# Chapter 4 Kinematics of Fluid Motion

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14 Νοε 2013 (πριν από 4 χρόνια και 8 μήνες)

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Chapter 4 Kinematics of Fluid Motion

motion

fluid
(pressure)

stress

normal

of

inbalance

e
appropriat
An
stresses
shear

of
slightest

The

cause
motion.

the
of

kinematics

the
consider

i.e.,

motion,

the
produce

to
necessary

force

actual

the
with

concerned

being
thout
motion wi

fluid

of

aspects

various
discuss
-

-
chapter

In the
1

motion.

its

of
ation
visualiz
and
n
descriptio

the
and

fluid,

the
of
on
accelerati

and

velocity
of

discussion

the
-
-

chapter)

(in this

-
-
motion

fluid

of

Kinematics

motion.

the

produce

to
necessary

forces

specific

the
of

analysis

the
-
-

chapter)
next
in

(

-
-
motion

the
of

Dynamics

2

§
4.1 The Velocity field

particles.

fluid

the

of
on
accelerati

&

velocity
the
of

in terms

described

be
can
motion

This

molecules.

individual
an
rather th

particles

fluid

of
motion

of

in terms

fluid

a

of

flow

the
describe
can

We
molecules.

numerous

contains

particle
Each
gs.
surroundin
eir
with th
and
other
each
with
interact
that
particles

fluid

of

up

made

be

to
fluids
consider

and

hypothesis

continuum

e
employ th

We
time.
of
function

a

as
point
another

to
space
in
point

one

from

molecules

of
motion
net

a

flows

fluid

general,
In

3

flow.

of
tion
representa

field
s)
coordinate

spatial

(

f

=

)
location

s
fluid'

(

f

=

T)
a
,

V

,

p

,

ρ

as
(such
property

fluid
any

of
n
descriptio

in time,
instant
given

a
At

t)
z,
y,
(x,

T

=

T

example,
For
4

n
accleratio
a
dt
V
d
speed
w
v
u
V
V
magntude
direction
V
V
V
V
A

2
2
2
A
ector
position v
=
r

where
dt
r
d

=

>
w
v,
u,
=<
k

w
+
j

v
+

i
u
=
t)
z,
y,
(x,

k

t)
z,
y,
w(x,
+

j

t)
z,
y,
v(x,
+
i

)
t
z,
y,
u(x,
=
)
t
z,
y,
(x,

field

velocity
the
is

s
variable
fluid

important
most

the
of

One
5

.
quadrant
first

in the

field

velocity
the
of
sketch

a

Make

(b)

V

to
equal

speed

the
is

field

flow

the
is
location
At what

(a)
:
Determine

constants

are

and

where
)
)(
(

:
Given
4.1

Example
0
0
0
l
V
j
y
i
x
l
V
V

)
(
tan
).
(
)
(
tan

of

Direction
)
(
)
(

of

Magnitude
;
:
Solution
1
0
0
1
2
2
0
2
0
2
0
2
2
0
0
0
0
x
y
x
l
V
y
l
V
V
V
y
x
l
v
y
l
v
x
l
v
v
u
V
y
l
V
v
x
l
V
u
j
y
l
V
i
x
l
V
V

6

§
4.1.1 Eulerian and Lagrangian Flow Description

time.
of
function

a

as

change

particles

ese
with th
associated

properties

fluid

the
how

determined

and
about

move
they
as

particles

fluid

individual

following

involves

-
-

move.
y
the

as

determined

properties
their
and

,

identified
or

"

tagged
"

are

particles

fluid

The

-
-

method

Lagrangian

(2)
points

e
past thos

flows

fluid

the
as

space

in

points

fixed
at

happens
what
of

in terms

flow

about the
n
informatio
obtain

-
-

time)
f(space,

etc.)
v,
,
(p,

properties

necessary

the
g
prescribin

completely
by
given

is
motion

fluid

the
-
-

above

introduced
concept

field

the
uses
-
-
-
method
Eulerian

(1)
problems

mechanics

fluid

analyzing
in

approaches

general

Two

7

on)
(informati

(data)

data
Eulerian

n
informatio

Lagrangian
method
Eulerian

mechanics

fluid
In
use

8

§
4.1.2 One
-

, Two
-

, and Three
-

Dimensional flows

)
(
)
,
(
)
(
)
,
(
flow

l
dimensiona
-
)
(
),
(
)
,
,
(

flow

l
dimensiona
-
two

components
other

two
to
relative

)

sense

some
(in

small

be
may

components

velocity
the
of

one

,

situations
many
In
wing.
airplane
an
past
air

of

flow

the
,

example
For
flow

l
dimensiona

Three
)
(
)
,
(
),
(
)
,
,
,
(

general,
In
t
u
t
r
V
V
or
t
u
t
x
V
V
one
or
t
v
t
u
t
y
x
V
V
t
w
t
v
t
u
t
z
y
x
V
V
r





