Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)
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Contents xiii
Problems ........................................................................................ 339
6.5.1 Flow over a Flat Plate .................................................... 340
6.5.2 Integral Analysis of Boundary Layer Equation.............. 346
6.5.3 Boundary Layer Separation............................................ 350
6.5.4 Integral Relation for Thermal Energy ............................ 353
Exercise.......................................................................................... 355
Problems ........................................................................................ 362
6.6 Turbulent Flow......................................................................... 364
6.6.1 Turbulence Models......................................................... 370
6.6.2 Turbulent Heat Transfer ................................................. 381
Exercise.......................................................................................... 385
Problems ........................................................................................ 390
Nomenclature................................................................................. 391
Bibliography .................................................................................. 394
7. Non-Newtonian Fluid and Flow................................................... 399
7.1.1 Rheological Classifications............................................ 400
7.1.2 Generalized Newtonian Fluid Flows.............................. 406
Exercise.......................................................................................... 408
Problems ........................................................................................ 415
7.2 Standard Flow and Material Functions.................................... 417
7.2.1 Simple Shear Flow......................................................... 418
7.2.2 Shearfree Flow ............................................................... 421
7.2.3 Oscillatory Rheometric Flow ......................................... 424
7.2.4 Viscometric Flow in Rheometry .................................... 428
Exercise.......................................................................................... 437
Problems ........................................................................................ 445
7.3 Viscoelastic Fluid and Flow..................................................... 447
7.3.1 Linear Viscoelastic Rheological Equations.................... 448
7.3.2 Linear and Nonlinear Viscoelastic Models .................... 456
7.3.3 Viscoelastic Models to Standard Flow and Application
to Some Engineering Flow Problems............................
465
7.3.3.1 UCM, CRM and Giesekus Equations................ 465
7.3.3.2 Unidirectional Basic Flow Problems 472
Exercise.......................................................................................... 478
Problems ........................................................................................ 489
Nomenclature................................................................................. 491
Bibliography .................................................................................. 493
Exercise ......................................................................................... 334
6.5 Laminar Boundary Layer Theory ........................................... 340
7.1 Non-Newtonian Fluids and Generalized Newtonian
Fluid Flow...................................................................................... 400
.................
xiv Contents
8. Magnetic Fluid and Flow ............................................................. 497
8.1 Thermophysical Properties ...................................................... 498
Exercise ......................................................................................... 504
Problems ........................................................................................ 506
8.2 Ferrohydrodynamic Equations ................................................ 507
Exercise ......................................................................................... 516
Problems ........................................................................................ 519
8.3 Basic Flows and Applications.................................................. 520
8.3.1 Generalized Bernoulli Equation..................................... 521
8.3.2 Hydrostatics ................................................................... 522
8.3.3 Thermoconvective Phenomena ...................................... 527
Exercise ......................................................................................... 532
Problems ........................................................................................ 537
Nomenclature................................................................................. 538
Bibliography .................................................................................. 540
Appendix............................................................................................ 543
Index .................................................................................................. 561
Contents
Introduction
Since the beginning of human civilization, communities have consistently
been established at locations that feature a viable source of flowing water.
Throughout history, people have continuously attempted to manipulate the
natural flow of water, in order to affect an improvement in such areas as
agricultural stability, living environment, and transportation. Indeed, even
opment of civilization’s ability to adapt, and adapt to, the natural environ-
ment. Along with a reliable source of water, weather prediction and aware-
ness of seasonal changes have been critical to basic social structures like
farming, animal hus
bandry, and housing.
As understanding of the natural world has grown, and modern tech-
nologies have emerged, we have become increasingly reliant on the fun-
damental principles of fluid flow. Humanity has come to depend upon the
development and design of modern transport, like cars and aircraft, which
are rooted in an essential understanding and knowledge of fluid flows.
