3mm RHEOLQGY: .
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SATEQN CONCEY‘KTRMEON ANS ACTIVE WM '

Twas ft)? the flame of Ph. D; »
mom-3m STA‘EE wvmsn‘v’
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This is to certify that the

thesis entitled

BLOOD RHEOLOGY:
PROPOSED MODELS AND THE EFFECTS OF
CATION CONCENTRATION AND ACTIVE HYPEREMIA

presented by

JOSEPH G . MASIN

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in CHEMICAL ENGINEERING

/L:c[rf:c’1“~ ////((9L .Ct(fl—‘,\ ,7

Major professor D ANDERSON

Date August 10, 1971

0-169

LIBRARY

Michigan State
University

 

 

ABSTRACT

BLOOD RHEOLOGY:
PROPOSED MODELS AND THE EFFECTS OF
CATION CONCENTRATION AND ACTIVE HYPEREMIA

By

Joseph G. Masin

Two rheological models were investigated for their ability
to describe the flow of blood, and the effects of both cation concen-
tration and active hyperemia on blood viscosity were studied.

A couple stress model was prOposed for blood by Valanis and
Sun, but their model lacks a yield stress. This rheological feature
has been demonstrated by many investigators in blood and must be
included in any model which is to describe blood flow. A yield stress
term was added to the couple stress model of Valanis and Sun and the
mathematical results were found to be physically impossible. Therefore,
their concept of a couple stress does not describe the rheology of
blood.

Turbulence theory was applied to the flow of blood and it was
concluded that momentum transport by the eddy-like rotation of red
blood cells (RBC's) was negligible when compared to momentum transport
by the laminar flow of a suspension of RBC's in plasma. At low shear
rates, there were more particle-particle interactions than preposed by
a simple yield stress fluid.

The best available rheological model for blood is the Casson
model. It was used to investigate the effects on viscosity and yield

stress of altered cation concentrations and of active hyperemia.

Joseph G. Masin

The concentration of the principal cations (sodium, potassium,
calcium, magnesium, and hydrogen) in canine blood was altered by 1:7
dilution with various isotonic salt solutions differing in their ionic
makeup. The resulting ionic concentrations in blood represented high,
normal, and low physiological values. Total tonicity remained constant.
Low pH (7.02) blood had a 0.019 dyne/crn2 higher yield stress than
control blood (pH 7.43) with a significance of less than 0.01. Control
blood was a paired aliquot of blood diluted 1:7 with dialysate solution
to maintain tonicity, other ionic concentrations, and hematocrit at the
same level as the test sample. Blood viscosity increased 15.2% at a
potassium ion concentration of 13.2 meq/1 when compared with control
blood (4.2 meq K+/1). This change is significant at the 0.05 level.
All other conditions tested produced no significant changes in either
viscosity or yield stress. Venous outflow from a canine gracilis under-
going active hyperemia was not different in viscosity from the venous
outflow prior to stimulation (6V, 6 sec'l, .06 msec). However, the
yield stress was an average of 0.028 dyne/cm2 higher, which change is

significant at the 0.01 level.

BLOOD RHEOLOGY:
PROPOSED MODELS AND THE EFFECTS OF

CATION CONCENTRATION AND ACTIVE HYPEREMIA

By ;/
‘9

Joseph Gerasin

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemical Engineering

1971

To my parents

and

my wife

ACKNOWLEDGMENTS

The guidance and encouragement of Dr. Donald K. Anderson
were sincerely appreciated throughout the course of this study.

The patience of the author's wife, Carol, and her aid in the
preparation of this manuscript were deeply appreciated.

The author is grateful for financial support received during
his studies to the National Science Foundation and to the Department of
Chemical Engineering and the Division of Engineering Research at
Michigan State University.

The craftmanship and labors of the Mechanical Sh0p of the
Division of Engineering Research were appreciated.

Photographic work was done by William D. Hamilton.

ii

TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS. . . . . . . . . . . . . . . . . . . . . . . . ii
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . v
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . vi
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1
BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Ionic Effects. . . . . . . . . . . . . . . . . . . . . . .

Previous Experimental Procedures . . . . . . . . . . . .
Fluid Modeling of Blood. . . . . . . . . . . . . . . . .

(DOC-p

FLUID MODELS . .... . . . . . . . . . . . . . . . . .

Fluid with Couple and Yield Stresses . . . . . . . . . . . 14
Fluid with Particulate Turbulence. . . . . . . . . . . . . 22

EXPERIMENTAL . . . . . . . . . . . . . . . . . . . . . .

Apparatus. . . . . . . . . . . . . . . . . . . . . . . . . 27
Equipment Calibration. . . . . . . . . . . . . . . . . . . 34
Procedure. . . . . . . . . . . . . . . . . . . . . . . . . 40

RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . .

Cation Effects . . . . . . . . . . . . . . . . . . . . . . 46
pH Effects . . . . . . . . . . . . . . . . . . . . . . . . 51
Active Hyperemia . . . . . . . . . . . . . . . . . . . . . 53

SWY. C O I O O O O O O O O O O O O O O O O O O O O O O O O 56
RECOMMENDATIONS. . . . . . . . . . . . . . . . . . . . . . . . 58
APPENDICES O O O O O O O O O O O O O O 0 O O O O O O O O O O O 59

A. Mathematical Derivations
l. Poiseuille Flow of a Casson Fluid . . . . . . . . . . 59
2. Poiseuille Flow of a Fluid with Couple and Yield
Stresses. . . 61
3. Viscosity of a Suspension of RBC's. . . . . . . . . . 73
4. The Fluid with Particulate Turbulence . . . . . . . . 75

iii

Page
APPENDICES (Cont'd.)
B. Computer Programs. . . . . . . . . . . . . . . . . . . 83
C. Experimental Results . . . . . . . . . . . . . . . . . 96
D. Notation . . . . . . . . . . . . . . . . . . . . . . . 105

BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . 109

iv

Figure

10
11
12
13
14

15

LIST OF FIGURES

Determinants of Local Blood Flow. .

Casson Equation for Blood Flow. .

Velocity Profiles for the Fluid with Couple and
Yield Stresses .

Excess Shear Needed for Couple Stress Model . . .

Comparison of Casson and Turbulence Models. . . .

Flow-Pressure Apparatus Diagram .
Flow—Pressure Apparatus Used. . . .

Apparatus Detail 0 O O C C . O O C 0

Some Precision Bore Capillaries Used.

Pressure Transducer Response. . . .
Syringe Calibration . . . . . . .

Cation Effect on Casson Viscosity .
Cation Effect on Yield Stress . . .

Forces Acting on a Fluid Element. .

Fluctuations Due to A Sphere Rotating

in A Shear
Field

Page

11

17
20
24
28
30
31
32
36
38
48
50

62

76

Table

LIST OF TABLES

Main Components of Normal Plasma.

Flow Rates Available. . . . . .

Tube Calibration Results. . . .

Results of Ionic Additives Experiments.

Results of Active Hyperemia Experiments

Summary of Treatment Effects. .

vi

0

Page

39
41
97
102

104

INTRODUCTION

The human body, its parts, and its functions have
increasingly become objects for engineering study. It is only natural
that the engineer, as a human being, should be concerned with trying
to eliminate the suffering and aging that befall other humans.
However, it has only been fairly recently that many engineers have
seen that they can make contributions which are as useful as those
made by the traditional healers. Several avenues of attack on health
problems have been used and are particularly well suited to engineering
study. One is modeling and computer studies, since the engineer is well
trained in the mathematical and physical sciences. Another is the
design of research equipment and laboratory apparatus. In this study,
both of these avenues were employed--an apparatus was designed to
measure pressure drOp as a function of flow rate for blood, and two
proposed rheological models were tested with the collected data.

The rheology of complex suspensions, such as blood, is not
well understood. Blood is a suspension of erythrocytes (red blood
cells or RBC's), leukocytes (white blood cells), and platelets in a
solution called plasma. The components of normal plasma are shown
in Table 1. The suspension usually consists of about forty-five
volume percent (hematocrit or Hct) red blood cells, fifty-five
percent plasma and small amounts of the other cells. Associated with
many diseased or abnormal states is an altered ability of the blood

to flow. Some cellular changes which affect viscosity are abnormal
l

TABLE 1

MAIN C(MPONENTS OF NORMAL PLASMA [14. 4. 16]

.82282

PROTEINS Total 6.0 - 8.0 gm/lOOml
ALBUMIN 3.3 - 5.4 gm/lOOml
GLOBULINS 1.6 - 3.4 gm/100ml
FIBRINOGEN 0.2 - 0.4 gm/lOOml

OTHER SUBSTANCES
LIPIDS 400 - 850 mg/lOOml
GLUCOSE 70 - 170 mg/lOOml
UREA 24 - 49 mg/lOOml
ORGANIC ACIDS 3.5 - 7.7 mg/lOOml

INORGANIC IONS
SODIUM 313-334 mg/100m1, 136-145 mEq/l
POTASSIUM 14-20 mg/100m1, 3.58-5.12 mEq/l
CALCIUM 9-11.1 mg/lOOml, 4.49-5.53 mEq/l
‘MAGNESIUM 1.8-3.0 mg/lOOml, 1.48-2.47 mEq/I
CHLORIDE 352-375 mg/lOOml, 99.3-106 mEq/l
BICARBONATE 130-170 mg/lOOml, 21.3-27.8 mEq/l

PHOSPHATE 9.0-14.0mg/100m1, 1.88-2.98 mEq/l

3

shape, size, flexibility, and concentration of cells. Alterations in
the plasma of the amounts of ions, proteins, or other substances are
also associated with viscosity changes. This study is an investigation
of the effects of abnormal concentrations of the potassium, magnesium,
calcium, and hydrogen ions on the viscosity and yield stress of whole
blood.

The organization of this study is to first consider the previous
work concerning:

1) local ionic effects upon the resistance to blood flow,

2) the equipment and procedures used to gather data on these
effects, and

3) the fluid models which aid in interpreting the data collected.

Next, the work done by this investigator will be described.
Presented are:

1) two new fluid models and their applicability to blood rheology,

2) the equipment and procedures used in this study, and

3) the effects of cation concentration and active hyperemia on

blood viscosity.

BACKGROUND

All animals above microscopic size rely on the circulation
of blood to supply necessary raw materials and to remove waste products
from each of the cells of that organism. There are many immediate
and remote determinants of blood flow and some of these are shown in
Figure 1. For almost one hundred fifty years, investigators have
tried to derive equations which quantitatively describe the dependence
of flow on each of the parameters shown. This science is now known
as hemorheology.

IONIC EFFECTS

One parameter listed in Figure l is ions. Ions can affect
viscosity by their individual presence (concentration) or by their
collective presence (tonicity). They can also affect flow geometry
via active radius alteration. The separation of in-vivo ionic effects
into viscosity and geometry components is not easy and many investi-
gators, for example, Roth [53], simply assumed any effects could be
attributed to geometry alteration. An example showing there is an
effect of ions on viscosity is the case of pH in vitro [48, 50, 69]
where increased pH causes a marked increase in viscosity.

In vivo effects are harder to interpret. Locally decreased
blood pH causes decreased resistance to flow through the mechanism of
vasodilation [31, 56] while locally increased blood pH causes

increased resistance to flow. However, non-blood perfusate yields

DETERMINANTS OF LOCAL BLOOD FLOW [30]

FIGURE

m._.<.._
V<Sm<4m

 

 

 

szSE
. C582?
mzo_luz_:o._u 20.23.53
:a
36.23 ma<zmoz< 35 333
0.29
295253
A. Go <zm
zoluaoomav m A I 32355:
Ihozmn
382$: 8.23%
3:33 25:03.
mmsmmmma 4<mazmz<me
30332 was:
.22: 33:3 :6:
3.3232: 353:: :22
T2223
3:: .2255
3:335: .282:
mmzozmo: I
23354
1.5::

6
slightly different results [9], and there are inconclusive results of
systemic pH effects [50].

The in vivo local effects of other ions have been summarized
by Haddy and Scott [31, 56]. Potassium depletion or hypokalemia causes
increased resistance to flow while mild hyperkalemia (16-32 mg/100 ml)
causes decreased resistance. Hyperkalemia above 40 mg/100 ml causes
increased resistance. Calcium depletion or hypocalcemia causes
increased resistance. Increased magnesium concentration or hyper-
magnesemia causes decreased resistance, but the results from hypo-
magnesemia are inconclusive. There appears to be no effects from an
excess or a depletion of sodium, chloride, bicarbonate, biphosphate,
or sulfate ions. The acetate and citrate ions decrease resistance,
perhaps via calcium binding.

The tonicity or osmolarity of blood also has an effect
[31, 56]. Hypotonic (diluted) blood exhibits increased resistance to
flow while hyperosmolar (concentrated) blood exhibits decreased resistance.
PREVIOUS EXPERIMENTAL PROCEDURES

To study the effects of ions and other parameters on the
resistance to flow of blood, many different procedures and equipment
designs have been used. The results and conclusions of much early work
on blood rheology have had to be modified as scientific understanding
has been applied to the original crude procedures.

Many early investigators, for example Fahraeus and Lindqvist
[2L], tried to mimic the system of interest, the body, by imposing the
same pressure drOp as the heart, 100 mm Hg, and measuring the resulting
flow through an experimental capillary. Several problems developed with

this system. Firstly, the apprOpriate pressure drOp is not 100 mm Hg

7

but that pressure corrected for kinetic energy effects [49], and end,
wall, and other effects [66]. Secondly, the meniscus movement used for
flow measurement exerts a resistance of its own [30] which is quite
considerable in small tubes and varies with direction of meniscus
movement and degree of wall soil. Thirdly, blood settling in the
apparatus may cause a varying hematocrit. Finally, the yield stress

of blood cannot be detected at the high shear rates of this design,

a factor which helped delay this important discovery.