9

.
sufficient
may

analysis

l
dimensiona

one

simple
A

river,

in the

rate

flow

the
know

to
wishes
one

If
flow.

of
effect

l
dimensiona

three
include

to
necessary

is
it

river,

the
of

meandering

study the

to
wishes
one

If
10

§
4.1.3 Steady and Unsteady Flows

properties

fluid

)

(

where
0
)
(
0
)
(
0
)
(
0
)
(
0
)
(
0
)
(
int

x
or
x
state
unsteady
t
x
or
x
state
steady
t
po
fix
at
11

§
4.1.4 Streamlines , Streaklines , and Pathlines

introduced

parameter

ds

where

//
instant
given

a
at
vector
velocity
the
to
tangent
everywhere

line
-A
-
or

point
at that

velocity
the
of

direction

in the

is
point
any
at

tangent
whose
line
-A
-
s
Streamline

Pathlines

and

s,
Streakline

s,
Streamline

use

fields,

flow

of

analysis

and
ation
visualiz
In the

ds
V
dr
w
dz
v
dy
u
dx
r
d
V
k
dz
j
dy
i
dx
r
d
k
w
j
v
i
u
V
12

.
of
n
Integratio
A

Method
etc
v
dy
u
dx
or
v
dy
u
dx

ds
V
dr
w
dz
v
dy
u
dx

13

.
streamline

the
ng
representi

t)
z,
y,
f(x,

function

desired

e
obtain th

to
s

eliminate
Then
0
s
at

)
t
,
z
,
y
,
(x
condition

initial

the

using
by

obtained

be
can

c

and

,

c

,

c
B

Method
0
0
0
0
3
2
1
3
0
2
0
1
0

c
wds
dz
z
c
vds
dy
y
c
uds
dx
x
wds
dz
vds
dy
uds
dx
z
y
x
n
Integratio
ds
V
dr
w
dz
v
dy
u
dx

14

Flow Patten:

Streamlines; Streaklines; Pathlines; and Timelines

introduced

parameter
//
instant)
given

a
at
vector
velocity
the
to
tangent
everywhere

line
A
(or

)
point
at that

velocity
the
of

direction

in the

is
point
any
at

tangent
whose
line
A

(

Streamline

ds
v
dr
w
dz
v
dy
u
dx
v
r
d

.
streamline

the
ng
representi

t)
z,
y,
f(x,
function

desired

e
obtain th

to
s

eliminate
Then
0
s
at

)
t
,
z
,
y
,
(x
condition

initial

the
using
by

obtained

be
can

c

and

,

c

,

c
0
0
0
0
3
2
1
0
0
0
3
0
2
0
1
0
0
0
0

s
z
z
s
y
y
s
x
x
z
y
x
wds
dz
vds
dy
uds
dx
or
c
wds
dz
z
c
vds
dy
y
c
uds
dx
x
n
Integratio
wds
dz
vds
dy
uds
dx
w
ds
dz
v
ds
dy
u
ds
dx
15

.

z)
y,
f(x,

,
function

pathline

the
give

to
time
eliminate

,
Then
.

s
at

)
t
,
,z
y
,
(x
condition

using

time
o
respect t
with
Integrate
)
,
,
,
(
)
,
,
,
(
)
,
,
,
(
as
such

,
nt
displaceme

and
elocity
between v
relation

the
of
n
integratio
by

defined

is
It

)

fluid

of

particle

single

a

of
ry
trajecto
(The

Pathline
0
0
0
0
0

t
z
y
x
w
dt
dz
t
z
y
x
v
dt
dy
t
z
y
x
u
dt
dx
or
wdt
dz
vdt
dy
udt
dx
16

pathline

e
obtain th

result to

the
from

eliminate

,
Then
wdt
0
at t

)
z
,
y
,
(x

through
pass

to
pathlines

the
cause
which
constants
n
integratio

the
Find
t
0
0
0
0
0
0
0
0
0
t
dz
vdt
dy
udt
dx
z
z
t
y
y
t
x
x

17

point)

prescribed

a

through
passed
earlier

have
which
particles

of

locus

(or the

point)

one
ugh
thro

passed
or

source

one

from

issued

have
which
particles

of

succession

a

of
position

ous
instantane

the
joining

line
(A

Streakline

1
2
1
)
,
,
,
(
)
,
,
,
(
)
,
,
,
(
eqs.

