Fundamental
Mechanics
of
Fluids
Third Edition
1.
G.
Currie
University
of
Toronto
Toronto, Ontario, Canada
MARCEL
MARCEL DEKKER,
INC.
DEKKER
NEW
YORK
.
BASEL
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To my wife Cathie, our daughter Karen, and our sons
David and Brian
Preface
This book covers the fundamental mechanics of fluids as they are treated at
the senior level or in first graduate courses. Many excellent books exist that
treat special areas of fluid mechanics such as ideal-fluid flow or boundary-
layer theory. However, there are very few books at this level that sacrifice an
in-depth study of one of these special areas of fluid mechanics for a briefer
treatment of a broader area of the fundamentals of fluid mechanics. This
situation exists despite the fact that many institutions of higher learning
offer a broad, fundamental course to a wide spectrum of their students
before offering more advanced specialized courses to those who are spe-
cializing in fluid mechanics. This book is intended to remedy this situation.
The book is divided into four parts. Part I, ‘‘Governing Equations,’’
deals with the derivation of the basic conservation laws, flow kinematics,
and some basic theorems of fluid mechanics. Part II is entitled ‘‘Ideal-Fluid
Flow,’’ and it covers two-dimensional potential flows, three-dimensional
potential flows, and surface waves. Part III, ‘‘Viscous Flows of Incom-
pressible Fluids,’’ contains chapters on exact solutions, low-Reynolds-
number approximations, boundary-layer theory, and buoyancy-driven
flows. The final part of the book is entitled ‘‘Compressible Flow of Inviscid
Fluids,’’ and this part contains chapters that deal with shock waves, one-

dimensional flows, and multidimensional flows. Appendixes are also inclu-
ded which summarize vectors, tensors, the governing equations in the
common coordinate systems, complex variables, and therm odynamics.
The treatment of the material is such as to emphasize the phenomena
associated with the various properties of fluids while providing techniques
for solving specific classes of fluid-flow problems. The treatment is not
geared to any one discipline, and it may readily be studied by physicists and
chemists as well as by engineers from various branches. Since the book is
intended for teaching purposes, phrases such as ‘‘it can be shown that’’ and
similar cliche
´
s which cause many hours of effort for many students have
been avoided. In order to aid the teaching process, several problems are
included at the end of each of the 13 chapters. These problems serve to
illustrate points brought out in the text and to extend the material covered in
the text.
Most of the material contained in this book can be covered in about 50
lecture hours. For more extensive courses the mate rial contained here may
be completely covered and even augmented. Parts II, III, and IV are
essentially independent so that they may be interchanged or any one or more
of them may be omitted. This permits a high degree of teaching flexibility,
and allows the instructor to include or substitute material which is not
covered in the text. Such additional material may include free convection,
density stratification, hydrodynamic stability, and turbulence with applica-
tions to pollution, meteorology, etc. These topics are not included here, not
because they do not involve fundamentals, but rather because I set up a
priority of what I consider the basic fundamentals.
For the third edition, I redrew all the line drawings, of which there are
over 100. The problems have also been reviewed, and some of them have
been revised in order to clarify and=or extend the questions. Some new

problems have also been included.
Many people are to be thanked for their direct or indirect contribu-
tions to this text. I had the privilege of taking lectures from F. E. Marble,
C. B. Millikan, and P. G. Saffman. Some of the style and methods of these
great scholars are probably evident on some of the following pages. The
National Research Council of Canada are due thanks for supplying the
photographs that appear in this book. My colleagues at the University of
Toronto have been a constant source of encouragement and help. Finally,
sincere appreciation is extended to the many students who have taken my
lectures at the University of Toronto and who have pointed out errors and
deficiencies in the material content of the draft of this text.
vi Preface
Working with staff at Marcel Dekker, Inc., has been a pleasure. I am
particularly appreciative of the many suggestions given by Mr. John J.
Corrigan, Acquisitions Editor, and for the help he has provided in the
creation of the third edition. Marc Schneider provided valuable information
relating to software for the preparation of the line drawings. Erin Nihill, the
Production Editor, has been helpful in many ways and has converted a
patchy manuscript into a textbook.
I. G. Currie
Preface vii
Contents
Preface v
Part I. Governing Equations 1
1. Basic Conservation Laws 3
1.1 Statistical and Continuum Methods
1.2 Eulerian and Lagrangian Coordinates
1.3 Material Derivative
1.4 Control Volumes
1.5 Reynolds’ Transport Theorem