Problems have also been discovered from the use of a
rotational viscometer for blood rheology. First, blood settles in the
cup since it is not adequately stirred. Second, blood proteins are
attracted to the surface, form a layer, and cause varying results
depending on the length of time the blood is in the apparatus. However,
this difficulty can be overcome by the use of a guard ring[18, 2S].

Third, since the centers of red blood cells physically cannot be within
l.5f((one RBC width) of the rotating bob, there is a skimming effect which
can be circumvented only through the use of a notched bob [13].

Since the systemic circulation contains roughly cylindrical
capillaries and geometry dependent effects have been noticed, and since
all objections to the capillary type apparatus can be removed by apprOpriate
equipment design and mathematical treatment of the raw data, this type
rheometer was used here. The treatment of pressure drOp corrections can
easily be accomplished by standard techniques using a calculator or computer.
The problem of meniscal resistance was avoided by changing from a system
in which flow was the measured or dependent variable and pressure was the

determined or independent variable (a pressure-flow system). Instead a

8
a constant flow rate was established by a displacement type pump and the
resulting pressure drop was measured (a flow-pressure system). Blood
settling was avoided by stirring. Finally, selection of a low flow rate
pump resulted in low pressure drOps which allowed the yield stress to be
determined.

Blood pH may be measured with any of the commercial meters.
However, there is a small hematocrit-dependent error due to RBC polarization
if bulb electrodes are used [17]. This was neglected in this study.

A major problem with experiments using blood is blood's tendency
to clot soon after removal from the body. There are many anticoagulants
available to eliminate this tendency. However some (like Red Cross acid-
citrate dextrose solution or ACD) must be added in large volumes and this
precludes study at physiological hematocrits. Studies [15, 37, 52] of other
anticoagulants have shown that they alter blood viscosity. Heparin
was used in this study because it is effective at low concentrations and
does not alter viscosity [23] .

FLUID MODELING OF BLOOD

Once data has been collected on a system of interest, fluid models
aid in interpreting the data. The first attempt to mathematically describe
data from the flow of blood in a tube was made by Poiseuille [22].
Experimental difficulties (clotting) forced a change of fluid to water
and the famous Hagen-Poiseuille equation for the flow of a Newtonian

fluid resulted.

Q = Aim/114 I 8/(L (1)
where Q is the flow rate,l§p is the pressure drOp, R is the tube radius,

[(is the fluid viscosity, and L is the length of tube considered.

9

Unfortunately, blood is not Newtonian and a few investigators
soon reported that Equation 1 did not fit blood data. However, there
was almost a century lapse before the first systematic study on blood
was done to try to explain the observed deviations (non-Newtonian behavior).
This study by Fahraeus and Lindqvist [2F] showed a geometric dependence
of viscosity in the form of a decrease in apparent viscosity as the
viscometer tube diameter decreased towards body capillary size. Divers
studies were undertaken thereafter, but, as late as 1959, Haynes and Burton
[32] complained of the lack of comprehensive work on the rheology of blood.
Even these researchers were largely empirical in their approach to the
problem.

An important feature of blood rheology is the presence of a yield
stress or non-zero intercept on a stress-strain rate recording. That is,
the stress on blood must be greater than some positive number, called the
yield stress, before there is any effect on the strain rate. This type
behavior is called Bingham plastic if the stress is linear with strain rate.
However, data for blood shows a non-linear relationship.

A constitutive equation based on the assumption of interacting
chains of particles was prOposed by Casson [lélfor India ink and was applied

to blood by Scott Blair [SE]. The equation is

=, . (2a)
7'3 T3+Szr25 77,43

3 ‘ 2b
() 3f 1“:5 7’? ( )
where quis the shear stress,’r'y is the yield stress, 82 is a coefficient

of viscosity, and 3’13 the strain rate. For flow in a tube, X- -dv/dr. '

If Equation 2a is squared,

7-72),... 32.5» +23(T'yj’)’5 (3)

10

Equation 3 is equivalent to a Bingham plastic model with an additive cross
term in the square roots. At high strain rates, the dominant term on the
right hand side is the second and behavior is seemingly Newtonian. Thus 32
may be called the limiting Newtonian viscosity for a Casson fluid.

Equation 2 has been found to describe the stress versus strain
rate behavior of blood very well in both capillary and rotational
viscometers. Current debate [11, 45] concerns whether there is a change
to Newtonian behavior at high strain rate or whether Casson's equation
is valid for all strain rates. Based on the acceptance of Casson's
equation to describe data at the strain rates used in this study, it was
used here to evaluate the effect of abnormal ion concentrations on blood
flow.

If'U = 4 Ql’l'lD3 is the average flow velocity in a tube in units
of tube diameters per time and’TQ BAPB/4L is the shear stress at the tube
wall, the Hagen-Poiseuille equation for flow of a Newtonian fluid can
be written

0 - We .4 (4)

while Bingham's model predicts

U - (rr’w/8 4,76 +1";/12 73,) lfl (5)

and Casson's model (see Appendix A) predicts

U - (4"w/8 - 27’3 4v;/7 -7’§/168q43+73/6)/32 (6)

Figure 2 shows the ability of each of these equations to fit data on the
flow of whole blood in the range investigated.
There are several minor drawbacks in applying Casson's equation

to blood flow. An extensive study by Benis[:3] has shown that model

11

FIGURE 2

 

CASSON EQUATION FOR BLOOD FLOW

to’

r

00*

,DY_NE/CM"' _
O

Tw

 

RUN 6C
0 02397.6
0 D=503.5

 

1‘;.0,|28 DYNE/CM"

5211: 0.0465 POISE

 

IO
U, SECONDS”

I00

 

12

parameters still vary Slightly with viscometer tube radius and that

not all samples fit Casson's equation well. Also, the yield stress is

a function of hematocrit [4L] and fibrinogen concentration [42, 43], and
therefore cannot be predicted without knowledge of blood composition.

One attempt to correct these drawbacks was a constitutive
equation prOposed by Valanis and Sun [63], which introduced the concept
of a couple stress for blood.

This model has a radius dependent parameter::;, which may help
explain the deviation noted by Benis. An unusual feature of the model is
that the viscosity is not hematocrit dependent [65]. Of course, the
ability of blood to flow changes extensively with hematocrit, as has been
noted and charted by Haynes and Burton [32] and Whittaker and Winton [70].
To explain this change, Valanis and Sun have allowed one parameter,;?, to
vary with hematocrit, so this feature is really no improvement over Casson's
equation. Some drawbacks of the model are that the dependence of flow rate
on wall shear rate is linear and no provision has been made for the existence
of a yield stress, both contrary to experimental data. An empirical
modification of the model, based on the shear rate dependence of the
Casson equation, removes the last two defects [38].

An equation utilizing both yield and couple stresses is developed

in the theory section and its applicability to blood flow is evaluated.

FLUID mDELS

FLUID MODELS

Many models have been prOposed for blood flow. The most
successful is the Casson equation, which describes blood rheology well
over a wide range of conditions. This model is based on the assumption
of a suspension of interacting rods. The length to diameter ratio of the
rods is assumed to be inversely prOportional to the square root of the
shear rate. Photographic studies of blood flow Show that rouleaux, or
stacked agglomerates of RBC's do exist in blood and are rodlike in
appearance. However, the amount and size of these rouleaux do not
correlate well with the prOposed shear rate dependence.

Thus, despite the availability of an accurate model, other
equations have been proposed, and will be prOposed, in order both to
explain blood rheology and to explain phenomena which are observed
microsc0pically. The two models which are considered in detail in
this section are

1. Poiseuille Flow of a Fluid with Couple and Yield Stresses and
2. The Fluid with Particulate Turbulence.

The first model investigated here utilizes the concept that a
couple stress exists in blood. Valanis and Sun prOposed this concept and
found that published velocity profiles could be fit well with it. However,
their equation involving the couple stress does not include a yield stress
term in it, and this feature has been observed by many investigators in
blood. Their model is modified here to include this feature so that both

a couple stress and a yield stress exist in the fluid model.
13

14

The other model investigated here is turbulence theory
applied to blood flow. Modifications are made to the normal turbulence
theory to fit the nature of turbulence which would be due to RBC rotation.
This allows the order of magnitude of momentum transport by this mechanism
to be calculated compared with transport by laminar shear.

POISEUILLE FLOW OF A FLUID WITH COUPLE AND YIELD STRESSES

A fluid model using a couple stress in addition to the usual
viscosity has been developed by Valanis and Sun [63]. They show in their
paper that data on the velocity profile of blood in a tube can be fit to
their equation if high shear rates are involved. Their model must fail
to describe blood at low shear rates since their shear stress is linear
and homogeneous with strain rate. A yield stress must be included in the
fluid model if it is to describe blood.

Explanation of the concept of a couple stress is needed since,
by the following reasoning, it cannot exist in a continuum [20]. If the
couple stress is confined in a volume element and the size of the volume
element is allowed to go to zero, the length of the lever arm of the
couple must go to zero. Then the existence of a finite couple requires
that the parallel forces of the couple must become infinite, which is
not tractable. However, in a fluid such as blood, which is discontinuous
even on the microscOpic scale due to the presence of red blood cells and
other particulate matter, this limiting process cannot be performed and
the concept of a couple stress is feasible. The rigidity of the formed
matter in blood causes a drag resisting the shear forces. This drag
varies over the surface of the rigid body and results in a net couple

which is finite and volume dependent, since as the volume goes to zero,

15

the drag also disappears. This intuitive picture of the couple stress
as a drag force which arises to resist shear is in accord with Stokes'
[6F] conclusion from analyzing the equations of motion that a shear
stress field is necessary in order for a couple stress to be possible.
Appendix A gives the mathematical derivation of the velocity
profile using the fluid model with couple and yield stresses. The

resulting equation is derived as A74.

- 2 -
v-2:wR [1 (r/R) +A(Io(okr/R) Io (00)]

 

-4’ R l-r/R+ 47’ (L (dr/R)-L(o()
:1?— [ “—2 ° °
-B(Io(o<r/R) - loam] for 44>qu (7a)
v = constant for ’IJ</IJY (7b)
where
A = A2 2 (1 -7l)- (7c)
(10(0<) " (1 +"() 11(4) /0<)
and

B = [Lo (4) - (1 +6>L1(o<'>/o<']/[ Io(o<) - (M)

(1 +311 coo lak].

16

It is desired to determine if this equation follows the
correct limiting process as the shear stress at the wall falls to the
yield stress. There have been several studies made, for example by
Merrill, et a1 [43, 45], which show that the yield stress as determined
by extrapolation of wall shear stress from pressure-flow data is identical
with the value of the yield stress as determined from static tests.
Therefore, flow must go to zero as the shear stress at the wall goes to the
yield stress. There are two possible ways for this to occur. Firstly,
the velocity might go to zero at all radii, as occurs when a Newtonian
fluid approaches its lower limit of shear stress = zero. Secondly, there
might exist a plug flow region which grows in size and the plug velocity
might go to zero, as occurs when a Bingham fluid approaches its lower limit
of shear stress = yield stress.

In this case, if the apprOpriate derivative is formed, from

Equation 7
dv/dr = 11 (acr/R) [(1 0i) fr’w/d +/I'/V’I‘ly(Lo (d) "

(1 +5) L1 (a) la) I2] / [s2 (100:6) - (1+7) 11 (4) /-<)] -

[fir/R + L1 (d r/R)7/"7Ԥ/2] Is2 (8)

it is clear that, in general, dv/dr is not zero at the yield radius,
ry -R my, ’I’w. Therefore, the same plug region does not exist for
this model as the Bingham model, and the regional distinction implied
by Equation 7a and b is incorrect since then the velocity gradient would
be discontinuous.

The velocity profile across the tube for some values ofo( and

for the range of permissible j'values is shown in Figure 3. An unusual

l7

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m maze;

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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83:82. > ”:35: N 3...... a8 zzozm ~53. 33:05 :33“; Ms.
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18
feature of the velocity profile is that, although the model contains
a yield stress, the velocity extreme occurs at the center of the tube.
This means that a plug flow region which exists in Bingham or Casson flow
does not exist in this model, the limiting procedure‘ using a plug region
is invalid, and that it is valid to use Equation 7a for the entire tube
cross-section.

Another important feature of Figure 3 is that negative flow
rates are mathematically predicted for some low values of wall shear.
Since negative flow rates are not physically possible for positive wall
shear rates, this feature needs further investigation. Predicted negative
flows are also a feature of Bingham or Casson flow uncorrected for the
plug flow region. Since there is no plug flow region in this model,
that cannot be used as an explanation here. The only realistic physical
description at this point is that there is actually no flow while negative
velocities are predicted and that flow begins only when positive velocities
are predicted. That is, the velocity goes to zero everywhere as the wall
shear stress approaches some lower limit, to be calculated next.

Since the velocity extreme predicted by Equation 7a occurs at
the center of the tube, the predicted shear stress at which flow begins

can be found by setting the velocity at r = 0 equal to zero. This gives
'T'w/fry =

1 - m’lZoC [Lo (00 + (1 - 10(4)) (Lo(o() - (1 “DIM-4 Was )]
10(4) - (1 +2?) 11(o()/o<

 

1 + 2 (1 Jim - mun/(«‘2 (10(4) - (1 +2?) 11(4) loo)

(9)

19
The only permissible values forcK and i are those for which the right hand
side of Equation 71 is unity, that is, the shear stress at which flow begins
equals the yield value.

