chemical engineering fluid mechanics 2nd edition revised and expanded (r darby)

ISBN: 0-8247-0444-4
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Preface
The objectives of this book are twofold: (1) for the student, to show how the
fundamental principles underlying the behavior of fluids (with emphasis on
one-dimensional macroscopic balances) can be applied in an organized and
systematic manner to the solution of practical engineering problems, and (2)
for the practicing engineer, to provide a ready reference of current informa-
tion and basic methods for the analysis of a variety of problems encountered

in practical engineering situations.
The scope of coverage includes internal flows of Newtonian and non-
Newtonian incompressible fluids, adiabatic and isothermal compressible
flows (up to sonic or choking conditions), two-phase (gas–liquid, solid–
liquid, and gas–solid) flows, external flows (e.g., drag), and flow in poro us
media. Applications include dimensional analysis and scale-up, piping sys-
tems with fittings for Newtonian and non-Newtonian fluids (for unknown
driving force, unknown flow rate, unknown diameter, or most economical
diameter), compressible pipe flows up to choked flow, flow measurement
and control, pumps, compressors, fluid-particle separation methods (e.g.,
iii
centrifugal, sedimentation, filtration), packed columns, fluidized beds, sedi-
mentation, solids transport in slurry and pneumatic flow, and frozen and
flashing two-phase gas–liquid flows. The treatment is from the viewpoint of
the process engineer, who is concerned with eq uipment operation, perfor-
mance, sizing, and selection, as opposed to the details of mechanical design
or the details of flow patterns in such situations.
For the student, this is a basic text for a first-level course in process
engineering fluid mechanics, which emphasizes the systematic application of
fundamental principles (e.g., macroscopic mass, energy, and momentum
balances and economics) to the analysis of a variety of fluid problems of
a practical nature. Methods of analysis of many of these operations have
been taken from the recent technical literature, and have not previously been
available in textbooks. This book includes numerous problems that illus-
trate these applications at the end of each chapter.
For the practicing engineer, this book serves as a useful reference for
the working equations that g overn many applications of practical interest,
as well as a source for basic principles needed to analyze other fluid systems
not covered explicitly in the book. The obj ective here is not to provide a
mindless set of recipes for rote application, however, but to demonstrate an

organized approach to problem analysis beginning with basic principles and
ending with results of very practical app licability.
Chemical Engineering Fluid Mechanics is based on notes that I have
complied and continually revised while teaching the junior-level fluid
mechanics course for chemical engineering students at Texas A&M
University over the last 30 years. It has been my experience that, when
being introduced to a new subject, students learn best by starting with
simple special cases that they can easily relate to physically, and then pro-
gressing to more generalized formulations and more complex problems.
That is the philosophy adopted in this book. It will certainly be criticized
by some, since it is contrary to the usual procedure followed by most text-
books, in which the basic principles are presented first in the most general
and mathematical form (e.g., the divergence theorem, Reynolds trans port
theorem, Navier Stokes equations, etc.), and the special cases are then
derived from these. Esoterically, it is very appealing to progress from the
general to the specific, rather than vice versa. However, having taught from
both perspectives, it is my observation that most beginning students do not
gain an appreciation or understanding from the very general, mathemati-
cally complex, theoretical vector expressions until they have gained a certain
physical feel for how fluids behave, and the laws governing their behavior, in
special situations to which they can easily relate. They also understand and
appreciate the principles much better if they see how they can be applied to
the analysis of practical and useful situations, with results that a ctually work
iv Preface
in practice. That is why the multi-dimensional vector generalizations of
the basic conservations laws have been eschewed in favor of the simpler
component and one-dimensional form of these laws.
It is also important to maintain a balanced perspective between funda-
mental, or theoretical, and empir ical information, for the practicing
engineer must use both to be effective. It has been said that all the tools

