CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
PHYSICAL INTREPRETATION OF MATHEMATICALLY
INVARIANT K(p,P) TYPE EQUATIONS OF STATE FOR
HYDRODYNAMICALLY DRIVEN FLOW
George M. Hrbek
Los Alamos National Laboratory, Los Alamos, NM 87545
In order to apply the power of a full group analysis(1) to the problem of an expanding shock in planar,
cylindrical, and spherical geometries, the expression for the shock front position R[t] has been modified to allow the wave to propagate through a general non-uniform medium. This representation incorporates the group parameter ratios as meaningful physical quantities and reduces to the classical
Sedov-Taylor solution for a uniform media.
Expected profiles for the density, particle velocity, and pressure behind a spherically diverging shock
wave are then calculated using the Tait equation of state for a moderate (i.e., 201TNT equivalent)
blast load propagating through NaCl. The changes in flow variables are plotted for Mach < 1.5
Finally, effects due to variations in the material uniformity are shown as changes in the first derivative of the bulk modulus (i.e., K0').
INTRODUCTION
DIMENSIONAL ANALYSIS APPLIED TO
THE EXPANDING SHOCK FRONT
In the companion paper(1) a general solution to the
ID hydrodynamic shock wave was given in terms
of its group invariance properties.
Self-similar profiles for the reduced density, particle velocity, and pressure behind the shock were
shown to be explicit functions of the Mach number
at the front, the equation of state, the shock formation time, and the uniformity of the material ahead
of the shock.
In order to illustrate how the group theoretic
method can be applied to the investigation of real
experiments, this study will consider a moderately
strong (M<1.5), spherically diverging shock wave
propagating through a solid block of NaCl using
the Tait equation of state
The traditional Taylor-Sedov dimensional expression for the expanding shock front only allows for
an ideal gas, power law non-uniformity.
In order to apply the present work into the nonuniform regime for a general material, it is necessary to incorporate a velocity dependence into the
expression for the expanding shock front, R[t].
This yields the following ordinary differential
equation;
R\f\ = (p~lERlt]%-l)(l~*)/(J+l)
(t - a}*' (1)
Where R'[t] is the velocity of the shock front, t is
time, p is the density, E is the energy of the blast, |i
is the uniformity of the material ahead of the
115
∆µ
1
−∆λ
∆2µ∆µ 2(1+J ) ( E1)1 (3+J ) (t −σ−−∆
)
∆λλ 2 ×
E
E
3+JJ) )
2(21(+1J+)J )
(3(+
22
R [t] =
] =22
RRR[i\=
[t[]t=
( ( )ρ)
((tt−−σσ))
ρρ −2(λ−1)
(2)
0.4
Q 3
0.3
0.3
/
0.2
0.3
0.2
0.1
0.2
0.1
0.1 000 - j
-0.1
0 00t
-0.1
«
-0.1 0
is
n j>
«
*
0.1
0.1
R[t]
——R[t]
R[t]
R[t]
«
0.2
0.2
0
0.3
0.3
A
0.40.4
0.5
—————
Q0.5
5
0.5
«0
«
0.1
0.2
0.3
0.4
Figure
Shock
Front
Position
Figure11Shock
ShockFront
FrontPosition
Position
Figure 1 Shock Front Position
Time
(sec.)
Time
Time (sec.)
(sec.)
Time (sec.)
\
^-^. ———=
R'[t]
R'[t]
B
*•
HI"
HI-
0.1
0.2
0.3
0.4
fl 1
01
Q3
0A
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
Figure 2 Shock Front Velocity
Figure 2 Shock Front Velocity
0.3
00.3
3
0.3
0.4
4
0 0.4
0.4
0.5
0.5
05
0.5
R''[t]
R''[t]
—R''[t]
R"W
example,
consider
the
Tait
Equation
AsAs
anan
example,
consider
the
Tait
Equation
ofof of
As
an
example,
consider
the
of
(2)
As
an
consider
theTait
TaitEquation
Equation
(2)example,
(2)
State
with
a
shock
propagating
through
NaCl
State(2)with
withaaashock
shock
propagating
through
NaCl
State
propagating
through
State
with
shock
propagating
throughNaCl
NaCland
(ρ=2.17
gm/cc,
c=2440m/s,
Ko=23.81GPa
(p=2.17gm/cc,
gm/cc,
c=2440m/s,Ko=23.81GPa
Ko=23.81GPaand
and
(ρ=2.17
c=2440m/s,
(ρ=2.17
gm/cc,
(3) (3). c=2440m/s, Ko=23.81GPa and
(3)
Ko’=5.68)
Ko'=5.68)
Ko’=5.68)
(3)..
