49
CHAPTER 4. THEORY OF SCANNING CAPILLARY-TUBE RHEOMETER
Chapter 4 presents the theory of scanning capillary-tube rheometer (SCTR).
Mathematical procedures for both viscosity and yield-stress measurements were
demonstrated in detail using power-law, Casson, and Herschel-Bulkley (H-B) models.
Section 4.1 provides a brief introduction to the SCTR. In section 4.1.1, the
description of a U-shaped tube set is reported. In addition, this section shows how the
dimensions of the disposable tube set were determined. Section 4.1.2 demonstrates
the equations for the energy balance in the disposable tube set.
Section 4.2 provides the mathematical details of data reduction for both
viscosity and yield-stress measurements. Sections 4.2.1, 4.2.2, and 4.2.3 deal with
the mathematical modeling in the data reduction by using the power-law, Casson, and
H-B models, respectively. Especially, in sections 4.2.2 and 4.2.3, the yield stress as
well as the viscosity of blood was considered in the data reduction.
4.1 Scanning Capillary-Tube Rheometer (SCTR)
One of the drawbacks of using conventional capillary viscometers is that one
needs to change the pressure in the reservoir tank in order to measure the viscosity at
a different shear rate. Viscosity can only be measured at one shear rate at a time in
the conventional system. Similarly, in other types of viscometers such as rotating
viscometers and falling object viscometers, the rotating speed has to be changed or
50
the density of the falling object has to be changed in order to vary shear rate as
mentioned in Chapter 3. Such operations can make viscosity measurements time
consuming and labor intensive. Because of the time required to measure viscosity
over a range of shear rates, it is necessary to add anticoagulants to blood to prevent
clotting during viscosity measurements with these conventional viscometers. The
present study introduces an innovative concept of a new capillary tube rheometer that
is capable of measuring yield stress and viscosity of whole blood continuously over a
wide range of shear rates without adding any anticoagulants.
4.1.1 U-Shaped Tube Set
Figure 4-1 shows a schematic diagram of a U-shaped tube set, which consists
of two riser tubes, a capillary tube, and a stopcock. The inside diameter of the riser
tubes in the present study is 3.2 mm. The inside diameter and length of the capillary-
tube are 0.797 and 100 mm, respectively. The small diameter of the capillary tube,
compared with that of the riser tubes, was chosen to ensure that the pressure drops at
the riser tubes and connecting fittings were negligibly small compared to the pressure
drop at the capillary tube [Kim et al., 2000a, 2000b, and 2002].
Furthermore, the inside diameter of the capillary tube was chosen to minimize
the wall effect which is often known as Fahraeus-Lindqvist effect [Fahraeus and
Lingqvist, 1931]. The details of the wall effect will be discussed in Chapter 5. In the
present study, the wall effect was found to be negligibly small.
51
The length of the capillary tube (i.e., cL = 100 mm) in the U-shaped tube set
was selected to ensure that the end effects would be negligible [Kim et al., 2000a,
2000b, and 2002]. The end effects at the capillary tube will be also reported in
Chapter 5. In addition, the capillary-tube dimensions in the SCTR were selected to
complete one measurement within 2-3 min, a condition that is desirable when
measuring the viscosity of unadulterated whole blood in a clinical environment.
Figure 4-2 shows sketches of the fluid levels in the U-shaped tube set as time
goes on. The fluid level in the right-side riser tube decreases whereas that in the left-
side riser tube increases. As time goes to infinity, the two fluid levels never become
equal due to the surface tension and yield stress effects as shown in Fig. 4-2(c) (i.e.,
∞=∆ th > 0). While a test fluid travels through the capillary tube between riser tubes 1
and 2, the pressure drop caused by the friction at the capillary tube can be obtained by
measuring the fluid levels at riser tubes 1 and 2. In Fig. 4-3, a typical fluid-level
variation measured by the SCTR is shown. Points (a), (b), and (c) represent the three
moments indicated in Fig. 4-2 (i.e., at 0=t , t > 0, and ∞=t , respectively).
4.1.2 Energy Balance
Figure 4-4 shows the liquid-solid interface condition for each fluid column of
a U-shaped tube. A falling column (right side) always has a fully wet surface
condition, while a rising column (left side) has an almost perfectly dry surface
condition at the liquid-solid interface during the entire test. Therefore, the surface
52
tension at the right side was consistently greater than that at the left side since the
surface tension of a liquid is strongly dependent on the wetting condition of the tube
at the liquid-solid interface [Jacobs, 1966; Mardles, 1969; Kim et al., 2002]. The
height difference caused by the surface tension at the two riser tubes was one order of
magnitude greater than the experimental resolution desired for accurate viscosity
measurements. Thus, it is extremely important to take into account the effect of the
surface tension on the viscosity measurement using the disposable tube set.