of mathematics and physics in the world are not sufficient to calculate how
much water will flow in a given time from a kitchen tap when it is opened.
However, by proper formulation and utilization of certain experimental
observations, this is a routine problem for the engineer. The engineer
must be able to solve certain problems by direct application of theoretical
principles only (e.g., laminar flow in uniform conduits), others by utilizing
hypothetical models that account for a limited understanding of the basic
flow phenomena by incorporation of empirical parameters (e.g., :turbulent
flow in conduits and fittings), and still other problems in which important
information is purely empirical (e.g., pump efficiencies, two-phase flow in
packed columns). In many of these problems (of all types), application of
dimensional analysis (or the principle of ‘‘conservation of dimensions’’) for
generalizing the results of specific analysis, guiding experimental design, and
scaling up both theoretical and experimental results can be a very powerful
tool.
This second edition of the book includes a new chapter on two-phase
flow, which deals with solid–liqui d, solid–gas, and frozen and flashing
liquid–gas systems, as well as revised, updated, and extended material
throughout each chapter. For example, the method for selecting the proper
control valve trim to use with a given piping configuration is presented and
illustrated by example in Chapter 10. The section on cyclone separators has
been completely revised and updated, and new material has been incorpo-
rated in a revision of the material on particles in non-Newtonian fluids.
Changes have made throughout the book in an attempt to improve the
clarity and utility of the presentation wherever possible. For example, the
equations for compressible flow in pipes have been reformulated in terms of
variables that are easier to evaluate and represent in dimensionless form.
It is the aim of this book to provide a useful introduction to the
simplified form of basic governing equations and an illustration of a con-
sistent method of applying these to the analysis of a variety of practical flow

problems. Hopefully, the reader will use this as a starting point to delve
more deeply into the limitless expanse of the world of fluid mechanics.
Ron Darby
Preface v

Contents
Preface iii
Unit Conversion Factors xvi
1. BASIC CONCEPTS 1
I. FUNDAMENTALS 1
A. Basic Laws 1
B. Experience 2
II. OBJECTIVE 2
III. PHENOMENOLOGICAL RATE OR TRANSPORT
LAWS 3
A. Fourier’s Law of Heat Conduction 4
B. Fick’s Law of Diffusion 5
C. Ohm’s Law of Electrical Conductivity 5
D. Newton’s Law of Viscosity 6
vii
IV. THE ‘‘SYSTEM’’ 9
V. TURBULENT MACROSCOPIC (CONVECTIVE)
TRANSPORT MODELS 10
PROBLEMS 11
NOTATION 13
2. DIMENSIONAL ANALYSIS AND SCALE-UP 15
I. INTRODUCTION 15
II. UNITS AND DIMENSIONS 16
A. Dimensions 16
B. Units 18

C. Conversion Factors 19
III. CONSERVATION OF DIMENSIONS 20
A. Numerical Values 21
B. Consistent Units 22
IV. DIMENSIONAL ANALYSIS 22
A. Pipeline Analysis 25
B. Uniqueness 28
C. Dimensionless Variables 28
D. Problem Solution 29
E. Alternative Groups 29
V. SCALE-UP 30
VI. DIMENSIONLESS GROUPS IN FLUID
MECHANICS 35
VII. ACCURACY AND PRECISION 35
PROBLEMS 40
NOTATION 52
3. FLUID PROPERTIES IN PERSPECTIVE 55
I. CLASSIFICATION OF MATERIALS AND FLUID
PROPERTIES 55
II. DETERMINATION OF FLUID VISCOUS
(RHEOLOGICAL) PROPERTIES 59
A. Cup-and-Bob (Couette) Viscometer 60
B. Tube Flow (Poiseuille) Viscometer 63
III. TYPES OF OBSERVED FLUID BEHAVIOR 64
A. Newtonian Fluid 65
B. Bingham Plastic Model 65
C. Power Law Model 66
viii Contents
D. Structural Viscosity Models 67
IV. TEMPERATURE DEPENDENCE OF VISCOSITY 71

A. Liquids 71
B. Gases 72
V. DENSITY 72
PROBLEMS 73
NOTATION 83
REFERENCES 84
4. FLUID STATICS 85
I. STRESS AND PRESSURE 85
II. THE BASIC EQUATION OF FLUID STATICS 86
A. Constant Density Fluids 88
B. Ideal Gas—Isothermal 89
C. Ideal Gas—Isentropic 90
D. The Standard Atmosphere 90
III. MOVING SYSTEMS 91
A. Vertical Acceleration 91
B. Horizontally Accelerating Free Surface 92
C. Rotating Fluid 93
IV. BUOYANCY 94
V. STATIC FORCES ON SOL ID BOUNDARIES 94
PROBLEMS 96
NOTATION 104
5. CONSERVATION PRINCIPLES 105
I. THE SYSTEM 105
II. CONSERVATION OF MASS 106
A. Macroscopic Balance 106
B. Microscopic Balance 107
III. CONSERVATION OF ENERGY 108
A. Internal Energy 110
B. Enthalpy 112
IV. IRREVERSIBLE EFFECTS 113