following

the
of
result

integrated

the
take
,

streakline

the
compute

To
c
wdt
dz
c
vdt
dy
c
udt
dx
t
z
y
x
w
dt
dz
t
z
y
x
v
dt
dy
t
z
y
x
u
dt
dx
streakline

e
obtain th

result to

the
from

eliminate

,
Then
wdt

t.
<
ξ

times
of

sequence

a
for

)
z
,
y
,
(x

through
pass

to
pathlines

the
cause
which
constants
n
integratio

the
Find
t
0
0
0
0
0
0

z
z
t
y
y
t
x
x
dz
vdt
dy
udt
dx
18

instant
given
at

line

a

form

that

particles

fluid

of
set
A

-
-

Timeline
19

ally
experiment

generate

Easier to
ally
mathematic

calculate

Easier to

-

Streamline

(a)

Timeline
Streakline
Pathline
line

of

Passage
.

lines

ous
Instantane
)
(

streakline
Pathline
Timeline
Streamline
b

s
Streakline

Pathlines

s
Streamline

flow
steady
In

(c)

20

21

0
at t

0
z
2,
y
1,
x
is

three
all
point
common

t the
Given tha

.

Streakline

(c)

and

,

Pathline

(b)

Streamline

(a)

the
of
equation

the
Find

)
2
(
)
1
(

velocity
Given the

)
Granger

R.A.
in

(P.425

8.1

Example

j
y
t
i
x
t
V

22

y
t
v
x
t
u
j
y
t
i
x
t
V
)
2
(
)
1
(
)
2
(
)
1
(
:
Solution

ds
t
y
dy
ds
t
x
dx
ds
y
t
dy
ds
x
t
dx
ds
v
dy
ds
u
dx
ds
dz
v
dy
u
dx
)
2
(
)
1
(
)
2
(
)
1
(
:
Streamline

(a)

)
(
)
1
(
ln
)
1
(
)
(
)
2
(
ln
)
2
(
)
2
(
ln
)
1
(
ln
)
2
(
)
(
ln
)
1
(
)
(
ln
)
(
)
2
(
ln
)
(
)
1
(
ln
2
1
2
1
2
1
1
2
1
1
t
c
t
y
t
t
c
t
x
t
t
c
y
t
c
x
t
t
c
y
s
t
t
c
x
s
t
c
s
t
y
t
c
s
t
x
n
Integratio

23

)
1
(
1
)
2
(
)
2
(
1
)
1
(
)
2
(
4
2
2
2
1
)
(
2
1
0
at t

0
z
2,
y
1,
x
Condition

Initial
t
t
t
t
t
t
x
y
x
y
y
x
t
c

)
(
)
(
ln
ln
)
(
ln
ln
)
(
)
2
(
)
(
)
1
(
ln
)
1
(
ln
)
2
(
4
)
1
(
)
2
(
4
)
1
(
)
2
(
3
)
1
(
)
2
(
1
2
t
c
y
x
t
c
y
x
t
c
y
x
t
c
t
t
c
t
y
t
x
t
t
t
t
t
t
t

24

0
=
at t

0
z
2,
y
1,
x
Condition

Initial
)
(
ln
2
1
2
)
(
ln
2
1
1
)
2
(
1
)
1
(
)
2
(
)
1
(
:
Pathline

(b)
2
2
1
2

jj
c
y
t
t
j
c
x
t
t
dy
y
dt
t
dx
x
dt
t
dt
dy
y
t
dt
dx
x
t
dt
dy
v
dt
dx
u
)
(
ln
)
2
ln(
)
.(
)
.(
)
(
)
2
ln(
2
1
2
)
(
ln
2
1
ln
0
2
2
2
1
iii
x
y
t
i
eq
ii
eq
ii
y
t
t
i
x
t
t
y
c
c

0
)
2
ln(
2
1
ln
2
)
2
ln(
ln
)
2
ln(
2
1
ln
)
2
ln(
ln
ln
)
2
ln(
2
1
ln
)
2
ln(
)
.(
)
.(
2
2
2

x
y
x
y
x
x
y
x
y
x
x
y
x
y
iii
eq
and
i
eq
25

2
ln
ln
2
2
2
2
ln
2
2
1
ln
2
2
ln
2
1
)
2
(
1
)
1
(
1
)
2
(
1
)
1
(
)
2
(
)
1
(

Streakline

(c)
2
2
2
2
2
2
1
2
1

y
t
t
x
t
t
y
t
t
x
t
t
dy
y
dt
t
dx
x
dt
t
dy
y
dt
t
dx
x
dt
t
dt
dy
y
t
dt
dx
x
t
dt
dy
v
dt
dx
u
y
t
x
t
y
t
x
t

