NLG~
The LincoKn Ventmeter
—
Going with the flow
Charlie Page and Eva Chan, Harrison Manufacturing Company Pty Ltd., Brookvale, NSW, Australia
Presented at NLGI’s 80th Annual Meeting, June, 2013, Tucson, Arizona, USA
1. Introduction
A vacuum would form at the upstream end of the
capillary as soon as grease began to move down it,
eliminating the driving force.
An earlier paper (Ref 2) gave a more comprehensive
analysis. It attributed the driving force to a combina
tion of the relaxation of the initially compressed grease
sample itself and the relaxation of the initially expanded
capillary tubing (along its length). This grease compres
sion and tube expansion both occur when the device is
initially pressurized and the relaxation occurs once the
exit valve is opened. This is a far more complex mecha
nism but the analysis ultimately assumed a longitudinal
pressure driving force and used average flow velocities
which were calculated from the amount of grease dis
charged during a timed test run.
So what is happening in the ventmeter during a test?
As will be explained later, the answer is probably several
things, simultaneously. In this paper, as a starting point,
it is simply assumed that the quantity of fluid that is dis
charged in the course of a test run is proportional to the
change that occurs in the initial test pressure, that is
The Lincoln Ventmeter (Figures 1 and 2) is a longstand
ing utilitarian test method for grease properties relevant
to centralized lubrication systems. A capillary tube is
filled with grease and pressurized. One end of the tube
is opened to atmosphere and the residual pressure
after 30 seconds is recorded.
While it is generally accepted that the final pressure
reading in a veritmeter run relates to the yield stress
of the grease being tested, it remains something of a
moot point as to what is actually occurring in the test.
It is also of interest as to whether a test could perhaps
give more information about the flow properties of
the grease sample, in addition to its apparent yield
stress value.
In a recent paper (Ref 1) it has been suggested that
the ventmeter is a version of a capillary viscometer.
However, at first sight it appears unlikely that the ventmeter can be modelled simply as the flow of grease
through a capillary under a driving pressure differential.
The ventmeter is completely filled with grease when
it is initially pressurized. Continuity means that the
grease cannot flow along the tube as a “plug” once the
inlet valve is closed and the outlet valve is opened.
GREASE GUN
02000 PSI.
GAUGE
0
~.—
r
25 ~. COIL
¼” OD. TUBING
GR~SE FIfING
VALVE #1
~ENTING VALVE)
Figure 1
—
VALVE #2
Figure 2 — Internal detail (ex Ref. 2)
Lincoln ventmeter
—32
—
VOLUME 78, NUMBER 2
NLGA
dP/dV
k (a constant)
=
A plot of the natural logarithm (Ln) of ventmeter pres
sure against time should give a straight line with a
slope which is dependent only on the fluid’s viscosity,
provided k (that is, dP/dV) is a constant. From
Equation 6, this slope, B, is given by
It is shown in Section 4 that this assumed relation
ship holds true for a variety of physical mechanisms
that may plausibly be considered to be occurring in a
ventmeter test.
It is also assumed that the flow of the fluid is one
dimensional, along the axis of the tube, and is driven
by the difference between the test pressure at any
instant and the external atmospheric pressure. It is also
assumed that the viscosity of the fluid is sufficiently
high to result in laminar flow of the fluid.
B=-’wH4/(8r1A)k
The viscosity is related to B by
11
~P(t)
R~ z~P(t)
=
-~tR4/(8BL) k
….(2)
P(t) upstream end P atm
–
P(t)~~
…
(3)
FR=2nRL’r
(4)
KS~
/
/
connected to closed end,
Indicating the ventmeter
pressure P which varies
with time
where t is the applied stress, i~ is the absolute
viscosity of the fluid and S is the resulting rate
of strain or shear. In the Appendix it is shown
that, for steady one-dimensional flow in a tube,
Equations 1 to 5 lead to the result
i~
Capillary tube,
Radius A
Length L
(5)
-3tR4/(8
(9)
Pressure gauge)
For a Newtonian fluid
=
.. . .
—
2.1 A Newtonian fluid
Ln P(t)
+
where ~tyed is the yield stress and K and n are con
stants, all the values being dependent on the fluid
concerned.
At practical flow rates the yield stress can be ignored
it is very small in comparison with the second term
The resisting force, FR, which is caused by the drag
at the wall, is
t=~S
.(8)
–
t = t
=
. . .