Ko’=5.68)
.
We
will
assume
shock
wave
is produced
We
will
assume
thatthat
thethe
shock
wave
produced
We
will
assume
that
the
shock
wave
isis produced
We
will
assume
thatequivalent
the shock blast
wavepropagating
is produced
a 20
TNT
byby
20
tonton
TNT
equivalentblast
blastpropagating
propagating
by
aa20
ton
TNT
equivalent
by
a 20 ton
equivalent
blast
propagating
through
a TNT
uniform
block
of
NaCl
and
through
uniform
block
ofNaCl
NaCl
and
thatthat
the the
through
aauniform
block
of
and
that
the
through
a
uniform
block
of
NaCl
and
that the
shock
takes
5ms
to
form.
shock
takes
5ms
toform.
form.
shock
takes
5ms
to
shock
takes
5ms
toaaform.
The
choice
of
such
a long
formation
The
choice
of
such
long
formation
timetime
tois to
The
choice
of
such
long
formation
time
isisto
The
choice
of
such
a long of
formation
time
isformation
to
allow
graphical
illustration
ofthe
the
shock
formation
allow
graphical
illustration
of the
shock
allow
graphical
illustration
shock
formation
allow
graphical
illustration
of
the
shock
formation
delay
onon
thethe
same
graph
asthe
the
characteristic
delay
same
graph
as the
characteristic
delay
on
the
same
graph
as
characteristic
delay
on
same
graph
as
the
characteristic
shapes
ofthe
thethe
shock
front
position
over
shapes
of
the
shock
front
position
over
aa a
shapes
of
shock
front
position
over
shapes
of the
shock
front
position
over
a formation
reasonably
large
range
oftime.
time.
Actual
formation
reasonably
large
range
of
Actual
formation
reasonably
large
range
of
time.
Actual
reasonably
large
range
of
time.
Actual
formation
times
would
of
course
be
much
shorter.
times
would
of
course
be
much
shorter.
times would of course be much shorter.
times
would
of
course
be
much
shorter.
Figure
4
shows
the
speed
of
the
shock
front
asaaas a
Figure
4
shows
the
speed
of
the
shock
front
as
Figure 4 shows the speed of the shock front
Figure
4
shows
the
speed
of
the
shock
front
as a
function
of
time
for
such
a
blast.
function
of
time
for
such
a
blast.
function of time for such a blast.
function of time for such a blast.
2^10.0-
H
——R'[t]
w
"X,
¥
0.2
0.2
02
0.2
Time
(sec.)
Time
(sec.)
Time (sec.)
Time (sec.)
20.0
20.0
20.0
1 15.0
15.0 15.0
15.0
10.0
10.0
it JSC 10.0
cn
5.0
*^^—-—*.-—————
5.0
•* i i i i i i ini i i in....
5.0
0.0
1 a°'
0.0
0(
0.1
0.2
0.3
0.4
01
02
03
04
0.0 1---------_5.0
. . . . . .0.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^^ 0.4
-5.0
0
-5.0
0
0.1
0.2
0.3
0.4
Figure
Figure44u/g[x]
u/g[x]For
For20
20tons
tonsTNT
TNT
-5.0
1
\
0.1
0.1
1
0.1
ILLUSTRATIVE
EXAMPLE
ILLUSTRATIVE
EXAMPLE
ILLUSTRATIVE
ILLUSTRATIVEEXAMPLE
EXAMPLE
Shock
Front
Speed
Shock
Front
Speed
(km/s)
(km/s)
Shock
Front
Speed
(km/s)
Un-Scaled Shock
Un-Scaled
Shock
Un-Scaled
Velocity Shock
Velocity
Velocity
6
6
65
5
54
4
43
3
32
2
21
1
10
00
0-1 01
-1
-1 0
*^*~^
j__rf-—
^***^^
^
00a - 0
-2000
00
-2000
-2000
-4000
Figure
Shock
Front
Acceleration
Figure
Figure333Shock
ShockFront
FrontAcceleration
Acceleration
———""
Un-Scaled Shock
Un-Scaled
Shock
Un-Scaled
Position Shock
Position
Position
I
1
-10000
-10000
-10000 - —————————————————————————
Figure 3 Shock Front Acceleration
Time(sec.)
(sec.)
Time
Time
(sec.)
Time (sec.)