A. Kinetic Energy Correction 116
V. CONSERVATION OF MOMENTUM 120
A. One-Dimensional Flow in a Tube 121
B. The Loss Coefficient 123
C. Conservation of Angular Momentum 127
Contents ix
D. Moving Boundary Systems and Relative Motion 128
E. Microscopic Momentum Balance 130
PROBLEMS 134
NOTATION 146
6. PIPE FLOW 149
I. FLOW REGIMES 149
II. GENERAL RELATIONS FOR PIPE FLOWS 151
A. Energy Balance 151
B. Momentum Balance 152
C. Continuity 153
D. Energy Dissipation 153
III. NEWTONIAN FLUIDS 154
A. Laminar Flow 154
B. Turbulent Flow 155
C. All Flow Regimes 164
IV. POWER LAW FLUIDS 164
A. Laminar Flow 165
B. Turbulent Flow 166
C. All Flow Regimes 166
V. BINGHAM PLASTICS 167
A. Laminar Flow 168
B. Turbulent Flow 169
C. All Reynolds Numbers 169
VI. PIPE FLOW PROBLEMS 169

A. Unknown Driving Force 170
B. Unknown Flow Rate 172
C. Unknown Diameter 174
D. Use of Tables 177
VII. TUBE FLOW (POISEUILLE) VISCOMETER 177
VIII. TURBULENT DRAG REDUCTION 178
PROBLEMS 184
NOTATION 192
REFERENCES 193
7. INTERNAL FLOW APPLICATIONS 195
I. NONCIRCULAR CONDUITS 195
A. Laminar Flows 195
B. Turbulent Flows 198
x Contents
II. MOST ECONOMICAL DIAMETER 200
A. Newtonian Fluids 203
B. Non-Newtonian Fluids 205
III. FRICTION LOSS IN VALVES AND FITTlNGS 206
A. Loss Coefficient 207
B. Equivalent L=D Method 207
C. Crane Method 208
D. 2-K (Hooper) Method 209
E. 3-K (Darby) Method 209
IV. NON-NEWTONIAN FLUIDS 214
V. PIPE FLOW PROBLEMS WITH FITTINGS 215
A. Unknown Driving Force 216
B. Unknown Flow Rate 217
C. Unknown Diameter 218
VI. SLACK FLOW 221
VII. PIPE NETWORKS 225

PROBLEMS 228
NOTATION 237
REFERENCES 238
8. PUMPS AND COMPRESSORS 239
I. PUMPS 239
A. Positive Displacement Pumps 239
B. Centrifugal Pumps 240
II. PUMP CHARACTERISTICS 241
III. PUMPING REQUIREMENTS AND PUMP
SELECTION 243
A. Required Head 244
B. Composite Curves 245
IV. CAVITATION AND NET POSITIVE SUCTION
HEAD (NPSH) 247
A. Vapor Lock and Cavitation 247
B. NPSH 248
C. Specific Speed 249
D. Suction Specific Speed 250
V. COMPRESSORS 252
A. Isothermal Compression 254
B. Isentropic Compression 254
C. Staged Operation 255
D. Efficiency 256
Contents xi
PROBLEMS 256
NOTATION 265
REFERENCES 266
9. COMPRESSIBLE FLOWS 267
I. GAS PROPERTIES 267
A. Ideal Gas 267

B. The Speed of Sound 268
II. PIPE FLOW 270
A. Isothermal Flow 271
B. Adiabatic Flow 273
C. Choked Flow 273
D. The Expansion Factor 275
E. Ideal Adiabatic Flow 277
III. GENERALIZED EXPRESSIONS 279
A. Governing Equations 279
B. Applications 281
C. Solution of High-Speed Gas Problems 283
PROBLEMS 286
NOTATION 290
REFERENCES 291
10. FLOW MEASUREMENT AND CONTROL 293
I. SCOPE 293
II. THE PITOT TUBE 293
III. THE VENTURI AND NOZZLE 295
IV. THE ORIFICE METER 304
A. Incompressible Flow 305
B. Compressible Flow 306
V. LOSS COEFFICIENT 308
VI. ORIFICE PROBLEMS 310
A. Unknown Pressur e Drop 311
B. Unknown Flow Rate 311
C. Unknown Diameter 312
VII. CONTROL VALVES 312
A. Valve Characteristics 313
B. Valve Sizing Relations 314
C. Compressible Fluids 327