Greases are non-Newtonian, their viscosities are not
independent of their rate of movement or strain. They
also often exhibit a yield stress at very low rates of
applied stress they become essentially elastic solids.
Their viscosity is infinite.
The nature of the relation between applied stress
and the consequent rate of strain of a grease is
shown schematically in Figure 4. A common model
(e.g. Ref 3) for this non-Newtonian behaviour is to
replace Equation 5 with
A simplified schematic of the assumed model is shown
in Figure 3 which also defines some of the symbols
used. Noting that all pressures are true pressures
(gauge pressure plus atmospheric pressure) and that
the ventmeter or upstream pressure varies with time,
then the driving force FD is given by
~t
=
2.2 A non-Newtonian fluid
2. Derivation of the model
FD=
(7)
L) k t
+
Ln P(O)
. . .
(6)
/
Open end
closed end
Tube is full of
test fluid,
Viscosity n
Figure 3 — 5chematic of flow model
—33 —
NLGI SPOKESMAN, MAY/JUNE 2014
NLGII
in Equation 14. However at very low or creeping flow
rates, or at flow initiation from rest, it is the yield stress
that dominates. It is an approximation to this value that
the standard Lincoln ventmeter test which records
just the final pressure after 30 seconds, starting with an
initial test pressure of 1 800psi is measuring.
To simplify analysis, the two cases yield stress not
relevant and yield stress dominant will be considered
separately.
where
a
=
n/(n+1)
(3
= 3t
(14)
—
R3(R/2KL)1”~ (1 -n)/(3n-1) k (p(Q))(l~)/n
(15)
–
Residual least squares techniques can be used to find
the value of a that gives the best linear relation between
the P(t)/P(0)data from a ventmeter test run and time, in
accord with Equation 13. The slope of this best-fit line
gives (3 and the value of a that gave the best-fit also
allows n to be calculated from Equation 14. Finally, K
can be calculated from Equation 15.
It can be shown by expanding both Equation 6
and Equation 13 as infinite series and then letting
n approach 1, that Equation 13 asymptotes to
Equation 6 the Newtonian result as expected.
—
–
2~2.1 No yield stress case
In situations where the yield stress can be ignored,
Equation 9 becomes
-u=KS~
(10)
—
Equation (10) can also be written as
=
2~2.2 Yield stress case
(KS~)S
At
which, when compared with Equation (5), shows that
a non-Newtonian fluid can be considered to have an
apparent viscosity, i~, given by
=
KS~1
—
very low flow rates or rates of shear, Equation 9
approaches
(16)
t = tyj~
When a grease is flowing in a tube under a driving force
that reduces with time, as is the case in the ventmeter,
the rate of flow or strain S asymptotes to zero. A point
will eventually be reached where the driving pres
sure cannot overcome the inherent yield stress of the
(11)
This apparent viscosity is seen to vary with the rate of
strain, S. If n is greater than 1 the fluid thickens with
increasing applied shear, if it is less than 1 it thins and
if n=1 the fluid is Newtonian and its
thickness or viscosity is a constant
1600
(K= Ti), independent of shear. Most
1400
greases demonstrate shear thinning,
n being typically less than 1 (Ref 3).
1200
Repeating the previous Newtonian
1000
analysis but beginning with Equation 10
800
instead of Equation 5 leads to the out
come (see Appendix)
600
400
P(t)/P(0) = (1 + (3 t)~
(1 2)
200
Idealized
Typical
.. .
or
(P(t)/P(0))l/a= 1 +
OJ~
(3t
1~
200
400
Figure 4— Non-Newtonian grease
—34—
VOLUME 78, NUMBER 2
600
800
1000
-~1
1200
Rate of strain (s1)
NLG~
grease, presuming it has one. At this point, substituting
from Equation 16 in Equations 2, 3 and 4 gives
P(oc)
=
2L/R
tyjed
..
In accord with Equation 8, the viscosity of each
polybutene at the test temperature can be calculated
from these slopes, using the relevant dimensions of the
ventmeter listed in Table 1. The calculated viscosities
for each run are given in Table 2, together with the vis
cosity values calculated from the information provided
by the supplier of the polybutenes. These latter were
obtained by extrapolating the supplier’s viscosity values
at 100 and 40°C to the test temperature. In the case of
the mixture, the mixed viscosity at 100 and 40°C was
calculated prior to the extrapolation.
The PIB-1 polybutene was tested at three differ
ent pressures and the agreement between the three
viscosity values obtained was quite good (Table 2).