2
ln
2
1
2
2
1
2
ln
2
1
2
1
ln
2
1
0
,
2
,
1
)
(
ln
2
1
2
)
(
ln
2
1
Eq.(jj)

and

Eq.(j)

From
or
2
2
2
1
2
2
1
2
2
2
1
2

c
c
c
c
t
z
y
x
jj
c
y
t
t
j
c
x
t
t
26

)
(
2
1
2
2
ln
2
1
2
)
(
2
1
ln
2
1
2
ln
2
1
2
ln
2
1
2
2
1
ln
2
1
Eq.(jj)

and

Eq.(j)

into

above

c

and

c

ng
Substituti
2
2
2
2
2
2
2
2
2
1
hh
y
t
t
h
x
t
t
y
t
t
x
t
t

)
(
2
ln
ln
2
ln
ln
2
ln
Eq.(h)

-
(hh)

Eq.
hhh
y
x
t
x
y
t
x
y
t

0
)
2
(ln
2
1
2
ln
ln
0
0
)
2
(ln
2
1
2
ln
2
ln
ln
)
2
(ln
2
1
2
ln
2
1
2
ln
ln
2
1
)
2
(ln
2
ln
2
2
1
2
ln
ln
2
1
)
2
ln
(
2
1
2
ln
ln
2
1
Eq.(h)
with
Eq.(hhh)

ng
Substituti
2
2
2
2
2
2
2
2
2
2

y
x
y
x
x
t
For
y
x
y
x
t
y
x
x
y
x
y
x
t
t
y
x
x
t
y
x
y
x
t
t
y
x
x
x
y
x
t
y
x
t
x
t
t
27

28

.

(0,0)
origin

the
through
passes
that
streakline

(c)

/2

=
t
&

0
=
at t

(0,0)
origin
at

s
that wa
particle

the
of

pathline

(b)

/2
=
t
&

0
=
at t

(0,0)
origin

through
pass

Streamline

(a)

:
Determine
constants

are

and

,

where
)
(
sin

:
Given
4.3

Example

0
0
0
0
0

v
u
j
V
i
v
y
t
u
V

0
0
0
0
0
0
)
(
sin
)
(
sin
:
Solution
v
v
v
y
t
u
u
j
v
i
v
y
t
u
V

29

0
0
0
;
)
(
sin
v
v
v
y
t
u
u

)
sin(
)
2
(
cos
2
1
)
cos(
0
cos
)
(
cos
cos
)
(
cos
)
(
)
(
cos
)
(
sin
)
(
sin
)
(
sin
A

method

-
-

Streamline

(a)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v
y
u
v
y
u
x
t
v
y
u
x
t
t
v
y
t
u
x
t
v
u
v
y
t
v
u
x
v
v
v
y
t
u
x
v
dy
v
y
t
u
dx
v
dy
v
y
t
u
dx
v
v
dy
v
y
t
u
dx
v
dy
u
dx
y
x
y

30

)
sin(
2
1
)
cos(
0
cos
)
cos(
cos
)
1
(
)
1
(
sin
)
sin(
sin
)
3
(
&
)
1
.(
)
3
........(
..........
..........
..........
..........
)
2
(
2
..
..........
..........
1
......
)
(
sin
B

method

-
-

Streamline

(a)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
v
y
u
x
t
v
y
u
x
t
t
s
t
u
t
u
d
ds
ds
d
s
t
d
u
ds
s
t
u
ds
v
s
v
t
u
dx
x
eq
s
v
y
from
ds
v
dy
vds
dy
ds
v
y
t
u
dx
uds
dx
ds
v
dy
u
dx
s
t
t
s
t
t
s
s
x
s
y
s
x

31

)
2
(
)
1
(
)
(
sin
Pathline

(b)
0
0
0

dt
v
dy
dt
v
y
t
u
dx
vdt
dy
udt
dx
dt
dy
v
dt
dx
u

0
0
..
..
0
....
0
)]
3
(
&
)
3
.(
)
3
(
0
...
...
0
0
)
(
sin
)
(
sin
(3)

&

(1)

eq.
)
3
(

eq.(2)

From

0

For t
0
0
0
0
0
0
0
0
0
0
0
0
y
x
t
for
t
v
y
x
a
eq
from
pathline
a
x
dx
dt
v
t
v
t
u
dt
y
y
t
u
dx
t
v
y
t
v
dy
y
dt
u
dy
x
t
y