1.6 Conservation of Mass
1.7 Conservation of Momentum
1.8 Conservation of Energy
1.9 Discussion of Conservation Equations
1.10 Rotation and Rate of Shear
1.11 Constitutive Equations
1.12 Viscosity Coefficients
1.13 Navier-Stokes Equations
1.14 Energy Equation
1.15 Governing Equations for Newtonian Fluids
1.16 Boundary Conditions
2. Flow Ki nematics 4 0
2.1 Flow Lines
2.2 Circulation and Vorticity
2.3 Stream Tubes and Vortex Tubes
2.4 Kinematics of Vortex Lines
3. Special Forms of the Governing Equations 55
3.1 Kelvin’s Theorem
3.2 Bernoulli Equation
3.3 Crocco’s Equation
3.4 Vorticity Equation
Part II. Ideal- F luid Flow 69
4. Two - Di mensional Potential F lows 73
4.1 Stream Function
4.2 Complex Potential and Complex Velocity
4.3 Uniform Flows
4.4 Source, Sink, and Vortex Flows
4.5 Flow in a Sector
4.6 Flow Around a Sharp Edge
4.7 Flow Due to a Doublet

4.8 Circular Cylinder Without Circulation
4.9 Circular Cylinder With Circulation
4.10 Blasius’ Integral Laws
4.11 Force and Moment on a Circular Cylinder
4.12 Conformal Transformations
4.13 Joukowski Transformation
4.14 Flow Around Elli pses
x Contents
4.15 Kutta Condition and the Flat-Plate Airfoil
4.16 Symmetrical Joukowski Airfoil
4.17 Circular-Arc Airfoil
4.18 Joukowski Airfoil
4.19 Schwarz-Christoffel Transfor mation
4.20 Source in a Channel
4.21 Flow Through an Aperture
4.22 Flow Past a Vertical Flat Plate
5. Three-Di mensional Potential Flows 161
5.1 Velocity Potential
5.2 Stokes’ Stream Function
5.3 Solution of the Potential Equat ion
5.4 Uniform Flow
5.5 Source and Sink
5.6 Flow Due to a Doublet
5.7 Flow Near a Blunt Nose
5.8 Flow Around a Sphere
5.9 Line-Distributed Source
5.10 Sphere in the Flow Field of a Source
5.11 Rankine Solids
5.12 D’Alembert’s Paradox
5.13 Forces Induced by Singularities

5.14 Kinetic Energy of a Moving Fluid
5.15 Apparent Mass
6. Su rface Wave s 201
6.1 The General Surface-Wave Problem
6.2 Small-Amplitude Plane Waves
6.3 Propagation of Surface Waves
6.4 Effect of Surface Tension
6.5 Shallow-Liquid Waves of Arbitrary Form
6.6 Complex Potential for Traveling Waves
6.7 Particle Paths for Traveling Waves
6.8 Standing Waves
6.9 Particle Paths for Standing Waves
6.10 Waves in Rectangular Vessels
6.11 Waves in Cylindrical Vessels
6.12 Propagation of Waves at an Interface
Contents xi
Part III. Viscous F lows of Incompres sible Fluids 249
7. Exac t S olu t i o n s 2 53
7.1 Couette Flow
7.2 Poiseuille Flow
7.3 Flow Between Rotating Cylinders
7.4 Stokes’ First Pro blem
7.5 Stokes’ Second Problem
7.6 Pulsating Flow Between Parallel Surfaces
7.7 Stagnation-Point Flow
7.8 Flow in Convergent and Dive rgent Channels
7.9 Flow Over a Porous Wall
8. Low-Reynolds-Number Solutions 288
8.1 The Stokes Approximation
8.2 Uniform Flow

