fluid k2opt
TRANSCRIPT
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Fluid Mechanics
Richard
Fitzpatrick
Professor of
Physics
The
University
of
Texas at
Austin
Overview
1.1
Intended Audience
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1.2 Major Sources
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1.3
To
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Mathematical
Models o f
Fluid Motion
Introduction
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isa Fluid?
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Volume and Surface Forces
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2.4
General
Properties of
Stress
Tensor
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2.5
Stress
Tensor
in
a
Static
Fluid
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2.6
Stress
Tensor ina Moving Fluid
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Viscosity
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Laws
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15
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Conservation .
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2.10 Convective
Time
Derivative
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2.11 Momentum Conservation
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Navier-Stokes
Equation
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2.13
Energy Conservation
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Equations
of Incompressible
Fluid
Flow
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2.
of Compressible Fluid Flow
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2.1
Numbers
in
Incompressible
Flow
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mensionless Numbers in Compressible Flow
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2.
Equations in
Cartesian
Coordinates
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2.19
Fluid Equations in Cylindrical Coordinates
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2.20
Fluid Equations in
Spherical
Coordinates
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2.21
Exercises
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Hydrostatics
In troduction
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31 3
Pressure
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3.3 Buoyancy
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Equilibrium of Floating Bodies
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32 3
Stability
of Floating Bodies
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Stability of Floating Bod ies
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3
of
Metacentric
Height
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3.8 Energy of a
Floating
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3.9 Curve of Buoyancy
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Rotational Hydrostatics
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MECHANICS
1
1
Equilibrium of a Rotating
Liquid
Body
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3.13
Jacobi
Ellipsoids
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3.14
Roche Ellipsoids
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3.15
Exercises
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Surface Tension
1
Introduction
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Equation
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Interfaces .
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Length
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4.5 Angle of Contact
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4.6 Jurin
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Law
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4.7 Capillary
Curves ......
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4.8
Axisymmetric
Soap-Bubbles
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Exercises
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Incompressible
Inviscid
Fluid Dynamics
Introduction
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77 5
reamlines, Stream Tubes, and Stream Filaments
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Lines,
Vortex Tubes,
and
Vortex Fi laments ..
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rcula tion and
Vorticity
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80
5
Circulation
Theorem
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Flow
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5.8 Two-Dimensional Flow
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Two-Dimensional Uniform Flow
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Sources
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5.
Vortex
Filaments
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5.12 Two-Dimensional Irrotational Flow
Cylindrica l Coord ina tes .
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Flow Past a Cylindrical Obstacle
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5.14
Inviscid Flow Past a Semi-Infinite Wedge
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5.15 Inviscid Flow
Over
a Semi-Infinite Wedge
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Velocity Potentials
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Stream Functions
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5.17
Exercises
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2D
Potential Flow
1
Introduction
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01 6
Funct ions
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Relations
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Velocity
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Maps
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9 6.8 Complex
Line
Integrals
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Theorem of
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Incompressible
Boundary Layers
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7.1
Introduction
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7.2 N o Slip
Condit ion
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7.3
Boundary
Layer Equat ions
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Self-Similar Boundary Layers
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Wake Downstream of
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Flat
Plate
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Momentum Integral
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Boundary Layer Separation
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Solutions of Boundary Layer Equations .
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Incompressible Aerodynamics
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Introduction
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of
Kutta
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Zhukovskii
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Airfoils .
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s Hypothesis
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Sheets
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Flow
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Airfoi ls
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Forces
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8.1
Flight
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Incompressible
Viscous
Flow
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Introduction
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Between Parallel Plates
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Down an Inclined
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Taylor-Couette
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9.6 Flow inSlowly-Varying Channels
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Lubrication Theory
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Stokes
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Waves in Incompressible Fluids
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10.3
Gravity
Waves in
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Water
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Waves in
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Energy of
Gravity
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Atmosphere
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Stabil i ty
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Vectors
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Vector Fields
3
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Algebra
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Transformations
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Triple
Product
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Triple Product
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Calculus
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Line Integrals
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Integrals
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Surface Integrals
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Integrals
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Vector Identities
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Cartesian
Tensors
3
1 Introduction
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and Tensor
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Transformation
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Tensors
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Non-Cartesian
Coordinates
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1 Introduction
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Curvil inear Coordinates
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Calculus of
Variations
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D .1 Euler-Lagrange Equat ion
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D.2
Conditional
Variat ion
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D.3 Multi-Function Variation
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Ell ipsoidal Potential
Theory
6
MECHANICS
verview
1 Intended Audience
book
presents a
single
semester course on fluid
that is intended
primarily
for
advanced
students
majoring
inphysics. A thoroug
of
physics
a t the lower-division level ,
a basic
working knowledge of the laws of
is assumed.
It
is
a lso taken for
granted
tha
are familiar
with
the fundamentals of
integral and
differential calculus,
complex
and
ordinary differential
equations.
On
the
othe
vector analysis
p lays such
a central role in the
of
f luid mechanics
that
a
brief,
but fairly
review of this subject area is provided
in
A.Likewise,
those aspects
of
cartesian
tensor
orthogonal
curvilinear coordinate
systems,
and
the
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of variations, that
are
required in
the
study of
flu
are
outlined in
Appendices
B,C,and D,
Majo r
Sources
e material appearing inAppendix A is largely
based
on
e authors
recollections of a
vector
analysis course
given
Dr.
Stephen
Gull
at
the
University
of
Cambridge.
sources for
the
material
appearing
in
other
chapter
d appendices
include:
Including
Hydrostatics and
the
Elements of
the
of
Elasticity
H.Lamb,3rdEdit ion(Cambridge
versity
Press,
Cambridge UK,
1928).
H.Lamb,
6th
Edition
(Dover,
New York
, 1945).
oretical Aerodynamics L.M.
Milne-Thomson,
4th
Revised
and enlarged
(Dover,
New York
N Y ,
Ellipsoidal
Figures
of
Equil ibr ium S.
(Yale
University Press, New
Haven
C T,
Boundary
Layer
Theory
H.
Schlichting,
7th
Editio
New
York N Y , 1970).
Methods
for
the
Physical
Sciences
Riley (Cambridge University
Press,
Cambridge U K ,
Fluid Mechanics L .D. Landau,
and E.M.