32

)
4
(
)
2
(
/2

=
For t
0
2
0
0
0

t
v
y
dt
v
dy
y
dt
v
dy
t
y

0
0
2
0
0
0
0
0
0
0
0
)
5
(
&
)
4
.(
.
)
5
(
)
2
(
...
...
...
...
2
sin
)
2
sin(
)
2
(
sin
)
)
2
(
(
sin
(1)

&

(4)

eq.
u
v
x
y
eq
from
pathline
t
u
x
dt
u
dx
dt
u
dx
dt
u
dx
dt
u
dt
t
t
u
dx
dt
v
t
v
t
u
dx
o
t
x
o

33

)
2
(
)
1
(
)
(
sin
Streakline

(c)
0
0
0

dt
v
dy
dt
v
y
t
u
dx
vdt
dy
udt
dx
dt
dy
v
dt
dx
u

equation

streakline

the
is

This
sin

(4)

&

(3)

eq.

from
)
4
(
)
)(
(sin
)
sin(
)
sin(
)
(
sin
)
)
(
(
sin
(3)

&

(1)

eq.

Form
)
3
(
)
(
......
(2)

eq.

From
0
0
0
0
0
0
0
0
0
0






v
x
y
t
x
dt
dx
x
dt
u
dx
dt
t
t
u
dt
v
t
v
t
u
dx
t
v
y
t
for
dt
v
dy
y
t
x
t
y

34

35

§
4.2 The Acceleration Field

a
m
F
t
z
y
x
a
t
z
y
x
V
t
V
tion
this
in
discussed
Conversion
)
,
,
,
(
)
,
,
,
(
)
(
n
descriptio
Eulerian
n
descriptio

Lagrangian
motion

Fluid
sec
36

§
4.2.1 The Material Derivative

time)
,
(position

f
),
(
),
(
),
(
)
,
(

particle
moving
of
Location
A
A
A
A
A
A
A
A
A
z
y
x
t
t
z
t
y
t
x
V
t
r
V
V
z
w
w
y
w
v
x
w
u
t
w
Dt
Dw
a
z
v
w
y
v
v
x
v
u
t
v
Dt
Dv
a
z
u
w
y
u
v
x
u
u
t
u
Dt
Du
a
z
V
w
v
V
v
x
V
u
t
V
Dt
V
D
a
particle
any
For
z
V
w
y
V
v
x
V
u
t
V
dt
V
d
a
dz
z
V
dy
y
V
dx
x
V
dt
t
V
V
d
z
y
x
A
A
A
A
A
A
A
A
A
A
A
A
A
A

37

.
steady

is

flow

the

if

space
in
point

fixed

a
at

parameters

flow
in

change

No

0
.
,
)
6
.
4
(
.

derivative

l
substantia
or

derivative

material

the
termed
is

This

)
5
.
4
(
flow
steady
for
t
derivative
local
or
on
accelerati
Local
t
n
descriptio
Lagranigan
Dt
D
on
accelerati
the
just
not
parameters
fluid
any
where
V
t
Dt
D
z
v
v
x
u
t
t
D

38

space
through
particle
the
of
motion
or
convective
the
to
due
derivative
Convective
V
,
,

39

?

:

Determine
tan

:
Given

4.5

Example
0
0

a
ts
cons
are
l
and
v
where
j
y
i
x
l
V
V

2
2
0
2
2
0
2
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
;
0
0
:
,
,
:
Solution
l
y
v
a
l
x
v
a
j
y
l
v
i
l
v
j
y
l
v
i
x
l
v
y
y
l
v
j
y
l
v
i
x
l
v
x
x
l
v
w
t
v
note
y
v
v
x
v
u
z
v
w
y
v
v
x
v
u
t
v
dt
v
d
a
y
l
v
v
x
l
v
u
note
y
l
v
x
l
v
j
y
l
v
i
x
l
v
V
y
x

40

x
y
u
v
direction
y
x
l
v
v
u
V
Velocity
x
y
a
a
a
of
direction
y
x
l
v
a
a
a
a
a
of
magnitude
x
y
z
y
x

tan
tan
2
2
0
2
2
2
2
1
2
2
2
0
2
2
2

41

§
4.3 Control volume and system Representations

approach.

volume
control

the
-
-

and

approach,

system

the
-
-

including

fluid,

a

to
applied

be
can

laws

governing

ese
that th
ways
various
are

There
namics
thermody
of

Law

-
-

motion

of

laws

s
Newton'