8.3 Doublet
8.4 Rotlet
8.5 Stokeslet
8.6 Rotating Sphere in a Fluid
8.7 Uniform Flow Past a Sphere
8.8 Uniform Flow Past a Circu lar Cylinder
8.9 The Oseen Approxi mation
9. Boun dary Layers 313
9.1 Boundary-Layer Thicknesses
9.2 The Boundary-Layer Equations
9.3 Blasius Solution
9.4 Falkner-Skan Solutions
9.5 Flow Over a Wedge
9.6 Stagnation-Point Flow
9.7 Flow in a Convergent Channel
9.8 Approximate Solution for a Flat Surface
9.9 General Momentum Integral
9.10 Ka
´
rma
´
n-Pohlhausen Approximation
9.11 Boundary-Layer Separation
9.12 Stability of Boundary Layers
xii Contents
10. Buoyancy-Driven F lows 3 63
10.1 The Boussinesq Approximation
10.2 Thermal Convection
10.3 Bound ary-Layer Approximations
10.4 Vertical Isothermal Surface

10.5 Line Source of Heat
10.6 Point Source of Heat
10.7 Stability of Horizontal Layers
Part I V. Compres sible Flow of I nviscid Flu id s 39 5
11. Shock Waves 39 9
11.1 Propagation of Infinit esimal Disturbances
11.2 Propagation of Finite Disturbances
11.3 Rankine-Hugoniot Equations
11.4 Conditions for Normal Shock Waves
11.5 Normal Shock-Wave Equations
11.6 Oblique Shock Waves
12. O ne-Dimensi onal Flows 430
12.1 Weak Waves
12.2 Weak Shock Tubes
12.3 Wall Reflection of Waves
12.4 Reflection and Refraction at an Interface
12.5 Piston Problem
12.6 Finite-Strength Shock Tubes
12.7 Nonadiabatic Flows
12.8 Isentropic-Flow Relations
12.9 Flow Through Nozzles
13. Multidimensional Flows 4 61
13.1 Irrotational Motion
13.2 Janzen-Rayleigh Expansion
13.3 Small-Perturbation Theory
13.4 Pressure Coefficient
Contents xiii
13.5 Flow Over a Wave-Shaped Wall
13.6 Prandtl-Glauert Rule for Subsonic Flow
13.7 Ackert’s Theory for Supersonic Flows

13.8 Prandtl-Meyer Flow
Appendix A.VectorAnalysis 495
Appendix B.Tensors 50 0
Appendix C. Governing Equations 505
Appendix D. Complex Variables 510
Appendix E.Thermodynamics 516
Index 521
xiv Contents
I
GOVERNING EQUATIONS
In this ¢r st part of the book a su⁄cient set of equ ations wil l be derived, based
on physical laws and post u lates, gover ni ng the de pe nde nt variables of a £ u id
that is moving. T he dependent var iables a re the £uid -velocit y components,
pressure, de n sit y, te mpe rat u re, an d i nte r n al ene rgy or some sim ilar set of
var iables. The equations governi ng these variables will be d e r ived from the
pri nc iples of mass, momentu m , and ener gy conse rvat ion and f rom equations
of state. Hav i ng established a su⁄cient set of go ve r ni ng equations, some
purely k inematical aspects of £uid £ow are discussed, at which time the
conce pt of vort icity is int ro d uc ed . The ¢ n a l sec t ion of this pa rt of the book
introduces certain relationships that can be derived from the governing
equations u nder ce rtain simplif y ing conditions. These relationships may be
used i n c onju nct ion with the ba sic gove r ni ng equations or as a lte r nat ives to
them.
Taken as a whole, this part of the book establishes the ma thematical
equationsthatresultfrominvokingcertainphysicallawspostulatedto bevalid
for a moving £uid. These equat ions may assume d i¡ere nt forms, depend i ng
uponwhich var iables are chosen andu pon wh ich si mpl i fyi ng a ssumpt ions are
made.The remaining p a rts of the book are devoted to solving these gove rni ng
equations for di ¡e re nt classes of £uid £ows and thereby explaini ng qu ant it a -
tively some of the phenomena that are observed in £uid £ow.