Lifshitz,
Edition
(Butterworth-Heinemann,Oxford
UK,
1987).
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Fluid Dynamics D.J. Tritton, 2nd Edition
University
Press, Oxford UK,1988).
Dynamics
fo r
Physicists T.E.
Faber,
t Edition (Cambridge University
Press,
Cambridge UK,
Schaum
s
Outline
of
Fluid
Dynamics
W .
Hughes
d
J.Brighton,
3rd
Edition (McGraw-Hil l , New York N Y
An Introduction to Fluid Dynamics G.K .
(Cambridge
University
Press,
Cambridge
UK,
Theoretical Hydrodynamics L.M.Milne-Thomson
h
Edition
(Dover, New
York
N Y ,
2011).
To D o
List
1.Add chapter on vortex
dynamics.
2.Add chapter on3Dpotential flow.
3.
Add
appendix
on
group
velocity
and Fourier
4.
Add chapter on incompressible
flow in
rotating
5.Add chapter on instabilities.
FLUIDMECHANI
6.
Add
chapter on
turbulence.
7.Add chapter on
1Dcompressible
flow.
8.Add chapter on sound waves.
9.
Add chapter on compressible
boundary layers.
10.
Add
chapter
on
supersonic
aerodynamics.
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11.Add chapter on convection.
Models ofFluid Motion
Mathematical Models of Fluid
1
Introduction
this
chapter,
we
set
forth
the
mathematical
mode
used to
describe
the equilibrium
a
of
fluids. Unless
stated
otherwise,
all of
t
is performed using a standard right-hande
coordinate
system: x1,2,3.oreover, t
summation
convention
is
employed
(so repeat
subscripts are
assumed
to
be
summed
from
1
see Appendix
B).
Wh a t isa
Fluid?
definition, a
solid material
is
rigid. N o w , although
material tends
to
shatter when
subjected
to ve
stresses, it can withstand a
moderate
shear stress (i.
stress that tends to deform
the
material by
changing
without necessarily changing its volume) for
period.
To
be more exact,
when
a
shear
stress
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applied
to
a rigid material it
deforms
slightly, but th
back to
its
original shape
when
the
stress
A
plastic
material,
such
as clay, also possess som
of
rigidity.
However, the crit ical shear stress
it
yields is
relatively small ,
and once this stress
the
material
deforms continuously a
and does
not recover its
original shape
wh
e stress
is relieved.
By definit ion, a
fluid material possesses
no rigidi
all. In other words,
a
smal l fluid element is
unab
withstand
any tendency
ofan
applied shear stress
its
shape. Incidentally,
this does not
preclude
t
that
such
an element may offer resistance
stress. However, any resistance must be
incapab
preventing
the
change
in
shape
f rom
eventua
which implies
that
the force of resistan
with the rate
of
deformation. An obvio
is that
the
shear stress must
be
zero everywhe
a
fluid
that
is in
mechanical
equilibrium.
Fluids
are
conventionally
classified
as
either
l iquids
The
most
important
difference
between
these tw
of fluid lies in their relative
compressibil i ty:
i .
can be compressed much more easily than l iquid
sequently, any motion
that
involves
significant
pressu
is
generally accompanied
by
much larg
inmass density in the case of a gas than in the ca
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a l iquid.
Of course,
a
macroscopic fluid ultimately
consists
huge
number
of individual molecules. However, mo
applications of fluid mechanics
are
concerne
behavior on length-scales
that are far
larger th
typical
intermolecular spacing.
Under the
it
is reasonable to suppose
that
the bu
of a
given
f luid
are the
same as if it
we
continuous in structure.
A corollary of th
is that
when, in the
fol lowing,
we talk abo
in itesima l vo lume elements,
we
really
mean
elemen
are sufficiently small
that
the
bulk fluid propertie
as
mass
density,
pressure, and velocity)
a
constant across them, but are s
large
that
they con ta in a
very
great number
(which
implies
that
we
can
safely
neglect
a
variations in the bulk
properties).
T
hypothesis also requires infinitesimal
volum
to be
much
larger than the
molecular
n-free-path between collis ions.
In
addition
to
the
continuum
hypothesis,
our
study
mechanics
is
premised
on
three major
assumptions:
.Fluids
are
isotropic media: i .e., there is
no
preferred
in
a fluid.
.
Fluids
are
Newtonian:
i.e.,
there
is
a
l inear
relationsh
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the local
shear
stress
and
the local rate of strain,
as first postulated by Newton.
It
is also
assumed th
isa l inear
relationship between the local
heat flux
density
and
the local temperature gradient.
.
Fluids
are
classical:
i .e.,
the
macroscopic
motion
of
fluids
is
well-described
by
Newtonian dynamics,
and
both
quantum
and
relativistic effects
can
be safely
should be noted
that
the above
assumptions
are
n
for
all
fluid types (e.g., certain
l iquid
polyme
are
non-isotropic;
thixotropic
f lu ids,
such
as je
pa in t, wh ich
are
non-Newtonian;
and
quantum
f luid
as l iquid
helium, which exhibit non-classical effects
length-scales).
However, most practic
10
MECHANICS
f luid
mechanics involve the equ ilib rium and motion
of
water or
air,extending over macroscopic lengt
and
situated
relatively
close
to
the
Earth
s
surfac
bodies are very well-described
as isotrop
classical flu ids.
Volume and Surface Forces
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speaking, f luids
are
acted upon
by
tw o distin
of
force.
The
f irst
type is long-range in nature
such that
it decreases relatively
slowly w
reas ing d istance
between
in te racting e lements
and
of
completely
penetrating
into
the
interior
of
Gravity
is
an obvious example of a
long-rang
One
consequence
of
the relatively slow variati
long-range forces with
position
is
that
they act equa
all of
the
fluid
contained
within
a
sufficiently
sm
element .
In
this
situation,
the
net
force acti
the element
becomes
directly proportional
to
For this reason,
long-range forces are often call
forces. In the fol lowing,
we
shal l
write
the to
fo rce acting at
t ime
t on the
fluid
contained with
smal l
volume element
of magnitude dV,
centered
ixed point whose position vector is
r,as
The second
type offorce
is short-range
in
nature,
and
conveniently modeled
as
momentum
transport
with
e
fluid.