-
-

mass

of
on
Conservati

-
-

laws

physical

l
fundamenta

of
set

a
by

governed

is
behavior

fluid
A
42

flow.
may

fluid
which
through
mass)

of

t
independen

entity,

geometric

a

(

space
in

volume
a

-
-

volume
Contorl

gs.
surroundin

its
with

interact

and

flow,

move,
may
which
,
particles)

fluid
or

atoms

same

the
(always
identity

fixed

of
matter

of

collection

a

-
-

System

:
Definition
]
location

fixed

a
at
behavior

s
fluid'

the
observe

and

stationary
remain

we
[

n
descriptio

flow
Eulerian

-

volume
Control
]
about

moves
it

as
behavior

its

observe

and

fluid

the
follow

we
[

n
descriptio

flow

Lagrangian

-
-
-
-
-

System

43

chapter)

(This

volume
Control

--
-
-
-
-

System
"

system.

on the

acting

force

the
all

of

sum

the
to
equal

is

system

a

of

momentum

of

change

of

rate

time
The

"
or
"
constant

remains

system

a

of

mass

The

"
example,
For
approach.

system

a

of

in terms

form

basic

in their

stated

are

fluid

a

of
motion

the
governing

laws

the
of

All

44

§
4.4 The Reynolds Transport Theorem.

)
ume
given vol

a
consider
(
concept

volume
control

and

fluid)

the
of

mass
given

a

(consider

concept

system
both

using
motion

fluid

governing

laws

the
describe

to
need

We
t theorem
transpor
Reynolds

The
.
concept

volume
control

&

system
between

ip
Relationsh

45

e

E/m

=

b

==>
)
energy

(

E
B

If

V
=

b

==>
)

momentum

(

V
m
B

If

V

1/2

=

b

==>
energy)

kinetic

(

mV

1/2

=

B

If

1

=

b

==>
)

mass

(

m

=

B

If

)

s
or vector

scales

be
may

B

&

b

(

bm

=

B
B/m

=

b

mass,
unit
per

parameter
that
of
amount

=

b
......
,
,
,
,

as
such

),

properties

fluid
(or

parameters

fluid

B
2
2

v
m
T
m
a
v

sys
i
i
i
i
v
sys
dV
b
V
b
B

)
(
lim
dV
b
m
b
B

dV
m

0
δV

δV

size,

of

particle

fluid

mal
infinitesi
For
)

mass

(

f

property

intersive

=

b
momentum
angular

energy;

momentum;

mass;

)

considered

being

mass

(

f

=
property

extensive

=

B
0
46

dt
dB
theorem
transport
ynolds
dt
dB
dt
dV
b
d
dt
dB
dt
dV
b
d
dt
dB
V
C
sys
V
C
V
C
sys
sys
.
.
.
.
.
.
..
..
Re
)
9
.
4
(
)
(
)
8
.
4
(
)
(

mass
B
if
dt
dB
and
dt
dB
V
C
sys

.
.
between

s
difference

the
Discuss

4.7

Example
volume.
control

the
within
mass

of

change

of

rate

time
)
(
system

the
within
mass

of

change

of

rate

time
)
(
m

B
:
Solution
.
.
.
.
.
.

dt
dV
d
dt
dm
dt
dB
dt
dV
d
dt
dm
dt
dB
V
C
V
C
V
C
sys
sys
sys

47

0
)
(
0
)
(
fixed.

remains

volume
control

he

t

C.V.

the
of

outside

moved

has

system

the
of
part

opened)

is

valve
(the

0

t
itself.
tank
the
be

to
volume
control

0)
(t

opened

was

valve
the
time
at the
tank
the
within
fluid

the
be

to
system

Choose

0
t
.
.

dt
dV
d
dt
dV
d
V
C
sys

48

§
4.4.1 Derivation of the Reynolds Transport Theorem

"

-
C.V.

=

Sys

"
right

the
to
slightly

moving

system

The

t
+
t
=
At t
"

C.V.

=

Sys

"

volume
control

the
occupying

system

The

t
=
At t
1
1
2
2

t
V
l
t
V
l

(2)

and

(1)
section

across

constants

are

v
and

that v
(B)

and

surfaces

these
to
normal

direction

a
in

(2)

and

(1)
section

across

flows

fluid

that the
(A)

Assume
2
1
49

Theorem
Transport
ynolds
the
is
This
B
B
dt
dB
dt
dB
t
t
t
B
t
t
t
B
dt
dB
dt
dB
t
t
B
t
B
t
t
t
B
t
t
t
B
t
t
B
t
t
B
t
t
B
t
t
B
t
t
B
t
t
B
t
t
B
t
t
B
t
B
t
t
B
t
t
B
t
t
B
t
t
B
t
B
t
B
out
in
cv
sys
II
t
I
t
cv
sys
cv
sys
II
I
sys
cv
sys
II
I
cv
sys
sys
sys
II
I
cv
sys
cv
sys
Re
)
(
lim
)
(
lim
0
limit

In the

))
(
)
(

:
(note
)
(
)
(
)
(
)
(
)
(
)]
(
)
(
)
(
[
)
(
)
(
)
(
)
(
)
(
)
(
t
+
t
=
At t
)
(
)
(
t
At t
0
0

"

-
C.V.