Figure 4 shows the ratio’7<wfl7;’predicted from Equation 9 for
selected values och and if. Since all values are greater than unity, the
yield stress obtained from static tests will never agree with the yield
stress obtained from dynamic measurements. Therefore, this model will
never be in agreement with the data obtained from whole blood.

The model can be forced into predicting a consistant yield
stress by altering the boundary conditions stated by Valanis and Sun,
Equation A 63 and A 64. Equation A 64 was invoked because A 63 was
satisfied identically. Instead of Equation A 63, the second non-trivial
boundary condition establishes a plug flow region at radii below the

yield radius.
av/ax = 0 at x = ”Ivy/4V“, (10)

Since the flow rate is given by

Q=27TR [S xvdx+ xvdx=//[vy(7VyR/7Vw) +
0 vT§l7Jw

2112;, vxdx
’Tylflrw

as x goes to one, the second term should go to zero as the range of

(11)

integration goes to zero and the first term should go to zero as vy
approaches the velocity at the wall, which is zero. This should give
a consistant value forfir’.

Applying Equations A 62 and 11 to Equation A 61, the velocity

is given by

20

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21

v = 01.11/st (l-x2) -7’yR/32 [l-x- W/zx (Lo(o()-L0(0( x)-
(10(o() - 100410) (ma fry/72W) + 2/rf512)/Ile«7’y/7/w))]

for /;/> ’f’y (12)

It can be immediately seen that the velocity does not go to zero as
the wall shear stress approaches the yield stress and that, therefore,
this approach will also not yield an equation applicable to the flow of
whole blood. The process of the velocity everywhere going to zero cannot
be used here since a plug flow region has been forced.
Another possible approach to the model is to force the gradient
of vorticity to be zero at the centerline.
l/x d/dx(x dv/ dx) = 0 at x = O (13)
Then the velocity profile resulting from Equation A 62, A 63, and 13 is
given by
v =//_<,R/232 [1 - x2 - 4/a( 2(Io(<>(') " IO(O<‘x) ]‘
Wyn/32 [1 - x47/2oC(L0 (a) - Lo (ax))] (14)
Again for the static and dynamic yield stresses to agree, either the

velocity must go to zero for all x as 41, approaches/[J

y’ which, it can

be seen, is not the case, or the velocity gradient must go to zero at
the yield radius, r =‘7VyRI’T;, and there be plug flow below this radius.

If the derivative is formed, this is the case for
11 («fly/fin nary/7’.) = 7774047773) (15)
which is satisfied only if
(,2 ”fly/7w = 1.900 (16)
The "constant",cx', is then a function of the shear stress at

the wall and not a true constant. Therefore, this approach is also not valid.

22

No combination of realistic boundary conditions allows the
concept of a couple stress in the form prOposed by Valanis and Sun to
deseribe the flow of blood. It was necessary to change their model by
the inclusion of a yield stress in the description of the stress tensor
because many investigators have demonstrated its existence. The yield
stress must be simply additive so that the strain rate goes to zero as
the stress approaches the yield value. However, if the yield stress is
included in this form, the velocity profile then derived fails to exhibit
features which have been observed in data using blood. Therefore, it
must be concluded that these features are due to some phenomenon other
than a couple stress.
THE FLUID WITH PARTICULATE TURBULENCE

Cinematographic studies of the flow of blood in cylindrical
vessels [8,55] show that RBC's rotate and do not maintain a constant
radial position. This movement of the particulate matter in whole blood
precludes the establishment of a smooth velocity profile. Rather, the
velocity distribution can more prOperly be described in terms of an
average profile with a superimposed series of small, irregular
fluctuations. Fluctuations superimposed on a steady velocity profile
have long been noted in the turbulent flow of fluids and a branch of
rheology called turbulence theory has grown to try to describe flow
exhibiting these characteristics. The basic assumption of turbulence
theory is that momentum is transferred during flow by two mechanisms.
The first is laminar shear, the mechanism used in Newtonian flow. The
second is the action of eddies or vortices. A vortex of diameterAf

can transfer momentum by physically displacing some fluid (with a

 

III)

23
Inomentum characteristic of its radial position) a distance <?to a new
radial position (with a different characteristic momentum).

In Appendix A a model is derived which assumes that blood
acts as a continuous laminar suspension of viscosity,/(, with eddies
of rotating RBC's in it. The apprOpriateness of the assumption that
a suspension can be described as a continuum in laminar flow is also
discussed in Appendix A.

The equation which results for flow in a tube of radius R
is (A 107)

0/247’113 = U -[’F’w/8 'flvy/O +7§I244v3 + (432, (b5/80R5-b2/48R2
-b4/48R4 - b3/48R3) + ’1‘; 7‘5, (b3/24R3
+ b2/24R2 + b/24R) 17;, (b3/48R3 + b2/48R2
+ b/48R + l/48) 4.737120 raw/4+
(4": (-b6/48R6 + b5/48R5 + 154/4811.4 +b3/48R3)
+73, 7‘; (3b5/80R5 - b4/16R‘I - b3/16R3
- b2/16R2) + flvf, (b3/16R3 + b2/16R2
+ b/16R) 4’2, (b3/48R3 + b2/48R2 + b/48R

+ 1/48) +7’SI2407'W )IC/ 2,42 +....] /4 (17)
where Q is the volumetric flow rate,9’isJ(/2)o H2126 , fis the
fluid density, H is hematocrit,45 is a constant of prOportionality
ii 1, and b is the distance from the wall at which the laminar boundary
layer begins.

Figure 5 shows Equation 17 for various values of 4 and q) .
.At high values of Q}, the model is indistinguishable from a Bingham

Inodel with equivalent yield stress and viscositywparametera. At a

24

 

 

 

 

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25

given value of wall shear stress, as 9” decreases, the flow rate
decreases. The magnitude of the decrease in flow rate grows as the
value of the wall shear stress increases. At fl) = 750,000 and‘7"w =
1,000 the difference from the Bingham model is about 1%. This value
of 4/ corresponds to individual RBC's rotating and a normal hematocrit.
At 9)- 75,000 and 7”,, a 1,000, the decrease is about 8%. This value
of QV corresponds to a grouping or rouleaux of three red blood cells.
These results are for no marginal layer or b/R = 1.00. At b/R = 0.90,
the discrepancy between the two models is decreased by approximately
25%. This value of b/R corresponds to a marginal layer one RBC wide
in a tube 100‘? in diameter.

Also shown in Figure 5 for comparison is the profile
predicted by Casson's model, which is known to describe the flow of
blood with only minor discrepancies. Identical yield stress values
are used in both the Casson and turbulence models because the yield
stress in a dynamic model must agree with the value obtained from an
independent static test. This was discussed in the section on the
couple stress model. The Casson model and turbulence model agree at
wall shear stresses above 20 dynes/cm2 only if identical viscosity
coefficients are chosen for the two models. Then it is also necessary
that the deviation due to turbulence be small so that the models agree
at wall shear stresses above 300 dynes/cmz. At wall shear stresses
below 20 dynes/cmz, the turbulence model and the Bingham model agree
and both differ significantly from the Casson model.

This analysis of Figure 5 implies that momentum transport

by the rotation of RBC's is negligible. In the range of wall shear

26

rates above 30 dynes/cmz, momentum transport is indistinguishable
from the amount predicted for a suspension of non-interacting
particles. The linearity of the QTW versus U plot in this region
reflects the Newtonian nature of such a suspension.

Below 1 dyne/cm2 significant non-linearity appears in the
turbulence model. This is due to particle-particle adhesion due to
the presence of a yield stress in the fluid. However, such a simple
form of interparticle interaction does not describe the flow of blood
for the Casson equation deviates from this model at wall shear rates
below 30 dynes/cmz. The flow rate for a given wall shear stress is
always lower for the Casson model than for the model with the yield
stress only. Therefore, the flow of blood must involve more complex
interparticle interactions or other hydrodynamic features, such as
doublet or larger size rouleaux formation. The inadequacy of the
correlation of the square root of the shear rate with rouleaux size
and concentration indicates that rouleaux formation is not the only
important phenomenon. The shear rate induced orientation of RBC's
is one possibility as is flushing of the cell biconcavity at higher
shear rates. Both of these phenomena result in an altered hydro-
dynamic length-to-diameter ratio. Thus, the ability of Casson's
equation to describe blood flow may rest with its ability to describe
the combined effects of the hydrodynamics of flow around a single RBC
and rouleaux formation rather than the effect of rouleaux formation
alone.

Since it has been concluded that the Casson equation describes
blood data better than any other model, it will be used to analyze the

data collected in this study.

EXPERIMENTAL

EXPERIMENTAL

The two parameters in the Casson equation which can be
determined experimentally are yield stress and high Shear viscosity.
An experimental apparatus and procedure was utilized which allowed a
reliable extrapolation of these two parameters.
APPARATUS

The apparatus used to gather flow pressure data was designed
after the apparatus used by Merrill et. a1., [5] at Massachusetts
Institute of Technology. The system, shown in Figure 6, uses a syringe
drive to displace the experimental fluid from a sealed, constantly
stirred reservoir through a precision bore capillary into another reservoir
vented to the atmosphere. The pressure drOp resulting from this forced
flow is measured by a differential pressure transducer connected between
both reservoirs.

A syringe drive (Model 940, Harvard Apparatus Company,
Millis, Massachusetts) maintains a precise flow rate with a high torque,
synchronous motor. An all gear, variable reduction drive train allows
a series of twelve precise flow rates to be obtained. The syringes are
also interchangeable to allow more flexibility in the variety of flow
rates available. Most syringes used were Yale Multifit Tuberculin
(Becton, Dickinson, and Company, Rutherford, New Jersey, or B & D)
equipped with a Luer-Lok fitting although a 10A syringe (Hamilton
Company, Whittier, California) with Luer-Lok fitting attached with epoxy

was also used for very low flow rates.
27

28

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29

The syringe is connected to the sealed reservoir with a 14
gauge stainless steel needle, a Touhy-Borst adapter (B & D) and
stainless steel Swagelok 0 seal adapter (Crawford Fitting Company,
Solon, Ohio).

The upstream reservoir is a 3/4 inch diameter by 1% inch
cylinder machined from a two inch block of Plexiglas with a removable
Plexiglas cover sealed by a one inch 0 ring. The cover is beveled
towards the center vent to aid in the removal of bubbles. The vent is
sealed with a silicone greased brass screw. The reservoir contents
are stirred with a Teflon coated ball or rod rotated by a magnetic
stirring motor located below the reservoir (See Figure 7). The air
gap between the stirring motor and reservoir is flushed by an air
stream to prevent heating of the experimental Sample. In the block
face Opposite the syringe drive inlet is machined a one to ten or
standard taper (S) for insertion of the interchangeable precision bore
capillaries. In the other faces at the same height as the capillary
port are Swagelok fittings for the bypass and the transducer circuit.

The capillaries were made from precision bore glass tubing
(Fischer Porter Company, Warminster, Pennsylvania) and ground at both
ends to standard taper. Some are shown in Figure 9.

The downstream reservoir is a 1 inch diameter by 2% inch
cylinder machined from a two inch block of Plexiglas with a removable
Plexiglas cover sealed by a 1% inch 0 ring. The cover is vented to
the atmosphere through a two way valve (B & D). One face has a hole

with a standard taper machined into it to accept the capillary.

30

 

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33

Adjacent faces are equipped with Swagelok adapters leading to the
bypass and transducer circuit. While the sealed reservoir is attached
to the Plexiglas base of the system, the downstream reservoir is
movable in a track so various length capillaries can be fixed between
the two reservoirs.

A seetion of one-eighth inch Tygon tubing is used as a bypass
so the two reservoirs can be quickly brought to the same pressure. The
bypass is also valved to allow filling of either reservoir with a syringe.

The transducer circuit is valved to permit either of two
variables reluctance differential pressure transducers (Model KPlS,
Pace Engineering Company, North Hollywood,California) to be connected
between the reservoirs or to the calibration system or to be zeroed.
The transducer signal is amplified (Model CD 25, Pace) and recorded
(Model MR, E. H. Sargent and Company, Chicago, Illinois). The
calibration system consists of two reservoirs containing dialysate
solution connected to the valving assembly by polyethylene surgical
tubing, the end of which is immersed under the liquid level. While
the downstream reservoir is always Open to the atmosphere, the air
space over the upstream reservoir can be pressurized by a hand bulb.
The air space is also connected to one leg of a water manometer so
the pressure can be determined. The level of the manometer legs are
read with a cathetometer accurate to 0.005 cm. (Gaertner Scientific
Company, Chicago, Illinois). The transducer circuit is also valved to
allow any portion to be flushed with dialysate solution.

Hematocrits were determined after high speed

centrifugation in heparinized capillary tubes (A. H. Thomas, Philadelphia,

34
Pennsylvania) and read on a special chart (Thomas). pH was determined
on an expanded scale pH meter (Model 10, Corning Scientific Instruments,
Corning, New York).

Several changes were made in the design of Merrill and co-
workers. The entire system except the precision bore capillaries was
constructed from plexiglas, polyethylene, and stainless steel since
these materials do not induce clotting. This allowed lower levels of
anticoagulent to be used. In addition, the feed reservoir was
constantly stirred so change of hematocrit with time was minimized.

The blood was not elaborately centrifuged to ensure uniformity of
cells and thus was in a more natural state.