D. Viscosity Correction 330
PROBLEMS 333
xii Contents
NOTATION 338
REFERENCES 339
11. EXTERNAL FLOWS 341
I. DRAG COEFFICIENT 341
A. Stokes Flow 342
B. Form Drag 343
C. All Reynolds Numbers 343
D. Cylinder Drag 344
E. Boundary Layer Effects 345
II. FALLING PARTICLES 347
A. Unknown Velocity 348
B. Unknown Diameter 349
C. Unknown Viscosity 3 49
III. CORRECTION FACTORS 350
A. Wall Effects 350
B. Drops and Bubbles 351
IV. NON-NEWTONIAN FLUIDS 352
A. Power Law Fluids 352
B. Wall Effects 357
C. Carreau Fluids 358
D. Bingham Plastics 358
PROBLEMS 361
NOTATION 363
REFERENCES 364
12. FLUID–SOLID SEPARATIONS BY FREE SETTLING 365
I. FLUID–SOLID SEPARATIONS 365
II. GRAVITY SETTLING 366

III. CENTRIFUGAL SEPARATION 367
A. Fluid–Solid Separation 367
B. Separation of Immiscible Liquids 371
IV. CYCLONE SEPARATIONS 375
A. General Characteristics 375
B. Aerocyclones 376
C. Hydrocyclones 382
PROBLEMS 385
NOTATION 389
REFERENCES 390
Contents xiii
13. FLOW IN POROUS MEDIA 391
I. DESCRIPTION OF POROUS MEDIA 391
A. Hydraulic Diameter 392
B. Porous Medium Friction Factor 393
C. Porous Medium Reynolds Number 394
II. FRICTION LOSS IN PORO US MEDIA 394
A. Laminar Flow 394
B. Turbulent Flow 395
C. All Reynolds Numbers 395
III. PERMEABILITY 395
IV. MULTIDIMENSIONAL FLOW 396
V. PACKED COLUMNS 398
VI. FILTRATION 401
A. Governing Equations 401
B. Constant Pressure Operation 405
C. Constant Flow Operation 406
D. Cycle Time 406
E. Plate-and-Frame Filters 407
F. Rotary Drum Filter 408

G. Compressible Cake 408
PROBLEMS 409
NOTATION 417
REFERENCES 418
14. FLUIDIZATION AND SEDIMENTATION 419
I. FLUIDIZATION 419
A. Governing Equations 420
B. Minimum Bed Voidage 421
C. Nonspherical Particles 421
II. SEDIMENTATION 423
A. Hindered Settling 423
B. Fine Particles 425
C. Coarse Particles 428
D. All Flow Regimes 428
III. GENERALIZED SEDIMENTATION/
FLUIDIZATION 430
IV. THICKENING 430
PROBLEMS 436
NOTATION 441
REFERENCES 442
xiv Contents
15. TWO-PHASE FLOW 443
I. SCOPE 443
II. DEFINITIONS 444
III. FLUID–SOLID TWO-PHASE PIPE FLOWS 447
A. Pseudohomogeneous Flows 447
B. Heterogeneous Liquid–Solid Flows 449
C. Pneumatic Solids Transport 454
IV. GAS–LIQUID TWO-PHASE PIPE FLOW 459
A. Flow Regimes 459

PROBLEMS 474
NOTATION 475
REFERENCES 477
Appendixes
A. Viscosities and Other Properties of Gases and Liquids 479
B. Generalized Viscosity Plot 499
C. Properties of Gases 501
D. Pressure–Enthalpy Diagrams for Various Compounds 505
E. Microscopic Conservation Equations in Rectangular,
Cylindrical, and Spherical Coordinates 513
F. Standard Steel Pipe Dimensions and Capacities 519
G. Flow of Water/Air Through Schedule 40 Pipe 525
H. Typical Pump Head Capacity Range Charts 531
I. Fanno Line Tables for Adiabatic Flow of Air in a Constant
Area Duct 543
Index 553
Contents xv
Unit Conversion Factors
xvi
1
Basic Concepts
I. FUNDAMENTALS
A. Basic Laws
The fundamental principles that apply to the analysis of fluid flows are few
and can be described by the ‘‘conservation laws’’:
1. Conservation of mass
2. Conservation of energy (first law of thermodynamics)
3. Conservation of momentum (Newton’s second law)
To these may also be added:
4. The second law of thermodynamics