(17)
where P(oc) is the ventmeter pressure at infinite (long)
times. Equation 17 can be rearranged to
y~eId =
R/(2L) P (cc)
. . .
(1 8)
If a grease has a yield stress then the ventmeter pres
sure will not drop to zero in an extended test. It will
asymptote to a finite value in accord with Equation 17.
The yield stress of the grease can then be calculated
from Equation 18.
Table 1
Data relevant to ventmeter tests
3. Ventmeter Tests
3.1 Tests with Newtonian fluids
Radius of capillary tube
0.3 cm
A series of Lincoln Ventmeter tests was run using two
Length of capillary tube
762.5 cm
viscous Newtonian polybutene fluids (PIB-1 and PIB-2)
at a variety of initial test pressures. Table 1 gives some
Test temperature
22°C
~
relevant data about the ventmeter used,
Table 2
Table 2 shows the properties of the two
Ventmeter tests with viscous Newtonian fluids
polybutenes and the various test pres
sures, and Figures 5a and 5b show the
I :{Ui~~ ~ ;~iiiici
-i
pressure vs time results obtained. The
50/50 mix
mass of fluid ejected in each run was
Fluid tested
Polybutene PIB-1
PIB-1 and
also measured and used (along with the
PIB-2
density of the polybutene under test) to
Initial pressure (psi gauge) 1800
800
400
1800
calculate z~V The ratio of AP to t~V was
Weight of polybutene
found to be approximately constant, in
discharged
1.2
0.6
0.31
1.2
accord with Equation 1 (see Table 2).
(volume x density = gms)
Converting the pressure values to
Ratio initial pressure (abs)
logarithmic form and plotting them
to volume discharged
1360
1210
1170
1360
against time, as suggested by
(psi/gm)
Equation 6, gave good linear relation
Slope of best fit straight
ships in all runs. An example is given in
-1.097 -0.949 -0.812
-0.246
line (Eqn 6 and Fig 6)
Figure 6 and the values of the slopes
Viscosity calculated from
of the best-fit straight lines in each run
35,380 36,600 41,380 175,000
slope (cSt)
are listed in Table 2. The linear regres
Viscosity measured by
sion correlation coefficient, r2, was of the
gravity viscometry and
order of 0.99 in all cases (e.g. 0.987 in
35,600
190,000
extrapolated to test
the example plotted in Figure 6).
.—
~ ~
temperature of 22°C (cSt)
~
—35
—
NLGI SPOKESMAN, MAY/JUNE 2014
~
NLG~
Because of difficulty in handling, the PIB-1/PIB-2 mix
was tested at only one pressure. The agreement of the
viscosity values found for
the straight PIB-1 and the
PIB-1/PIB-2 mix with the
2000
relevant viscosity values calp
culated from data provided
by the supplier (Table 2) was
also quite good, being within
1400
the relevant 95% confidence
ranges and giving some
1200
credence to the flow model
1000
assumed.
3.2 Tests with a
non~Newtonian fluid
A ventmeter test was run
on an NLGI#2 lithium grease
at three different starting
pressures (1820, 820 and
400 psi). The test results
are plotted in Figure 7.
Note that the run times
are of the order of several
thousand seconds, com
pared with the 30 seconds
of the standard test.
Figure 8 shows the test
data for the 1 82Opsi run
plotted in the form sug
gested by Equation 13, with
n set equal to 0.3. There
appear to be two distinct
behaviours or periods within
the test run
1. The relatively “linear”
period up to about 500
seconds (8.3 minutes)
2. The asymptotic period
from 500 to about 3,000
seconds (50 minutes)
There are actually two more
periods. One is the period
beyond 3,000 seconds where the pressure stabilizes
at a finite asymptotic value (1 Oopsi, Figure 7) and the
Pressure vs Time (PIB-1)
–
“‘4~’1800P5I
“41”” 800P51
400PSI
800
600
400
200
0
0
1
2
Figure 5a
3
5
4
6
Time, sec
Pressure vs Time PB95012400
–
2000
1800
800PSI
~ 1600
1400
1200
1000
800
600
400
200
0
0
5
15
10
Time, 5
Figure 5b
—36—
VOLUME 78, NUMBER 2
20
NLGfl
other is the first 1 .3 seconds or so of the run, where
stress balanced the driving pressure. If the ventmeter
pressure at 30 seconds had been used as is the case
flow appears to be becoming established, overcoming
in the standard ventmeter test then a much higher
the grease’s yield stress which dominates at rest. This
value would have been calculated for the yield stress,
period is shown in Figure 9, which is simply the first
higher by a factor of about three.
portion of the graph in Figure 8, plotted on a much finer
scale (note the values on the
two time axes).