EQUIPMENT CALIBRATION

Transducer

The transducer was valved to the calibration system, the
positive pressure side connected to the reservoir which may be
pressurized and the negative pressure side connected to the
atmospheric reservoir. One hour was allowed for the transducer
amplifier to warm up. After this time, the manometer leg levels were
read with the cathetometer to ensure there was no pressure in the
upstream reservoir. The zero adjustment on the transducer amplifier
was then turned until the same recorder reading was obtained on both
the positive and negative polarity modes of the transducer amplifier.
An offset from the edge of the recording paper was established with the
displacement control on the recorder to facilitate this adjustment.

Then various pressure levels were established in the upstream

reservoir by infusion or bleeding through the hand pump. Corresponding

35
manometer leg levels and recorder readings were recorded. Room
temperature was monitored to ensure its constancy. The zero pressure
reading was checked for constancy at the end of pressure series and
its value subtracted from the other readings. The pressure was
determined as p = f7gh where fgis the density of water (the manometer
fluid), g is the local value of the gravitational acceleration, and h is
the difference in manometer leg levels. The almost linear correspondence
between p and the recorder reading, r, is shown in Figure 10. Of the
various simple algebraic expressions, the best correlation coefficient
is obtained with
p = ar + br2 (18)
and this curve is shown in Figure 7 with best fit values for a and b.
It was observed that the best values of a and b did not
remain constant if determined on different days, probably due to
equipment drift. However, the ratio of the small non-linear coefficient
to the linear coefficient did remain essentially constant. Therefore,
during each day's experimentation, the transducer was calibrated at one
pressure and recorder reading by
p = a(r + 0.0005462 :2) (19)
Mes
Each syringe to be calibrated was filled with clean mercury,
fitted with a small bore needle, and mounted in the syringe drive
assembly. Lines were scribed on the syringe drive screw and an adjacent
stationary bracket. Infusion was started until mercury flowed from the
needle tip and was stOpped when the scribed lines became coincident.
An empty beaker or watch glass was weighed to 0.0001 gm on a balance

(Model EA-lAP, Torsion Balance Company, Clifton, New Jersey) then

PRESSUREJfiPSI

 

36

FNGURE IO
PRESSURE TRANSDUCER RESPONSE

 

 

 

 

RECORDER SCALE,nVOLTS

 

37

placed under the syringe needle. Mercury was forced into the beaker,
in increments of screw revolutions and the increased weight of the
beaker was recorded with the corresponding number of revolutions. Room
temperature was monitored for constancy. Figure 11 shows the volume
displacement with syringe drive screw revolution. The relation was
assumed linear and the SIOpe was determined and is Shown on the figure.
Syringe Drive

The syringe drive screw rotation rate was checked by timing
one hundred rotations at the highest rate with a laboratory timer
accurate to 0.01 second. The scribe marks used for syringe calibration
were the reference points. The rotation rate was 1 revolution per
second with less than 0.1% variation whether or not the drive was
loaded with a water filled syringe emptying through a small bore needle.
The rotation rate at the next lowest rate was checked by timing fifty
revolutions of the screw. Since the rotation rate was exactly one
quarter the highest rate as anticipated from the rate table affixed
to the syringe drive, and all rates were the result of direct gear
drive, the lower rotation rates were derived by rate table ratio from
the timed high rotation rate. The result of syringe drive and
syringe calibration is shown in Table 2.
Capillary Tubes

The capillary tubes used were calibrated in the eXperimental
apparatus. After the length of the tube to be calibrated was measured
with a micrometer, it was mounted between the two reservoirs which were
then filled with water. The stirrer was usually not used so that the

recorder fluctuations due to its rotation would not be present and more

38

FIGURE II
SYRINGE CALIBRATION

 

 

 

 

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7..
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NUMBER OF SCREW DRIVE REVOLUTIONS (MARCH)

39
TABLE 2

FLOW RATES AVAILABLE

Gear Box Syringe

 

 

Setting 10 cc 2 cc 1 cc 107*-
1 10.70 3.96 1.11 .0119
2 4.28 1.585 .444 .00478
3 2.14 .793 .222 .00239
4 1.07 .396 .111 .00119
5 .535 .198 .0555 .000597
6 .214 .0793 .0222 .000239
7 .107 .0396 .0111 .000119
8 .0535 .0198 .00555 .0000597
9 .0214 .00793 .00222 .0000239
10 .0107 .00396 .00111 .0000119
11 .00535 .00198 .000555 .00000597
12 .00214 .000793 .000222 .00000239

Flow rates are in ml/min. The flow rates above represent the
results of calibration calculations on the particular syringes used
in this study and thus do not agree with the flow rate chart supplied

with the syringe drive.

40

precise readings could be obtained. A series of pressure-flow points
were obtained using the calibrated pressure transducer and syringe drive.
The pressure drop was corrected for entrance and kinetic energy effects
by subtracting 1.17 velocity heads according to the recommendations
of Van Wazer, et a1 [67]. Since water is Newtonian and therefore
obeys Poiseuille's Law and its viscosity is known as a function of
temperature, all that is now unknown in Equation 1 is tube radius.
This is calculated from the flow-pressure data. The results are shown
in Table 3.
Other Equipment

The pH meter was calibrated daily with pH 7.00 buffer
(Mallinckrodt Chemical, New York). In addition the span was checked
once with pH 4.01, pH 7.00, and pH 10.00 and assumed constant.

The centrifuge head was fabricated by the Machine ShOp of
the Division of Engineering Research and thus needed standardization.
This was accomplished by measuring its speed (6500 rpm) with a tachometer,
calculating the time for centrifugation [29], and running duplicate blood
samples at various hematocrits in a standard laboratory centrifuge
(from the Department of Physiology) and in the fabricated centrifuge.
Samples from twenty to seventy hematocrit agreed to within one unit.
EXPERIMENTAL PROCEDURE

For the ionic experiments, blood was drawn from dogs
maintained by the Center for Laboratory Animal Resources at Michigan
State especially as blood donors. The dogs were heparinized (5 mg/kg)
to prevent blood coagulation and the blood sample was treated with

additional heparin (35 cc 1% heparir/lOO ml blood/day) if storage for a

41

 

 

 

 

 

 

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42
prolonged period was anticipated. Further treatment of the samples is
described in the sections on the ionic effects and active hyperemia effects
of the Results and Discussion chapter of this work.

Since the blood from the donors was usually obtained in a one
liter polyethylene bottle, but the sample Size for the rheometer was
forty cubic centimeters, preparation of aliquots was the first step.

The liter bottle was gently rolled to thoroughly mix the contents while
preventing hemolysis from excessive shearing. The blood was then
immediately poured into fifty cc. polyethylene bottles. These bottles
were capped and refrigerated at 32°F. For the hyperemia experiments,
samples were directly collected in the 50 cc. bottles.

Before using one of the aliquots, it was removed from the
refrigerator and allowed to stand at room temperature for about forty-five
minutes. When it was almost at room temperature, thirty-five cc. were
poured into a glass beaker and five cc. of the desired isotonic diluent
were added to the beaker. About fifteen minutes time was allowed for
buffering and for transmembrane and thermal equilibrium to establish.

The pH meter was calibrated with buffered pH 7.00 solution
and then the blood mixture pH was measured.

A 14 gauge stainless steel needle was attached to a syringe,
the barrel of which was siliconized to prevent binding. The syringe
was filled with the blood mixture and was then mounted on the syringe
drive carriage and attached to the upstream reservoir. The syringe
drive carriage was advanced until the fittings into the reservoir were
filled. The drive was then turned on at the lowest speed so the syringe

plunger would not bind.

43

The tapered ends of desired capillary were greased and mounted
into the two reservoirs for a leakproof seal. The transducer circuit
lines were flushed with dialysate solution and excess solution was
removed from the reservoirs. The 0 rings on the reservoir covers were
lightly siliconized and the covers were mounted and sealed. Agitation
with the stirring ball and air gap cooling flow were started. The
reservoirs were then filled. All bubbles were removed from the upstream
reservoir which was then sealed with a siliconized brass bolt. The
capillary was filled with blood so that meniscal effects would not
affect the pressure drOp results.

The pressure transducer was valved to the calibration
reservoirs as in the calibration procedure for the transducer. The
no pressure reading of the transducer was read on the recorder and the
equal manometer leg heights were read by the cathetometer. The sealed
reservoir was then pressurized with a hand bulb until the recorder
reading almost went off the tap of the chart. When the pressure reading
became steady, the manometer leg levels were read by the cathetometer.
The pressurized reservoir was then vented, and it was confirmed that the
no pressure reading reestablished at the former level. Room temperature
was recorded so the manometer fluid density would be known.

The pressure transducer valving was then turned so that the
positive pressure tap connected to the upstream reservoir and the
negative pressure tap connected to the vented reservoir. A pressure
drOp reading was then obtained for the lowest flow rate. The syringe

drive was then stOpped and a no flow reading obtained.

44

Next a flow rate was selected which gave a pressure reading
of ten scale units or more. The syringe drive motor was started and
the resulting pressure drOp was recorded.

Then the flow rate was increased one gear stOp and a new
pressure reading was obtained. Progressively higher flow rates were
selected until the pressure recorder reading exceeded chart capacity
(ascending series). Then progressively lower flow rates were selected
until the flow rate selected by the criterion cited in the previous
paragraph was reached (descending series). The drive was then shut off
and a new zero flow reading was obtained. Often this was not identical
with and generally increased from the original zero flow reading. This
change was found to occur when the pressure drop exceeded approximately
25 cm. of water. Thus, ascending pressures were compared with the
original zero while descending pressures were compared with the final
zero. When ascending and descending pressure drops were not similar
at any flow rate, that flow rate was repeated and the inconsistent
pressure drop was discarded.

The upstream reservoir sealing bolt was removed and the
syringe drive carriage was advanced until the sealable vent was filled.
Then a heparinized microhematocrit capillary tube was inserted into the
reservoir until the lower end was at the same level as the precision
bore capillary, and a sample was removed from this level. Hematocrits
were determined in duplicate. Data for the sample was discarded if
hemolysis was evident in the plasma portion of the tube after

centrifugation.

45

The stirring motor and air gap flow were stOpped. Blood
was removed from the reservoirs by syringe and placed in a glass
beaker. Blood pH was again determined and compared to the original
value. Sets of data were discarded if the sample pH varied more than
0.1 pH from the control sample value.

The transducer circuit was flushed with dialysate solution.
The rest of the apparatus was cleaned with water. In addition, the
precision bore capillary was flushed with cleaning solution and then
distilled water.

Blood pH, hematocrit, days Since withdrawal, transducer
calibration readings, room temperature, and flow-pressure data were
entered on IBM cards for use by Fortran program TYNS (in Appendix B).
Statistically best values for 7} and s were obtained from a CDC 6500
computer by this program. The pressure drOp correction used was 1.25
velocity heads recording to the recommendations of Van Wazer,
et a1 [67] using the power law exponent of 1.22 derived by Hershey
and Sung [33] for 35 hematocrit blood. The statistically best values
for’ry and s from both the experimental sample and its paired aliquot
were then compared using a two-tailed Student's t-test. Significance
of treatment effect (p value) and 95% confidence limits are reported
on this basis. Because nonhomogeneous variance was noted in comparing
the viscosity parameters (i.e. residuals increased with S), results
are reported as % change from control viscosity to eliminate this

defect.

RESULTS AND DISCUSSION

CATION EFFECTS

To study the effects of various individual cations, blood
samples collected by the method described in the proceeding sections
were mixed with isotonic (309 mOsm) solutions of sodium, potassium,
magnesium and calcium chloride. A paired procedure was used whereby
one aliquot of blood diluted with a chosen ionic solution was compared
with another aliquot of the same blood diluted with dialysate solution.
Five cubic centimeters of the above solutions were combined with thirty-
five cubic centimeters aliquots of whole canine blood prior to viscosity
determination. A constant 1:7 dilution was chosen so that hematocrit
variation between samples was minimized and so that the hematocrit
level was near normal. Isotonic solutions were used so that tonicity
effects were eliminated as a source of error.

This procedure markedly increases the concentration of the
cation studied, but to a lesser degree, the concentrations of the other
ions are also affected. For example, dilution with isotonic calcium
chloride raises the calcium ion level from 5.0 meq/l to 17.3 meq/l.
However, the concentration of the other cations are reduced. Magnesium
falls from 2.0 meq/l to 1.8 meq/l, sodium level is reduced from 140
meq/l to 122 meq/l, potassium from 4.35 meq/l to 3.81 meq/l. Chloride
concentration is increased from 102.5 meq/l to 109.0 meq/l. Bicarbonate
decreases from 24.6 meq/1 to 21.5, phosphate from 1.2 meq/l to 1.1 meq/l.
Changes resulting from this treatment may be ascribed to the calcium
ion change alone if the Casson viscosity and yield stress do not vary
significantly with the concentration of the other ions. It will be

seen that this is the case.

46

47

This dilution treatment yields only one ion concentration
higher than the control value. The low ionic concentration was
achieved by using saline as diluent. Since it was desired to collect
data relevant to the different resistance results which have been
noted for mild and severe hyperkalemia, an alteration of the standard
procedure was used to produce these conditions. If isotonic solutions
of potassium chloride and sodium chloride are mixed in various ratios,
any concentration of potassium ion is possible from zero to the isotonic
value. A mixture was used which gave potassium ion concentration of
one-half the maximum. Thus, counting the standard solutions, potassium
was available as none, half, or all of the added cation concentration.
Three runs were made with potassium as one quarter of the added cation
but, due to the small sample size, the results from these runs are not
analyzed here.