Not only are fluid flows critical for solving aerodynamics problems, but
also for a plethora of engineering problems concerning energy conserva-
tion and transmission. Time and again, methodological engineering, and
even bio-medical studies, have proven the universally accepted tenant that
understanding fluid flow is critical to the development of applied knowl-
edge. Furthermore, it is clear that they are all, in the end, derived from the
field of fluid engineering, which is key to opening the mental door to vari-
ous forms of inspiration.
Topics covered in fluid engineering are quite diverse. However, the
theoretical background of fluid engineering is based upon fluid mechanics
(or hydrodynamics), which assumes that all basic equations relating to the
conservation law, i.e. mass, linear momentum, angular momentum and en-
ergy, are derived from the concept of continuum mechanics. From the
characteristics, it is seen that both the gaseous and liquid phases of matter
can customarily be qualified as fluids. In dealing with the mechanics of
fluid flow, a continuum concept has to be introduced before commencing
discussion on the kinematics of fluid. We can treat fluids as continuum if
they are homogeneous, uniform and of macroscopic volume, in which only
1
sensitivity to air currents and cloud flow has been important to the devel-
common definition of a fluid, avoiding complicated discussions on fluid
2 Introduction
bulk properties are interested by taking mean molecules, atoms and aggre-
gations of the like, which consist of the fluid.
When deriving governing fluid mechanics equations, particularly for
fluids of low molecular weight, for instance water or air, the Navier-Stokes
equation can be obtained directly through Newton’s law of viscosity. These
have been called Newtonian fluids in consideration of their conservation of
linear momentum. Based on the Navier-Stokes equation together with con-
tinuity and energy equations, all the practical equations and formula in
fluid engineering dealing with conventional hydraulic and air machineries
can be satisfactorily obtained. Moreover, decades of experiments and engi-
neering practices have demonstrated the dependability of this theory.
However, there exist fluids, which do not necessarily obey Newton’s
law of viscosity, and those fluids, so-called non-Newtonian fluids, include
polymeric fluids – polymer solutions, polymer melts and multi-phase sys-
tems, and electro-magnetic fluids – magnetic fluids, plasmas, and so forth.
All basic equations of the conservation law derived from continuum me-
chanics can still be upheld, but in each case for non-Newtonian fluid the
relationship between the internal stress and the applied strain, namely the
constitutive equation (or relation) must be specified, instead of Newton’s
law of viscosity. Due to growing interest in industrial applications, flows
of non-Newtonian polymeric fluids and magnetic fluids are introduced in
this book as advanced topics in fluid engineering, which may serve to cata-
lyze interest in very challenging subjects for readers who wish to further
extend their knowledge.
In science and engineering, when converting from absolute to engi-
neering unit and vice versa, some confusion occasionally arises. In 1960,
the metric system of units (Système International d’Unités or more com-
monly known as the S.I. system) was introduced to overcome this problem.
The S.I. system, is dimensionally consistent as it uses the absolute M.K.S
system (M for Meter [m]; K for Kilogram [kg]; and S for Second [s]).
Other fundamental units include the Ampere [A] for electric current; mol
[mol] for molecular weight; and Candela [Cd] for brightness of light. It
also includes the two supplementary units radian [rad] for angle and stera-
dian [st rad] for solid angle, as well as degrees Kelvin [K] for temperature.
With the S.I. system the units for heat, work and energy are the same (i.e.
Joule [J], which is defined as the work done when a force of 1N is dis-
placed through 1m along its direction). This is one of the advantages of the
S.I. system.
Furthermore, it is noted that in the S.I. system of units, some of the
units are named after scientists, such as Newton [N] for force; Kelvin [K]
for temperature; Stokes [St] for dynamic viscosity; Poise [P] for kinematic
viscosity; Watt [W] for power; Pascal [Pa] for pressure; Hertz [Hz] for
Introduction 3
frequency; Joule [J] for energy: Coulomb [C] for electric charge; Ampere
[A] for electric current; and Tesla [T] for magnetic flux density. These
units, and combinations thereof, are the fundamental units of the S.I. sys-
tem. The typically combined units frequently used in fluid engineering are
listed in Table 1.1. Throughout this book, the numerical examples and
problems are given in the S.I. system of units.