The mathematical model of the flow analysis began with the equation of the
conservation of energy in the form of pressure unit, where the surface-tension effect
was considered between the two top points of the fluid columns at the riser tubes (see
Fig. 4-4). Assuming that the surface tension for the liquid-solid interface at each riser
tube remains constant during the test, one may write the governing equations as [Bird
et al., 1987; Munson et al., 1998]:
dstVhgPghVPghVP
s
stc ∫ ∂∂
+∆+∆+++=++ ∞=2
1
22
2212
11 21
21 ρρρρρρ , (4-1)
where
1P and 2P = static pressures at two top points
ρ = density of fluid
g = gravitational acceleration
1V and 2V = flow velocities at two riser tubes
1h and 2h = fluid levels at two riser tubes
)(tPc∆ = pressure drop across capillary tube
∞=∆ th = additional height difference
53
V = flow velocity
t = time
s = distance measured along streamline from some arbitrary initial point.
In Eq. (4-1), the energy emitted from LEDs was ignored since the energy transferred
from the LEDs, which can affect the temperature of a test fluid, was negligible small.
In order to ensure that the amount of the heat emitted from the LEDs is very small,
the temperature of bovine blood was measured during a room-temperature test. The
results showed no changes in temperature during the test, indicating that the energy
emitted from LEDs might be negligibly small.
For the convenience of data-reduction procedure, the unsteady term in Eq. (4-
1), dstVs
s∫ ∂∂2
1
ρ , may be ignored under the assumption of a quasi-steady state. In order
to make the assumption, one should make sure that the pressure drop due to the
unsteady effect is very small compared with that due to the friction estimated from
the steady Poiseuille flow in a capillary tube.
The unsteady term can be broken into three integrations that represent the
pressure drops due to the unsteady flow along the streamlines at riser tube 1, capillary
tube, and riser tube 2 as [Munson et al., 1998]:
++=
∂∂
∫ ∫∫∫′
′
′
′
1
1
2
2
2
1
2
1
s
s
s
srs
s
crs
sds
dtVdds
dtVd
dsdtVdds
tV ρρ , (4-2)
where rV and cV are mean flow velocities at riser and capillary tubes, respectively.
Since the term of tV∂∂ is independent of streamlines, one can simplify the equation as:
54
( )dtVd
lldtVd
LldtVd
LdtVd
ldtVd
dstV rc
cr
ccrs
s 2121
2
1
++=
++=
∂∂
∫ ρρρρ , (4-3)
where 1l and 2l are lengths of the liquid columns whereas cL is the length of the
capillary tube as shown in Fig. 4-5. Using the mass conservation, rrcc VRVR ⋅=⋅ 22 ππ ,
the pressure drop due to the unsteady effect can be reduced as:
dtVdll
RRLds
tVP r
c
rc
s
sunsteady
++
=
∂∂
=∆ ∫ 21
2
2
1
ρρ , (4-4)
where
unsteadyP∆ = pressure drop due to the unsteady flow
rR and cR = radii of riser and capillary tubes, respectively.
In the present experimental set up, 1l , 2l , and cL are measured to be
approximately 12, 4, and 10 cm, respectively. Since )(1 th and )(2 th are strongly
dependent on each other by the conservation of mass for incompressible fluids, rV
must be equal to dt
tdh )(1 and dt
tdh )(2 . In order to calculate the term of dtVd r from the
experimental values, one could use the following central differential method:
[ ] [ ]2
2222
111 )()(2)()()(2)(t
tththttht
tththtthdtVd r
∆∆−+−∆+
=∆
∆−+−∆+= . (4-5)
For the comparison of unsteadyP∆ with cP∆ , unsteadyP∆ was estimated through a curve-
fitting process. In order to obtain a smooth curve from raw data, the following
exponential equation was used.
2
⋅−= −btr ea
dtVd
Error . (4-6)
55
Two constants, a and b , were obtained through a curve-fitting process, a least-
square method, which minimized the sum of error for all experimental data points
obtained in each test.
Typical results showed that the magnitude of the pressure drop due to the
unsteady flow, unsteadyP∆ , was always less than 1% of that of pressure drop at capillary
tube, cP∆ , over the entire shear-rate range. This confirms that the assumption of a
quasi-steady state could be used for the present data procedure. The details of
experimental results will be discussed in Chapter 5.