1
1
Basic Conserv ation Laws
Theessentialpurposeofthischapteristoderivethesetofequationsthat
results from invok i ng the physical laws of conse rvation of ma ss, moment um,
and energy.In order to realize this objective, it is necessary to discuss certain
preliminary topics. The ¢rst topic of discussion isthe t wobasic waysin which
the conse rvat ion equations may be der i ved: the statist ical method a nd the
continuum method. Having selecte d the basic method to be used in deriving
the equations, one is then faced with the choice of reference frame to be
employed , euler ia n or la gra ngian. Next, a g e ner a l theorem , cal led Rey nolds’
tra n sport theorem, is de rived, sinc e this theorem relates derivat i ves in the
lagrangian f ramework to derivatives in the eulerian framework.
Havi ng established the basic method to be employed and the tools to be
used, the basic conse rvat ion laws are then der i ved.The con se rvat ion of mass
yields the so- called continuity equation. The conservation of momentum
leads ultimately to the Navier-Stokes equations, whi le the conservation of
thermal energy le ads to the ener gy equatio n. The d erivation is followed by a
discussion of the set of equations so obtained, and ¢nal ly a summary of the
basic c on se rvation laws is gi ve n.
3
1.1 STATISTICAL AND CONTINUUM METHODS
There are basically two ways of deriving the equations that govern the
motion of a £ u id . One of these methods app roaches the question from the
molecular point of view. That is, th is method treats the £ u id as con sist i ng of
molecules whose motion is governed by the laws of dynamics. The macro-
scopic phenomena are assumed to arise from the mole cular motion of the
molecules, a nd the theory a t te mpts to pred ict the macroscopic behav ior of
the £uid from the laws of mechanics and probability theory. For a £ui d that is
in a st ate not too far re moved from equ il ibr iu m , this app roa ch yields the

equations of mass, momentum, and energy conservation. The molecular
approa ch a lso yields expressions for the tran s por t coe⁄cients, such as the
coe⁄cient of viscosity and the thermal conductivity, in terms of molecular
quantities such as the forces acting between molecules or molecular dia-
meters. The theory is well developed for light gases, but it is incomplete for
polyatomic gas molecules and for liquids.
The alter n at i ve method used to derive the equat ions gover ning the
motion of a £ u id uses the cont inu u m c onc e pt. In the c ont i nuu m approach,
indi v i d ua l molecu les are ig nored a nd it is assumed that the £ui d consists of
cont in uous matte r. At ea ch point of th is conti nuous £uid the re is supposed to
be a u nique value of the velocit y, pressure, de nsity, an d other so -c alled ¢eld
var iables. The cont inuo us matter is the n required to obey the conserv ation
laws of mass, momentum, and energy, which give rise to a set of di¡erential
equations governi ng the ¢eld va r iables. The solut ion to these di ¡e re ntia l
equations then de¢nes the variation of each ¢eld variable with space and time
which correspond s to the mean va lue of the molecula r ma g nitude of that ¢eld
var iable at each c orrespondi ng posit ion and ti me.
The statistical method is rather elegant, and it may be used to treat gas
£ows in situ ations where the continu u m conc e p t is no longer val id. Howeve r,
as was mentioned before, the theory is incomplete for dense gases and for
liquids. The conti nuu m ap proach requires that the mean free p ath of the
molecules be very small compared with the smallest phy sical-length scale o f
the £ow ¢eld (such as the dia meter of a cylind e r or other body about wh ich
the £uid is £owing). Only in this way can meaningful averages over the
molecules at a ‘‘point’’ be ma de and the molecular str uctu re of the £uid be
ignored. However, i f this condit ion is satis¢ed, there no d istinc t ion among
light gases, dense gases, or even l iquidsöthe results apply equally to al l.
Since the va st ma jori t y of phenomena encou nte red i n £u id mecha nics fall
well within the c ont i nu u m domain and may invol ve liquids as well as gases,
the conti nu u m method wil l b e used in this book. With this background, the

mean i ng and val id it y of the continuu m concept w il l now be explored in some
deta il.The ¢eld variables, such as the density r and t he velocity vector u,will
4Chapter1
in general be functions of the spatial coordinates and time. In symbolic form
this is written a s r ¼ rðx; tÞ and u ¼ uðx; tÞ,wherex is the position vec tor
whose ce rtesian coord i n ates are x, y,andz. At any particular point in space
these cont i nu u m variables are de¢ned in terms of the prope rties of the var-
ious molecu les that occupy a smal l volu me in the neighborhood of that point .
Consider a small volume of £uid DV cont a i ni ng a larg e number of
molecules. Let Dm and v be the mass and velocit y of any individu a l molecule
contained within the volume DV, as indicated in Fig. 1.1. The density and the
velocity at a poi nt in the c ont i nu u m are then de¢ ned by the follow i ng li m its:
r ¼ lim
DV !e
P
Dm
DV

