation 1 embodies
certain basic and familiar facts concerning fluids at rest. It shows
that, if h = 0, the pressure difference is zero and thus pressure is con-
stant over horizontal planes
a fluid. Conversely, if

Piezometer m
columns

20.35 in.

of
mercury

Manometer

Mercury

Water

the pressure is constant over
a horizontal plane the height

23.ift of fluid above that plane is

of i i

water constant, resulting in the

tendency of liquids to "seek
their own levels. "

Equation 1 also indicates
the fact that pressure at a
point in a liquid of given
density is dependent solely
upon the height of the liquid
above the point, allowing

this vertical height, or " head," of liquid to be used as an indication
of pressure. Thus pressures maybe quoted in "inches of mercury,"
" feet of water/' etc. The relation of pressure and head r is illus-
trated numerically by the " manometer " and "piezometer columns"
of Fig. 11.

1 For use in problem solutions it is advisable to keep in mind certain pressure and
head equivalents for common liquids. The use of " conversion factors," whose physi-
cal significance is rapidly lost, may be avoided by remembering that standard atmos-
pheric pressure is 14.70 lb/in. 2 , 29.92 in. of mercury (32 F), or 33.9 ft of water (60 F).

10 Ib. per sq. in.
FlG. 11.

ABSOLUTE AND GAGE PRESSURES

ILLUSTRATIVE PROBLEMS

A closed tank is partially filled with carbon tetrachloride. The pressure on the
surface of the liquid is 10 lb/in. 2 . Calculate the pressure 15 ft below the surface.

w = 1.59 X 62.4 = 99.1 lb/ft 3
p l - pz wk, pi - 10 X 144 - 99.1 X 15
pi = 2927 lb/ft 2 = 20.3 lb/in. 2

If the atmospheric pressure at the earth is 14.70 lb/in. 2 and the air tempera-
ture there 60 F, calculate the pressure 1000 ft above the earth, assuming that the
air temperature does not vary between the earth's surface and this elevation.

Isothermal condition, therefore, pi/wi = pz/wz = p/w.

14.70 X 144
53.3(60 + 460)

pi 14.70 X 144

0.0763

. r 1 ^ =

Jpz W

1000

1000
27,750

0.0361

i pl
In

-- 0.0763 lb/ft 8
= 27,750ft

27,750
14.70

In-

14.70
log = 0.0361 X 2.303 = 0.083

Therefore

14.70

= 1.21,

12.15 lb/in. 2

9. Absolute and Gage Pressures. Although
pressures are measured and quoted by two
different systems, one relative, the other
absolute, no confusion will result if the rela-
tion between these systems is completely FlG 12 .
understood.

" Gage pressures " may be best understood by examination of the
common Bourdon pressure gage, a diagrammatic sketch of which is
shown in Fig. 12. A bent tube (^4) is held rigidly at B and its free
end connected to a pointer (C) by means of the link (D). When pres-
sure is admitted to the tube at -B, the tube tends to straighten, thus
actuating the lever system which moves the pointer over a graduated

FLUID STATICS

scale. When the gage is in proper adjustment the pointer rests at
zero on the scale when the gage is disconnected, and in this condi-
tion the pressure inside and outside of the tube will be the same, thus
giving the tube no tendency to deform. Since atmospheric pressure
usually exists outside of the tube, it is apparent that pressure gages
are actuated by the difference between the pressure inside and that
outside of the tube. Thus, in the gage, or relative, system^of pressure
measurement, the atmospheric pressure becomes the zero of pressure.
For pressure greater than atmospheric the pointer will move to the
right ; for pressure less than atmospheric the tube will tend to contract,

moving the pointer to the left.
The reading in the first case is
positive and is called "gage pres-
sure" or simply "pressure," 2 and
is usually measured in pounds
per square inch ; the reading in
Gageo the second case is negative, is
designated as " vacuum," and is
usually measured in inches of
mercury.