1800 psi test run. PB mix
Concentrating on the
period from 1 .5 to 500 sac8
onds, the best least-squares
P
fit of Equation 13 was found
whena=0.219(n=0.28,
Equation 14), The associated
~
fit to the data is shown in
4
====Ln Pressure
Figure 10 and is good, with
Linear (Ln Pressure)
r2 = 0.995. However, the
2
—
–
—
relation of the least squares
1
minimum to the value of a,
and therefore n, is reason0
10
15
20
ably broad and the best-fit
0
5
Time (s)
n could potentially lie in the
range 0.25 0.45. Such a
Figure 6
value indicates that
the grease shear thins
Pressure Dron vs Time. NLGI #2 Grease
substantially in flow.
2000
The grease also has
1900
a readily-observable
1800
yield stress. The
1700
ventmeter pressure
pU 1600
1500
does not drop to zero
~ 1400
but instead asympi~3~
totes to 100 psi at
1200
about 3,000 seconds
1100
and remains there
1000
900
(Figures 7 and 8).
800
From Equation 18, this
700
J
residual pressure value
600
corresponds to a
500
yield stress of 135 Pa.
400
However, note that
it took about 3,000
100
seconds (50 minutes)
0
1000
1500
2000
for the grease studied
0
500
to reach this final equi
Time, sec
Figure 7 Extended ventmeter tests
libnum where the yield
—
—
I
J
—
-~
—37 —
NLGI SPOKESMAN, MAY/JUNE 2014
=‘=—1800PSI
800 PS I
400PSI
NLG~
The same “four-flow-periods”
(P(t)/P(0))(~a) 700
type of behaviour was found in
the other two ventmeter runs,
600
Plotted with n=0.3_F
at the 820 and 400psi starting
500
pressures.
The results obtained from all
400
three runs obtained by analysing
300
the periods from the initial estab
lishment of flow, which occurred
200
in less than 1 .5 seconds in all
100
cases, up to the break signifying
the start of the asymptotic
0
period are summarized in
0
500
1000
1500
2000
Table 3. The best-fit values found
Figure 8— 1820 psi test
for n in the three runs fell in the
range 0.28 0.38 and the “linear”
(P(t)/P(0))(l/al
fits were good in all cases
26
r2 values of 0.995 or higher. This
Plotted with n=0.3
range of n value is comparable
21
with some values in the literature
(e.g. Ref 4) but the reality is that
16
many values have been found.
It is noteworthy that the authors
11
of Ref 1 assumed the value of
n to be 0.3.
6
However, the accuracy of the
values of n obtained in the work
0
2
4
6
8
reported here was not as high as
there was not a distinct minimum
Figure 9 Early part of Figure 8
in the residual sum of squares
when n was varied over a 15%
(P(t)/P(0))(l/al 700
range around the best-fit values.
600
Best-fit n=0.28
It was fairly broad. However, in
all cases, n was definitely sig
500
nificantly lower than 1 and so
400
the grease was certainly nonNewtonian, with shear thinning
300
characteristics.
200
The yield stress found was the
same in all three runs and was
100
135 Pa.
0
If the theoretical analysis
0
300
100
200
holds then the values obtained
for K and n should be the same
Figure 10— 1 82Opsi test
-~
–
–
2500
3000
3500
Time (s)
—
—
1~-
10
12
Time (s)
14
—
F
–
–
-r
400
Time (s)
—~—4~,q—
—38—
VOLUME 78, NUMBER 2
500
~tw_
NLGD
for all three test runs. Table 3 shows that there is a
reasonable agreement. Using these n and K values in
Equation 11 allows the apparent viscosity of the grease
at different strain rates to be calculated. The results,
at a strain rate of 20 sec-1, are included in Table 3
and the full relationships are plotted in Figure 11. The
agreement between the three runs is again considered
reasonable for this non-Newtonian property.