Such transport is generally due
to
a
combinatio
the
mutual
forces
exerted
by contiguous
molecules, a
fluxes caused by relative molecular motio
that
x
(r,
t
is
the
net
flux
density of x-directe
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momentum
due to short-range forces at position r a
t.In other words, suppose
that, at
position rand
time
a direct consequence
of short-range
forces,
x-momentu
f lowing
at the rate of |x |newton-seconds per
met
per second in
the
direction of vector
x
.
Consid
infinitesimal plane surface
element, dS
=
ndS,
locate
point
r.Here,
dS
is the area of the e lement, and
n
its
u
(See Section
A.7.) The
fluid
which
lies
on th
of the element toward which n points is
said
to
lieon
side, and vice versa. The net flux of x-momentu
the element (in the direction of n) is x
which
implies (from
Newtons second
law
that the fluid
on the positive
side of
the surfa
experiences a
force
x dS in
the
x-direction
d
short-range interaction
with
the
f luid
on the
negati
According
to
Newton
s
third
law
of
motion,
t
on
the
negative side of the surface experiences a for
x dS
in
the x-direction due to interaction with the
flu
urfac
e
ex perien ces
the
positive side.
Short-range
forces are often call
forces
because
they are directly
proportional
idon the po sitive
si
de.Short -range
e
area
of
the
surface e lement
across
which
they
act.
L
(r,
t
and
z
(r,
t be
the net f lux
density
of y- and
respectively, at position r and t ime t.
B y
extension
of above argument, the n
force
exerted
by the
fluid on the positive
side
planar
surface element, dS , on the fluid on
side is
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( x
dS,
y dS,
z dS).
tensor notation (see Appendix B), the above
equation
be
written
= % ij dS
j
%
11
=
(
x
)x
,
%
12
=
(
x
)y
,
%
21
=
(
y
)x
,
e
that,
since
the
subscript
j is repeated, it is assum
be summed
from
1o 3.Hence,
%
ij dS
j
is
shorthand
f
j=1,3
% ij dS
j
.
Here, the %
ij
(r,
t are
termed
the
loc
in
the
flu id at position rand t ime
t,
and have un
P
force per unit
area. Moreover,
the
%ij are t
of a
second-order
tensor
(see Appendix
B
as
the stress tensor. [This
fol lows
because the
the
components of a first-order tensor (since all
forc
proper
vectors),
and the
dS
i
are the components of
first-order tensor (since surface elements are al
vectors&see
Section A.7
&
and
(2.3] holds
f
elements whose normals point in
any
direction),
of the quotient
rule
(see Section
B.3)
(2.3)
reveals that
the %
ij trans form under
rotati
the
coordinate
axes as the
components of
tensor.] W e
can
interpret
% ij
(r,
t
as
t
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of the fo rce
per
unit
area
exerted, at positi
nd t ime t,across a plane surface
element
normal
to
t
The three diagonal components
of
ij a
normal stresses, since each of them gives
t
component of
the
force per unit area acting acro
plane surface
element parallel
to one
of
the Cartesia
planes.
The s ix non-diagonal components a
shear
stresses,
since they d rive shearing m o tio n
parallel
layers
of
fluid
slide
relative to
one another.
Models of
Fluid Motion
General
Properties o f
Stress
Tensor
e
i-component of the total force acting on a
flu
consisting
of
a
f ixed
volume
V enclosed by
S is written
fi
=
ZV
Fi dV +
IS
ij
dS
(2.
the
f irst
term on the
right-hand side
is t
volume force acting throughout
V,
whereas t
term is the
net
surface force acting across S.
akin
of the tensor divergence
theo rem (see Section
B.4),
t
expression
becomes
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fi
=
ZV
Fi dV +
ZV
ij
x
j
d
(2.
the
l im it
V % 0, it is reasonable to suppose that the
d
ij
/x
j
are approximately
constant across
t
n
the l
im it V
%
0 , it
i
s
reasonable
to
su
pp
In this
situation,
both contributions
on
t
side
of the above equation scale as V.N o
to
Newtonian
dynamics, the
i-component of
t
t force acting on the
element
is
equal
to
the
of
the
rate
o f
change
of
its
l inear momentu
in
the
l im it
V
% 0 ,
the
linear acceleration a
density of
the
fluid are
both approximately
consta
the element. In this case, the
rate
o f change o f t
mass
d
ensity of the fluid
&s
l inear
momentum
also
scales
as
V.
In
oth
the net volume
force,
surface
force,
and rate
of l inear momentum of an infinitesimal
f lu
all scale
as
the
volume of
the element,
a
remain
approximately the same
order
as the volume shrinks
to
zero. W e conclud
the
l inear
equation
of motion of
an
infinitesimal
flu
places
no particular restrictions on the
stre
The
i-component
of
the total
torque, taken
about t
O
of
the
coordinate system,
acting on
a
flu
that
consists
of
a
f ixed
volume
V
enclosed
by
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S
is
written
[see Equations (A.46)
and
(B.6)]
=
ZV
ijk
x
j
Fk
dV +
IS
ijk
x
j
%kl
dS
l
th e firs t and second terms on the right-hand si
e due
to volume
and surface
forces, respective
ijk is
the third-order permutation tensor. S
(B.7).]
Making use of the tensor divergen
(see
Section B.4), the above expression becomes
=
ZV
ijk
x
j
Fk dV +
ZV
ijk
&
(x
j
%kl )
&
xl
dV,
reduces to
i =
ZV
ijk x
j
Fk
dV +
ZV
ijk % kj
dV +
ZV
ijk x
j
&%kl
&xl
d
(2.
&
xi /&x
j
=
+
ij
.
[Here, +ij is
the
second-orde
tensor. See Equation
(B.9). ] Assuming
that po
lies
within the fluid element, and taking the lim it V 0
which the Fi,%
ij
,and &%
ij
/&x
j
are
all
approximate
ithin
the fluid ele men
t,
and
ta king th
e li
across the element,
we deduce that
the fir
and
third
terms
on the
right-hand
side o f the abo
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scale as
V
4/3,
V,and V
4/3,
respectively (since x
N o w ,
according to
Newtonian
dynamics,
t
of the
total torque acting on the fluid eleme
equal
to
the i-component of the
rate
of change of its
n
momentum
about
O. Assuming
that the
l ine
of
the
f luid is approximately constant acro
e element,
we
deduce
that
the
rate of
change of
momentum scales as
V
4/3
(since the
net
l ine
scales as
V,
so the
net
rate o f change
momentum
scales
as xV,and x
V
1/3).