Table 1 Named-combined unit
Unit Abbreviation Combined relation
Ampere
A
sC1A1
Gauss
G
T10G1
4
Hertz
Hz
1Cycle s1Hz
Joule
J
mN1J1
Newton
N
2
smKg1N1
Oersted
Oe
mAʌ41000
Pascal
Pa
2
mN1Pa1
Poise
P
sPa0.1cmsdyn1
2
Stokes
St sm10scm1St1
242
Tesla
T
mANT1
Watt
W
sJ1W1
Non-S.I. system of units C ;electric charge, Coulomb
Introduction
1. Fundamentals in Continuum Mechanics
description of the motion of flow is given in three dimensional space with
respect to the reference frame or relative to the rotational frame. Dynamics
involves the frequent use of derivatives of scalars, vectors and tensors. Ve-
locity and acceleration are the time derivatives, which are important kine-
matic parameters necessary to set an equation of motion in Newtonian me-
chanics, on which continuum mechanics is based. Also given are forces
connected with space derivatives in regard to displacement gradient and
relative strain, which are other important aspects of continuum mechanics.
In this chapter, definitions of stress tensors and strain tensors are also pro-
vided, and will be developed in more detail in the chapters to follow.
1.1 Dynamics of Fluid Motion
When we deal with fluid motion, in many fluid engineering cases the dy-
namics of a molecule, or the molecular structure of the fluid body, does not
explicitly come into effect. At the scale of molecular motion, properties of
the fluid body, such as density, are typically subject to extreme variation
with respect to the instantaneous distance of the frame. While, for the mo-
tion of fluid flow, the macro-motion with the scale of flow channel or ex-
ternal object takes place, thus we may apply the “continuum hypothesis”,
with which the fluid body has a continuous structure in the instantaneous
frame of space, as schematically indicated in Fig. 1.1. In Fig. 1.1, let us
denote
m
L as the small scale (molecular scale), which can be taken as the
mean free path of the molecule;
l
L as the large scale, which can be the
characteristic length of the geometric configuration of fluid motion. More-
over, there may exist an intermediate length scale
i
L , where a certain ef-
fect of a molecule or the molecular structure retains the properties of fluid.
In order to quantify the effect of the scale in the properties of fluid, and
consequently to the dynamics of fluid motion, we will take the ratio be-
tween the actual characteristic length of flow geometry, typically
l
L , and
Certain concepts and definitions are basic to the study of continuum
mechanics, and they should be thoroughly understood at the outset. Below, a
5
6 1 Fundamentals in Continuum Mechanics
the mean free path of the molecules (the correlation length of the mole-
cules)
m
L
, such that we get the following formula where is called the
Knudsen number.
l
m
L
L
(1.1.1)
“Continuum Mechanics” in a general sense, and “Fluid Mechanics” in
particular, is normally valid when
1 , where the continuum hypothe-
sis can be assured. Henceforth we shall make two assumptions: first, that
in every case, the flow of fluid has a small Knudsen Number, with which
the scale of momentum of flow is far longer than the correlation length of
the molecules; and second, that the fluid body has a continuous structure.
Fig. 1.1 Property variation with scale (as typically seen with properties
such as density
U
)
It is interesting to mention, although we will not deal with the problem
in this text, that there is a field of study that deals with small scaled flow
phenomena in continuum mechanics. These phenomena have been dubbed
fluid flows in a micro-channel, or micro-fluid-mechanics as it is more
commonly referred to. The reader may wish to refer to a more detailed de-
scription provided by Kim and Kavila, 1991, and Tabeling, 2005.
The motion of a fluid can be perfectly determined, when the velocity at
every point of the space is occupied by fluid motion. Therefore, to express
the velocity with independent variables, there are two distinct methods, the
so-called Eulerian and Lagrangian specifications.