5. Conservation of dimensions (‘‘fruit salad’’ law)
6. Conservation of dollars (economics)
These conservation laws are basic and, along with appropriate rate or trans-
port models (discussed below), are the starting point for the solution of every
problem.
Although the second law of thermodynamics is not a ‘‘conservation
law,’’ it states that a process can occur spontaneously only if it goes from a
1
state of higher energy to one of lower energy. In practical terms, this means
that energy is dissipated (i.e., transformed from useful mechanical energy to
low-level thermal energy) by any system that is in a dynamic (nonequili-
brium) state. In other words, useful (mechanical) energy associated with
resistance to motion, or ‘‘friction,’’ is always ‘‘lost’’ or transformed to a
less useful form of (thermal) energy. In more mundane terms, this law
tells us that, for example, water will run downhi ll spontaneously but cannot
run uphill unless it is ‘‘pushed’’ (i.e., unless mechanical energy is supplied to
the fluid from an exterior source).
B. Experience
Engineering is much more than just applied science and math. Although
science and math are important tools of the trade, it is the engineer’s ability
to use these tools (and others) along with considerable judgment and experi-
ment to ‘‘make things work’’—i.e., make it possible to get reasoable answers
to real problems with (sometimes) limited or incomplete information. A key
aspect of ‘‘judgment and experience’’ is the ability to organize and utilize
information obtained from one system and apply it to analyze or design
similar systems on a different scale. The conservation of dimensions (or
‘‘fruit salad’’) law enables us to design experiments and to acquire and
organize data (i.e., experience) obtained in a lab test or model ssytem in
the most efficient and general form and apply it to the solution of problems
in similar systems that may involve different properties on a different scale.

Because the vast majority of problems in fluid mechanics cannot be solved
without resort to experience (i.e., empirical knowledge), this is a very impor-
tant principle, and it will be used extensively.
II. OBJECTIVE
It is the intent of this book to show how these basic laws can be applied,
along with pertinent knowl edge of system properties, operating conditions,
and suitable assumptions (e.g., judgment), to the analysis of a wide variety
of practical problems involving the flow of fluids. It is the author’s belief
that engineers are much more versatile, valuable, and capable if they
approach the problem-solving process from a basic perspective, starting
from first principles to develop a solution rather than looking for a
‘‘similar’’ problem (that may or may not be applicable) as an example to
follow. It is this philosophy along with the object ive of arriving at workable
solutions to practical problems upon which this work is based.
2 Chapter 1
III. PHENOMENOLOGI CAL RATE OR TRANSPORT LAWS
In addition to the conservation laws for mass, energy, momentum, etc.,
there are additional laws that govern the rate at which these quantities are
transported from one region to another in a continuous medium. These
are called phenomenological laws because they are based upon observable
phenomena and logic but they cannot be derived from more fundamental
principles. These rate or ‘‘transport’’models can be written for all conserved
quantities (mass, energy, momentum, electric charge, etc.) and can be
expressed in the general form as
Rate of transport ¼
Driving force
Resistance
¼ Conductance ÂDriving force
ð1-1Þ
This expression applies to the transport of any conserved quantity Q, e.g.,

mass, energy, momentum, or charge. The rate of transport of Q per unit
area normal to the direction of transport is called the flux of Q. This trans-
port equation can be applied on a microscopic or molecular scale to a
stationary medium or a fluid in laminar flow, in which the mechani sm for
the transport of Q is the intermolecular forces of attraction between mole-
cules or groups of molecules. It also applies to fluids in turbulent flow, on a
‘‘turbulent convective’’ scale, in which the mechanism for trans port is the
result of the motion of turbulent eddies in the fluid that move in three
directions and carry Q with them.
On the microscopic or molecular level (e.g., stationary media or lami-
nar flow), the ‘‘driving force’’ for the transport is the negative of the gradient
(with respect to the direction of transport) of the concentration of Q. That
is, Q flows ‘‘downhill,’’ from a region of high concentration to a region of
low concentration, at a rate proportional to the magnitude of the change in
concentration divided by the distance over which it changes. This can be
expressed in the form
Flux of Q in the y direction ¼ K
T
À
dðConc. of Q
dy