The absolute zero of pressure
will exist only in a completely
Abs. o evacuated space (perfect vacu-
um). Atmospheric pressure as
measured by a barometer will
be seen to be pressure in excess of this absolute zero and is, there-
fore, an "absolute pressure." 2 The magnitudes of the atmospheric
pressure in both the absolute and gage system, being known, the
following equation may be written

Gage
pressure A

\ Movable

t
Vacuum B

datum |

^ ! .

Atmospheric

Absolute B

pressure

pressure A
Absolute

(vanes with
weather and

pressure B

altitude)

, 1

Fixed 1

datum

FIG. 13.

Absolute pressure = Atmospheric pressure

Vacuum

+ Gage pressure

which allows easy conversion from one system to the other. Possibly
a better picture of these relationships can be gained from a diagram
such as that of Fig. 13 in which are shown two typical pressures, A and
5, one above, the other below, atmospheric pressure, with all the rela-
tionships indicated graphically.

2 Throughout the remainder of the book "pressure" should be understood to mean
"gage pressure"; when "absolute pressure" is meant, it will be designated as such.

MANOMETRY 27

ILLUSTRATIVE PROBLEM

A Bourdon gage registers a vacuum of 12.5 in. of mercury when the barometric
pressure is 14.50 lb/in. 2 Calculate the corresponding absolute pressure.

14.70
Vacuum = 12.5 X = 6.15 lb/in. 2

Absolute pressure = 14.50 - 6.15 = 8.35 lb/in. 2

10. Manometry. Bourdon pressure gages, owing to their inevi-
table mechanical limitations, are not in general satisfactory for precise
measurements of pressure; when greater precision is required, measure-
ments of the height of liquid columns of known density are commonly
used. Such measurements may be accomplished by means of "manom-
eters" like those of Fig. 14, pressures being obtained by the application
of the pressure-density-height relationships.

Consider the elementary manometer of Fig. 14a, consisting of a
glass tube connected to a reservoir of liquid. With both the reservoir
and the tube open to the atmosphere the liquid surfaces will stand on
the horizontal line 0-0. If the reservoir is now connected to a volume
of gas having an unknown pressure, p x , the surface of the reservoir
liquid will drop to the line 1-1 and that in the tube will rise to the
point 2. Since pressures over horizontal planes in a fluid are constant,
the pressure p x existing on the reservoir surface will also exist as in-
dicated in the tube. Applying equation 1

Px - p2 = *
and with the tube open to the atmosphere, p 2 = and

p x = wh

thus p x may be obtained by measurement of the distance h.

Figure 146 illustrates the measurement of a pressure less than
atmospheric with the same type of manometer. Here the reservoir-is
open to the atmosphere and the unknown pressure, p xt admitted to the
tube. Applying the same principles to the liquid column of height h,

pi - Px = wh
in which

Pi =0
giving

p x = wh

FLUID STATICS

the negative sign indicating the pressure to be less than the atmos-
pheric pressure.

The familiar mercury barometer is shown in Fig. 14c. Such a
barometer is constructed by filling a glass tube, closed at one end, with

|M_

h
^^P

FIG. 14. Manometers.

mercury and inverting the tube, keeping its open end below the surface
of the mercury in a reservoir. The mercury column, hj will be sup-
ported by the atmospheric pressure, leaving an evacuated space, con-
taining only mercury vapor, in the top of the tube. Using the absolute

MANOMETRY 29

system of pressures, equilibrium of the mercury column, h, will be
expressed by

P*tm. - Pv = Wh

in which p v for mercury has been seen to be negligible at ordinary
temperatures (Table V) resulting in

thus allowing atmospheric pressure to be easily obtained by measure-
ment of the height of the mercury column.