Assuming a quasi-steady flow behavior, one may rewrite Eq. (4-1) as follows
[Bird et al., 1987; Munson et al., 1998]:
∞=∆+∆+++=++ tc hgtPtghVPtghVP ρρρρρ )()(21)(
21
22
2212
11 . (4-7)
Since atmPPP == 21 and 21 VV = , Eq. (3-7) can be reduced as:
[ ]∞=∆−−=∆ tc hththgtP )()()( 21ρ . (4-8)
Note that h∆ at ∞=t contains a height difference due to the surface tension, sth∆ ,
and an additional height difference due to the yield stress, yh∆ , for the case of blood
(i.e., see Fig. 4-3). The next section addresses the mathematical procedure of
handling the yield stress.
56
Fig. 4-1. Schematic diagram of a U-shaped tube set.
3.2 mm
0.797 mm
100 mm
Riser tubes
Capillary tubeStopcock
Open to air
57
(a) at 0=t (b) at 0>t (c) at ∞=t
Fig. 4-2. Fluid-level variation in a U-shaped tube set during a test.
Riser tube 2 Riser tube 1
∞=∆ th
58
Fig. 4-3. Typical fluid-level variation measured by a SCTR. (a) at 0=t , (b) at 0>t , and (c) at ∞=t .
Hei
ght
Time(a) (b) (c)
)(1 th
)(2 th
∞=∆ th
59
Fig. 4-4. Liquid-solid interface conditions for fluid columns of a U-shaped tube set.
2
2' 1'
1
Dry surface condition
Wet surface condition
1l
cL2l
•
•
60
4.2 Mathematical Procedure for Data Reduction
In Chapter 2, we discussed the non-Newtonian characteristics of whole blood.
This section deals with non-Newtonian constitutive models for blood and their
applications to the SCTR. Since blood has both shear-thinning (pseudo-plastic) and
yield stress characteristics, three different constitutive models were used for the
viscosity and/or yield-stress measurements of blood in this study. Power-law model
was chosen to demonstrate the shear-thinning behavior of blood. Casson and
Herschel-Bulkley (H-B) models were selected to measure both shear thinning
viscosity and yield stress of blood.
For the purpose of clinical applications, disposable tube sets can be used for
the viscosity and yield-stress measurements of blood. Since the disposable tube sets
have different surface conditions at riser tube 1 and 2 during the test, one needs to
mathematically handle surface tension and yield stress effects in order to measure the
viscosity and yield stress of blood using Casson or H-B model. The details of
mathematical method of isolating those two effects are shown in this section.
4.2.1 Power-law Model
It is well known that power-law model does not have the capability to handle
yield stress. As provided in Chapter 2, the relation among shear stress, shear rate, and
viscosity in power-law fluids may be written as follows:
61
nmγτ &= , (4-9)
1−= nmγη & . (4-10)
Since n < 1 for pseudo-plastics, the viscosity function decreases as the shear rate
increases. This type of behavior is characteristic of high polymers, polymer solutions,
and many suspensions including whole blood.
We consider the fluid element in the capillary tube at time t as is shown in
Fig. 4-5. The Hagen-Poiseuille flow may be used to derive the following relationship
for the pressure drop at the capillary tube as a function of capillary tube geometry,
fluid viscosity, and flow rate [Fung, 1990; Munson et al., 1998]:
dtdh
RRL
RQL
RL
RL
rlP
c
rc
c
c
c
wcw
c
cc 4
2
4
88222 µπµγµ
ττ =====∆&
, (4-11)
where
r = radial distance
l = length of fluid element
τ and wτ = shear stress and wall shear stress, respectively
3
4
cw R
Qπ
γ =& = wall shear rate
µ = Newtonian apparent viscosity
tubecapillary oflength =cL
dtdhR
dtdhR
dtdhRQ rr ⋅=⋅=⋅= 22212
r πππ = volumetric flow rate.
The above relationship is valid for Newtonian fluids whose viscosities are
independent of shear rate. For non-Newtonian fluids, the viscosities vary with shear
62
rate. However, the Hagen-Poiseuille flow within the capillary tube still holds for a
quasi-steady laminar flow. When applying a non-Newtonian power-law model to
whole blood, the pressure drop at the capillary tube can be described as follows
[Middleman, 1968; Bird et al., 1987; Fung, 1990]:
n
c
r
c
c
n
cc
c
c
nwc
c
wcwc
dtdh
RR
nn
RmL
RQ
nn
RmL
RmL
RL
P
⋅
⋅
+
=
+
===∆
3
2
3
132
13222π
γγη &&
, (4-12)
where
wη = power-law apparent viscosity
3
13 c
w RQ
nn
πγ
+
=& .