u ¼ lim
DV !e
P
v Dm
P
Dm

where e is a volume which is su⁄cient l y small that e
1=3
is sma ll compa re d with
the smallest sig ni¢cant length scale in the £ow ¢eld but is su⁄ciently large
that it conta i ns a la r ge number of molecules. The summations in the above

expressions are taken over all the molecules contai ned withi n the volume DV.
FIGURE 1.1 An individual molecule in a small volume DV having a mass Dm and
velocity v.
Basic Conservation Laws 5
The other ¢eld variables may be de¢ ned in terms o f the molecula r properties
in an analogous way.
A su⁄cient condition, though not a necessar y c ondition, for the con -
tin u um ap proa ch to be va l id is
1
n
( e ( L
3
where n is the number of molecules per un it volume and L is the smallest
signi¢cant length scale in the £ow ¢eld, which is usually called the macro-
scopic length scale.The characteristic microscopic length scale is the mean free
path bet ween c ollisions of the molecules. Then the above condit ion st ates
that the cont inuu m concept will ce rt a inly be valid if some volume e can be
found that is much larger than the volume occ u pied by a sing le molecule of
the £uid but much smaller tha n the cube of the sma llest mac rosc o pic length
scale (such a s cyl i nde r diameter). Since a c ube of ga s, at norma l tempe ra t u re
and pressure,whose side is 2 micrometers contains about 2 Â10
8
molecules
and the cor respond i ng ¢g u re for a liquid is about 2 Â 10
11
molecules, the
continuum condition is readily met in the vast majority of £ow situations
encou nte red in physics and engi neering. It may be expected to brea k down in
sit uat ions where the smallest macrosc o pic length scale approaches micro-
scopic dimensions, such as in the structure of a shock wave, and where the

microsc opic length scale approaches macrosco pic dimen sions, such as when
a rocket passes throu g h the ed g e of the atmosphere.
1.2 EULERIAN AND LAGRANGIAN COORDINATES
Having selected the continu u m approach as the method th at w ill be used to
deri ve the basic conservation laws, one is next faced with a cho ice of refer-
ence frames in which to formulate the conservation laws.There are two basic
coordi n ate systems that may be employed, these being eu lerian a nd la g r a n -
gian coordinates.
In the euler ian fr a mework the i nde pe nde nt variables are the spat ial
coordinates x , y,andz and time t.Thisisthefamiliarframeworkinwhich
most problems are solved. In order to derive the basic conservation equa-
tionsinthisframework,attentionisfocusedonthe£uidwhichpasses
th rou g h a contro l volume that is ¢xed in space. The £u id inside the cont rol
volume at any instant in time wi ll consist of di¡erent £uid particles from that
which was there at some previous insta nt i n ti me. If the pr inc iples of con-
servation of mass, momentum, and energy are applied to the £uid passing
th rou g h the control volume,the basic conservation equations are obtained in
eulerian coordinates.
6Chapter1
In the lagrangian approach, attention is ¢xed on a par ticular mass of
£uid as it £ows. Suppose we could color a sma ll portion of the £uid without
changing its density. Then in the lagr angian framework we fo llow this
colored por t ion a s it £ows a n d cha ng es its shape, b ut we are a l ways c on-
sider i ng the s a me pa rticles o f £uid . T h e pr i nc iples of ma ss, moment u m , and
energy conservation are then applied to this particular element of £uid as it
£ows, resulting i n a set of c on se rvat ion equ ations in la gra ng ia n coordinates.
In this refere nc e f r a me x,y, z,andt are no longer indepe nde n t va riables, sin c e
if it is known that our colored portion of £uid passed th rou g h the coordinates
x
0

, y
0
,andz
0
at some time t
0
, then its position at some later ti me m ay be ca l-
culated if the velocity components u, v,andw are known.That is, as soon as a
time interval (t À t
0
) is spec i ¢ed , the velocit y compone nts uniquely deter-
minethe coordinate changes (x À x
0
), ( y À y
0
), and (z Àz
0
)sothatx,y, z,andt
are no longer independe n t. T he i nde pe ndent variables i n the la gra ngian sys-
tem are x
0
, y
0
, z
0
,andt,where x
0
, y
0
,andz