Calculation of the pressure, p x , measured by the U-tube manometer
of Fig. 14J may be obtained easily by noting that

Pi - p2
and that

Pi = Px +

giving

p x + wl =
resulting in

p x = wih wl
allowing p x to be calculated. 3

U-tube manometers are frequently used to measure the difference
between two unknown pressures p x and p y , as in Fig. 14e. Here,
as before,

P* = Pd
and

p4 = Px +

p5 = py +
giving

Px + v>il = Py +
and

Px Py = ^2^2 +

thus allowing direct calculation of the pressure difference, p x p y .
" Differential manometers*' of the above type are frequently made with
the U-tube inverted, a liquid of small density existing in the top of the
inverted U; the pressure difference measured by manometers of this

3 The use of formulas for manometer solutions is not recommended until experi-
ence has been gained in their limitations.

30 FLUID STATICS

type may be readily calculated by application of the foregoing prin-
ciples.

When large pressures or pressure differences are to be measured a
number of U-tube manometers may be connected in series. Several
applications of the above principles will allow solution for the unknown
pressure or pressure difference.

There are many forms of precise manometers, two of the most
cpmmon of which are shown in Figs. 14/ and 14g. The former is the
"ordinary " draft gage" used in measuring the comparatively small
pressures in drafts of all types. Its equilibrium position is shown at
A, and when it is submitted to a pressure, p x , a vertical deflection, h,
is obtained in which p x = wh. In this case, however, the liquid is
forced down a gently inclined tube so that the manometer " deflect ion,"
/, is much greater than h and, therefore, more accurately read. The
draft gage is usually calibrated to read directly in inches of water.

The principle of the sloping tube is also employed in the alcohol
'micromanometer of Fig. 14g, used in aeronautical research work. Here
the gently sloping glass tube is mounted on a carriage, C, which is
moved vertically by turning the dial, D, which actuates the screw, S.
When p x is zero the carriage is adjusted so that the liquid in the tube
is brought to the hair line, X, and the reading on the dial recorded.
When the unknown pressure, p x , is admitted to the reservoir the alcohol
runs upward in the tube toward B and the carriage is then raised until
the liquid surface in the tube rests again at the hair line, X. The
difference between the dial reading at this point and the original one
gives the vertical travel of the carriage, h, which is the head of alcohol
equivalent to the pressure p x .

Along with the above principles of manometry the following prac-
tical considerations should be appreciated: (1) manometer liquids, in
changing their specific gravities with temperature, will induce errors
in pressure measurements if this factor is overlooked ; (2) errors due to
capillarity may usually be canceled by selecting manometer tubes of
uniform size; (3) although some liquids appear excellent (from density
considerations) for use in manometers, their surface-tension effects
may give poor menisci and thus inaccurate readings ; (4) fluctuations of
the manometer liquids will reduce accuracy of pressure measurement,
but these fluctuations may be reduced by a throttling device in the
manometer line, a short length of small tube proving excellent for this
purpose; (5) when fluctuations are negligible refined optical devices
and verniers may be used for extremely precise readings of the liquid
surfaces.

FORCES ON SUBMERGED PLANE SURFACES

ILLUSTRATIVE PROBLEM

This vertical pipe line with attached gage and manom-
eter contains oil and mercury as shown. The manometer
is open to the atmosphere. What will be the gage reading,

Since

Pi
Px

Px + (0.90 X 62.4)10
(13.55 X 62.4) yf

Oil (.90)

Mercury
(13.55)"

10'

505 lb/ft 2 =3.51 lb/in. 2

11. Forces on Submerged Plane Surfaces. The calculation of
the magnitude, direction, and location of the total forces on surfaces
submerged in a liquid is essential in the design of dams, bulkheads,
gates, tanks, etc.

For a submerged, plane, horizontal area the calculation of these
force properties is simple, because the pressure does not vary over the

FIG. 15.

area; for non-horizontal planes the problem is complicated by pressure
variation. Pressure in liquids, however, has been shown to vary
linearly with depth (equation 1), resulting in the typical pressure
diagrams and resultant forces of Fig. 15.

Now consider the general case 4 of a plane submerged area A B, such
as that of Fig. 16, located in any inclined plane X-X. Let the center of

4 A general solution for the magnitude, direction, and location of the resultant
force on this area will allow easy calculation of the forces on areas of more regular
shape.

