Studies of arterial branching in models using flow birefringence.

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Title:
Studies of arterial branching in models using flow birefringence.
Uncontrolled:
Arterial branching in models using flow birefringence
Physical Description:
xix, 134 leaves. : ill. ; 28 cm.
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English
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Crowe, William Joseph, 1941-
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Subjects / Keywords:
Arteries -- Models   ( lcsh )
Refraction, Double   ( lcsh )
Fluid dynamics   ( lcsh )
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bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 128-132.
Statement of Responsibility:
By William Joseph Crowe.
General Note:
Manuscript copy.
General Note:
Vita.

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Source Institution:
University of Florida
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All applicable rights reserved by the source institution and holding location.
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notis - AFA6847
oclc - 18000657
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Full Text














STUDIES OF ARTERIAL BRANCHING
IN MODELS USING FLOW
BIREFRINGENCE










By

WILLIAM JOSEPH CROWE, JR.


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA
1969



















CIRCLE OF WILLIS


INTERNAL CAROTID ARTERY
-EXTERNAL CAROTID ARTERY

COMMON CAROTID ARTERY


ARCH OF AORTA -




THORACIC AORTA--




/ / i ,
SUPERIOR MESENTERIC ARTERY -

ABDOMINAL AORTA --



COMMON ILIAC ARTERY ----






FEMORAL ARTERY--




POPLITEAL ARTERY





















Reprinted with permission.
by Scientific American, Inc.


VERTEBRAL ARTERY


- SUBCLAViAN ARTERY







SBRACH:AL ARTERY

S- COELiAC AXiS

S.---- PENAl ARTERY


Copyright (c) 1961
All rights reserved.

















Copyright by
William Joseph Crowe, Jr.
1969





















To my lovely wife Doreen
whose enthusiasm is matched
only by her imagination














ACKNOWLEDGMENTS


Inevitably, in any work such as this there are those people who

through their continuing support make it both possible and meaningful.

It is a pleasure to acknowledge my indebtedness to them.

I should like to express my genuine appreciation to Dr. R. W.

Fahien, who provided much of my fundamental training and introduced me

to the field of biomedical engineering, and to Dr. L. J. Krovetz, who

provided most of the training, support and facilities for this project.

The logistic support of Dr. G. L. Schiebler has been invaluable.

Technical assistance provided by Mr. Horace Brown was the major

reason why the experiments proceeded smoothly.

Thanks are also extended to Dr. Michael DeBakey and Scientific

American, Inc. for permission to use their figure as a frontispiece.

Grateful appreciation is extended to the Chemical Engineering

Department and National Science Foundation for the opportunities to pur-

sue this work under first an Engineering College Fellowship and later a

National Science Foundation Traineeship.

The Departments of Medical Illustrations and Photography deserve

much credit for their prompt and courteous service of high quality.

Finally, I should like to thank Mrs. Lawrence Herrin for her

careful checking and typing of this manuscript.















TABLE OF CONTENTS


ACKNOWLEDGMENTS .

LIST OF TABLES .. ..

LIST OF FIGURES .

LIST OF SYMBOLS .

ABSTRACT. .

SECTION


I. INTRODUCTION .........

A. Problem Statement .

B. Prior Branching Studies.

II. THEORETICAL CONSIDERATIONS .

A. Fundamental Equations .

B. Flow Birefringence .

III. EXPERIMENTAL EQUIPMENT .

A. The Fluid-Perfusion System

B. The Flow Test Sections .

C. The Optical System .

IV. EXPERTMENTAL TECHNIQUE .

A. Preparation and Storage of
Suspension .


....

. .

. .

* .

* .

. .

e .

* .









the Dye


B. Determination of Fluid Physical Properties


Flow Regulation and Collection .

Photography. . .


Page

iv

vii

viii

xi

xvii


*


. .


""' '














SECTION

V. THE ANALYSIS OF FLOW-PATTERN PHOTOGRAPHS
AND EXPERIMENTAL RESULTS . .

A. Isochromatic Shear-Stress Calibration for
Laminar Tube Flow . .

B. Steady-Flow Experiments . .

C. Pulsatile-Flow Experiments .. ... ..

VI. CONCLUSIONS AND RECOMMENDATIONS .


APPENDICES............. ...........

A. THE STEWART-HAMILTON TECHNIQUE. .

B. APPARENT-VISCOSITY AND REYNOLDS-NUMBER DATA
FOR 1.4% MILLING-YELLOW DYE . .

C. PHOTOGRAPHIC DATA . .


Page


. 52

. 67

. 104

. 111


BIBLIOGRAPHY . . .


BIOGRAPHICAL SKETCH . . .


. .