Hence,et rate of c han
clear that the rotational equation of motion of a f lu
surrounding
a
general
point
O, becom
dominated by
the second
term
on t
side of
(2.8)
in the limit that
the
volume of
t
approaches
zero (since this term is a
factor V
1
than the other terms). It fol lows that the
seco
must be
identically
zero (otherwise an infinitesim
element
would acquire an absurdly
large angul
This is
only
possible, for all choices
of
t
of point O,
and
the
shape
of the element, if
%
ijk
&
kj
=
(2.
the f luid. The above relation
shows that
t
tensor must be
symmetric:
i.e.,
= &
ij
12
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MECHANICS
immediately fo llow s that the
stress tensor
only
has s
components
(i.e.,
11
,
22
,
33
,
12
,
1
d
23
).
N o w ,
it is always
possible to choose
the orientatio
a
set
of Cartesian axes
in
such
a
manner that t
diagonal
components
of a
given
symmetric
tensor
field
are
all
set to
zero at a
given
poi
space. (See Exercise B.6.) W ith reference to
su
axes, the diagonal components of the stre
ij
become so-
called principal
stresses
%
11
,
33
,
say. Of course, in general, the
orientation of
t
axes varies with posit ion. The normal stress
across a surface element perpendicular
to the
fi
incipal axis corresponds
to
a
tension (or
a
compressio
%11
is
negative) in the direction of that axis. Likewis
%
22
and
%
33
.
hus,
the
general
state o f
the f luid,
a
point in space,
can
be regarded as a superpositio
tensions, or compressions, in three
orthogonal
directions
The
trace
of
the
stress
tensor,
ii
=
11
+
22
+
3
a
scalar, and, therefore, independent
of
the orientatio
the coordinate axes. (See Appendix B.) Thus, it fol low
irrespective of the orientation of the
principal axe
trace
of
the
stress
tensor
at
a given
point is
always
equ
the
sum
of
the princ ipal
stresses:
i.e.,
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=
11
+
22
+
33
.
Stress Tensor ina Static
Fluid
the surface
forces
exerted on
some
infinitesim
volume
element of a static
fluid.
Suppose
that
t
of
the
stress tensor are
approximate
across
the
element. Suppose,
further, that t
of the cube
are
aligned parallel
to
the princ ipa l
ax
the
local stress tensor.
This
tensor, which now has ze
can
be regarded as the sum of two tenso
13
ii 0 0
0
13
ii 0
0 0
13
ii
022222222223
d
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%
&
+
11
0
1
3
&
ii 0 0
0 &
+
22
0
13
&
ii 0
0 0
&
+
33
0
1
3
& ii
(2.1
The
first of the above tensors is
isotropic
(see Secti
and
corresponds
to
the same normal
force per un
acting
inward
(since
the sign
of
&
ii
/3
is invariab
on each face of
the
volume element. Th
compression acts to change the
element5s volum
t not its
shape, and can
easily
be withstood by the flu
the element.
The
second
o f
the
above
tensors
represents
t
of the stress tensor from an isotropic f o rm. T
components of this tensor have
zero
sum,
of
(2.11),
and thus represent equal and oppos
per unit area, acting on opposing faces of t
element, which
are
such that the forces on at lea
e pair of
opposing
faces constitute
a tension,
and
t
on
at
least one pair constitute a
compression. Su
necessarily tend to change the shape of
the
volum
either
elongating or
compressing it along
oneof
axes. Moreover,
this
tendency
cannot be offs
any
volume force acting
on
the element, since su
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become
arbitrarily
small
compared to surface forc
the limit that the elements
volume tends to ze
the
ratio of
the net volume
force to the n
force
scales
as
the
volume to the surface area
e element, which tends to zero
in
the limit that t
tends to
zerosee
Section 2.4).
Now,
we
ha
defined a fluid
as
a
material
that
is
incapable
any tendency of
app lied fo rces
to change
(See
Section
2.2.)
It fol lows that
if
the
diagon
of the tensor (2. 1
3) are
non-zero anywhe
the fluid then it is impossible for the fluid at
th
to
be
at rest. Hence, we conclude that
the
princip
%
&11
,
%
&22
,
and
%
&33
,
must be
equal
to one another
l
points
in a static
fluid.
This
implies
that
the stre
takes
the isotropic form
(2.12)
everywhere in
fluid.
Furthermore,
this
is
true
irrespective
e orientation of the coordinate axes, since t
of an isotropic tensor
are
rotational
(See Section
B.5.)
Models of
Fluid Motion
Fluids at
rest
are generally
ina state o f
compression,
is
convenient
to write the
stress tensor
of
a
static fluid
e
form
%
ij
= +p0
(2.1
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p = ii /3 is termed the staticfluidpressur
d
is generally a function
of
r
and
t.
follows that, in
a
stationary f luid, the force
per
unit ar
across
a plane surface element
with
unit normal
luid,
the force
per
uni
t
ar
p
n.
[See
Equation
(2.3).]
Moreover,
this
normal for
s
the
same
value
for
all
possible orientations ofn.
Th
result%namely,
that
the
pressure
is the same
ldirections at a
given
point
in
a static
fluid
%
is
known
&
s law, and is
a
direct
consequence of
the fact
tha
element
cannot
withstand
shear
stresses,
any tendency of
applied forces to
change
Stress
Tensor ina Moving
Fluid
e have seen
that in
a static fluid the
stress tensor
takes
e
form
=p+ ij
p
=
ii /3
is
the static pressure: i.e.,
minus
t
stress acting in any direction. Now, the norm
at a
given
point
in
a moving
fluid
generally
vari
direction:
i.e., the principa l stresses are not
equal
e another. However,
-
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can
still define the mean
principal
stress
(
11
+
22
+
33
)/3 =
ii
/3.
given that
the
principal
stresses
are
actua
stresses
(in a coordinate frame aligned
with
t
axes), we
can
also
regard
ii
/3
as the me
stress.