It is of note that if 1=n , Eq. (4-12) yields to Eq.(4-11). Applying Eqs. (4-8), (4-11),
and (4-12), one can rewrite the energy conservation equation as follows:
{ }dtdh
RRL
hththgc
rct 4
2
218
)()(µ
ρ =∆−− ∞= for Newtonian fluids, (4-13)
{ } 132)()( 3
2
21
n
c
r
c
ct dt
dhRR
nn
RmL
hththg
⋅
⋅
+
=∆−− ∞=ρ
for power-law fluids. (4-14)
For convenience, one may define a new function, ∞=∆−−= thththt )()()( 21θ so that
Eqs. (4-13) and (4-14) become as follows:
dtd αθθ
−= for Newtonian fluids, (4-15)
63
dtd
n
1 βθ
θ−= for power-law fluids, (4-16)
where
dtdh
dtdh
dtdh
dtd 221 2−=−=θ
2
4
4 rc
c
RLgRµρ
α =
⋅
+
=
3
2
1
213
2
c
r
n
c
c
RR
nn
mLgRρ
β .
The above equations are the first-order linear differential equations. Since α and β
are constants, these equations can be integrated as follows:
)0()( tet αθθ −= for Newtonian fluids, (4-17)
1)0()(11 −−
−
−=nn
nn
tn
nt βθθ for power-law fluids, (4-18)
where ∞=∆−−= thhh )0()0()0( 21θ : initial condition.
Equation (4-18) can be used for curve fitting of the experimental data (i.e.,
)(1 th and )(2 th ) to determine ∞=∆ th , the power-law index, n , and the consistency
index, m . A least-square method was used for the curve fitting. The data reduction
procedure adopted is as follows:
1. Conduct a test and acquire all data, )(1 th and )(2 th .
2. Guess values for m , n , and ∞=∆ th .
3. Calculate the following error values for all data points:
64
{ } { }[ ]2 )()( valuelTheoreticavaluealExperiment ttError θθ −= . (4-19)
4. Sum the error values for all data points.
5. Iterate to determine the values of m , n , and ∞=∆ th that minimize the sum of
error.
6. Let the computer determine whether a test fluid is Newtonian or not.
7. Calculate shear rate and viscosity for all data points as follows:
)(22
tL
gRP
LR
c
cc
c
cw θ
µρ
µγ =∆=& for Newtonian fluids, (4-20)
n
c
cn
cc
cw t
mLgR
PmLR
11
)(22
=
∆= θ
ργ& for power-law fluids. (4-21)
When n becomes 1 (± 0.001), µ is equal to m , whereas when 0< n <1, the viscosity
is calculated from Eq. (4-10).
In order to obtain the velocity profile at the capillary tube, which changes with
time, using a power-law model, Eq. (4-21) can be used to derive it. Since drdV
−=γ& ,
the velocity profile can be expressed as follows:
n
cc
tPmLr
drrtdV
1
)(2
),(
∆−= ,
Crn
nmL
tPdrr
mLtP
rtV nnn
c
cnn
c
c +⋅
+⋅
∆−=⋅
∆−=
+
∫1
11
1
12)(
2)(
),( , (4-22)
where C is a constant. Using no-slip condition on the capillary wall, 0),( =cRtV ,
the constant can be obtained as:
65
nn
c
n
c
c Rn
nmL
tPC
11
12)( +
⋅
+⋅
∆= . (4-23)
Finally, the velocity profile within the capillary tube can be expressed as follows:
−⋅
∆−−⋅
+=
−⋅
∆⋅
+=
++∞=
++
nn
nn
c
n
c
t
nn
nn
c
n
c
cc
rRmL
hththn
n
rRmL
tPn
nrtV
111
21
111
2)()(
1
2)(
1),(
(4-24)
where [ ]∞=∆−−=∆ tc hththgtP )()()( 21ρ . Note that if power-law index becomes zero,
1=n , then the above equation yields to the equation for the Newtonian velocity
profile as:
( )22
4)(
),( rRL
tPrtV c
c
cc −⋅
∆=
µ. (4-25)
In order to determine the mean flow velocity at the riser tube, one has to find
the flow rate at the capillary tube first. The flow rate can be obtained by integrating
the velocity profile over the cross-sectional area of the capillary tube as follows:
[ ] nn
c
n
c
t
nn
c
n
c
c
R
c
RmL
hththgnn
RmL
tPnn
rdrrtVtQ c
131
21
131
0
2)()(
13
2)(
13
),(2)(
+∞=
+
∆−−⋅
+=
∆⋅
+=
= ∫
ρπ
π
π
(4-26)
Since )()( 2 tVRtQ rrπ= , the mean flow velocity at the riser tube can be determined by
the following equation:
66
[ ]2
131
21
2
131
2)()(
13
2)(
13)(
r
nn
cn
c
t
r
nn
cn
c
cr
RR
mLhththg
nn
RR
mLtP
nntV
+
∞=
+
∆−−⋅
+=
∆⋅
+=
ρ
(4-27)
where rR is the radius of the riser tube.