^*-jrf**'
<^*^~~
2000
2000 0
2000
1 -6000
-6000
-8000
-8000
-8000
-10000
Where ∆=2(J+1)/[(J+3)(λ-1)] and the shock posiWhere∆=2(J+1)/[(J+3)(λ-1)]
A=2(J+1)/[(J+3)(X-1)]and
andthe
the shock
shock posiposiWhere
∆=2(J+1)/[(J+3)(λ-1)]
and
the
shock
posiWhere
tion has
been shifted so that R[t]=0
at t=σ (the
initionhas
hasbeen
beenshifted
shiftedso
sothat
thatR[t]=0
R[t]=0at
t=a(the
(the iniinition
has
been
shifted
so
that
R[t]=0
atatt=σ
t=σ
(the
inition
tial point where the shock wave forms).
tial
pointwhere
wherethe
theshock
shockwave
waveforms).
forms).
tial
point
where
the
shock
wave
forms).
tial
point
The
group
parameters,
λ and
and (0,,
µ,are
arerelated
relatedthrough
through
The
group
parameters,
K
The
group
parameters,
λ
and
µ,
are
related through
through
The
group
parameters,
λ
and
µ,
are
related
the
invariance
of the
the energy
energyintegral
integral(i.e.,
(i.e.,µ=2µ=2theinvariance
invarianceofof
of
the
energy
integral
(i.e.,
(1=2the
invariance
the
the
energy
integral
(i.e.,
µ=2(J+3)λ)
and
both
are
ultimately
functions
of
the
(J+3)X)and
andboth
bothare
areultimately
ultimately functions
functions of
of the
the
(J+3)λ)
and
both
are
ultimately
functions
of
the
(J+3)λ)
properties
of
the
propagating
medium
propertiesofof
ofthe
thepropagating
propagatingmedium
medium
properties
the
propagating
medium
properties
Figures
11through
through
3are
areplots
plotsof
ofthe
theun-scaled
un-scaled(i.e.,
(i.e.,
Figures
through
3
are
plots
of
the
un-scaled
(i.e.,
Figures
1
3
Figures
1 term
through
3been
are plots
of
theshock
un-scaled
(i.e.,
the
E/ρ
has
taken
out)
front
posithe E/ρ
E/p term
term has
has been
been taken
taken out)
out) shock
shock front
front posiposithe
the
E/ρvelocity,
term has
been
taken out)
shock
front position,
and
acceleration
respectively.
Note
tion,velocity,
velocity, and
and
acceleration
respectively.
Note
tion,
acceleration
respectively.
Note
tion,
velocity,
acceleration
respectively.
Note
that
the shock
shockand
wave
takesaaafinite
finite
amountof
time
that the
wave
takes
finite
amount
ofoftime
time
that
wave
takes
amount
that
the
shock wave takes a finite amount of time
to
form.
to form.
form.
to
to form.
a
0.60.6
0.5
0.55
1§
°'
0.5
0.4
0.4
Time(sec.)
(sec.)
Time
(sec.)
Time
-4000
1 -4000
-4000
-6000
××
−λ(1+J )
(2)
(2)
1+∆
2
∆
∆
((J + 3)(t −σ − (2σ−)−2(2λ(λ−1−)1µ) µµ (t −σ−−λ)λ(1(1++JJ))µµ µ11))
++
t −σσ−−(2(σ
3)(
2σ))
)) 22
J J++3)(
t−
((((
(t(t−−σσ))
))
0.8
0.8
n fl
0.8
0.7
0.7
0.7
0.6
0.6
Time (sec.)
Un-Scaled Shock
Un-Scaled
Shock Acc.
Acc.
Un-Scaled Shock Acc.
shock, λ is the expansion rate of the shock, σ the
shock,
σ
shock,λformation
theexpansion
expansion
rate
of
the2shock,
shock,
the
shock
time, andrate
j=0,of
1,the
or
for rectangushock,
λXisisisthe
the
expansion
rate
of
the
shock,
σ0 the
the
shock
formation
time,
and
j=0,
1,
or
2
for
rectangushock
formation time,
time,
and j=0,
j=0, 1,1,geometries
or22for
forrectangurectangular,
cylindrical,
and and
spherical
respecshock
formation
or
lar,
and
spherical
geometries
lar,cylindrical,
cylindrical,
andthis
spherical
geometries
respeclar,
cylindrical,
and
spherical
geometries
respectively.
Note that
equation
reducesrespecto the
tively.
Note
that
this
equation
reduces
to
tively.