It
is convenient
to define pressure in
fluid as
minus
the mean normal stress:
i.e.,
=%
13
ii
.
we can write the stress tensor in a moving fluid
e
sum
o f an isotropic
part,
%p
& ij
,which
has
the sam
as
the
stress tensor
in
a static f luid, and
a
remainin
part, dij
,
which
includes
any
shear stresse
d
also
has diagonal components
whose
sum is zero.
words,
=
%
p
&
ij
+
dij
dii
=
(2.1
since
ij and &
ij
are both
symmetric
tensors,
it
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that dij is
also
symmetric: i.e.,
=dij
.
It is
clear
that
the so-called deviatoric stress tensor, d
a
consequence
of fluid motion, since it is zero in a
sta
Suppose, however, that we were
to view
a sta
both in its rest
frame
and in a
frame
of
referen
at
some
constant
velocity
relative
to
the re
N o w , we would expect the
force distributio
the
fluid to be the
same in
both
frames
since the fluid does not accelerate in ei th
in
the
f irst
frame, the fluid appears stationa
d
the
deviatoric
stress
tensor
is therefore
zero,
whi
the second
it
has a spatially
uniform velocity
field
a
e
deviatoric stress tensor is also zero (because it is
t
as
in the rest
f rame).
We,
thus, conclude that t
stress tensor is zero
both
in a stationary
f lu
d
in
a
moving
f luid
possessing
no
spatial
veloci
This
suggests that the
deviatoric
stress
tensor
by
velocity gradients within the fluid. Moreover, t
sor must
vanish
as
these
gradients
vanish.
Le t
the
vi (r,t) be the Cartesian
components of
t
velocity a t point r and t ime
t.
The vario
gradients within
the
f luid
then take
the
for
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/x
j
.
The simplest possible assumption, which
with the above discussion, is that
the componen
the deviatoric
stress
tensor
are
l inear functions
of
the
gradients:
i.e.,
dij
=
Aijkl
v
x
14
MECHANICS
Aijkl
is
a fourth-order
tensor (th is
fol lows f rom t
rule because
dij
and vi /x
j
are both
prop
order
tensors). A ny
fluid in which the deviator
tensor
takes
the above
form
is termed a
f luid, since
Newton
was
the first
to
postulate
relationship between
shear
stresses and veloci
Now, in an isotropic fluidthat is, a
f luid
in whi
is no preferred
directionw e would
expect
t
order
tensor Aijkl to be isotropic that is, to have
in which all physical
distinction
between
differe
is
absent.
As
demonstrated
in
Section
B.5,
t
general
expression
for
an isotropic
fourth-orde
is
=%& ij &kl ++&
ik
&
jl
+0& il &
jk
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,
,
and
% are
arbitrary scalars
(which can
of position and
t ime).
Thus,
it follows fro
and (2.21) that
dij
=
&vk
&
xk
+
ij
+
&
vi
&
x
j
+
%
&v
&
x
(2.2
ver, according to
Equation
(2.19), dij isa
symmetric
which implies that =%
,
and
=
ekk + ij +
2
eij
eij =
1
2
&vi
&
x
j
+
&v
&
x
(2.2
called
the
rate o f strain tensor. Finally,
according
(2.18), dij
is
a
traceless
tensor,
which yields
3
0
2
, and
dij
=2
eij 0
1
3
ekk
+
ij
(2.2
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=
.
W e, thus, conclude that the most gener
for the
stress tensor
in
an isotropic
Newtonia
is
ij
=
%p&ij +2
eij %
13
ekk & ij
(2.2
p(r,t)
and
(r,t)
are arbitrary scalars.
Viscosity
e
significance
of
the parameter
appearing in
t
expression for the
stress tensor, can
be
se
the form
taken by
the
re la tion (2 .25)
in the
spec
of simple shearing mot ion. W ith +v1 /+x2 as t
non-zero
velocity
derivative,
allof
the components
are
zero apart
from
the shear stresses
=
d21
=
+v1
+
x2
.
is
the
constant
of
proportionality between the
ra
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shear
and
the tangential force per unit area
when
parall
e layers
of
fluid
slide over one
another.
This consta
proportionality
is
genera lly referred to
as viscosity.
It
matter of experience that the force between
layers
undergoing relative
sliding motion
always
tends
the
motion,
which
imp lies that
>
0.
The
viscosities of dry air and pure water at 20
d atmospheric
pressure
are about
1.8
10
5
kg/(m
d
10
10
3
kg/(m s),
respectively. In
neither
case
do
viscosity exhibit
much variation
with pressu
the
viscosity of a ir increases by
about
0
and that of
water
decreases
by
about 3 perce
r degree Centigrade rise in temperature.
Models
of
Fluid Motion
Conservation
Laws
that %(r, t is
the
density of
some bulk
flu
(e.g., mass,
momentum, energy)
at
position r
a
t.
In other words, suppose that, at t ime t,
fluid
element
o f
volume
dV ,
located
r,
contains an
amount
%(r,
t
dV of the property
Note, incidenta lly , that
%
can
be
either
a
scalar,
of a
vector,
or
even
a
component
of a tens
e total
amount of
the
property contained
with
fixed
volume V
is
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8/10/2019 Fluid k2opt
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=
ZV
d
(2.2
the
integral
is
taken
over
all
elements
of
V.
Let
an outward
directed
element of
the
bounding surface
.
Suppose that this element is located at point r.
T
of
fluid that flows
per
second
across
the eleme
d
so out of V,
is
v(r,t)dS. Thus, the amount of the
flu
under consideration
that
is convected
across
t
per second is
(r, t v(r,t)
dS. It
follows
that
t
t amount
of
the property
that
is
convected
out
of
volum
by
fluid flow across its bounding surface S is
% =
ZS
v d
(2.2
the integral
is taken
over all
outward
directe
of
S
.
Suppose, f inally, that the property
is
created
within
the volume
V
at
the
rate
S
p
The
conservation
equation for
the fluid proper
the form
d
dt
=
S
&
%
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(2.3
other
words, the rate of increase in the amount of t
contained within V
is the
difference between
t
rate o f the
property inside V,
and the
rate
at whi
property
is
convected
out
of
V
by
fluid
f low.