ð1-2Þ
where K
T
is the transport coefficient for the quantity Q. For microscopic
(molecular) transport, K
T
is a property only of the medium (i.e., the
material). It is assumed that the medium is a continuum, i.e., all relevant

physical properties can be defined at any point within the medium. This
means that the smallest region of practical interest is very large relative to
the size of the molecules (or distance between them) or any substructure of
the medium (such as suspended particles, drops, or bubbles). It is further
assumed that these properties are homogeneous and isotropic. For macro-
Basic Concepts 3
scopic systems involving turbulent convective transport, the driving force is
a representative difference in the concentration of Q. In this case, the trans-
port coefficient includes the effective distance over which this difference
occurs and consequently is a function of flow conditions as well as the
properties of the medium (this will be discussed later).
Example 1-1: What are the dimensions of the transport coefficient, K
T
?
Solution. If we denote the dimensions of a quantity by brackets, i.e.,
[x] represents ‘‘the dimensions of x,’’ a dimensional equation corresponding
to Eq. (1-2) can be written as follows:
½Flux of Q¼½K
T

½Q
½volume½y
Since ½flux of Q¼½Q=L
2
t, ½volume¼L
3
, and ½y¼L, where L and t are
the dimensifons of length and time, respectively, we see that [Q] cancels out
from the equation, so that
½K

T
¼
L
2
t
That is, the dimensions of the transport coefficient are independent of the
specific quantity that is being transported.
A. Fourier’s Law of Heat Conduction
As an example, Fig. 1-1 illustrates two horizontal parallel plates with a
‘‘medium’’ (either solid or fluid) between them. If the top plate is kept at
a temperature T
1
that is higher than the temperature T
0
of the bottom plate,
there will be a transport of thermal energy (heat) from the upper plate to the
lower plate through the medium, in the Ày direction. If the flux of heat in
the y direct ion is denoted by q
y
, then our transport law can be written
q
y
¼À
T
dðc
v
TÞ
dy
ð1-3Þ
where 

T
is called the thermal diffusion coefficient and ðc
v
TÞ is the ‘‘con-
centration of heat.’’ Because the density () and heat capacity ðc
v
Þ are
assumed to be independent of position, this equation can be written in the
simpler form
q
y
¼Àk
dT
dy
ð1-4Þ
4 Chapter 1
where k ¼ 
T
c
v
is the thermal conductivity of the medium. This law was
formalized by Fourier in 1822 and is known as Fourier’s law of heat con-
duction. This law applies to stationary solids or fluids and to fluids moving
in the x direction with straight streamlines (e.g., laminar flow).
B. Fick’s Law of Diffusion
An analogous situation can be envisioned if the medium is stationary (or
a fluid in laminar flow in the x direction) and the temperature difference
ðT
1
À T

0
Þ is replaced by the concentration difference ðC
1
À C
0
Þ of some
species that is soluble in the fluid (e.g., a top plate of pure salt in contact
with water). If the soluble species (e.g., the salt) is A, it will diffuse through
the medium (B) from high concentration ðC
1
Þ to low concentration ðC
0
Þ.If
the flux of A in the y direction is denoted by n
Ay
, then the transport law is
given by
n
Ay
¼ÀD
AB
dC
A
dy
ð1-5Þ
where D
AB
is the molecular diffusivity of the species A in the medium B.
Here n
Ay

is negative, because species A is diffusing in the Ày direction.
Equation (1-5) is known as Fick’s law of diffusion (even though it is the
same as Fourier’s law, with the symbols changed) and was formulated in
1855.
C. Ohm’s Law of Electrical Conductivity
The same transport law can be written for electric charge (which is another
conserved quantity). In this case, the top plate is at a pot ential e
1
and the
bottom plate is at potential e
0
(electric potential is the ‘‘concentration of
charge’’). The resulting ‘‘charge flux’’ (i.e., current density) from the top
plate to the bottom is i
y
(which is negative, because transport is in the Ày
Basic Concepts 5
FIGURE 1-1 Transport of energy, mass, charge, and momentum from upper to
lower surface.
direction). The corresponding expression for this situati on is known as
Ohm’s law (1827) and is given by
i
y
¼Àk
e
de
dy
ð1-6Þ
where k
e