Data were collected in the flow-pressure apparatus
previously described and analyzed by computer program TYNS (listed in
Appendix B) for viscosity and yield stress. The results of these
calculations are listed in Table 4 in Appendix C. A summary of the
average effects is given in Table 6. The tube diameter used most for
these runs was 200£(.

Figure 12 shows the effects of cation concentration on the
viscosity parameter of the Casson equation. This is the viscosity at
a high shear rate. One to seven dilution of blood with isotonic
potassium chloride produces blood with 22.6 meq KF/l, if it is assumed
that the undiluted blood contains the normal average of 4.4 meq KI/l.

Actual blood potassium level should not be far from the normal, since

48

FIGURE l2
CATION EFFECT ON CASSON VISCOSITY

j
a
fi

IOS

 

 

E—'_‘_:

JN=IO

 

 

951% MAGNESIUM

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

85
_l I05. 'I
O ..
‘3‘ E.
g z 95 "'5 CALCIUM ”“0
O E I
U o
u. 21 85v
0 > I30~
N S I-
«II— r I
>- z
I— - / — '
_ IZO‘
U) 3 POTASSIUM
O 5 "/ i F «P
o 9 NN=IO
_U_) 1; MO“
> 8 _ 1. J
11.: L [lNzII
as I00 U. 1
N=6
.I.
90L .1
O 5 IO IS 20 35

CATION CONCENTRATION MEQ/L
o INDICATES CONTROL CATION CONCENTRATION

49

blood was collected from dogs maintained by the Michigan State
University Center for Laboratory Animal Resources especially as blood
donors. Average blood viscosity at this potassium level is elevated
4.1% over the corresponding control, which is another aliquot diluted
with dialysate solution. The increase is not significant at the .05
level. Blood viscosity is elevated 15.2% above control at a potassium
ion concentration of 13.2 meq/l. This is significant at the 0.01 level.
Dilution of blood with saline results in a 2.1% decrease in viscosity,
which is not significant at the .05 level. This corresponds to a
potassium ion concentration of 3.9 meq/1.
Elevation of blood calcium level from 4.7 meq/1 to 17.3 meq/1
results in a 4.5% decrease in viscosity. This is not significant at
the 0.05 level. Elevation of blood magnesium from 2.0 meq/l to 14.7 meq/l
results in a 5.0% decrease which is not significant at the 0.05 level.
Figure 13 shows the effect of various cation concentrations
on the yield stress. None of the changes are statistically significant.
The lack of any significant change of viscosity or yield stress
with dilution by saline or magnesium chloride suggests that the slight
increase in chloride ion concentration or decrease in any other anion
concentration accompanying these dilutions also have no effect on
either the Casson viscosity or yield stress over the range which the
anions varied. Otherwise, any change due to anion concentration would
have to be counterbalanced exactly by the effects of each of these

cations, and this would be highly unlikely.

50

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

NE FIGURE l3
3 CATION EFFECT ON YIELD STRESS
LLI 0'04I
Z
(>3- 0.02+ _
__;~ [MAGNESIUM «r
O O f
I’ I
Z ~02
O b
O 04
2 .
O 0.04» T
‘I E CALCIUM
LL ff} 0.02-
U) 2 r 1
U) o I
LIJ _J o  
0‘ § . U
{ID—E "-°2I J

2 — .
SE 004.. [POTASSIUM
9.“ 5 I T
>‘E O.Ozi T
Z Z I‘ [IL IL
--—- o

u 0 ». a
‘83 S , J_ I +
E m -.02
I .I.
U 6 5 IO IS 20 2:5

CATION CONCENTRATION MEQ/L

O INDICATES CONTROL CATION CONCENTRATION

N

SAME AS FIGURE I2

pH EFFECTS

To study in a viscometer the effect of pH on the Casson flow
parameters, it is necessary to increase and decrease the pH of some
aliquots from the normal value. There are three possible methods to
do this. Two are aeration with varying mixtures of oxygen and carbon
dioxide and admixture with isotonic solutions of differing pH. The
results from these two procedures are complicated by the effect of pH
on hematocrit [48, 50] and the resulting effect of hematocrit on the
Casson parameters. Decreasing pH is associated with swelling of RBC's
because of the water movement accompanying bicarbonate ion diffusion,
and a resulting increased hematocrit. The third method of adjusting pH
is admixture with non-isotonic solutions of differing pH. If the
tonicity of these solutions are properly adjusted, RBC size and
therefore hematocrit will not change. Since it is not the purpose of
this study to determine the effects of altered tonicity, and since
altered hematocrit is the natural response of whole blood to pH change,
admixture with isotonic solutions is used here.

Control procedure for this series of experiments consisted of
admixture of 35 cc. of whole blood with 5 cc. of dialysate solution as
in the ionic section. The yield stress and viscosity of the collected
data are listed in Table 4 of Appendix C and are summarized in Table 6.
Average hematocrit of eleven control samples was 33.94. Average pH of
controls was 7.428 for comparison with low pH treatment and was 7.444
for comparison with high pH treatment. Average yield stress of controls
was 0.0306 dyne/cm2 for comparison with low pH treatment and was 0.0273

for comparison with high pH treatment. The tube diameter used was 400/{.

51

52

Low pH was attained by admixtures of blood with a 1:1
mixture of isotonic HCl and dialysate solution. This mixture was
used because pure isotonic HCl gave values of pH far outside natural
limits. Dilution was the same as with control so that there was no
viscosity change due to different protein levels [40]. Average blood
pH was 7.024. Average hematocrit was 37.10. Blood viscosity increased
1.41%, but this change is not significant at the .05 level. Average
yield stress increased by 0.019 dyne/cmz, and this change is
significant at the 0.01 level.

High pH was attained by admixture of blood with a 1:3
mixture of isotonic NaOH and dialysate solution. Dilution was the
same as with control. Average blood pH was 7.831. Average hematocrit
was 32.27. Average blood viscosity increased 0.53%, but this change
is not significant at the .05 level. The average yield stress fell
by 0.0042 dyne/cmz, but this change is not significant at the .05
level.

The observed increase in yield stress at decreased pH might
be attributed to pH induced swelling. However, since the viscosity
does not increase, this hypothesis must be considered suspect. Since
RBC's are positively charged particles, it is possible that a change
in pH toward making their environment more positively charged (more
H+ or lower pH) would enhance their self-attraction, and thus result

in an increased yield stress.

ACTIVE HYPEREMIA

A natural process during which ionic and pH changes occur
is exercise or active hyperemia. For example, hydrogen and potassium
ion concentrations increase during exercise [5%]. It is desired to
determine if the altered resistance to flow observed during active
hyperemia can be explained by viscosity changes due to ionic effects.

The organ used in this study was the canine gracilis muscle.
Preparation involved anesthetizing the dog with sodium pentobarbitol
(30 mg/kg) and ventilating with a mechanical respirator. The right
gracilis was surgically exposed and freed from connective tissue. All
blood vessels connected to the muscle except the major artery and vein
were ligated. Heavy occlusive cord ligatures were placed at each end
of the muscle to completely eliminate collateral flow. A short section
of the gracilis nerve was freed from investing fascia and encircled
with a loose ligature. The gracilis vein was cannulated and its outflow
diverted to a polyethylene collection bottle. Sodium heparin (5 mg/kg)
was injected intravenously to prevent coagulation.

In small dogs, the leg vein was cannulated instead of the
gracilis vein so a larger cannula could be used. Lower back pressure
results and thus the transcapillary pressure rise accompanying increased
flow was minimized.

A sample was taken under natural flow conditions. This was
used as control. Electrical stimulation was then applied to the gracilis
nerve (6-10 volts, 0.06 msec duration, 6/sec frequency). After several
minutes, venous outflow was collected as the active hyperemia sample.
Mbst samples were collected in the late morning, refrigerated, and run

within four hours. Several runs were collected from the cannula under
53

54

mineral oil to minimize gaseous exchange with the atmosphere. These
samples are designated in the Appendix. The most frequently used tube
diameter was 400/(.

During active hyperemia, blood hematocrit increased to an
average of 44.33 from the control average of 41.27. A viscosity increase
of 12% is predicted from viscosity versus hematocrit charts for this
change. Despite this predicted increase, the Casson viscosity as
determined in the experimental apparatus did not change. A statistically
insignificant decrease of 3.3% was observed. Average pH changed from
7.384 to 7.249 during active hyperemia, and the pH results from the
previous section imply no effect on viscosity. Blood potassium increases
during active hyperemia,but this also may result in an increased viscosity
depending on the potassium level attained. While it has long been
known that vasoactive agents are released during active hyperemia, the
fact that the observed lack of viscosity change cannot be explained on
the basis of altered hematocrit, pH, and ionic concentrations suggests
thatviscoactive agents may also be released, or the results may be due
to some other ionic effect not investigated.

Although the observed lack of change in viscosity cannot be
explained in terms of the variables studied, the lack of increase in
resistance to flow expected from pH and hematocrit changes is fortunate
since increased blood flow is needed to supply oxygen and to remove
metabolic by-products from exercising muscle.

An increase in the yield stress by 0.028 dyne/cm2 from an
average of 0.0374 dyne/cm2 was also Observed. This is statistically
significant at the 0.01 level. Decrease in pH is associated with a

marked increase in yield stress so most of this increase could be due

55

to a pH effect. Increase in blood potassium has no effect on yield
stress. Also, yield stress is roughly proportional to the hematocrit
cubed [44,, so an increase in Hct also implies an increased yield
stress. Another factor is pertinent here since this is an in vivo
experiment.

Siegel and Kolmen [58] have found that during metabolic
or respiratory acidosis, the fibrinogen level in plasma or lymph is
inversely proportional to pH. Thus, decreased pH should yield
elevated fibrinogen levels. Merrill, et a1., [43, 46] and Walder,
Weaver, and Evans [67] have found that yield stress is proportional
to fibrinogen concentration. Therefore, an increase in yield stress
should result from the pH induced increase in fibrinogen.

Thus, the increase in yield stress may be explained as the

sum of effects of pH, hematocrit, and fibrinogen.

SUMMARY

Two rheological models were investigated - a couple and
yield stress model and a turbulence model.

The couple stress model prOposed by Valanis and Sun fits
some blood velocity profiles well, but the model lacks a yield stress.
Since a yield stress has been demonstrated by many investigators in
blood, one must be included in any model which is to describe blood
flow. If this is done, negative velocities are predicted for some
wall shear rates above the yield value using a variety of boundary
conditions. Since this is physically impossible, the concept of a
couple stress does not truly describe blood rheology.

If turbulence theory is applied to the flow of blood, it is
concluded that momentum transport by the eddy-like rotation of RBC's
is negligible when compared to momentum transport by laminar flow of
a suspension of RBC's in plasma. At low shear rates, there are more
particle-particle interactions than prOposed by a simple yield stress
fluid.

The best available rheological model for blood flow is the
Casson model. It was used to investigate the effects of altered
cation concentration and of active hyperemia.

The concentration of the principal cations (sodium,
potassium, calcium, magnesium, and hydrogen) in canine blood was

altered by 1:7 dilution with isotonic salt solutions differing in their

56

57

ionic makeup. Blood viscosity increased 15.2% at a potassium ion
concentration of 13.2 meq/l when compared with control blood
(4.2 meq K+/l). Control blood was a paired aliquot of blood diluted
1:7 with dialysate solution to maintain tonicity, hematocrit, and
other ionic concentrations at the same level as the test sample. This
viscosity increase is significant at the 0.05 level. Low pH (7.02)
blood had a 0.019 dyne/cm2 higher yield stress than control blood
(pH 7.43) with a significance of less than 0.01. Other cation
concentration levels produced no significant changes.

Venous outflow from a canine gracilis during active
hyperemia was not different in viscosity from the venous outflow prior
to stimulation. However, the yield stress was an average of 0.028

dyne/cm2 higher, which change is significant at the 0.01 level.

RECOMMENDATIONS

To improve the accuracy and utility of the experimental
apparatus used here, several modifications might be made. Periodic
variations in the pressure drop readings have been noticed, and the
period decreases with increasing flow rates. This might be due to
imperfections or non-uniformity in the drive screw or to gear lash in
the drive train. These defects can be corrected by regrooving the
drive screw and movable carriage and by changing the drive train to
square tooth gears. The latter modification will, of course, make it
more difficult to change speeds. However, the pressure drOp readings
will then be more precise due to a more uniform flow rate supply.

There are also fluctuations in the pressure reading due to the
physical impact of the stirring ball on the reservoir walls. These
could be eliminated by allowing the ball to rotate on a pin or by
making a dishshaped floor in the reservoir so that the ball would not
climb up and come in contact with the walls. The brass seats on the
B&D valves wear out rather rapidly and the valves could be replaced
by a more durable type to eliminate this problem.

This study has found an elevated yield stress in blood of
low pH and in blood from active hyperemia. An elevated level of fibri-
nogen has been implicated as the causative mechanism. The concentration
of fibrinogen in active hyperemia blood should be determined to confirm

this hypothesis.