The
abov
law can also be
written
d
dt
+ = S
(2.3
is
termed the flux o f the property out o f
S
is called the
net generation
rate o f the proper
Mass
Conservation
t % (r, t and v(r,t be the
mass density
and velocity o
flu id at
point r and
t ime
t.Consider
a
fixed volum
, surrounded by a surface S
.
The
net
mass
containe
V
is
M
=
ZV
%d
(2.3
dV is an element of
V. Furthermore,
the mass f l
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S,
and
out of
V,
is
[see
Equation (2.29)]
M
=
ZS
v
d
(2.3
dS is
an outward directed
element
of S. Ma
requires that the rate of increase of t
contained
within
V,
plus the net
mass f lux
out
of
equal
zero:
i.e.,
dM
dt
+ M
=
(2.3
Equation
(2.31)].
Here,
we
are
assuming
that there
mass generation (o r destruction) within V (sin
ua l
molecules
are effectively
indestructible).
that
ZV
%
%t
dV
+
ZS
v
dS
=
(2.4
FLUIDMECHANI
V
is non-time-varying.
Making
use
of the divergenc
(see Section A.20), the above equation becomes
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ZV
t
+
% (v)
#
dV =
(2.3
this result
is true irrespective of
the
size, shap
location of volume V,which
is
only possible if
t
+ % (v)
=
(2.3
the f luid.
The
above expression is
known as t
of fluid continuity,
and is
a direct consequence
conservation.
Convective Time
Derivative
e
quantity
(r, t)/ t, appearing in Equation (2.3
the tim e derivative of the fluid
mass
density
fixed
point r.
Suppose that v(r,t
is the instantaneou
velocity at the
same
point. It follows that the t im
of
the
density, as
seen
in a
frame
of referen
is
instantaneously co-moving
with
the
fluid at poi
is
lim
& t+0
(r +
v& t,t
+
&t)
0
(r,t)
&
t
=
t
+ v
%
=
D
D
(2.3
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we have Taylor expanded (r + v
t,
t + t) up
order in
t, and where
D
Dt
=
%
%t
+ v& =
%
%
t
+v i
%
%
(2.3
the so-called convective t ime
derivative,
D/D
the tim e
derivative seen in
the local rest
fram
the
fluid.
The continuity equation
(2.37)
can be rewritten in t
1 D
Dt
=
D ln
Dt
=
+&v
(2.4
&
(
v)
=
v& +& v[see (A.174)].
Consider
element V that is
co-moving
with the
f luid.
as the element
is convected by
the fluid
its volum
In
fact, it is
easily seen that
DV
Dt
=
ZS
v
dS
=
ZS
v i dS
i
=
ZV
%
vi
%xi
dV =
ZV
& vdV
(2.4
S
is
the bounding surface
of
the element, and u
s
been
made
of
the
divergence
theorem.
In the
lim
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V
0 , and
v
is
approximately
constant
across
t
we obtain
1
V
DV
Dt
=
DlnV
Dt
=
(2.4
we conclude
that
the divergence
of
the
f lu
at a
given
point
in
space
specifies
the
fraction
of
increase
in the volume
of an
infinitesim
fluid element
at
that point.
Momentum
Conservation
a f ixed volume V
surrounded by a
surface S
.
of the
tota l
l inear momentum contained with
is
Pi
=
ZV
%
vi
d
Models
of
FluidMotion
the
flux
of
i-momentum
across
S
,
nd
out
of
[see
Equation (2.29)]
&i
=
ZS
%
vi v
j
dS
(2.4
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the i-component of the
net
force acting on the
flu
V is
fi
=
ZV
Fi
dV
+
IS
ij
dS
(2.4
the first and second
terms
on the
right-hand
side
a
e contributions f rom
volum e and
surface
forces,
respe
Momentum conservation
requires that
the rate
of the net i-momentum of the
f luid containe
V,
plus the f lux
of
i-momentum out of
V,
is
equ
the
rate of i -momentum generation
within
V. O
from Newton
s
second
law of motion, the
latt
is
equal to the i-component of the net for
on the fluid contained
within V.
Thus, we obta
Equation (2.31)]
dPi
dt
+
%
i
=
(2.4
can
be
written
ZV
&(+vi )
&
t
dV
+
ZS
+vi v
j
dS
j
=
ZV
Fi
dV
+
IS
ij dS
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(2.4
the
volume V is non-time-varying.
Making
use of
t
divergence theorem, this becomes
ZV
(
vi
)
t
+
(
vi
v
j
)
x
j
#
dV
=
ZV
Fi +
%
ij
x
j
d
(2.4
the above result
is valid irrespective of
the
si
or
location
of volume V,which
is
only possible if
(vi
)
t
+
(vi
v
j
)
x
j
=
Fi +
%
x
(2.4
inside
the
fluid. Expanding
the derivatives,
a
we
obtain
t
+v
j
x
j
+
v
j
x
j
v i +
vi
t
+v
j
vi
x
j
=
Fi +
%
x
(2.5
in
tensor notation,
the
continuity
equation
(2.37)
t
+ v
j
x
j
+
v
j
x
j
=
(2.5
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, combining Equations
(2.50) and
(2.51),
we obtain
t
fluid
equation
o f motion,
v i
t
+v
j
v i
x
j
=Fi +
%
x
(2.5
alternative form of this equation
is
Dvi
Dt
=
Fi
+
1
%
x
(2.5
e
above
equation describes how
the net
volume
a
forces per unit
mass
acting on a co-moving
f lu
determine its acceleration.
FLUIDMECHANI
Navier-Stokes
Equation
(2 .2 4), (2 .2 6), a nd (2.53)
can
be
combined
the equation of motion of
an
isotropic, Newtonian
fluid: i.e.,
Dvi
Dt
=
Fi
&
p
xi
+
x
j
v i
x
j
+
v
j
xi
#
&
xi
23
v
j
x
j
(2.5
equation
is generally known as the Navier-Stoke
N o w , in situations in
which
there
are
temperature gradients in the f luid, it
is
a go
to
treat
viscosity
as
a
spatially unifor
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in which case the Navier-Stokes
equatio
somewhat to give
Dvi
Dt
=Fi
%
p
%
xi
+
%
2
v i
%x
j
%x
j
+
13
%
2
v
j
%
xi
%x
j
(2.5
expressed in vector fo rm, the above expressio
Dv
Dt
=
%v
%t
+
(v
&
)v
#
=
F
&
p
+
&
2v
+
13
&
(
&
v)
(2.5
use
has been made of Equation (2.39). Here,
[(a
&
)b]i =
a
j
%
b
%
x
(2.57)
(&
2v)i
= &
2
(2.5
however,
that
the above identities are only valid
esian coordinates. (See Appendix C.)