FLUID STATICS

gravity of this area be located as shown, at a depth h g and at a distance
l g from the line of intersection, 0-0, of plane X-X and the liquid surface.
Calculating the force, dF y on the small area, dA,

dF = pdA = whdA

but h = / sin a, and substituting this value for h

dF = wl sin a i

~ W AODC

T ^tp

If 1 1 .424r-2.12'-fi

kr.5'
-WAOB

.166

t 1 ! _

1F H

Location:

AA*B* = 8 X5 = 40ft 2
8 X5 3

83.2
12.5 X 40

83.2 ft 4

= 0.166ft

Vertical component , Fv

Direction: Vertically downward
Magnitude:

Fy = WAODC + WAOB
F v = 10 X 5 X 8 X 62.4

X 8 X 62.4 - 34,750 Ib

Location: e = distance, between FV and line DB, and taking moments about
point B

34,750 X e - 24,950 X 2.5 -f 9800 X 2.12

e - 2.40 ft.

FLUID STATICS

31,200 Total force F:

Direction: Downward to right
48 with horizontal

34,750

Magnitude: F = 1000 A/31.2 2 + 34.7S 2 = 46,600 Ib
Location: Through a point located 2.334 ft above B
and 2.40 ft to the right of B

13. Buoyancy and Flotation. The familiar principles of buoyancy
(Archimedes' principle) and flotation are usually stated Respectively:
(1) A body immersed in a fluid is buoyed up by a force equal to the
weight of fluid displaced by the body;
and (2) a floating body displaces its
own weight of the fluid in which it
floats. These principles may be easily
proved by the methods of Art. 12.

A body, A BCD, suspended in a
liquid of specific weight, w, is illustrated
on Fig. 19. After the vertical lines AE
and CF are drawn, it is obvious that
the force F\, acting vertically downward
on the upper surface A DC, is given by

F l = -^(Volume ADCFE)

and F 2 , the force upward on the lower
surface ABC, by

F 2 = ^(Volume ABCFE)

The net vertical force, F&, exerted by the liquid on the body is upward
and given by

FB = 7*2 FI
or

F B = ^(Volume ABCFE - Volume ADCFE)

Performing the indicated subtraction results in

FB = ^(Volume A BCD) = w( Volume of body)

thus "the buoyant force is equal to the weight of fluid displaced by the
body." The vertical equilibrium of the body is expressed by

F + F B - W =

For a floating body (Fig. 20) vertical equilibrium is expressed by

F B - W =

BUOYANCY AND FLOTATION

and the vertical component of force on the immersed area, A BCD,
will (Art. 12) be given by

thus

F B = ^(Volume ABCD)
W = ^(Volume ABCD)

1W of body
e.g. of body

and the body "displaces its own weight of the fluid in which it floats."
The above principles find many applications in engineering, such as
calculations: of the draft of sur-
face vessels; of the weight of a
ship's cargo from the increment
in depth of flotation; of the lift
of balloons; etc. \ F

The stability of submerged or P 20

floating bodies is dependent upon

the relative location of the buoyant force and the weight of the body.
The buoyant force acts upward through the center of gravity of the
displaced volume; the weight acts downward at the center of gravity

FIG. 21.

of the body. Stability or instability will be determined by whether
a righting or overturning moment is developed when the center of
gravity and center of buoyancy move out of vertical alignment. Obvi-
ously, for the submerged bodies, such as the balloon and submarine
of Fig. 21, stability requires the center of buoyancy to be above the

40 FLUID STATICS

center of gravity. In surface vessels, however, the center of gravity
is usually above the center of buoyancy, and stability exists because
of movement of the center of buoyancy to a position outboard of the
center of gravity as the ship "heels over," thus producing a righting
moment. An overturning moment, resulting in capsizing, occurs if
the center of gravity moves outboard of the center of buoyancy.

ILLUSTRATIVE PROBLEM

A ship has a cross-sectional area of 4000 ft 2 at the water line when the draft
is 10 ft. How many pounds of cargo will increase the draft 2 in.? Assume salt
water.