4. Possible flow mechanisms
and some complications
• ;fI1I~
Fluid tested
Initial pressure (psi gauge)
Weight of grease discharged (gms)
Ratio initial pressure (abs) to volume
discharged
Value of n (Eqns 13 and 14) that
gives best fit straight line
Slope f3 of best fit straight line
(Eqns 13, 15 and Fig 10)
r2 of linear fit
Calculated value of K (Eqn 15)
Calculated apparent viscosity
(Poise) at strain rate of 20 s~1
Yield stress calculated from final
constant pressure, Pa
4.1 Tubing expansion/contraction
Both the radius and the length of the
tubing will increase in response to applied
internal pressure. The two effects tend
to counteract each other a little. An
increase in length will cause the radius
to contract and an increase in
I 3000W
radius will cause a contraction
~
in the length. This situation
2500
occurs in all cylindrical
pressure vessels with closed
2000
ends and, provided the wall
thickness is small compared
~ 1500
with the diameter, the change
$.
1000
of internal volume is given
by (Ref 5)
500
~u PR3L (5
–
4v)/(2bE)
. .
(19)
~
~
~11T;T~~ :11Th!c1
NLGI #2 lithium grease
1820
820
400
1.1
0.5
0.25
1655
1640
1600
0.33
0.38
0.28
1.603
0.0857
0.0188
0.995
7521
0.998
6862
0.997
8148
870
1071
1095
135
135
135
~
~_w~
P = 182OPSi
~—P =820PS1
P = 400PSI
01
500
where v is the Poisson’s ratio
and F is the Young’s modulus
—
Table 3
Ventmeter tests with non-Newtonian fluid
The pressure (1 800psi) that a standard
ventmeter test is run at is relatively high.
When subjected to this level of pressure,
expansion of the capillary tube and
physical compression of the test fluid
are potentially significant. This is the
source of driving force assumed in Ref 2.
Also, the test starts with the entire mass
of grease at the test pressure. The
starting situation does not correspond
to a linear one-dimensional driving force.
=
of the material that the tubing is constructed from and
b is its wall thickness.
For P=1 800 psi and assuming E=30 x 1 Q6 psi, v
0.25 (typical values for stainless steel) and b = 0.06cm,
the volume change of the capillary tubing would be
about 0.13 cm3. This equates to about 10% of the
actual volume change (volume of fluid discharged) of
approximately 1 .3 cm3 found in the 1 800psi tests.
500
500
Rate of strain
Figure 11 —NLGi #2 grease
—39 —
NLGI SPOKE5MAN, MAY/JUNE 2014
500
(~~1)
500
500
NLGÜ
It can be noted from Equation 19 that, for the tube
expansion/contraction case, dV/dP (or dP/dV) remains a
constant. This is the assumption made in the theoretical
analysis (Equation 1) and so this mechanism complies
in this important respect. However, in the tube contrac
tion case, the volume change is along the entire tube
length. It is a squeezing action, not a force applied just
at the upstream end. Two dimensional flow should be
considered. However, as noted in Ref 2, the flow would
be predominantly one dimensional and so the analysis
here may still apply to a reasonable approximation.
4.2 Sensitivity to tube radius
In the Newtonian case the viscosity calculation involves
raising the tube radius to the power of 4 while for
the Non-Newtonian case it is raised to the power of
(3 + 1/n) corresponding to a power of about 6 for
the value found for n in the grease studied. This means
that the calculated viscosity is extremely sensitive to
the value assumed for the tube radius.
The tube is curved and, as explained above, may
change slightly in radius in the course of a test run.
Viscosity values obtained by means of the ventmeter
will always be subject to significant uncertainty from
this source but calibration with Newtonian fluids of
known viscosities could reduce it.
In fluid compression it is again the case that dP/dV is
constant and so the analysis here may generally apply.
However, again, the flow will be two-dimensional, with
the pressure driving force distributed through the fluid
body, not just applied at the upstream end.
4.4 Trapped air
Air may be trapped in the ventmeter with the fluid
sample. This air will be compressed to the initial test
pressure and, once the outlet valve is opened, the
air will expand and expel some fluid from the tube.
Assuming the trapped air has volume V and that the
ideal gas laws under isothermal conditions apply then,
for a small change in initial pressure of AP and a
corresponding volume change of zW,
P(0) V(0)
–
=
=
(P(0)
+
AP) (V(0)
P(0) V(0) +t~P V(0)
+
+
AV)
AV P(0)
+
LPAV
The last term is very small and can be ignored.
Therefore, after rearrangement
AP/zW
=
(20)
-P(0)/V(0)
which is a constant in a given situation. So for this sce
nario it is also the case that dP/dV is constant.