58

APPENDICES

APPENDIX A

APPENDIX A

MATHEMATICAL DERIVATIONS

l. Poiseuille Flow of a Casson Fluid
For steady state axial flow of a fluid in a tube, the

Navier - Stokes equation reduces to

dp/dz + l/r d/dr (r’T) = 0 (A 1)
Since the first term is a function of 2 only, it is constant. Let
dp/dz - -Ap/L where L is the length of the tube considered. If
Equation A l is multiplied by r and integrated

+ (A p/2L) r2 + r 'r .. cl (A 2)
From symmetry‘Y is zero at the centerline. Equation A 2 thus requires
that C1 be zero at r = 0. Thus

’1’ = -Ap r/2L (A 3)

Equation 2 on page 9 can be rearranged to give

1/2 1/2 1/2

s( -dv/dr) = ’7’ — 1‘}, (A 4)

Combining Equations A 3 and A 4, and squaring,

1/2
-dv/dr - [-Ap r/2L - (-Ap r21’y/L) + ’T’y] /s2 (A 5)

Integrating with respect to r

1/2 3/2 2
-v ... [-Ap r2/4L - 2/3 (-2Ap Ty/L) r +7’yr] /s + 02 (A 6)

From the lack of slip at the wall, v = O at r = R.

1/2 3/2
0 - [-Ap 18/41 - (-8Ap q’y/9L) R + 13R] II;2 + cz (A 7)

Equations A 6 and A 7 imply the velocity profile is

- 3/2 3/2
v - [(-Ap/4L)(R2-r2) “-V -8Ap ’T'y/9L (R - r ) + ’T'y(R-r)] /sz (A 3)

 

59

60

Recall that the Casson model is valid only for shear stress greater
than the yield value. Since the shear stress is linear with radius
in Poiseuille flow, Equation A 8 is valid only radii greater than

the radius, r at which the shear stress reaches the yield stress.

y,
The flow rate of a Casson fluid can be obtained by integrating

the velocity across the tube.

R
Y A 9
Q = 2‘7’ vrdr = Zifvy rdr + 217 vrdr ( )
0 O ry

where vy denotes the constant velocity at radii at or below ry.

.."2 -AP _ _-8AP 32_32 _
Q (845,. [—41.4112 :3) ”Tina / ry/ ) + ’50: ry)] (A 10)

I '“_‘ 3/2 5/2
+ fig. [:ZAf‘£(R2r-r3) . L8.§A_ITB.'_YY(R r-r ) + Ty(Rr-r2)] dr
r
y

From Equation A 4, the shear stress at the wall, 7;, is given as
-£5pD/4L. Also from Equation A 4 may be obtained the expression for

the yield radius.

If Equation A 11 is substituted into the integrated Equation A 10, like

terms in powers of T; can be collected to give

Q = 7D3[Tw/4-4\/TyTw/7-T§t /841',3, wry/3] /832 (A 12)
Expressing the average velocity in tube diameter per time,
" . . - ..«A 3 2
U $93.3 [TV/4 a 'ry'rwn Iy/am'wwyla] /2s (A 13)

is the desired expression for Poiseuille flow of a Casson fluid.

61
2. Poiseuille Flow of a Fluid with Couple and Yield Stresses

To solve the problem of Poiseuille flow Of a fluid with
both couple and yield stresses, the apprOpriate differential equations
must first be derived from the principles of continuum mechanics.

Figure 14 shows a fluid element in a velocity field and the
forces acting upon it. Forces acting to change the linear momentum
of the element are the two traction forces, pressure and shear. By
the definition of a couple as two equal forces acting in opposite
directions, but separated by a lever arm, the net effect of a couple
upon the linear momentum of the element is zero.

Forces acting to change the angular momentum of the element
are the couple stress and the shear stress. Since the pressure stress
acts only perpendicularly to the faces of the element, the net effect
of the pressure upon the angular momentum of the element is zero.

The first equation of motion can be obtained by considering
the linear momentum of the fluid element. At any point in time, t,
the linear momentum in the ith coordinate direction, P1, of all

particles in the domain, D, is

p, = jDflvidv (A 14)

where/0 is the fluid density, and vi is the instantaneous velocity
in the ith coordinate direction. If the domain is subjected to surface

tractions, T1, the resultant force, F1, on the body is

F1 =§T1ds (A 15)

where dS is a surface element of D. If the tractions are decomposed
by Cauchy's formula [24] into components

VOLUME ELEMENT IS DOMAIN D

62

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PRESSURE
m4-

 

WHERE Y IS THE POSITION VECTOR

SHEAE_._._._..
ads Y
: @dv
V < ,."‘J.\'
% I. x,
% - COUPLE
A ‘-
c———-- "’
#—
{—
SHEAR+ PRESSURE = TRACTION
Ix" Tkj Ti]
COUPLE + TORQUE : MOMENT
FROM THE CENTER OF D
Xi
xi
FIGURE I4

FORCES ACTING ON A FLUID ELEMENT

63
where nj is an orthonormal vector directed outward in the jth
coordinate direction, Equation A 15 can be transformed by the Green-

Gauss theorem [25] into the volume integral

Pi = In 6131/an dV (A 17)

Newton's Second Law states that the rate of change of
linear momentum of a body is equal to the force acting on that body.

This can be expressed in the Eulerian reference system as

DC

Combining Equations A 14, A 17, and A 18.
D - .
D D

Since Equation A 19 is true for any choice of domain, the kernels of

I

the integrals must be equal
,0 Dvi/Dt =3’T’j1/QXj (A 20)
This is the first equation of motion or Cauchy's First Law.l:7]

When considering angular momentum it is customary to use the
vorticity vector, wi, which is defined [60] as half the curl of the
velocity vector, instead of the velocity vector,

W1 = 1/2e11mW1m '- 1/4e11m(a vm/ 6x1 -aV1/3xm) (A 21)
where eilm is the cyclic permutation number [26], or the permutation
tensor, and Wij is the vorticity tensor. Considering the angular
momentum instead of the linear momentum the equivalent of Equation A 19

is

fiJIfl/Owidv- f<IYIaflj1/axj+Yx‘r'ji)dV (A22)
D n

6.4,.
where,£¥ij is the couple stress tensor and [Y' is the volume element
half-length. Again equating kernels, and dividing by IYI , [2i].

This is the second equation of motion or Cauchy's Second Law.

The equation of continuity is
2... -"0 A24
Dtjvl ( >

For simple shear of an incompressible fluid, Equation A 24 can be

simplified to

H
C)

which has the analogy

fl

dwj/dxj O (A 26)

The constitutive equations for the fluid with couple and

yield stresses are defined after Valanis and Sun [63] as

”I“ = - P513 ' 32(3V1/3 "j +3VJ/3xi)

iJ
+"\'y(1 ..Jij) for Tijatriy (A 278)
Tij - inj’ avi/axj =Ofor 7’11(’f’y (A 27b)
/c1j=4})8wj/axi+4)1'awi/axj (A28)

where irij is the symmetric part of the stress tensor, p is the static
pressure, a ij is Kronecker's delta, 82 is the coefficient of viscosity,
79y is the yield stress, andiz and)? ' are the couple stress
coefficients.

From Equation A 26, it can be seen that the left hand side

of Equation A 23 is zero. Rephrasing Equation A 23 in terms of some

65

dummy indices,

(\-
341:1 /<9Xr+ e lrk ’rk = 0, (A 29)
premultiplying by eijl and converting the cyclic permutation number

product into equivalent Kronecker's deltas [26]

eijl 9/(1’1/‘9 Xr + (fricflcj 'Jrjé'ki)’l"rk = 0 (A 30)

OI'

frll

eijla/rrI/«er = 2! ji (A 31)

where’T'A = % (GVij "Tji) is the antisymmetric part of the stress

11
tensor.
. . . . ,c A+
Combining Equations A 20, A 24, and A 31, with 0 ji = ji
8
(V
’ ji.
Av 3
<9 ’ji/3xj + 35 eijlaz ,‘(rl/axraxj = O (A 32)

Substituting Equations A 27a, b, and A 28.

2
-3P/axi - sZaZVj/ axiaxj - 32&2v1/axj +

Zeij1(’l 93.1,.) 2., as - ’2' a3wrlaxrax1

9 Xj) = 0 for Tij 2;; (A 338.)
a we xj = O for 7",“ try (A 33b)

The second and fifth terms in Equation A 33a may be eliminated by
rearranging the order of differentiation and using Equations A 25 and

A 26. Substituting the vorticity tensor, wij’ from Equation A 21

-ap/axi - 5292v1/9xi + 2 )2 eij1(35 eir3<93wrs /ax293j) = o
r

(A 34)

66

or in terms of the Kronecker's deltas

-ap/axi - szazvi/8x§ +7(erfls -
stcfrl)<93 ”rs/69X: 3 xj = o (A 35)
-ap/axi - szazvi/ax? +7(<93\Ij1/ax2<9xj -

J

63 2 . =
wlj/a xr ax] O (A 36)
Since Wij = - wji: Equation A 36 becomes
'ap/axi - szézvi/axg + 2733 {43-1/3 x3, an (A 37)

Substituting Equation A 21 to obtain the differential equation in
velocity,

_ _ 2 2 2 w 2 2- a4.

ap/axi s a vi/c9xj +)z¥‘v]z/c9xrc9xJ ? vJ/

axgaxjaxl = O (A 38)

where the fourth term may be removed by the use of Equation A 25. A
convenient way to convert Equation A 38 into the cylindrical coordinates
needed to solve the problem of Poiseuille flow is to first express it

in vector notation

-vp- #vw+nvw=o (Aws
-Vp + v2 (nvzv - 82v) = O (A 39b)

Now formally stating the conditions for Poiseuille flow,
vr=vg=0, vz=v (A 40)
ap/ar asp/as = o, ap/a. = dp/dz (A 41)

¢r9= 792 =0: Trz"l for 7r? Ty (A 42)

67

and applying this to Equation A 39 in cylindrical coordinates,
-dp/dz + l/r d/dr [r d/dr ()zV72v -szv) ] = O
for G” 2.7%y (A 43a)
dv/dr = O for Q” (Ty (A 43b)
For steady state flow, the pressure drop is independent of
axial location. Let
-dp/dz = Ap/L = k (A 44)
Integrating Equation A 43a with respect to r, followed by division
by r, and substituting Equation A 44,

'82 dv/dr + ’7; + 7? d/dr V72v = -kr/2 + cl/r

A)“

for I / I y (A 45a)

’I“?

dv/dr = O for y (A 45b)

Since the right hand side of Equation A 45a may not become infinite at
the centerline, c1 must equal zero.

d/dryvzv - szdv/dr = - kr/2 '4; for T)? (A 46a)

Y

v = constant for flx< II; (A 46b)
Equation 46a may be integrated with respect to r, and the final nabla

expanded to give

Ef- [d/dr (r dv/dr)] - 82V = -kr2/4 “'Tyr + C2 (A 47)

Let x = r/R and form the indicated derivatives to get, upon some

rearrangement,

x2 $324+ x-gl-oézxzv-flx4+3x3-orx2=0 (A48)
x x

where 0(2 I 82R2/7 , ,5 =ApR4/47 L, 7= flw’yR3/7z , and

J - 021120? .

68

Assume a series solution of the form:

v = aixi (A 49)

m a
dv/dx = E (1+1) a1“ X1 (A 50)

(i+2)(i+l) ai+2 Xi (A 51)

80

dzv/dx2 a

M,

i 0
When these Equations are substituted into Equation A 43,

OO

1'0 flx‘I +D’x3 «fo - O (A 52)
Let j a i + 2. be

alx -flx4 +D/x3 «fxz + E (jzaj -0(2aj-2) xi + 0 (A 53)
i=7-

Expanding some of the series terms to allow evaluation of like

coefficients.

alx + (4a2-O(2ao- d") x2 + (9a3-O(2a1+ a”) x3 + (l6a4-0( 2a2- ,8) x4
0Q

+ E (jzaj -o(2aj-2) xi = O (A 54)

j=s

The unknown coefficients can now be obtained:

a1 - O (A 55a)
a2 - (42% + 5V4 (A 55b)
a3 = («2:31 - 5.),32 = - W32 (A 55c)
a4 - (0(2a2+/6)/42 = [42(o(2ao+ 5w. +5] /42 (A 55d)

an .. 0(2an-2/n2 ngs (A 55e)

69

v - a0 + (aC'zao+=S') x2/4 - J‘ x3/32 +42x5/325M4x7/325272+...J

+[o(2<oc2a.+a'>/a +5] [x4/42—I-o(2x6/4262+o( 4x6/426232+. . .]

for 4‘”?y (A 56a)
v = constant for ’IV‘T’}, (A 56b)

The infinite series in Equation A 56a, b can be expressed
in terms of more familiar functions. Consider the modified Bessel

function of zero order [1] :

GD
10““) ‘ i._[_).o<x 2 2k (A 57a)
k 1 k 1
k B 0
42x2 44x4 d6x6 a< 8x8
" + —— + __ + __ + A
I 22 2242 224262 22426282 ( 571))
(744 4 x2 x4 0R2x6 Gangs
‘- _ —+"_+—+ +_+"° (A57c)
Therefore
fi+¢<2x6+¢<4x8+ = 4 » -4_2

Consider the modified Struve function of zero order [2]:

 

00
L, (.(x) E (XX/Zflkfl 2 (A 59a)
[r' (k+3/2)]

(X +d3x3 +115x5 + DC 7x7 + ... (A 591))
32 3252 325272

 

 

=2a(3(x +l3+°‘2"5+’<4x7+ > (AS9c)

70

 

Therefore
3 2 "‘
3 35 357 0< o(2

Combining Equations A 56, A 58, and A 60
v = a0 + (...(Zao +5) x2/4 -&/[7/V/20(3L0(o(x) -
x/o( 2] + [o( 2(o(2ao +5)/4 +fl]

[II/(x410 (0(x) - 4M4 - x2/o(2] (A 61)

Still undetermined are ax>constants, a0 and ci'which contains the
unknown quantity, c2. Consider the appropriate boundary conditions.