is the ‘‘electrical conductivity’’ of the medium between the plates.
D. Newton’s Law of Viscosity
Momentum is also a conserved quantity, and we can write an equivalent
expression for the transport of momentum. We must be careful here, how-
ever, because velocity and momentum are vectors, in contrast to mass,
energy, and charge, which are scalars. Hence, even though we may draw
some analogies between the one-dimensional transport of these quantities,
these analogies do not generally hold in multidimensional systems or for
complex geometries. Here we consider the top plate to be subject to a force
in the x direction that causes it to move with a velocity V
1
, and the lower
plate is stationary ðV
0
¼ 0Þ. Since ‘‘x-momentum’’ at any point wher e the
local velocity is v
x
is mv
x
, the concentration of momentum must be v
x
.If
we denote the flux of x-momentum in the y direction by ð
yx
Þ
mf
, the trans-
port equation is
ð
yx

Þ
mf
¼À
dðv
x
Þ
dy
ð1-7Þ
where  is called the kinematic viscosity. It should be evident that ð
yx
Þ
mf
is
negative, because the faster fluid (at the top) drags the slower fluid (below)
along with it, so that ‘‘x-momentum’’ is being transported in the Ày direc-
tion by virtue of this drag. Because the density is assumed to be independent
of position, this can also be written
ð
yx
Þ
mf
¼À
dv
x
dy
ð1-8Þ
where  ¼  is the viscosity (or sometimes the dynamic viscosity). Equation
(1-8) applies for laminar flow in the x direction and is known as Newton’s
law of viscosity. Newton formulated this law in 1687! It applies directly to a
class of (common) fluids called Newtonian fluids, which we shall discuss in

detail subsequently.
1. Momentum Flux and Shear Stress
Newton’s law of viscosity and the conservation of momentum are also
related to Newton’s second law of motion, which is commonly written
F
x
¼ ma
x
¼ dðmv
x
Þ=dt. For a steady-flow system, this is equivalent to
6 Chapter 1
F
x
¼
_
mmv
x
, where
_
mm ¼ dm=dt is the mass flow rate. If F
x
is the force acting in
the x direction on the top plate in Fig. 1. to make it move, it is also the
‘‘driving force’’ for the rate of transport of x-mom entum ð
_
mmv
x
Þ which flows
from the faster to the slowerfluid (in the Ày direction). Thus the force F

x
acting on a unit area of surface A
y
is equivalent to a ‘‘flux of x-momentum’’
in the Ày direction [e.g., Àð
yx
Þ
mf
]. [Note that þA
y
is the area of the surface
bounding the fluid volume of interest (the ‘‘system’’), which has an outward
normal vector in the þy direction.] F
x
=A
y
is also the ‘‘shear stress,’’ 
yx
,
which acts on the fluid—that is, the force þF
x
(in the þx direction) that acts
on the area A
y
of the þy surface. It follows that a positive shear stress is
equivalent to a negative momentum flux, i.e., 
yx
¼Àð
yx
Þ

mf
. [In Chapter 3,
we defi ne the rheological (mechanical) properties of materials in terms that
are common to the field of mechanics, i.e., by relationships between the
stresses that act upon the material and the resulting material deformation.]
It follows that an equivalent form of Newton’s law of viscosity can be
written in terms of the shear stress instead of the momentum flux:

yx
¼ 
dv
x
dy
ð1-9Þ
It is important to distinguish between the momentum flux and the shear
stress because of the difference in sign. Some references define viscosity (i.e.,
Newton’s law of viscosity) by Eq. (1-8), whereas others use Eq. (1-9) (which
we shall follow). It should be evident that these definitions are equvialent,
because 
yx
¼Àð
yx
Þ
mf
.
2. Vectors Versus Dyads
All of the preceding transport laws are described by the same equation (in
one dimension), with different symbols (i.e., the same game, with different
colored jerseys on the players). However, there are some unique features to
Newton’s law of viscosity that distinguish it from the other laws and are