Note
that
this
equation
reduces
the
tively.
that for
this
equationmaterial.
reduces to the
classicalNote
solution
a uniform
classical
solution
for
a
uniform
material.
classical
solution
for
a
uniform
material.
classical
solution
for
a
uniform
material.
This differential equation has the solution
This
differential
This
differentialequation
equationhas
hasthe
thesolution
solution
This
differential
equation
has
the
solution
0.5
0
0.5
0.5
\V
Figure 4 u/g[x] For 20 tons TNT
Figure 4 u/g[x] For 20 tons TNT
Figure 2 Shock Front Velocity
Figure 2 Shock Front Velocity
116
0.5
05
0.5
0.5
Fromthis
this calculationa acorrespondence
correspondencebetween
between
From
From
thiscalculation
calculation
a correspondence
between
the
shock
speed
and
time
can
be
determined
for the
the
speed and
time
can
be
theshock
shock
andas
time
can1.
bedetermined
determinedfor
forthe
the
blast
andisisspeed
shown
Table
blast
and
shown
as
Table
1.
blast and is shown as Table 1.
Particle Velocity (km/s)
Particle Velocity (km/s)
1.20
1.20
1.00
1.00
0.80
0.80
0.60
0.60
0.40
0.40
0.20
0.20
0.00
0.00
0.6
0.6
0.6
Table1.1.Shock
ShockSpeed
Speedfor
for a 20tTNT
TNT Equiv.Wave
Wave
Table
Table 1.Through
Shock Speed
foraa20t
20t TNTEquiv.
Equiv. Wave
Passing
NaCl
Passing
PassingThrough
ThroughNaCl
NaCl
Time after
Time
Timeafter
after(s)
Initiation
Initiation
(s)
Initiation
0.20 (s)
0.20
0.20
0.28
0.28
0.28
0.33
0.33
0.33
Mach No.(M)
(M)
Mach
MachNo.
No. (M)
1.5
1.5
1.5
1.2
1.2
1.2
1.1
1.1
1.1
6.06.0
0.9
0.9
0.9
Density (gm/cc)
Density (gm/cc)
1
Pressure (GPa)
Pressure (GPa)
2.02.0
1.01.0
0.00.0
0.60.6
0.70.7
0.80.8
0.8
0.90.9
0.9
Profiles
NaCl
Figure
7 Pressure
NaCl
Rgure
7 Pressure
Profiles
forfor
Nad
Figure
11 1
CHANGESIN
INKo’
Ko’FOR
FORNaCl
NaCl
CHANGES
Ko'
CHANGES
Grouptheory
theorymay
mayalso
alsobebeused
usedtotoaid
aidininthetheinterinterGroup
theory
Group
interpretationof
experiments.
pretation
ofofexperiments.
experiments.
pretation
(3)
(3)(3) reported
Asaaacase
casein
point,Chbabildas
Chbabildasand
andRuoff
Ruoff
As
case
ininpoint,
and
reported
As
reported
valuesfor
forthe
theKo’
Ko’ofofNaCl
NaClbetween
between5.68
5.68and
and5.98.
5.98.
values
for
the
Ko'
values
and
5.98.
By
performing
the
procedure
outlined
in
this
paper,
By performing
performing the procedure outlined in this
By
this paper,
paper,
researchermay
maybebeable
abletotofind
findananinconsistency
inconsistency
researcher
may
aaaresearcher
theirexperiment
experiment bybydetermining
determiningthe
thedensity,
density,
inintheir
their
experiment
in
density,
particle
velocity,
and
pressure
profiles
behind
particle
velocity,
and
pressure
profiles
behind
thethe
particle velocity,
shock
for
a
specified
Mach
number
using
the
group
shock
for
a
specified
Mach
number
using
the
group
shock for a specified
the group
theoryrepresentation
representationof
theflow.
flow.
theory
representation
ofofthe
the
flow.
theory
Thisisisisbecause
becausedifferences
differencesin
thenumerical
numericalvalue
value
This
because
differences
ininthe
the
numerical
This
value
ofofthe
the
material
parameters
and
the
thematerial
materialparameters
parametersand
andthe
theuniformity
uniformityof
of
uniformity
ofof
the
material
ininfront
front
ofofthe
the
shock
show
thematerial
materialin
frontof
theshock
shockshow
showup
quite
the
upupquite
quite
clearly
asasvariations
variations
ininthe
the
calculated
clearlyas
variationsin
thecalculated
calculateddensity,
density,parparclearly
density,
particle
velocity,
and
pressure
profiles.
ticlevelocity,
velocity,and
andpressure
pressureprofiles.
profiles. These
Thesedifferdifferticle
These
differences
are
unique
signatures
that
modify
encesare
areunique
uniquesignatures
signaturesthat
thatmodify
modifyeach
eachproproences
each
profile
as
the
shock
slows
down.
file
as
the
shock
slows
down.
file as the shock slows down.