Assuming no slip at the wall,

v = 0 at x = l (A 62)
From symmetry considerations.

dv/dx = O at x = 0 (A 63)

From the vanishing of the couple stress at the wall [63] ,

 

d2v - )z'lx dv/dx = 0 at x = l (A 64)
dx2 '

The first and second derivatives of v arel:2, 47]

dv/dx - 1/2 (0(2ao+a’)x -&’ [’57le (L1(o(x) + 2/;,\-) - 1/a( 2]+
[.(2(.< 2ao+c§)/4 +/3] [RI/431161") - 2x/o(2] (A 65)
dZv/de = (0(2ao+J)/2 -3’ 67/204 Lowx) - L1(.(x)/.(x]

+ [o( 2(a(2ao+J)/4 +,6’] [MC—(Iowa) - I1 (dx)/MM 3

-2/0( 2] (A 66)

71

Applying Equation A 62 to Equation A 61,
o = a0 + (orzaowm -a*[1rL.<o<)/2.( 3-1/A2 ]

+ [m 4ao/4+.( Ulla/9] [410w )/.< 4-4/o< 4-1/o(2] (A 67)
Multiplying byGK 2 and combining terms involving a0 and cf;

Azaoromn (Iowa-1). =J[O-Lo(o(>/2o<-1] +,6’[1-4(I.(q>-1)/o(2]
(A 68)
Equation A 63 is solved identically. Letlfi;= 7Z’IIZ , and apply

Equation A 64 to Equation A 66 and A 67,
(dzao+J)/2-7/\J[Lo(o()-L1(o()/w( ] /20( + [6(4ao/4+0(2J/4+fl]
[41.(o<>/.(2-411(o<>/.<3-2/a<2] a? (0(2ao+f)/2 +r2rfL1(oo/2.<2
+[.<4ao/4+o<ZJ/4+fl] [2 770(2-4 7743 11(4)] = o (A 69)

or, combining terms in ac and dfl,

x 2.0+; . WE...“ 1-(1+4')11(.( 11¢) _ 4
20((Io(°()'(1+’i)11(0()/0() ‘7‘

,2(IO(¢I)'(1+7)I1(0()/0’\)

 

Multiply Equation A 70 by Io(o() and subtract form Equation A 68,

.5 -br[1 -’/7’(Lo(.;() _ Lo(o<>-(1+i)L1(a<)/a< ,M
1 - (1+7)11(o()/o( Io(o()

 

-/[1+4/.(2- Zfl'i) — ] (A71)
0< 2 (1- (l+’Z)I]_(a{)/0& 10(4))

72

Combining Equation A 70 and A 71, and solving for so,

a0 .../0% [1 + 2(1-77) (1-Io(:2()) J
42(Io(o(>-(1+rz>11(.x>/.x>

- 0’ 7r (I‘IO(<‘<)(L0(c{)‘(1+”T)L1(<\’)/d)
—2- [1 - ._ Lo(o() + ](A 72)
2d ( Io(o()-(1+)E)I1(o\)/d\ )

The velocity profile can now be written in terms of known
parameters

.129

2.2 [1 - x2 + 2 (1-"»;>(Io(.ér>-Io(a<>) ]

“(2(Io(a()'(1+’z)11(0()/0()

 

- BE [l-x + .11? Lo(o(x)-Lo(o()-(Io(c(x)-Io(o(>
20K

Lo(o()-(1+§)L1(o()/o( )]
10({)-(1+/z‘)11(o<)/a<
for ”Iva/IV (A 73a)

v = constant for ’T< (A 73b)

Let 4"W = A pD/4L, and substitute system parameters for Band 3'";

v'f-%1[1£2 +A( o(a(r/R)' I.(x>)]
1%.?[1 - 1% __7_2 (Lo(a( Ir/R)-Lo(o()-B(Io(o(r/R)-Io(o()))]
for “"7773, (A 74a)
v - constant for ’T’< ’1'), (A 74b)
where A and B are given by
A - 2(1-fi)/(o(2(10(oc) - (Minnow/om (A 74c)

B =- (10(4) - (1+fz‘>L1(o<>/o<)/(I.(o<)~(1+'i)11(o<)/oc> (A 74d)

73
3. The Viscosity of A Suspension of RBC's

The presence of red blood cells in blood is the chief
factor which makes blood viscosity higher than plasma viscosity. The
presence of suspended material is known to elevate viscosity. Assuming
that the inertia terms in the Navier-Stokes equation are negligible,
it can be shown [37] that a random suspension of rigid spheres
elevates viscosity according to Einstein's equation

«app = ,4{liquid(1 + 2'5 g spheres) (A 75)
without altering the non-Newtonian behavior at moderate shear rates.
The supposition made is equivalent to assuming that the particle
density is equal to the liquid density and that particle migration
in the velocity field is negligible. These assumptions are
approximately correct for blood flow.

However, the assumption of rigid spheres is quite
inappropriate. Taylor [E2] has shown that for non-rigid spheres,
Einstein's equation must be modified to

v«app " «liquid (1 + 2-5 Qspheres (A drop + .4

«4 liquid)/ (4drop +’6(liquid)) (A 76)
Since the internal viscosity of an RBC is reported to be in the range
of 1-6 cp. [l9] this correction is not negligible.

The biconcave shape of the RBC should not effect the
applicability of the analysis since Kynch [37] has shown that spheres
with holes drilled through them behave as rigid spheres of the same
diameter. However, the orientation of the non-spherical RBC's in
the velocity field would make it seem as if a suspension of different

size spheres were present. Roscoe [51 has shown that a mixture of rigid

74

sphere sizes gives

“4 app = Aliquid/ ( 1 ' ¢ spheres)2.5 (A 77)
where ¢ spheres must be 1.35 times the actual volume fraction if the
spheres are large enough to entrap a significant amount of liquid in a
closest packed structure.

Numerous other assumptions can be made and Rutgers [54] has
compiled about a hundred different equations for suspensions which
have been derived or been empirically fit to data.

A form of the Einstein's equation which has been found
appropriate for blood is in the reciprocal of concentration, which

implies a mixture of non-spherical particles. The equation is

,1(app = .1‘p1asma / ( 1 "17'§ZS) (A 78)
where
JL= 0.07 e2-49Q) + ( 1107/12) e‘1-65@ (A 79)
and T is the absolute temperature in 0 Kelvin. This has been found
by Charm and Kurland [12] to have an average accuracy of within 1.5%
between Q) -= 0.1 - 0.6 and 10°C to 37°C.

Equation A 78 and all other equations above do not consider
particle-particle interactions nor the ability Of the particles to
rotate. The equations merely represent the increased resistance to
flow of aiiuid in which micro-regions of plug flow are established.

In the turbulence model, particle-particle interactions will
be accounted for by a yield stress and the rotation of the RBC's will

be modeled as constant diameter eddies.

75
4. w’ a r e
The equation of motion must be derived which describes a

fluid with particulate turbulence in it. Following turbulence theory,
it is assumed that the instantaneous velocity at any point can be
mathematically separated into a time-averaged component, 3, and a
fluctuating component, v',

v='\7+v' (A80)

Equation A 83 can be transformed into (see Hinze [B4], for example)

-4—4-
49! + f vrvz= - Apr/2L (A 81)
I-+—+
whereiqv is the shear stress of the laminar flow and - ' vrvz

is a statistical correlation called the Reynold's stress in
turbulence theory. This equation is applicable to flow with any
type of disturbance in it. Assumptions made in the derivation of
Equation A 81 are angular symmetry and that neither the magnitude of
'32 nor its fluctuations vary in the axial direction. These
assumptions are valid for flow in a constant radius viscometer tube
and are not far in error for flow in vessels with low taper.

For whole blood, it is assumed that RBC's rotate in the
velocity field in the tube at an angular velocity,LU , which varies
with radial position (see Figure 15). Velocity fluctuations are
produced tangential to the rotating RBC's and are therefore equal to

the radius of an RBC,,{ , times its angular velocity

v'(r) =,( w (r) (A 82)

76

FIGURE I5

FLUCTUATIONS DUE TO A SPHERE
ROTATING IN A SHEAR FIELD

(A)

“3" V’2
_. __ __ ' ______ R0
.9 v2
“-7 I

0)
Ar

 

 

77

The fluctuations in the radial direction at radius R0 are proportional
to the cell tangential velocity at radius R0 and to the percentage of

cells

v; (R0).( H (742010) (A 83)

while the fluctuations in the axial direction at radius R0 are
prOportional to the tangential velocities at radii Ro-+«('and

R0 —,€ and to the percentage of cells
v; (Ro)o<H((w(Ro - 4) + w(Ro+ Inn (A 84)

It is assumed that the rate of rotation is prOportional

to the mean velocity difference on the Opposite sides of the cell

a) (110)0(07 (Robb -Tr (R0 we) /2 C (A 85)
Next, the mean velocity is expanded in a Taylor's series

since the dimensions of a red blood cell are small

3 (R0+ 5043 (R0) + y 83 (R0) lar + 33/282; (Rowan? +...

(A 86)
Therefore, from Equation A 85 and A 86, I
w (110)05 c); lar (A 87)

And, from Equations A 84, A 85 and A 86,
dam/(”.93 lat (A 88)

Equations A 83 and A 87 combine to give
(4.012 ld'x'r/drl
since the sign of v; depends on whether,€’is considered in a positive

or negative direction and on the magnitude of dv/dr only.

78
Therefore, the equivalent of the turbulent Reynold's stress,
que’ is given by

/
{IV Re = - Pvév; = +FH2/2f (dv/dr)2 (A 90)

where ('is a constant of proportionality and where the absolute value
sign has been removed utilizing the fact that dv/dr is everywhere
negative for axial flow in a tube. This result is identical with
Prandl's mixing length theory or Taylor's vorticity transport theory,
except that this model states that X?is not a function of position
from the wall as is assumed in the other models.

Now it is assumed that the shear stress due to laminar flow
is linear with the strain rate, except that the yield stress must be

exceeded before strain begins.

r~y(= _ f“
I fldV/dr + I y {(42731 (A 913)
d'G/dr = O “(WY (A 91b)

where /?is the viscosity of a suspension of RBC's in plasma. See the
section on the viscosity of a suspension of RBC's in this appendix for
further explanation. Then Equation A 81, A 90, and A 91a can be

combined to

/,vy -/(d'x7/dr + ,91121 26 (dV/dr)2 = Apr/2L (A 92)

This can be solved by the quadratic equation for dV/dr.

 

dv/dr - (A - f/(Z - 4/0H2/(26 (7“), - Apr/2L?)

 

2 pnzfze (A 93)
where the negative value for the square root has been chosen because it

is desired that dv/dr go to zero as -£5pr/2L goes to the yield stress.

79
Equation A 93 predicts that the maximum velocity gradient will exist
at the wall boundary.

However this implies that the maximum turbulence, which is
proportional to the velocity gradient squared, also occurs at the wall
boundary. This is impossible since a RBC physically cannot be centered
at the wall and the model depicted in Figure 15 requires one at that
position to impart a velocity fluctuation in radial direction. In
order to correct for this inconsistency, a region which is approximately
one red blood cell in width will be included next to the wall in the
model. In this region, the turbulence will be presumed to fall
linearly from the value given by Equation A 93 to zero at the wall.

The model then consists of three regions expressed mathematically as

-Apr/2L - 4"), = - /((dv/dr) + fl(R-r) for R) r) b (A 94a)

-Apr/2L - ’7"), a - fl (dv/dr) +an 2g (dv/dr)2

for b g r a 7311/7"W (A 94b)
0 = dv/dr 4‘yR/fivw,rzo (A 94c)
where

fl(r-b) =/oH7-,(2é (dv/dr)2 r=b (A 95)

so that the turbulence is a continuous quantity. Integration of

Equation A 94a yields

v - (Ap/ZL-fl) r2/2/{ + (4"), +,5R) r l/( + 02 for Rgr>b (A 96)
Since the velocity goes to zero at the wall, the constant

C2 can be determined.

v - (Ap/2L-fi)(r2-R2)/2,q + (If), + flR)(r-R)//( for Rgr>b
(A 97)

80

From Equation A 93, which was derived from the equivalent

of Equation A 94b,
(dv/dr)3=b=2y2 - 29092 - 24yW/( _
AWN/14101” - 2’I"yw/‘(-
Apb'V/HL (A 98)
where Wan/{l2 f 117-1263

Therefore from Equations A 95 and A 98,
,6 = (WI-MW - 24”,, w/fl-AWb/mfl/z -
43y - Apb/zL)/(R-b) (A 99)

so thatfl can be expressed in terms of system constants.

Integration of Equation A 93 yields

VBVI' +yLL—_ (vz .. ZT/y {Pl/t- Aper/(Lfi/Z
3w A.
+03 for ogra/va/fla (A 100)

It is necessary that the velocity at radius b given by Equation A 100
agrees with the velocity given by Equation A 97 so that the velocity is

not discontinuous. This allows the constant C3 to be determined.

v=V(r-b) +2/1L (4)2 - 24~ W _
39)Ap I y 1!