Energy
Conservation
a f ixed volume V surrounded by a surface S
.
energy
content
of the
fluid
contained
within V is
E
=
ZV
E dV +
ZV
1
2
vi
v i d
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(2.5
the first and second terms on the right-hand si
the
net internal
and kinetic
energies,
respective
E(r , t
is the
internal (i.e., thermal)
energy
per u
o f
the
fluid.
The
energy
flux
across
S
,
nd
out
o f
[cf.,
Equation (2.29)]
=
ZS
E +
1
2
vi v i
v
j
dS
j
=
ZV
%
%
x
j
E +
12
v i v i
v
j
#
d
(2.6
use
has been made of the tensor divergenc
According to the firs t law of thermodynamic
rate of
increase
of the
energy
contained within
V,
pl
e net
energy
flux out
of
V, is
equal
to the net
rate
done on
the
fluid
within
V,minus
the net
heat
f l
t
ofV:
i.e.,
dE
dt
+ E
=
W &
(2.6
W
is
the net rate
of
work, and Q the net heat flu
can be seen that
W
& Q is the effective
energ
rate
within
V
[cf.,
Equation (2.31)].
Now, the net
rate
at which volume
and
surface forc
-
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work on the
fluid
within V is
=
ZV
vi
Fi
dV
+
ZS
v i
ij dS
j
=
ZV
vi
Fi +
(vi
ij
)
x
j
#
d
(2.6
use has been made of the tensor divergenc
Models ofFluid
Motion
Generally
speaking, heat
flow in fluids
is
driven
gradients.
Let
the
qi
(r,
t
be
the
Cartesi
ponents
of the heat
f lux
density at
position r
and
t im
It follows that the
heat f lux across a surface
eleme
,
located
at
point r, is q
dS = qi
dS
i
.Let T(r,t
be t
of the fluid at position r and t ime t.
Thus
temperature
gradient
takes
the
form
T/
xi
.
Let
that
there is a l inear relationship between t
of the local heat f lux density and the loc
gradient:
i.e.,
qi
=
Aij
T
x
(2.6
the Aij
are
the components
of
a
second-rank
tens
can be functions
of
position and t ime). Now, in
fluid
we
would
expect
Aij
to
be
an
isotrop
-
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(See Section B.5.) However, the most gene
order iso tropic tensor is simply a
multiple
of
we can write
Aij =
%
(2.6
%(r,
t
is
termed the thermal conductivity of t
It
follows
that the
most
general expression for t
f lux
density inan isotropic
fluid
is
qi = %
&
T
&
x
(2.6
equivalently,
q = %+T
(2.6
it is
a matter o f experience
that
heat
flow s dow
gradients:
i.e., % > 0.W e conclude that the n
flux out of volume
V is
Q=
ZS
%
&
T
&
xi
dS
i
=
ZV
&
&
xi
%
&
T
&
xi
d
(2.6
use has been made o f the tensor divergenc
Equations
(2.59)0(2.62)
and
(2.67)
can be combined
the fol lowing
energy conservation
equation:
-
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ZV
t
E +
1
2
vi v i
#
+
x
j
E
+
12
v i v i
v
j
#)
dV
=
ZV
vi
Fi +
x
j
v i % ij +
&
T
x
j
#
d
(2.6
this
result
is valid irrespective of the size, shap
location of
volume
V,
which
is
only possible
if
t
E +
12
vi v i
#
+
x
j
E
+
12
v i v i
= vi Fi +
x
j
v i %
ij
+
&
T
x
j
(2.6
inside
the
fluid.
Expanding
some
of
t
and
rearranging,
we
obtain
D
Dt
E +
1
2
v i v i
=
v i Fi +
x
j
v i % ij +&
T
x
j
(2.7
use
has
been made
of
the continuity equati
Now, the
scalar
product of v with the flu
of
motion (2.53) yields
vi
Dvi
Dt
=
D
Dt
12
vi v i
=
v i
Fi
+
v i
%
x
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(2.7
the
previous
two
equations, we get
DE
Dt
=
v i
x
j
% ij
+
x
j
&
T
x
j
20
MECHANICS
making
use
of
(2.26),
we deduce that
the
ener
equation
for
an
isotropic Newton ian
flu
the
general form
DE
Dt
= +
p
v i
xi
+
1
0
+
x
j
&
T
x
j
(2.7
0 =
i
j
dij
=
2
eij
eij
+
1
3
eii
e
jj
=
v i
x
j
v i
x
j
+
i
j
v
j
xi
+
2
3
v i
xi
v
j
x
j
(2.74)
the
rate
of
heat
generation per unit volume
due
When
written
in vector fo rm,
Equation
(2.7
-
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t
=
p
%
v
+
&
+
%
(
+%
T)
.
to
the above equation, the
internal energy
p
mass of
a co-moving
f luid e lement evo lves
in t ime
consequence of work done on
the
element by pressu
its volume
changes, viscous heat generation
due
shear, and heat
conduction.
Equations
o f
Incompressible Fluid
most situations ofgeneral
interest, the
f low of
l iquid,
such
as water, is incompressible to
degree of accuracy. Now, a
fluid
is said to
when the
mass
density of a co-movin
element does not
change
appreciably as t
moves
through
regions of varying pressure.
words,
for
an
incompressible
f luid,
the rate
of chan
fol lowing the
motion is
zero:
i.e.,
=
0.
-
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this case, the continuity equation (2.40)
reduces
to
v =
0.
e conclude that , as a consequence of mass
an incompressible fluid must have
or solenoidal,
velocity
field.
Th
implies,
f rom
Equation
(2.42),
that
t
of
a
co-moving fluid
element
is a
constant
of
t
In
most practical
situations, the
init ial
dens
inan incompressible fluid is uniform in spa
it
fol lows f rom (2.76)
that
the density distributio
uniform
in space and
constant
in
time. In oth
we
can generally treat the
density,
,
as
a unifor
in incompressible fluid
flow
problems.