To
illustrate
this
point,
calculations
reduced
Toillustrate
illustratethis
thispoint,
point,calculations
calculationsfor
forthe
the
reduced
To
for
the
reduced
flow
variables
have
been
calculated
for
the
range
flow
variables
have
been
calculated
for
the
range
flow variables have been calculated for the range
of
values
for
NaCl
stated
above.
of
values
for
NaCl
stated
above.
of values for NaCl stated above.
Figures
8-10
show
how
particle
velocFigures8-10
8-10show
showhow
howthe
thedensity,
density,
particle
velocFigures
the
density,
particle
velocity,
and
pressure
profiles
vary
due
to
changes
in
the
ity,
and
pressure
profiles
vary
due
to
changes
the
ity, and pressure profiles vary due to changes inin
the
2.35
2.35
2.25
2.25
0.8
0.8
0.8
11
M=1.1, t=0.33s
M=1.1, t=0.33s
M=1.2,
t=0.28s
M=1.2,
t=0.28s
M=1.5,
t=0.20s
M=1.5,
t=0.20s
3.03.0
M=1.1,
t=0.33s
M=1.1,
t=0.33s
M=1.2,
t=0.28s
M=1.2,
t=0.28s
M=1.5,
t=0.20s
M=1.5,
t=0.20s
0.7
0.7
0.9
0.9
0.9
Dimensionless
Position
Dimensionless
Position
(ξ)(ξ)
Dimensionless Position®
4.04.0
Dimensionless
Position
(ξ)
DimensionlessPosition
Position
Dimensionless
(ξ)(£)
2.15
2.15
2.15
0,6
0.6
0.6
0.8
0.8
0.8
5.05.0
Thisanalysis
analysishas
beencarried
carriedout
outinin
ingenerating
generating
hasbeen
been
carried
out
generating
This
figures55-7.
5- -7.7.
figures
2.55
2.55
2.45
2.45
0.7
0.7
0.7
Figure Velocity
6 Velocity Profiles
NaCl
Figure
forfor
NaCl
Rgure 6
6 Velocity Profiles
Profiles for
Nad
EachMach
Mach numbercorresponds
corresponds to a uniquecomcomEach
Each Machnumber
number correspondstotoaaunique
unique compression
ratio,
β,
[Ref.
1,
Eq.
30]
at
the
shock
pression
pressionratio,
ratio,β,P,[Ref.
[Ref.1,1,Eq.
Eq.30]
30]atatthe
theshock
shock
front.
Themagnitude
magnitudeofof
ofthe
thecompression
compressionisisisdedefront.
front. The
The
magnitude
the
compression
dependent
on
the
particular
equation
of
state
(e.g.,
pendent
on
the
particular
equation
of
state
(e.g.,
pendent on the particular equation of state (e.g.,
Tait).
Following
themethodology
methodologyoutlined
outlinedinininRefRefTait).
Tait).Following
Followingthe
the
methodology
outlined
Reference
1,
this
compression
ratio
is
used
to
generate
erence
erence1,1,this
thiscompression
compressionratio
ratioisisused
usedtotogenerate
generate
the
initial
conditionsatat
atthe
thefront
frontand
andthe
thereduced
reduced
the
theinitial
initialconditions
conditions
the
front
and
the
reduced
equations
forf[ξ],
f[ξ],g[ξ],
g[ξ],and
andh[ξ]
h[ξ]are
aresolved
solvednunuequations
equationsfor
for
f[£],
g[^],
and
h[^]
are
solved
numerically
from
the
coupled
set
of
equations
[Ref.
merically
mericallyfrom
fromthe
thecoupled
coupledset
setofofequations
equations[Ref.
[Ref.
1,
Eqs.
11-17].
1,l,Eqs.
Eqs. 11-17].
11-17].