4.3 Test fluid contraction/expansion
5. Conclusions
A typical compressibility factor for an oil or grease is
of the order of 0.45% per 1 000psi (Ref 6). Thus, at
1800 psi, the volume change (contraction) will be about
0.8%. For the fluid contained in the capillary tubing,
this equates to about 0.175 cm3 or 13% of the actual
volume change found.
On release of the pressure, the volume changes that
are associated with tube contraction and fluid expan
sion will be additive and so, in total, they may represent
up to nearly a quarter of the volume discharge that
was observed, Also, in the case of the fluid compres
sion/expansion, this will apply to all of the fluid in the
ventmeter, not just the portion contained in the capillary
tubing. The volume change from this source could be
larger than indicated.
The answer to the question “What does or could the
Lincoln Ventmeter measure?” appears to be:
a) If run on a grease in the standard way the test gives
a “pseudo” yield stress which, while relevant to the
design of centralized lubrication systems, is not
necessarily the true yield stress. The standard test
period of 30 seconds is not necessarily long enough
to reach a true equilibrium of forces.
b) If run on a relatively viscous Newtonian fluid the test
could measure the viscosity of the fluid.
c) For a relatively viscous shear-thinning non-Newtonian
fluid with a yield stress such as many greases
after allowing for a short period of flow establishment,
the early part of an extended test could measure the
—40—
VOLUME 78, NUMBER 2
—
—
NLGI
apparent viscosity of the fluid under an evolving continuum of medium to high strain rates, while the latter
part could measure the yield stress of the fluid.
u
i~
=
A) (P2
—
r2)
..
.(A2)
The volumetric flow rate 0 is given by
While the ventmeter cannot be viewed as a precision
test instrument for either the apparent viscosity or for
the non-Newtonian flow parameters of such a fluid, in
fairness, it was never designed to be.
R
Q=f 2~rudr
AP(t)/(8 ~ A)
=
~
.. .
.(A3)
All of the above are the well known Hagen-Poiseuille
relationships for a Newtonian fluid in laminar flow.
As the fluid flows out of the tube the volume that
remains decreases. The volume change is related to
the fluid’s flow rate as follows
References
.
1. He C., Conley R, Lincoln Ventmeter Reading could
be used to estimate Apparent Viscosity NLGI
Spokesman, 73, 6, Nov/Dec 2009
.
Q ~)
2. Rotter L.C., Wegmann J., The Lincoln Ventmeter and
its possibilities, NLGI Spokesman, November 1965
dV(t)/dt
3. Balan C. Compiler, The Rheology of Lubricating
Grease, ELGI May 2000
Substituting from Equations 1 and (A3) in (A4),
—
(A4)
—
dP(t)/P(t)
4. Flemming W., Sander J., Courtney S., A Rheological
Study: Do all #2 greases act the same?, NLGI
Spokesman, 76, 1, March/April 2012
= -~t
R~/(8 r~ L) k dt
. . .
.(A5)
Integrating gives
Ln P(t)
5. Collins J.A. et al, Mechanical Design of Machine
Elements and Machines, John Wiley and Sons 2009
=
-~tR4/(8
L) k t
i~
+
Ln P(0)
. . .
.(A6)
where P(t) is the ventmeter pressure at time t and P(0)
6. Oberg E. et al, Machinery’s Handbook 29th Edition,
Jan 2012
~5
the initial pressure.
Non-Newtonian fluid
Appendix
Repeating the analysis above, but using Equation
For one dimensional flow in a tube
10 rather than 5, leads to the result (compare with
Equation A5)
S= -du/dr
(Al)
dP(t)/(P(t))1’~’
where u is the longitudinal velocity at the radial position
r. All other symbols are defined in the main text.
= -~it
R3(R/2KL)1’~’ n/(3n+1) k dt
which, when integrated, rearranged and raised to the
power n/(n+1), gives eventually
Newtonian fluid
P(t)/P(0)
Equating, for the fully-developed steady-state flow
case, Equations 2 and 4 in the main text, for an
element of fluid at r, and substituting from Equation 5
and (Al), gives
where
du
a
=
AP(t)/(2
y~
=
(1
+
or (P(t)/P(0))~
A) P ~P
Integrating, assuming no wall slip, hence u=0 at r=0, gives
=
3 t)~
=
1
+
(3 t
n/(n+1)
~= ut
R3(R/2KL)1’~’ (1 -n)/(3n-1) k (P(0))(1~
—41—
NLGI SPOKESMAN, MAY/JUNE 2014