4.2.2 Casson Model
The Casson model can handle both yield stress and shear-thinning
characteristics of blood, and can be described as follows [Barbee and Cokelet, 1971;
Benis et al., 1971; Reinhart et al., 1990]:
γττ &ky += when yττ ≥ , (4-28)
0=γ& when yττ ≤ , (4-29)
where
τ and γ& = shear stress and shear rate, respectively
yτ = a constant that is interpreted as the yield stress
k = a Casson model constant.
Wall shear stress and yield stress can be defined as follows:
c
ccw L
RtP2
)( ⋅∆=τ , (4-30)
c
ycy L
trtP2
)()( ⋅∆=τ , (4-31)
67
where yr is a radial location below which the velocity profile is uniform as shown in
Fig. 4-6, i.e., plug flow, due to the yield stress. Now, for the Casson model, Eq. (4-8)
becomes [ ]stc hththgtP ∆−−=∆ )()()( 21ρ , indicating that the effect of the surface
tension is isolated from the pressure drop across the capillary tube. Using Eqs. (4-28)
and (4-29), one can obtain the expressions of shear rate and velocity profile at the
capillary tube as follows:
2
2)()(
2)(1
⋅∆−
⋅∆=−=
c
yc
c
cc
LtrtP
LrtP
kdrdV
γ& , (4-32)
−+−−−
∆⋅= ))((2))((
38)(
41),( 2
32
32
122 rRtrrRtrrRL
tPk
rtV cycycc
cc
for cy Rrtr ≤≤)( , (4-33)
))(31())((
)(41)( 3 trRtrR
LtP
ktV ycyc
c
cc +−
∆⋅= for rtry ≥)( . (4-34)
For the purpose of simplicity, one may define two new parameters,
cRrrC =)( and
c
yy R
trtC
)()( = , so that Eqs. (4-33) and (4-34) become as follows:
−+
−
−
−⋅
∆⋅=
cc
y
cc
y
cc
c
cc R
rRr
Rr
Rr
RrR
LtP
krtV 121
381
)(41),(
23
21
22
( )
−+
−−−
∆= )(1)(2)(1)(
38)(1
4)( 2
321
22
rCtCrCtCrCkL
tPRyy
c
cc
for cy Rrtr ≤≤)( , (4-35)
68
( )
( )
+⋅−
∆=
+⋅
−
∆=
)(311)(1
4)(
3111
4)(
)(
32
33
tCtCkL
tPR
Rr
RRr
RkL
tPtV
yyc
cc
c
yc
c
yc
c
cc
for rtry ≥)( , (4-36)
where )()(
)()(
)(tP
PtR
trtC
w
y
c
yy ∆
∞∆===
ττ
.
In order to determine the mean flow velocity at the riser tube, one has to find
the flow rate at the capillary tube first. The flow rate can be obtained by integrating
the velocity profile over the cross-sectional area of the capillary tube as follows:
−+−
∆=
⋅−⋅+⋅−
∆=
∆⋅−⋅+
∆⋅−
∆=
⋅=
−
∫
421
4
4214
34
21
214
0
211
34
7161
8
)(211)(
34)(
7161
8
]))(
()2
(211)
2(
34
))(
()2
(7
16))(
[(8
2)(
yyyc
cc
w
y
w
y
w
y
c
cc
c
c
c
y
c
y
c
c
c
y
c
cc
R
c
CCCkL
PR
kLPR
LtP
RR
LtP
RLtP
kR
drrVtQ c
π
ττ
ττ
ττπ
ττ
τπ
π
(4-37)
Since )()( 2 tVRtQ rrπ= , the mean flow velocity at the riser tube can be determined by
the following equation:
69
]211
34
7161[
8
]))(
()2
(211)
2(
34
))(
()2
(7
16))(
[(8
)(
421
2
4
34
21
21
2
4
yyycr
cc
c
c
c
y
c
y
c
c
c
y
c
c
r
cr
CCCLkRPR
LtP
RR
LtP
RLtP
kRR
tV
−+−∆
=
∆⋅−⋅+
∆⋅−
∆=
−ττ
τ
(4-38)
where rR is the radius of the riser tube.
For the purpose of simplicity, Eq. (4-38) can be rewritten to clearly display
the unknowns and the observed variables as:
]))()((211
34
)))()(((7
16))()([(8
)(
321
4
21
21212
4
−∆−−∆−∆+
∆−−∆−∆−−=
styy
stystcr
cr
hththhh
hththhhththLkRgR
tVρ
(4-39)
where st
yy hthth
htP
PtC∆−−
∆=
∆∞∆
=)()()(
)()(21
. Note that Eq. (4-39) contains two
independent variables, i.e., )(1 th and )(2 th , and one dependent variable, i.e., )(tVr .