After
calculatingthese
thesereduced
reducedprofiles,
profiles,the
theactual
actual
After
Aftercalculating
calculating
these
reduced
profiles,
the
actual
flow
variables
are
recovered
using
the
following
flowvariables
variablesare
arerecovered
recoveredusing
usingthe
thefollowing
following
flow
scales;
scales;
scales;
For
the
density
Forthe
thedensity
density
For
(3)
(
)
ρ
ξ
,
t
=
(3)
(ξ , t ) = ρPoρo(t)f(5)
ρp(5,t)=
(ot()t )f (fξ(ξ) )
(3)
andthe
thevelocity
velocity
and
(4)
R'[t'[]tg] g(ξ(ξ) )
(4)
uu«(!,*)
(ξ(ξ, t,)t )===RKM*(f)
(4)
and
finally
the
pressure
finallythe pressure
pressure
and finally
2
(5)
(5)
(5)
PP(ξ(ξ, t,)t )==ρρo (ot()tR) R' ['t[]t2]h[hξ[ξ] ]
2.75
2.75
2.65
2.65
Dimensionless Position (ξ)
Dimensionless Position (ξ)
M=1.1, t=0.33s
M=1.1, t=0.33s
1
——M=1.1,t=0.33s
M=1.2, t=0.28s
M=1.2, t=0.28s
——M=1.2,t=0.28s
M=1.5, t=0.20s
M=1.5,
t=0.20s
M=1.5,t=0.2Qs
11
Figure5 5Density
DensityProfiles
Profiles
for
NaCl
Profilesfor
forNaCl
NaCl
Figure
Figure
117
material uniformity
uniformity parameter (I,
µ, and the material
parameter Ko',
Ko’, for a general spherically diverging
shock wave for the asymptotically limiting compression ratio of 1.68 in NaCl.
To due this, the initial conditions on the backside
of the shock front are computed from the
Mach/compression correlation at the front.
front.
Experimentally, this correlation would be made
through regression of the shock front position with
time (i.e., R[t]).
The Mach number of the front is then used to estimate the compression ratio for a given equation of
state at a particular moment of time.
The flow variables can then be determined via
quadrature of the reduced equations and scaled.
Dimensionless Position
Position (g)
(ξ)
Dmensionless
Reduced Density f( )
1.7
1.6
µ=0.0∗
µ=0.5∗
µ=1.0∗
µ=0.0∗∗
µ=0.5∗∗
µ=1.0∗∗
1.5
1.4
1.3
1.2
1.1
1
0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
0.9
NaCl Variations for Ko' 568*31*1598**
5.68* and 5.98**
Figure 88 Nad
Rgure
REFERENCES
1
(1) Hrbek, G. M., in Proceedings Shock Compression of
Condensed
(HI.078), (2001).
Condensed Matter-2001
Matter -2001 (H1.078),
(2) Kushwah, S.S., and J. Shanker, Physica
Physica B
B 253, 9095 (1998).
(3) Chbabildas, L. C., and
and A.L. Ruoff, /.
ofAppl.
J. of
Appl. Phys.
47 (B13), 4182-4187 (1976).
Dimensionless Position (ξ)
Reduced Velocity g( )
0.405
0.4
µ=0.0∗
0.395
µ=0.5∗
0.39
µ=1.0∗
0.385
µ=0.0∗∗
0.38
µ=0.5∗∗
µ=1.0∗∗
0.37
0.9 0.91
0.91 0.92
0.92 0.93
0.93 0.94
0.94 0.95
0.95 0.96
0.96 0.97
0.97 0.98
0.98 0.99
0.99
0.9
NaCl Variations for Ko' fey
foy 568*
5.68* &
& 5.98**
5.98**
Figure 99 Nad
Rgure
0.375
1
Reduced Pressure h( ξ)
Dimensionless Position
(ξ)
Qmensiontess
Position (|)
0.4
µ=0.0∗ —
——— jjFQO*
0.38
£- 0.38
3
0.36
µ=1.0∗
—— M-ON
^
fi *3fi
| 0.34
0.34
µ=0.0∗∗
— - - jMltO**
1 0.32
0.32
1 0.3
0.3-
µ=0.5∗∗
tt
µ=1.0∗∗
——
^"^
s' ,
'-'
^
x3
'" ^
•^ !^x
•» — -,
µ=0.5∗
M*5*
„
'^*
^--
^^
r
^
//
x<X /
X
0.28
0.28
0.9
0
9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99
1I
NaCl Variations for Ko' for 5.68*
5.68* and
and5.98**
5.98**
Figure 10 Nad
Rgure
DISCUSSION AND CONCLUSIONS
The application of this method is straightforward
straightforward
and powerful.
powerful. It does not require that the total evolution of the flow
flow be determined.
118