=

Sys

"

t
+
t
=
At t
"

C.V.

=

Sys

"

t
=
At t

50

1
1
1
1
2
2
2
2
2
2
2
2
1
1
1
1
2
2
1
1
A
V
b
A
V
b
dt
dB
dt
dB
t
t
A
V
b
t
t
A
V
b
dt
dB
t
V
b
t
V
b
dt
dB
B
B
dt
dB
dt
dB
cv
sys
cv
cv
out
in
cv
sys

C.V.

the
into

passing

B

of
flux

The
)
(

C.V.

the
of
out

passing

B

of
flux

The
)
(

C.V.

the
within
B

of

change

of

rate

The

system

the
within
B

of

change

of

rate

The

:
Note
)
15
.
4
(
)
(
)
(
)
15
.
4
(
..
..
...
....
0
:
t theorem
transpor
Reynolds

the
is

This

1
1
1
1
2
2
2
2

in
out
cv
sys
in
out
cv
sys
cv
sys
cv
cv
cv
cv
cv
cv
cv
cv
VA
b
VA
b
t
B
dt
dB
a
VA
b
VA
b
t
B
dt
dB
or
A
V
b
A
V
b
t
B
dt
dB
volume
control
fixed
for
z
B
w
y
B
v
x
B
u
Note
z
B
w
y
B
v
x
B
u
t
B
dt
dB

51

.
tan
)
(
0
)
(
1
,
)
(
)
(
:
Solution
flow.

for the

t theorem
transpor
Reynolds

the
of

form

e
appropriat

the
Write
4.8

Example
out
flow
of
magnitude
equal
is
time
in
decreases
k
the
in
mass
The
VA
dv
t
VA
v
d
t
dt
dm
b
m
B
VA
b
VA
b
t
B
dt
dB
out
cv
out
cv
sys
in
out
cv
sys

52

V
C
the
o
carried
flow
n
v
V
C
of
out
flow
n
v
dA
n
v
b
d
b
dt
d
dt
dB
dA
n
v
b
dA
n
v
b
d
b
dt
d
dt
dB
dA
V
b
dA
V
b
d
b
dt
d
dt
dB
VA
b
VA
b
d
b
dt
d
dt
dB
or
a
VA
b
VA
b
dt
dB
dt
dB
cs
cv
sys
cs
cs
cv
sys
cv
cs
cs
in
out
sys
cv
in
out
sys
in
out
cv
sys
in
out
.
int
0
.
0
)
(
)
(
)
(
)
cos
(
)
cos
(
)
(
)
(
)
15
.
4
(
)
(
)
(
Eq.(4.15a)

From
Volume

Control

Arbitrary

53

54

V
C
the
o
carried
flow
n
v
V
C
of
out
flow
n
v
dA
n
v
b
d
b
dt
d
dt
dB
dA
n
v
b
dA
n
v
b
d
b
dt
d
dt
dB
dA
V
b
dA
V
b
d
b
dt
d
dt
dB
cs
cv
sys
cs
cs
cv
sys
cv
cs
cs
in
out
sys
in
out
.
int
0
.
0
)
(
)
(
)
(
)
cos
(
)
cos
(
Volume

Control

Arbitrary

.

ideas

system

and

ideas

volume
control
between
link

a

provide

to
is

purpose

-Its
-
tool.
use
-
to
-
easy

relatively

,

rward
straightfo
rather

a

is
it
that
show

will
involved

concepts

the
of

ing
understand

physical

a

,
However

-
-
.
expression

al
mathematic

formidable
rather

a

be

to
appears
it
first
At

-
-
55

od.
neighborho

the
within
cars

of
number

the
of

change

of

rate

time
the
discuss

to
t theorem
transpor
Reynolds

the
Use
figure.

as

indicated

is

as
city

a

of

od
neighborho
certain

a

of
out

and

into
driven

are

Cars

:
Given

:
4.9

Example
.

od
neighborho

in the

cars

of

#

N

)

time
initial

some
at

od
neighborho

in the

cars

with the

coincides

and
about

moves
that
cars

of

collection

a
or

(

system

a
in

cars

of

#

N

.

cars

of
number

the

N

Volume

Control

D
-
2

od"
neighborho
"