There are three unknown parameters to be determined through the curve fitting in Eq.
(4-39), namely sth∆ , k , and yh∆ . sth∆ is h∆ due to the surface tension, k is the
Casson constant, and yh∆ is h∆ due to the yield stress.
Once the equation for the mean flow velocity, )(tVr , was derived, one could
determined the unknown parameters using the experimental values of )(1 th and )(2 th .
A least-square method was used for the curve fitting. For the Casson model, there
were three unknown values, which were k , sth∆ , and yτ . Note that the unknown
values were assumed to be constant for the curve-fitting method. Since )(1 th and
70
)(2 th are strongly dependent on each other by the conservation of mass for
incompressible fluids, dt
tdh )(1 must be equal to dt
tdh )(2− . Therefore, it was more
convenient and accurate to use the difference between the velocities at the two riser
tubes, i.e., dt
ththd ))()(( 21 − , than to use dt
tdh )(1 and dt
tdh )(2 directly. In order to get
the difference between the two velocities from the experimental values, one could use
the central differential method as follows:
( ) [ ] [ ]t
tthtthtthtthdt
ththd∆
∆−−∆−−∆+−∆+=
−2
)()()()()()( 212121 . (4-40)
Using Eq. (4-39), the derivative of the velocity difference can be determined
theoretically as follows:
( )
]))()((211
34
)))()(((7
16))()([(4
)(2)()(
321
4
21
21212
4
21
−∆−−∆−∆+
∆−−∆−∆−−=
=−
styy
stystcr
c
r
hththhh
hththhhththLkRgR
tVdt
ththd
ρ
(4-41)
where )(tVr is the mean flow velocity at the riser tube.
In order to execute the curve-fitting procedure, one needs to have a
mathematical equation of rV for the Casson model. Eq. (4-40) and (4-41) were used
for the curve fitting of the experimental data to determine the unknown constants, i.e.,
k , sth∆ , and yh∆ . Note that Eq. (4-41) could be applicable for both Casson-model
71
fluids and Newtonian fluids regardless of the existence of the yield stress. The data
reduction procedure adopted is as follows:
1. Conduct a test and acquire all data, )(1 th and )(2 th .
2. Guess values for the unknowns, k , sth∆ , and yh∆ .
3. Calculate the following error values for all data points.
{ } { }[ ]2 )(2)(2 valueslTheoreticavaluesalExperiment tVtVError −= (4-42)
4. Sum the error values for all data points.
5. Iterate to determine the unknowns that minimize the sum of the error.
6. Calculate wall shear rate and viscosity for all data points as follows:
( ) 2
21 )()(2
)( ystc
cw hhthth
kLgR
t ∆−∆−−=ρ
γ& , (4-43)
[ ]cw
stcw Lt
hththgRt
)( 2)()(
)( 21
γρ
η&
∆−−= . (4-44)
Note that when yh∆ becomes approximately zero (i.e., ≤ resolution of 5103.8 −× ), the
non-Newtonian viscosity, η , is reduced to k , a Newtonian viscosity. Furthermore,
the relation between wall viscosity and shear-rate can be obtained from Eqs. (4-43)
and (4-44) as follow:
)(
2
)(2
)(
4
)()(
tL
hgRk
tL
hgR
k
t
k
tkt
w
c
yc
w
c
yc
w
y
w
yw
γ
ρ
γ
ρ
γ
τ
γτ
η
&&
&&
∆
+
∆
+=
++=
(4-45)
where k and yh∆ are the fluid properties to be determined using the Casson model.
72
Yield stress could be also determined through the curve-fitting method from
the experimental data of )(1 th and )(2 th by using the Casson model. Since the
pressure drop across the capillary tube, )(tPc∆ , could be determined using Eq. (4-8),
)(∞∆ cP represents the effect of the yield stress on the pressure drop. The relationship
between the yield stress, yτ , and )(∞∆ cP can be written by the following equation:
c
cy
c
ccy L
RhgL
RP22
)( ⋅∆=
⋅∞∆=
ρτ . (4-46)
Therefore, once yh∆ is obtained using a curve-fitting method, the yield stress can be
automatically determined.
