chapter 4. theory of scanning capillary-tube …

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

.




−+−

∆=

⋅−⋅+⋅−

∆=

∆⋅−⋅+

∆⋅−

∆=

⋅=

−

∫

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)




( ) ( ) ( )

( ) ( ) ]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

chapter 4. theory of scanning capillary-tube …

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