Since the ship floats, the weight of water displaced by the cargo equals the
weight of the cargo. Therefore

Weight of cargo = 4000 X A X 64.0 = 42,700 Ib

14. Stresses in Circular Pipes and Tanks. The circumferential
tension stresses in pipes and tanks under pressure may be readily

calculated if pressure variation is
neglected and if the thickness of
the pipe or tank is small com-
pared to the diameter.

A section of pipe of length /,
having an internal diameter d, is
shown in Fig. 22. This pipe con-
tains a fluid whose pressure is p;
the circumferential tension stress
in the walls is s t . Pass a vertical

plane through the center of the pipe and consider the horizontal
equilibrium of the forces acting on the section of pipe to the right of
this plane. This equilibrium is expressed by

2T - pdl =

in which T is the total tension force in the wall of length / and thick-
ness t due to the stress s^ and therefore is given by

T = s t tl

Substituting this in the equation above

2s t tl = pdl
\pd

FLUID MASSES SUBJECTED TO ACCELERATION

thus allowing the wall stress to be calculated when the internal pres-
sure and the dimensions of the pipe or tank are known. The final
equation also indicates that the stress caused by a given pressure may
be reduced by decreasing the diameter d, or increasing the wall thick-
ness t. Since increasing the wall thickness increases the cost it be-
comes evident why small-bore tubing is in general use in high-pressure
work.

A mechanical analysis similar to the above may be applied in the
design of wooden tanks or pipes where the tension is carried by external
circumferential hoops, and in the design of concrete structures of this
type where the tension stress is carried by circumferential reinforcing
rods. t

15. Fluid Masses Subjected
to Acceleration. Fluid masses
may be subjected to various
types of acceleration without
relative motion occurring be-
tween fluid particles or between
fluid particles and boundaries.
Such fluid masses will be found
to conform to the laws of fluid statics, modified to account for the
inevitable inertia forces which exist when acceleration occurs.

An open container of liquid subjected to a vertical upward accelera-
tion, a (Fig. 23), will contain greater pressures than if the liquid is
at rest, owing to the forces of inertia which act in the opposite direction
to that of acceleration. The general relationships for this type of
accelerated motion may be obtained by considering the vertical
equilibrium of a cylinder of fluid, of height A, cross-sectional area A,
and having its upper base in the liquid surface.

The force F, upward on the bottom of the cylinder due to pressure,
will balance the forces of inertia Fj and weight W, acting downward.
Therefore, __

in which ,-, . ,

r = pA

W = Mg
FI = M a
in which M, the mass of the liquid cylinder, is expressed by

^Static pressure pwk
FlG. 23.

M = phA =

FLUID STATICS

and substituting these values in the first equation

and solving for p

w w

pA = - hAg + - hAa
g g

p =

indicating that pressure variation with depth is linear and that the
pressure at any point will be given by the product of ( - ) and the

static pressure, wh, at the point. These facts are indicated graphically
in Fig. 23.

Similarly it may be shown that, for a fluid mass undergoing a ver-
tical downward acceleration a, pressure p, at a depth h, will be given by

p =

If a = g in this equation, the pressure becomes zero, showing that a
freely falling unconfined fluid mass exerts no pressure, a fact which
will have many applications in subsequent problems.

Horizontal acceleration of a liquid mass in an open container is

indicated in Fig. 24, acceler-
ation of this kind causing the
liquid surface to drop at the
front of the tank and to
rise at the rear. The forces
which a liquid particle at
the surface will exert on its
neighboring particles will be
its weight, W, acting down-
ward, and its inertia force,
Fj, acting horizontally and
in a direction opposite to
that of acceleration. How-

ever, for a liquid surface to be stable the resultant of these forces
must act normal to the liquid surface; thus, referring to Fig. 24, the
liquid surface will stand at an angle, 0, with the horizontal, and

FIG. 24.

Ma a

FLUID MASSES SUBJECTED TO ACCELERATION 43

proving that the liquid surface and other lines of constant pressure
are straight lines having a slope a/g.