very important when it is being applied. First of all, as pointed out earlier,
momentum is fundamentally different from the other conserved quantities.
This is because mass, energy, and elect ric charge are all scalar quantities
with no directional properties, whereas momentum is a vector with direc-
tional character. Since the gradient (i.e., the ‘‘directional derivative’’ dq=dy
or, more generally, rq) is a vector, it follows that the gradient of a scalar
(e.g., concentration of heat, mass, charge) is a vector. Likewise, the flux of
mass, energy, and charge are vectors. However, Newton’s law of viscosity
involves the gradient of a vector (e.g., velocity or momentum), which implies
two directions: the direction of the vector quantity (momentum or velocity)
and the direction in which it varies (the gradient direction). Such quantities
are called dyads or second-order tensors. Hence, momentum flux is a dy ad,
Basic Concepts 7
with the direction of the momentum (e.g., x) as well as the direction in which
this momentum is transported (e.g., Ày). It is also evident that the equiva-
lent shear stress ð
yx
Þ has two directions, corresponding to the direction in
which the force acts ðxÞ and the direction (i.e., ‘‘orientation’’) of the surface
upon which it acts ðyÞ. [Note that all ‘‘surfaces’’ are vectors because of their
orientation, the direction of the surface being defined by the (outward)
vector that is normal to the surface that bounds the fluid volume of interest.]
This is very significant when it comes to generalizing these one-dimensional
laws to two or three dimensions, in which case much of the analogy between
Newton’s law and the other transport laws is lost.
3. Newtonian Versus Non-Newtonian Fluids
It is also evident that this ‘‘phenomenological’’ approach to transport pro-
cesses leads to the conclusion that fluids should behave in the fashion that
we have called Newtonian, which doe s not account for the occurrence of
‘‘non-Newtonian’’ behavior, which is quite common. This is because the

phenomenological laws inherently assume that the molecular ‘‘transport
coefficients’’ depend only upon the thermodyamic state of the material
(i.e., temperature, pressure, and density) but not upon its ‘‘dynamic
state,’’ i.e., the state of stress or deformation. This assumption is not valid
for fluids of complex structure, e.g., non-Newtonian fluids, as we shall
illustrate in subsequent chapters.
The flow and deformation properties of various materials are dis-
cussed in Chapter 3, although a completely general description of the flow
and deformation (e.g., rheological) properties of both Newtonian and non-
Newtonian fluids is beyond the scope of this book, and the reader is referred
to the more advanced literature for details. However, quite a bit can be
learned, and many problems of a practical nature solved, by considering
relatively simple mod els for the fluid viscosity, even for fluids with complex
properties, provided the complexities of elastic behavior can be avoided.
These properties can be measured in the laboratory, with proper attention
to data interpretation, and can be represented by any of several relatively
simple mathematical expressions. We will not attempt to delve in detail into
the molec ular or structural origins of complex fluid properties but will make
use of information that can be readily obtained through routine measure-
ments and simple modeling. Hence, we will consider non-Newtonian fluids
along with, and in parallel with, Newtonian fluids in many of the flow
situations that we analyze.
8 Chapter 1
IV. THE ‘‘SYSTEM’’
The basic conservation laws, as well as the transport models, are applied to
a ‘‘system’’ (sometimes called a ‘‘control volume’’). The system is not actu-
ally the volume itself but the material within a defined region. For flow
problems, there may be one or more streams entering and/or leaving the
system, each of which carries the conserved quantity (e.g., Q) into and out of
the system at a defined rate (Fig. 1-2). Q may also be transported into or out

of the system through the system boundaries by other means in addition to
being carried by the in and out streams. Thus, the conservation law for a
flow problem with respect to any conserved quantity Q can be written as
follows:
Rate of Q
into the system
À
Rate of Q
out of the system
¼
Rate of accumulation of Q
within the system
ð1-10Þ
If Q can be produced or consumed within the system (e.g., through chemical
or nuclear reaction, speeds approaching the speed of light, etc.), then a
‘‘rate of generation’’ term may be included on the left of Eq. (1-10).
However, these effects will not be present in the systems with which we
are concerned. For example, the system in Fig. 1-1 is the material contained
between the two plates. There are no streams entering or leaving this system,
but the conserved quantity is transported into the system by microscopic
(molecular) interactions through the upper boundary of the system (these
and related concepts will be expanded upon in Chapter 5 and succeeding
chapters).
Basic Concepts 9
FIGURE 1-2 The ‘‘system.’’