0
are the c oordi n ates which a spe-
ci¢ed £uid element passed throug h at time t
0
.Thatis,thecoordinatesx
0
, y
0
,
and z
0
identify which £uid element is being con side red, and the time t ide n-
ti¢es its instantaneous location.
Thechoiceofwhichcoordinatesystemtoemployislargelyamatterof
taste. It is probably more convincing to ap p ly the conse rvat ion laws to a
control vol u me that always consists of the same £uid p a r t icles rather tha n
one through which d i¡erent £ uid particles pass. Th is is par ticularly true
when invoking the law of c on se rvat ion of ener gy,wh ich c onsists of applyi ng
the ¢rst law of thermodynamics, since the same £uid par ticles are more
readily justi¢ed as a thermodynamic system. For this reason, the lagrangian
coordin ate system w il l be used to der i ve the basic conser vation equations.
A lthough the lagr angian system w ill be used to derive the basic equat ions,
the eu ler ia n system is the prefe rred one for solv i ng the m ajor it y of problems.
In the next section the relation between the di¡erent derivatives will be
established .
1.3 MAT ER I A L DE R IVAT I VE
Let a be any ¢eld va riable such as t he de n sit y or t empe r at u re of the £u id .
From the e u lerian v iewpoint, a may be considered to be a fu nct ion of the
indep e nden t var ia bles x, y, z,andt. But if a speci¢c £uid element is observed
for a short period of time dt as it £ows, its positionw i ll change by amounts dx,

dy,anddz while its value of a wi ll change by an amount da.Thatis,if the£uid
element is observed i n the la gra ngian fr a mework , the i nde pe nde nt var iables
are x
0
, y
0
, z
0
,andt,wherex
0
, y
0
,andz
0
are initial coordinates for the £uid
element. Th us, x, y,andz are no longer i ndepe nde nt variables but a re
Basic Conservation Laws 7
functions of t a s de¢ned by the trajectory of the element . Du r i n g the time dt
the chan ge in a may be calculated from di¡erential calculus to be
@a
@t
dt þ
@a
@x
dx þ
@a
@y
dy þ
@a
@z

dz
Equat ing the prec ed i ng ch a nge in a to the observed ch ang e da in the lagran-
gian framework and dividing throughout by dt gives
da
dt
¼
@a
@t
þ
dx
dt
@a
@x
þ
dy
dt
@a
@y
þ
dz
dt
@a
@z
The left-h a nd side of this expression represe nts the total cha nge i n a as
obs erved in the lagrangian framework during the time dt, and in the limit it
represents the time derivative of a in t he lagrangian system, which will be
denoted by Da=Dt.Itmaybealsonotedthatinthelimitasdt ! 0theratio
dx=dt becomesthe velocitycomponentin the x direction,namely, u.Similarly,
dy=dt ! v and dz=dt ! w as dt ! 0, the expression for the change in
a becomes

Da
Dt
¼
@a
@t
þ u
@a
@x
þ v
@a
@y
þ w
@a
@z
Invectorformthisequationmaybewrittenasfollows:
Da
Dt
¼
@a
@t
þðu
Á
HÞa
A lte r n at i vely, using the Ein ste in su m mat ion convent ion where repeated
subscriptsaresummed,thetensorformmaybewrittenas
Da
Dt
¼
@a
@t

þ u
k
@a
@x
k
ð1:1Þ
The term Da=Dt in Eq. (1.1) is the so -ca l led materia l der i vat ive. It represents
the total change in the quantity a as seen by an observer who is following the
£uid and is watching a particular mass of the £uid.The entire right-hand side
of Eq. (1.1) represents the total change in aexpressed in euler ian coordinates.
The term u
k
ð@a=@x
k
Þexpresses the fac t th a t in a time-i nd e pe nde nt £ow ¢eld
in wh ich the £ u id properties de pe nd upon the spatial c oord in ates on l y,there
is a chang e i n a due to the fact that a g i ve n £u id element chang es its position
with time and therefore assumes di¡erent values of a as it £ows. The term
@a=@t is the famil ia r eulerian time de r i vative and ex presses the fact that at
any point in spa ce the £ u id prope r t ies may chang e w ith time. The n Eq. (1.1)
expresses the lagrangian rate of change Da=Dt of a for a g i ve n £ uid element in
terms of the eulerian derivatives @a=@t and @a=@x
k
.
8Chapter1