A fruitful means of examining this problem further is to imagine
the magnitude and direction of the acceleration due to gravity, g, to
be changed to those of g f . When this is done the pressure-variation
problem may be analyzed as one of simple fluid statics in which the
relation of pressure, p, to depth, /, along the direction of g f will be
given by

p = w'l

in which w' is the apparent specific weight of the liquid in a system
where acceleration due to gravity is g' instead of g. The density of the
liquid, p, is the same in both systems and, therefore,

w w f

from which

Substituting above

p = (^)wl (5)

but from similar triangles

and substitution of this relation in equation 5 gives the familiar
equation

p = wh

The fact that this equation applies to fluid masses while they are being
accelerated horizontally means that the total forces on vertical areas,
such as the ends of the container of Fig. 24, may be calculated by the
principles of Art. 11. These forces are indicated as F and F%, and if
the mass of fluid in the container is designated by M, it will be found
that

FI - F 2 = Ma

the unbalanced force, FI F 2j being equal to the product of mass and
acceleration, thus checking the Newtonian relationship, F = Ma.

FLUID STATICS

ILLUSTRATIVE PROBLEM

A rectangular tank 15 ft long, 5 ft high, and 8 ft wide is filled with water and
accelerated along the direction of its length, at 6 ft/sec 2 . Calculate the volume of

water spilled, and check the equal-
y ity of force to accelerate the final

I mass and the force exerted by

liquid on the ends of the tank.

15'

a =6 Ft. per sec. 2

32.2

X 15 = 2.80 ft

Volume spilled = J X 2.80 X 15 X 8 = 168 ft 3

Force, F, for accelerat ion = Ma

s 62.4
F = (5 X 15 X 8 - 168) X 6

o L ,L

5020 Ib

F l = 5 X 8 X 2.5 X 62.4 = 6230 Ib

2.2
F 2 = X 2.2 X 8 X 62.4 = 1210 Ib

p l _ F 2 = 6230 - 1210 = 5020 Ib (Check)

Fluid masses subjected to rotation at constant angular velocity
will contain pressure variations unlike those in fluids at rest, because
of the centrifugal forces exerted by fluid particles. The centrifugal
force exerted by a mass M, rotating about an axis with circumferential
velocity V, at a radius r, is given by

(6)

^ '

in which V 2 /r is termed the "centrifugal acceleration."

The simple case of fluid rotation about a vertical axis is indicated

FIG. 25.

in Fig. 25 where the forces on a small element of fluid are considered.
Let this element have dimensions dx, dy, and dz and rotate with the
circumferential velocity, v, at a distance x from the axis of rotation Z.

FLUID MASSES SUBJECTED TO ACCELERATION 45

Since the mass of fluid does not move in the radial direction an equi-
librium of forces exists in this direction which may be stated as

Centrifugal force == Centripetal force

in which the centripetal force can result only from pressure variation
along x. Taking the pressure on the inner vertical face of the element
to be p and assuming the pressure to increase with x, the pressure on
the outer face will be

p + dp or p + ~dx
dx

thus

Centripetal force = ( p H dx ) dy dz p dy dz

\ dx /

= dx dy dz
dx

The centrifugal force may be calculated from equation 6 in which,
for the fluid element,

M = p dx dy dz = dx dy dz
g

V = v cox

r = x
co being the angular velocity of rotation. Therefore

Centrifugal force dx dy dz

g x

Equating centrifugal and centripetal forces,

dp . , , w (ux) 2

dx dy dz dx dy dz

dx g x

results in

dp w 2

- = o> x
dx g

a differential equation expressing the variation of pressure in the radial
direction for fluid masses subjected to constant angular velocity. Since
dp/dx is a positive quantity it may be concluded directly that pressure
will increase as radius increases. This equation may be integrated for
practical use between the eixis of rotation where x = and p = p c

FLUID STATICS

and any point, x, where a pressure, p, exists. Separating the variables
and integrating,

{***= f X "


Search: Advanced Search	UploadAnonymous User (login or join us)