4.2.3 Herschel-Bulkley (H-B) Model
For a Herschel-Bulkley (H-B) model, the shear stress at the capillary tube can
be described as follows [Tanner, 1985; Ferguson and Kemblowski, 1991; Macosko,
1994]:
ynm τγτ += & when yττ ≥ , (4-47)
0=γ& when yττ ≤ , (4-48)
where
τ and γ& = shear stress and shear rate, respectively
yτ = a constant that is interpreted as yield stress
m and n = model constants.
73
Since the H-B model reduces to the power-law model when a fluid does not have a
yield stress, the H-B model is more general than the power-law model.
For the H-B model, wall shear stress and yield stress can also be defined as
follows:
c
ccw L
RtP2
)( ⋅∆=τ , (4-49)
c
cy
c
cc
c
ycy L
RhgL
RPL
trtP22
)(2
)()( ⋅∆=
⋅∞∆=
⋅∆=
ρτ , (4-50)
where yr is a radial location below which the velocity profile is uniform due to the
yield stress (see Fig. 4-7). Using Eqs. (4-47)-(4-50), one can obtain the expressions
of shear-rate outside of the core region as:
( ) ny
n
c
cc rrmLP
drdV 1
1
2−
∆=−=γ& for cy Rrtr ≤≤)( . (4-51)
The velocity profile outside of core region can be obtained by integrating Eq. (4-51)
as:
( ) ( )
−−−
∆⋅
+=
++n
n
yn
n
yc
n
c
cc trrtrR
mLtP
nnrtV
111
)()(2
)(1
),(
for cy Rrtr ≤≤)( . (4-52)
Since the velocity profile inside of the core region is a function of time, t , only, the
profile can be obtained using a boundary condition, )(),( tVrtV cc = at yrr = .
( ) nn
yc
n
c
cc trR
mLtP
nntV
11
)(2
)(1
)(+
−
∆⋅
+= for rtry ≥)( . (4-53)
74
Again, for the purpose of simplicity, one may define two new parameters, cR
rrC =)(
and c
yy R
trtC
)()( = , so that Eqs. (4-52) and (4-53) become as follows:
( ) ( )
−−−
∆⋅
+=
−−
−
∆⋅
+=
+++
+++
nn
yn
n
y
n
c
cnc
nn
c
y
c
nn
c
yn
c
cnc
c
tCrCtCmL
tPRn
n
Rtr
Rr
Rtr
mLtPR
nnrtV
111
1
1111
)()()(12
)(1
)()(1
2)(
1),(
for cy Rrtr ≤≤)( , (4-54)
( ) nn
y
n
c
cnc
nn
c
yn
c
cnc
c
tCmL
tPRn
n
Rtr
mLtPR
nntV
11
1
111
)(12
)(1
)(1
2)(
1)(
++
++
−⋅
∆⋅
+=
−⋅
∆⋅
+=
for rtry ≥)( . (4-55)
In order to determine the mean flow velocity at the riser tube, one has to find
the flow rate at the capillary tube first. The flow rate can be obtained by integrating
the velocity profile over the cross-sectional area of the capillary tube as follows:
( ) ( ) ( )
( ) ( ) ]13
212
2
[12
2)(
1312
1212
1
0
nn
ycnn
ycy
nn
ycycn
n
ycy
n
c
c
R
c
rRnnrRr
nn
rRrRrRrn
nmLP
drrVtQ c
++
++
−
+−−
+−
−⋅++−⋅
+
∆=
⋅= ∫
π
π
(4-56)
75
( ) ( ) ( )
( ) ( ) ]113
2112
2
111[21
]113
2112
2
111[12
1312
1212
113
1313
1213
1213
12131
nn
ynn
yy
nn
yyn
n
yy
n
c
cnn
c
nn
c
ynn
c
nn
c
y
c
ynn
c
nn
c
y
c
ynn
c
nn
c
y
c
ynn
c
n
c
c
CnnCC
nn
CCCCmLP
nnR
Rr
nnR
Rr
Rr
nnR
Rr
Rr
RRr
Rr
Rn
nmLP
++
+++
++
++
++
++
−
+−−
+−
−⋅++−
∆⋅
+=
−
+−
−
+−
−⋅
++
−
⋅
+
∆=
π
π
where st
y
w
y
c
yy hthth
htP
PtR
trtC
∆−−
∆=
∆∞∆
===)()()(
)()(
)()(
21ττ
.