Suppose
that
the
volume
force acting
on
the fluid
innature (see Section A.18):
i.e.,
=
% &
,
&
(r,
t
is the
potential energy per
unit
mass,
a
&
the
potential energy per
unit
vo lume. Assumin
the
fluid
viscosity
is
a
spatially
uniform
quanti
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is generally the case
(unless
there
are
stro
variations
within the f luid), the Navier-Stoke
for
an incompressible
fluid reduces
to
=
p
%
&
+
+
2v,
+
=
(2.8
termed the kinematic
v iscosity, and has
units of
mete
0+
t
per second.
Roughly speaking,
momentu
a
distance
of
order
0+
t
meters
in
t
seconds
as
of
viscosity. The kinematic
viscosity
at 20
2
C is
about 1.0
10
6
m
2/s.
It follow s th
momentum
diffusion in water
is a relatively slo
Models
of
Fluid Motion
e complete set of equations governing incompressible
is
=
0,
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=
p
%
&
+
+
2v.
%
and +
are
regarded as known
constants,
a
r,
t
as a known funct ion.
Thus,
we have
fo
0
nam ely, Equation
(2 .81), p lus the thr
of Equation
(2.82)0 fo r four
0
namely,
the
pressure,
p(r, t), plus the
thr
of the
velocity, v(r,
t .
Note that an ener
equation is
redundant
in the case
fluid
flow.
Equations ofCompressible Fluid
Flow
many
situations
of general interest, the f low of gas
compressible: i.e.,
there
are
significant changes
in
t
density as the gas f lows f rom
place to place.
For
t
of compressible f low,
the
continuity
equation
(2.40
d the
Navier-Stokes
equation (2.56), must
by
the
energy
conservation equation (2.7
well as
therm odynam ic relations
that
specify t
energy per unit mass, and the temperature
of
the
density
and
pressure.
For
an
ideal
gas,
the
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take the form
=
cV
T,
T =
MR
p
cV is the molar
specific
heat at constant
volume,
8.3145JK
1
mol
1
the molar ideal gas
constant, M
t
mass
(i.e., the
mass
of 1mole of
gas
molecules
d
T the temperature
in degrees Kelv in. Incidentally,
he
corresponds to 6.0221 10
24
molecules.
Here, w
assumed, for the sake of
s imp lic ity , that
cV isorrespon ds to6.0
221
10
constant.
It
is
also
convenient to assume that t
m al conductiv ity ,
%
,
is
a
uniform constant.
Making
u
these
approximations,
Equations (2 .40), (2 .75), (2 .8
d (2.84) can
be combined to
give
1
1
Dp
Dt
& p
D
Dt
= +
+
%
M
R
0
2
p
& =
c
p
=
cV +
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(2.8
cV cV
the
ratio of
the molar specific
heat at constant pressure
,
o that at
constant volume,
cV
.Incidentally,
the result
c
p
= cV
+R
fo r
an
ideal gas is a
standard
theorem o
The ratio
o f
specific
heats
o f
dry
a ir
at
C is1.40.
The complete set of
equations
governing compressible
gas
flow are
D
Dt
=
% & v ,
Dv
Dt
=
p
%
&+ +
&
2v
+
1
3
&(&v)
#
1
0 % 1
Dp
Dt
%
0 p
D
Dt
= 2 +
3M
R
&
2
p
the dissipation function2 is specified in
terms
o
d v
in
Equation (2.74). Here,
0
, 3, M,
and R a
as
known
constants,
and
+
(r,
t
as
a
know
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Thus,
we have
five
equationsnamel
(2.87)
and
(2.89), plus the three components
(2.88) for
f ive unknownsnamely, the densi
r, t , the
pressure,
p(r, t ,
and
the
three
components
e
velocity, v(r,t).
MECHANICS
Dimensionless
Numbers
in
Flow
is
helpful
to normalize the
equations
of
incompressib
f low, (2.81)
%
(2.82),
in the
following
manner: &
t is helpful
to
normalize th e equations
of
i
,
v
=v/V0
,
= (V0
/L)t,+
= +/(gL), and p
(
V
2
0
+
gL+
0
V0
/L).
L
is
a
typical
spatia l varia tion
length-
scale, V0
f luid
velocity, and
g
a typical gravitation
ce lera tion (assuming that +
represents
a gravitationa
energy
per unit mass). All barred quantities a
and
are designed to be comparable
w
e
normalized
equations
of
incompressible f luid
f low take the for
&
v = 0,
Dv
Dt
= 2 1+
1
Fr
2
+
1
Re
&p
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D/Dt
= / t + v
, and
Re
=
LV0
%
Fr
=
V0
(gL)
1/2
(2.90
2
+
2v
Re
(2.91
(2.92
(2.93
the
dimensionless
quantities Re
and F r
are known
Reynolds
number and
the Froude number,
respectively
e Reynolds number is the typical ratio of inertial
forces within
the f luid,
whereas the
square
o f
t
number is
the typical ratio
o f
inertial
to
gravitation
Thus, viscosity is relatively
important
compared
when R e + 1, and vice versa. Likewise,
gravity
important compared
to
inertia
when
Fr
+
d
vice
versa.
Note
that,
in principal,
Re
and
F r
are
t
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quantities in Equations (2.90)
and
(2.91)
that can
greater or
smaller
than
unity.
For the
case of
water
at
20
C,
located on the surfa
the Earth,
1.0 10
6
L(m)V0 (ms
%
1),
94) Fr 3.2 10
%
1
V0 (ms
%1)/[L(m)] 1/2.
if L & 1m and V0 & 1ms
%1,
as
is
often
the
ca
terrestrial water dynamics,
then the above expression
gest
that
Re
+
1
and Fr
& O(1).
In
this
situation,
t
on
the
right-hand side
of
(2.9
1)becomes negligible
d the
(unnormalized)
incompressible fluid flow
equation
to the
following
inviscid, incompressible, fluid flo
0v
=
(2.9
Dv
Dt
=
%
0
p
2
% 0
(2.9
For
the
case
of lubrication oil
at 20
C , located on
t
of the Earth ,
4
1.0
10
%4
m
2
s
%1
(i.e.,oi
100 t imes more
viscous
than water), a