Aer/(L)3/2 - (92 - 243, wa-
ApbW/(L)3/2] + (Ap/zL ~,6) (b2 - R2)/2/(

+ (’7'), +flR) (b-R)//( for b; r) OER/4”,, (A 101)

81
It is necessary that dv/dr goes to zero at the yield radius,
r a’r§R/¢{w. It can be seen from Equation A 93 that this condition
is identically satisfied. The (constant) velocity in the center of

the tube can then be found by using Equation A 101

v-Y(’I"yR/’I"w-b)+2_/1L [<73 - (4)2 -
39))Q,p

273W?” WWW/MW] + (AP/2L 73>
(ohm/24+ (7"y +511) (b-R)//( for ’fyR/y/wngo (A 102)
The flow rate in a tube can then be found by integration

of the velocity profiles

7’yR/7’w b R
Q - 277 ( vrdr + vrdr + [vrdr ) (A 103)
O wit/7", b

where the velocities are given by Equations A 102, A 101 and A 97,
respectively. By integration and collection of like terms,
Q/2'na3 - ’7", (1/8 - 11/2411 - b2/24R2 - b3/24R3)//¢+
7’). ('1/8 + b/24R + b2/24R2 + b3/24RB)//Q+
(’2 (7'9/6 9’3 - b3/24R3 - b2/24R2 - b/24R -
1’24 "f’Z/‘7;/6¢’3+9’3/127’y/15¢3 -
9’4/(3/105'7’3 + (1 - Z’ry/QJ/{+7’wb/

(pg 10% [9)(47'3/1057'3 - (uni/105 73,11 -

b2 ’ry/ss 7;,112 + b3/56R3 + b2/24R2 + b/24R +

2 2 2
1/24) +91 ”(475/3573 + 4 b Ty/los’l’wa +

b2/70’I’w112) 437% (2 T’y/as 7’3 + b/105’7’8R)
+9’4/(3/1057’3 ] (A 10“)

82
For the term involving the square root, the order of magnitude of
(AI/4 is estimated
(7,17 =/{2/2/° 11212 6 = (.06>2/(2>(1>(a>2(3x10-4)2(1>
= 8 x 104 dyne/cmz (A 105)
and it can be seen that 2(b’7;/R -’I;)/YZY is small in comparison
to unity for any reasonable values of'f;. Therefore, the square root

in Equation A 104 may be expanded in a Taylor's series
(1 + 2(‘1'wb/R -’I’y)/’//( )6 = 1 + (b7’w/R -7’y)/'//(-
(4,671: - 2;)2/29’2/(2 + (’I’wb/R “793/2 ’r/ 3H. 3 -
5(7’wb/R - 7/y)4/8</4/(4 + 7(7’wb/R -’r’y)5/8‘I’5/(5 -
21(’r(,b/R -’I’y)6/16'I)6/l6 +... (A 106)
Inserting Equation A 106 in Equation A 104 and collecting like terms,
0/2‘77R3 = ’11" = [’I’w/s -'I"y/6 +7’g/24 (re, + (7%(b5/80R5
-b4/481z4 - b3/48R3 - b2/48R2) +1”), ’T'w(b3/24R3 + b2/24R2
+ b/24R) 3'3, (b3/48R3 + b2/48R2 + b/48R
1/48) +73/120-7’3)/q’/( + (7’3(-b6/48R5 +
b5/48R5 + b4/48R4 + b3/48R3) +73, 7;,(3b5/80R5
-b4/16R4 - b3/16R3 - b2/16RZ) +71”? (b3/16R3

+ b2/16R2 + b/16R) -?’3(b3/48R3 + b2/48R2 +

b/48R + 1/48) +Yg/24O’r’3fl‘f’2flz +...] Ix, (A 107)

As the particle size approaches molecular dimensions, Equation A 107
should reduce to the fluid model without turbulence, that is, the
Bingham model. This is the case for, as ,( approaches zero,(}) becomes
infinite and all terms on the right hand side drop out except the

first three.

APPENDIX B

APPENDIX B

COMPUTER PROGRAMS

Several computer programs were used to analyze raw data

and to aid in

in FORTRAN IV

the preparation of graphs. The programs were written

compatible with a CDC 6500 computer. Use on other

computers requires deck modification.

The

first program below was used to get yield stress and

viscosity parameters from the flow-pressure data described in the

experimental section. This is program TYNS.

The
Figure 3. It
and STRUl for
The

Figure 4. It

second program listed, PROFILE, was used to generate
utilizes double precision functions BESO, BESl, STRUO,
the modified Bessel and Struve functions.

last program listed, ZERLI, was used to generate

also utilizes the double precision functions listed

with the previous program.

83

84

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APPENDIX C

APPENDIX C

EXPERIMENTAL DATA

Experimental data in the form of pressure drOps at certain
flow rates were gathered from the flow-pressure apparatus described
previously. These data can be converted to 1Uw versus ‘6 data such
as shown in Figure 2. The data in this form have been fit by least
squares analysis using computer program TYNS, which is listed in
Appendix B. The statistically best values for the Casson viscosity
and yield stress are listed below for each run. Table 4 gives the
results for the ionic additives experiments. Table 5 gives the

results for the active hyperemia experiments.

96

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APPENDIX D

APPENDIX D

NOTATION

ENGLISH UPPER CASE

IO, 11

L0, L1

term grouping

term grouping

constants of integration
tube diameter, domain
force on body

hematocrit

modified Bessel functions
torque, tube length
modified Struve functions
couple stress tensor
momentum

flow rate

tube radius

surface area

tra c tion

flow rate - 40 I471 D3
volume

vorticity tensor

volume element half-length

ENGLISH LOWER CASE

8

b

eijk

constant coefficient, power series coefficient

constant coefficient, marginal gap width

cyclic permutation number
105

f

 

106

ENGLISH LOWER CASE (Cont'd.)

fluidity

acceleration of gravity
difference in manometer leg levels
pressure gradient - -45p/L

RBC radius

normal vector

pressure

radial position, recorder reading
coefficient of viscosity
velocity vector, velocity
vorticity vector

coordinate axis direction

position vector

term grouping (C12 82 R2 /7?)

ARZ/ 7))

term grouping (512
term coefficient
term coefficient
term coefficient
Kronecker's delta
difference of
prOportionality constant
couple stress coefficient

couple stress coefficient

term grouping (ff!B 7(7’2)

GREEK (Cont'd.)

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SUBSCRIPTS

107

volume measure = 1 mm3
coefficient of viscosity
couple stress

3.14159265. ......

fluid density

summation of

stress, shear stress

volume fraction

term grouping I HIZfHZI 26

Einstein Law function

partial derivative of y
total derivative of y
Eulerian derivative of y
gradient of

nabla Operator

integral

surface integral

magnitude of vector

coordinate index, iteration index
coordinate index, iteration index
coordinate index
coordinate index

coordinate index

108

SUBSCRIPTS (Cont'd.)

s coordinate index

w wall value

y yield value

SUPERSCRIPTS

A antisymmetric

s symmetric

- time average

' fluctuating

IE velocity average (tube diameters/time)

1(- term grouping = 7th)

BIBLIWRAPHY

10.

11.

12.

13.

14.

15.

16.

17.

BIBLIOGRAPHY

Abramowitz, M., and Stegun, I. A. Handbook of Mathematical
Functions, U.S. Government Printing Office, Washington, D.C.,
375, 376 (1965).

Ibid., 496, 498.

Benis, A. M., Sc.D. Thesis, Massachusetts Institute of
Technology, 97, 99, 212 (1964).

Ibid., 17.
Ibid., 54, 55.
Bingham, E. C. and Roepke, R. R., J. Gen. Physiol., 28, 79 (1944).

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport
Phenomena, Wiley and Sons, New York, 85, (1966).

Bugliarello, G., and Hayden, J. W., Trans. Soc. Rheol., l, 209
(1963).

Carrier, 0., Ph.D. Thesis, U. Miss. Med. Center (1964).

Casson, N., in Rheology of Disperse Systems, ed. C. C. Mill,
Pergamon Press, New York, 84-104 (1959).

Charm, 8., and Kurland, G., Nature, 206, 617 (1965).
Charm, 8., and Kurland, G., Biorheology,.;, 163-164 (1966).
Cokelet, G. R., et al., Trans. Soc. Rheol.,,1, 303 (1963).

Conn, R. B., Current Therapy, 1969, W. B. Saunders, Philadelphia,
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COpley, A. L., in Flow Progerties of Blood and Other Biological
Systems, ed. Copley, A. L., and Stainsby, G., Pergamon Press,
New York, 110-113 (1960).

Davidsohn,I., and Henry, J. B., Clinical Diagnosis by Laboratory
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de Raedt, M., et. al., J. Appl, Physiol,, 26, 469 (1969).

109

18.
19.

20.

21.

22.

23.

24.

25.
26.
27.
28.

29.

30.

31.

32.

33.
34.

35.

36.

37.

110

BIBLIOGRAPHY (Cont'd.)

Dintenfass, L., Nature, 213, 179 (1967).
Dintenfass, L., Nature, 219, 956 (1968).

Eringen, A. C., Mechanics of Continua, Wiley and Sons, New York
98 (1967).

Fahraeus, R., and Lindqvist, T., Am. J. Physiol.,,2§, 562 (1930).

Fishman, A. P., and Richards, D. W},Circulation of the Blood,
Oxford U. Press, New York, 86, (1964).

Frasher, W. G., et al., J. Appl. Physiol.,,25, 751 (1968).

Fung, Y. C., A First Course in Continuungechanics, Prentiss-
Hall, Englewood Cliffs, New Jersey, 52 (1969).

Ibid., 190.

Ibid., 26.

Ibid., 40.

Gilinson, P. J. Jr., et. al., Trans, Soc. Rheol., 7, 319 (1963).

Guest, G. M., and Siler, V. E., J; Lab. Clin. Med,,‘12,
757-767 (1934).

Haddy, F. J., and Scott, J. B., Physiology 501 Class Notes,
Mich. St. 0., adapted from Annual Rev. Pharmacol., 6, 50 (1966).

Haddy, F. J., and Scott, J. B., in Electrolytes and Cardia-

vascular Diseases, ed. E. Bajusz, Karger, New York, 383-393 (1965).

Haynes, R. T., and Burton, A. C., Am. J. Physiol., 121, 943
(1959).

Hershey, D., and Sung, J. C., J. Appl. Physiol., 21, 27 (1966).
Hinze, J. 0., Turbulence, McGraw-Hill, New York, 14-19 (1959).

Hodgman, C. D., ed., Handbook of Chemistry and Physics,
Chem. Rubber Co., Cleveland, Ohio, 2251, 2257 (1963).

Jacobs, H. R., Biorheology,.§, 117 (1966).

Kynch, G. J., Brit. J. Appl. Physics, §,'§;, 55-511 (1954).

38.
39.
40.
41.
42.
43.

44.

45.

46.

47.

48.
49.
50.
51.
52.
53.
54.
55.
56.
57.

58.

111

BIBLIOGRAPHY (Cont'd.)

Masin, J. G., Biorheology, in process.

Mayer, G. A., and Kiss, 0., Am. J. Physiol.,_;Q§, 795 (1965).
Mayer, G. A., et. al., Biorheology,.3, 177 (1966).

Merrill, E. W., et. al., Bigghysics J., 3, 199 (1963).

Merrill, E. W., et. al., giggulatigg_Research,‘18, 437 (1966).
Merrill, E. W., et. al., J. A221. Physiol., 26, 1 (1969).
Merrill, E. W., in BiOphysical Mechanisms in Vascular Homeostasis
and Intravascular Thrombosis, ed. Sawyer, P. N., Appleton-

Century-Crofts, New York, 121-137 (1965).

Merrill, E. W., and Pel1etier, G. A.,‘J.4A221. Physiol., 23,
178-182 (1967).

Merrill, E. W., et. al., J. Appl. Physiol.,_29, 954-967 (1965).
Mickley, H. S., Sherwood, T. K., and Reed, C. B., Agglied
Mathematics in_§hgmica1 Engineering, McGraw-Hill, New York,
179 (1957).

Murphy, J. R., J.;Lab. Clin. Med.,_§2, 758 (1967).

Nielsen, A. V., et. al., J. A221. Physiol.,,gg, 1167 (1967).
Rand, P. W., et. al., J, A221, Physiol.,.25, 550 (1968).
Roscoe, R.,,figit. J. Appl. Physics, 3, 267-269, (1952).
Rosenblum, W. 1., filggg, 31, 234 (1968).

Roth, S. A., Ph.D. Thesis, Mich. St. U. (1970).

Rutgers, I. R., Rheol, Acta,‘§§2g_g, Heft 4, 305-348 (1962).
Schmid-Schonbein, H., and Wells, R., Science,_l§§, 288 (1969).
Scott, J. B. et. al., Federation Proc., 21, 1403 (1968).

Scott, J. B., et. al., Am. J. Physiol., 218, 338-345 (1970).

Scott Blair, G. W., Nature, 183, 613 (1959).

59.

60.
61.
62.
63.
64.
65.

66.

67.

68.

69.

70.

112

BIBLIOGRAPHY (Cont'd.)

Siegel, N. J., and Kolmen, S. N., Am. J. Physiol., 216, 707
(1969).

Stokes, V. R., Physics of Fluids,.2, 1709 (1966).

Ibid., 1710.

Taylor, G. I., Proc. Royal Soc., (London), A128, 41-48 (1932).
Valanis, K. C., and Sun, C. T., Biorheolggy,,fi, 86 (1969).
Ibid., 88.

Ibid., 97.

Van Wazer, J. R., et. al., Viscosity and Flow Measurement, Wiley
and Sons, New York, 201-215 (1963).

Ibid., 206.

Walder, D. N., Weaver, J. P. A., and Evans, A., Biorheology,
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Wells, R. E., et. al., Clin. Res., 11, 176 (1963).

Whittaker, S. R. F., and Winton, F. R., J. Physiol., 18, 331-369
(1933).

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