Since )()( 2 tVRtQ rrπ= , the mean flow velocity at the riser tube can be
determined by the following equation:
( ) ( ) ( )
( ) ( ) ]113
2112
2
111[21
)(
1312
1212
1
2
13
nn
ynn
yy
nn
yyn
n
yy
n
c
c
r
nn
cr
CnnCC
nn
CCCCmLP
nn
RR
tV
++
+++
−
+−−
+−
−⋅++−
∆⋅
+=
(4-57)
Equation (4-57) can be rewritten to clearly display the unknowns and the observed
variables as follows:
76
[ ]
])()(
113
2
)()(1
)()(122
)()(1
)()(1
)()(1
)()( [
2)()(
1)(
13
21
12
2121
12
2121
1
21
2
21
1
212
13
nn
st
y
nn
st
y
st
y
nn
st
y
st
y
nn
st
y
st
y
n
c
st
r
nn
cr
hththh
nn
hththh
hththh
nn
hththh
hththh
hththh
hththh
mLhththg
nn
RR
tV
+
+
+
+
+
∆−−
∆−⋅
+−
∆−−
∆−⋅
∆−−
∆⋅
+−
∆−−
∆−⋅
∆−−
∆++
∆−−
∆−⋅
∆−−
∆×
∆−−⋅
+=
ρ
(4-58)
Note that Eq. (4-58) contains two independent variables, i.e., )(1 th and )(2 th , and
one dependent variable, i.e., )(tVr . There are four unknown parameters to be
determined through the curve fitting in Eq. (4-58), namely m , n , sth∆ , and yh∆ .
Once the equation for the mean flow velocity, )(tVr , was derived, one could
determined the unknown parameters using the experimental values of )(1 th and )(2 th
by using the same curve-fitting method of determining unknowns as in the case of the
Casson model. In the case of the H-B model, there were four unknown values, which
were m , n , sth∆ , and yh∆ . Note that the unknown values were assumed to be
constant for the curve-fitting method.
Using Eq. (4-58), the derivative of the velocity difference can be determined
theoretically as follows:
77
( )
[ ]
])()(
113
2
)()(1
)()(122
)()(1
)()(1
)()(1
)()( [
2)()(
12
)(2)()(
13
21
12
2121
12
2121
1
21
2
21
1
212
13
21
nn
st
y
nn
st
y
st
y
nn
st
y
st
y
nn
st
y
st
y
n
c
st
r
nn
c
r
hththh
nn
hththh
hththh
nn
hththh
hththh
hththh
hththh
mLhththg
nn
RR
tVdt
ththd
+
+
+
+
+
∆−−
∆−⋅
+−
∆−−
∆−⋅
∆−−
∆⋅
+−
∆−−
∆−⋅
∆−−
∆++
∆−−
∆−⋅
∆−−
∆×
∆−−⋅
+=
=−
ρ
(4-59)
where )(tVr is the mean flow velocity at the riser tube. In order to execute the curve-
fitting procedure, one needs to have a mathematical equation of rV for the H-B model.
Eq. (4-58) and (4-59) were used for the curve fitting of the experimental data to
determine the unknown constants, i.e., n , m , sth∆ , and yh∆ . Note that Eq. (4-59)
could be applicable for H-B fluids, Shear-thinning fluids, and Newtonian fluids
regardless of the existence of the yield stress.
After iterations for the determination of the unknowns that minimize the sum
of the error, wall shear rate and viscosity for all data points can be calculated as
follows:
( )[ ] nyst
n
c
cw hhthth
mLgR
t1
21
1
)()(2
)( ∆−∆−−⋅
=
ργ& , (4-60)
78
[ ]cw
stcw Lt
hththgRt
)( 2)()(
)( 21
γρ
η&
∆−−= . (4-61)
Note that when yh∆ becomes approximately zero (i.e., ≤ resolution of 5103.8 −× ), the
H-B model is reduced to power-law model. In addition, when n becomes 1, the
mathematical form of the H-B model yields to Bingham plastic [Tanner, 1985], which
can be described as follows:
yBm τγτ += & when yττ ≥ , (4-62)
0=γ& when yττ ≤ , (4-63)
where
τ and γ& = shear stress and shear-rate, respectively
yτ = a constant that is interpreted as the yield stress
Bm = a model constant that is interpreted as the plastic viscosity.
Similar to the Casson model, the relationship between wall viscosity and
shear-rate using the H-B can be expressed as follows:
)(2
)(
)()()(
1
1
tL
hgR
tm
ttmt
w
c
yc
nw
w
ynww
γ
ρ
γ
γτ
γη
&&
&&
∆
+=
+=
−
−
(4-64)
where m , n , and yh∆ are the fluid properties to be determined using the H-B model.
79
(a) Motion of a cylindrical fluid element within a capillary tube.
(b) Free-body diagram of a cylinder of fluid.
Fig. 4-5. Fluid element in a capillary tube at time t .
Capillary tube
r cR
l
Flow direction
l
rlπτ 2
2rPπ 2)( rPP π∆−
80
Fig. 4-6. Velocity profile of plug flow of blood in a capillary tube.
Capillary tube
yr cR