*
*
*
1.1 Dynamics of Fluid Motion 7
One method is to trace the motion of a particle in space with time. In
particular, a particle in fluid is called “fluid particle”, which is a subdivi-
sion of the fluid around a specific point
x . This method is called Lagran-
gian specification. The position vector of the fluid particle, considering a
motion relative to a given frame of reference at time
t , can be expressed
as
t,
0
xxx
(1.1.2)
where
0
x and t are independent parameters, and
0
x is the original posi-
tion at
0 t , 0
00
,xxx . The velocity
u
and acceleration
a
at time t
can be written similarly as
t
t,
w
w
x
xuu
0
(1.1.3)
t
t,
w
w
u
xaa
0
(1.1.4)
As seen in Fig. 1.2(a), Lagrangian specification describes the motion of a
body of mass (fluid particle) and its variation of flow state along the parti-
cle path.
Another method is to give the spatial distribution of flow state as a
function of spatial coordinates
x ( zyx ,, x ; zy
x
,, are Euler variables)
and time
t , where
x
and t are independent variables. This method is
called Eulerian specification, and describes the variation of flow state in a
position
x (position vector in spatial coordinates x ) at a given timet , not
describing the behavior of each particle, see Fig. 1.2(b). In the Eulerian
specification the velocity
u can be expressed as
t,xuu
(1.1.5)
The expression of the acceleration
a in an Eulerian specification is given
by differentiation following the motion of a fluid particle and the rate of
change of the velocity of that particle with respect to time
)(
),(),(
u
xuxuxxu
a
ttt
ttt
ൺ0ǻtൺ0tǻ
lim
lim
(1.1.6)
Particularly in fluid mechanics,
a
in Eq. (1.1.6) is called the material de-
rivative or substantial derivative, and is often expressed as
)
(
)
(
()
()
ǻǻ
ǻ
ǻ
ǻ
8 1 Fundamentals in Continuum Mechanics
u
u
a u
¸
¹
·
¨
©
§
w
w
t
Dt
D
(1.1.7)
In general, the rate of change over time of material parameter
A, where A
can be a scalar, vector or tensor associated with the (material) point at time
t, can be expressed by the material derivative (the differential operator
DtD
is also called Lagrangian derivative) as
tAtA
tDt
tDA
,,
,
xux
x
w
w
0
(1.1.8)
(a) Lagrangian specification (b) Eulerian specification
Fig. 1.2 Description of fluid motion
The derivative defined in Eq. (1.1.7) can be further written, using vector
identities (see section of Appendix B-5), as follows
uuu
uu
uu
¸
¹
·
¨
©
§
w
w
2
2
1
tDt
D
(1.1.9)
The term
uu
or
Au
, operating in Eqs. (1.1.7) and (1.1.8) respec-
tively, are called the convective term, which express the fact that, for time
independent flow
0 ww t , the fluid properties, such u or A , depend only
upon the spatial coordinates
x
. Namely, the changes in fluid properties are
due to the changing spatial position of a given fluid particle as it flows.
The terms
tu ww , tA ww are the Eulerian time derivatives evaluated at a
position
x.
1.1 Dynamics of Fluid Motion 9
Having established the conceptual and mathematical representation of
the kinematic description of motion, we will now consider the decomposi-
tion of motion. This will be done in order to establish the concept and
mathematical description of that rate. An important concept in a motion of
continuum is to know how the deformation can be expressed from the ve-
locity field
t,xuu .
Fig. 1.3 Material line
L
change
Let us consider the configuration of an infinitesimal element
tL of a
straight material line. It undergoes translation, rotation, and stretching, as a
result of the nonuniform velocity field of
dtdxu
. As time elapses, the
tL becomes
dtt L . From the figure in Fig. 1.3, it is clear that
^ `
>@
>@
dttdttt
dtttdttttdtt
,,
,,)()(
xuLxuL
xuxLxuLxL
(1.1.10)
so that, the changing rate of
tL as dttt o can be written as
...
,
,,
w
w
L
xu
LxuLxu
LL
t
tt
dt
tdtt
(1.1.11)
where the right hand side of Eq. (1.1.11) is a Taylor series expansion for
t,xu around )(tL . Equation (1.1.11) can be further modified as
material line element
tL moves to the new positions P and Q, where