C.V

Solution;
cv
0
sys

t
56

acre))
per

cars

of
(number

z)
y,
(f(x,

od,
neighborho

ut the
througho
cars

of
density

n
(cars/hr)
neighbor

enter the

cars

at which

rate
net

the
(cars/hr)
neighbor

the
leave

cars

at which

rate
net

the
N

N

,
t
At t
cv
sys
0

i
in
i
i
out
i
cv
sys
in
out
in
out
cv
sys
N
N
ndA
t
dt
dN
N
N
N
N
t
N
dt
dN
boundary.

od
neighborho

the
cross

cars

at which

rate
net

the
plus

me,
with ti
changes

od
neighborho

in the

cars

of
number

the
at which

rate

the
to
equal

is

with time

changes

system

in the

cars

of

no.

the
at which

rate

e
that th
states

this
Physically
57

out.
driven

are
ey
that th
rate

the
and

od
neighborho

the
into
driven

are
they
at which

rate

he
between t

difference

the
equals

od
neighborho
given

a
within
cars

of

increase

of

rate

The
0
0

/dt
dN

,
destroyed)
nor

created
neither

are

cars

i.e.,

(

in time
constant

remains

system

in the

cars

of
number

the
If
sys

i
in
i
out
cv
i
in
i
out
cv
Ni
Ni
ndA
t
Ni
Ni
ndA
t
58

§
4.4.3 Relationship to Material Derivative

59

c.s.)

he
through t
rate

mass
net
(
0
)
(
)
mass

of
on
Conservati
(
0
)
(
1
..
)
(
0
)
(

cs
sys
cs
sys
cs
sys
dA
n
v
dt
dm
dA
n
v
dt
dm
b
m
B
for
dA
n
v
b
dt
dB
t
flow
steady
a
For

c.s)

the
across

momentum

of
flux
net
(
)
(
)
(
)
(

cs
cs
sys
dA
n
v
v
F
dA
n
v
v
dt
v
m
d
v
b
v
m
B
for

60

cs
cv
sys
cs
cv
sys
dA
n
v
if
dv
b
t
dt
dB
dA
n
v
b
dv
b
t
dt
dB
t
flow
unsteady
an
For
0
)
(
)
(
0
)
(

flow
unsteady
an
For

61

]
,
,
[

deforms

and

s
accelerate

moves,
that
volume
control
)
(
volumes
control

moving
For



w
v
u
v
dt
dB
dA
n
v
b
dt
dB
dt
dB
cv
cv
cv
cv
sys

system.

coordinate

fixed

a
in
observer

stationary

a
by
seen

as

velocity
fluid

e
th
C.S.

fixed

the
across

fluid

the
carries
that
,
velocity
Absolute

C.S.

moving

the
across

fluid

the
carries
that
,
velocity
Relative

Let

v
w
z
B
w
y
B
v
x
B
u
t
B
dt
dB
cv
cv
cv
cv
cv
62

)
23
.
4
(
]
)
[(

and

is

velocity
fluid
then
0

,

i.e.

volume
control
on

s
coordinate

observe
an
put

we
If

cv
cv
cv
sys
cv
cv
cv
cv
cv
cv
cv
dA
n
v
v
b
t
B
dt
dB
t
B
dt
dB
V
V
W
V
V
W
V
or
V
V
W

63

Selection of a Control Volume

velocity.
constant
with
moved

be

will
C.V.

cases,

some
In

volume.
ng
nondeformi

fixed,

a

be

will
C.V.

the

cases,
our

of
most
-In
-
size.
in

mal
infinitesi
or

size

finite

of

be
may

-C.V.
-
volume.
control

a

as

considered

be
can

space
in

me
-Any volu
-
others.
than
better"
much
"

are

some
but

,

wrong"
"

are

None

.

volume

control

best"
"

the
selecting
at

skill

develop

can we

practice
by
Only

used.

volume
control

the
of

choice

upon the

dependent

y
often var

is

problem

mechanics

fluid
given

a

solving

of

ease

-The
-
64

process.
solution

the
simplify
usually

will
This

.

180
or

0

be

will
s
flux term

in the

)
cos
(

angle

the
that
so

velocity
fluid

the
to
normal

be

should

surface

control

the
possible,

If

*
.
t theorem
transpor
Reynolds

the
of

integral)

surface

(the

term
convective

in the
appear
then
ill
unknown w

The

*
volume.
control

the
within
buried"
"
not

surface,

control

on the

located

is
point

is
that th
ensure

best to
usually

is
It

*
field.

flow

in the
point

some
at

force

and

pressure,

,
velocity
as

such

parameters

g
determinin

involve

will
problem

typical
a

of

-Solution
-

v
n
v

65