LIST OF TABLES


Table Page

1. FLOW TEST SECTIONS. .................. 41

2. LIGHT-PATH LENGTH AND AVERAGE SHEAR STRESS ALONG
THIS PATH AS DIMENSIONLESS FUNCTIONS OF FRACTIONAL
DISPLACEMENT IN THE FLUID MEDIUM. . .. 66

3. APPARENT-VISCOSITY AND REYNOLDS-NUMBER DATA FOR
1.4% MILLING-YELLOW DYE . ... .. 126

4. PHOTOGRAPHIC DATA ................... 127


vii












LIST OF FIGURES


Page


Figure


1. Dye-streamline experiment (Evans Blue in water) by
Krovetz for steady flow through a glass branch. .... 10

2. Dye-streamline experiment (Evans Blue in water) by
Krovetz for steady flow through a plastic branch
molded from a dog's artery. . .... ..11

3. Schematic diagram of the experimental setup .. 35

4. Flow wave generated by the rotary bellows pump
used for pulsatile-flow experiments . .. 39

5. Photograph of the flow test sections. .... ..42

6. Apparent viscosity and Reynolds number as functions
of flow rate for 1.4% Milling-Yellow dye. .. 48

7. Isochromatic-band distribution for fully developed
laminar flow through a circular tube (I.D. = 3.00 mm) 54

8. Schematic cross section of a circular tube, illustra-
ting the refraction of a typical light ray passing
from the fluid to the observer. . ... .56

9. Schematic flow cross section, illustrating a typical
light path along which the average shear stress
is required . . ... 63

10. Isochromatic-band distribution for steady flow past
a cardiac catheter (O.D. = 1.67 mm) inserted
axially into a circular tube (I.D. = 3.00 mm) ..... .70


11. Isochromatic-band distribution for steady flow past
a beveled needle (O.D. = 0.92 mm) inserted radially
through the wall of a circular tube (I.D. = 3.00 mm).

12. Isochromatic-band distribution for steady flow
through a 30 branch of circular cross section
(area ratio = 1.0) . .


. 75



S. 77


13. Flow ratio versus upstream Reynolds number, with area
ratio as a parameter, for 30 branches of circular
cross section . . .. .. 79


viii














14. Flow ratio versus upstream Reynolds number for Milling-
Yetllow dye and water in a 300 branch of circular cross
section (area ratio = 1.0). . . 82

15. Branch entrance lengths versus downstream Reynolds
number for steady flow through a 30 branch of circular
cross section (area ratio = 1.0). . 86

16. Isochromatic-band distribution for steady flow through
a 300 branch of rectangular cross section
(area ratio = 1.0). . ... ... 89

17. Flow ratio versus upstream Reynolds number for 300
branches of circular and rectangular cross section
(area ratios= 1.0). . .... ... 90

18. Isochromatic-band distribution for steady flow through
a 600 branch of circular cross section (area ratio = 1.0) 92

10. Branch entrance lZengths versus downstren Reyld
number for steady flow through a 600 branch of circular
cross section (area ratio = 1.0). . 93

20. Isochromatic-band distribution for steady flow through
a 900 branch of circular cross section (area ratio = 1.0) 95

21. Branch entrance lengths versus downstream Reynolds
number for steady flow through a 90 branch of
circular cross section (area ratio = 1.0) ... 97

22. Isochromatic-band distribution for steady unseparated
flow through a + 450 wye branch of circular cross
section (area ratio = 1.0). . . 99

23. Isochromatic-band distribution for steady separated
flow through a + 450 wye branch of circular cross
section (area ratio = 1.0). . ... 100

24. Flow ratio versus upstream Reynolds number, with
branching angle as a parameter, for branches of
circular cross section (area ratios= 1.0) .. 101

25. Branch entrance lengths versus flow-pulse phase angle
(time), with area ratio as a parameter, for pulsatile
flow through 300 branches of circular cross section 107


Figure


Page












Figure


Page


26. Branch entrance lengths versus flow-pulse phase
angle (time), with branching angle as a parameter,
for pulsatile flow through branches of circular
cross section (area ratios = 1.0). . .. 108

27. Branch entrance lengths versus flow-pulse phase
angle (time), with internal geometry and type of
branch as parameters, for pulsatile flow through
branches having an area ratio of unity .. 109

28. The Stewart-Hamilton experiment, illustrating a
typical indicator-dilution curve . .... 118











LIST OF SYMBOLS


A identifies the main downstream stem of a branch

denotes the arbitrary position of an isochromatic band seen in

a flow photograph

indicator injection site

mass of indicator injected, (mg)

AA cross-sectional area of the flowing stream in the main downstream

stem A

Ag cross-sectional area of the flowing stream in the side arm B

a perpendicular displacement of the light path in the fluid

from the center of the tube

amplitude of the component vibration parallel to the optic axis

of the fluid.

amax maximum value of the perpendicular displacement a, (aax = R)
B the side arm of a branch

the point on the outer surface of the circular tube where the

observed light ray emerges

b lateral displacement of an isochromatic band in the flow

photograph, (b = ang/n1)

amplitude of the component vibration perpendicular to the optic

axis of the fluid

bma, maximum value of the band displacement b, (bma = Rn3/n1)

C the point on the inner surface of the circular tube where the

observed light ray emerges from the fluid











C' position of point C after rotation

Ci mass concentration of indicator at point i, (mg/1)
D inside diameter of a circular tube

the point on the inner surface of the circular tube where the

observed light ray enters the fluid

D' position of point D after rotation

e subscript denoting the entrance to a branch

F arbitrary function of integration

Fbody total body force acting onthe macroscopic branching-flow system

drag total drag fcrce acting on the macroscopic branching-flow system
G arbitrary function of integration

g vector sum U of bdy forces per uni.t mus of fluid
i subscript denoting an arbitrary downstream point

subscript denoting an arbitrary inlet to the macroscopic

branching-flow system

j subscript denoting an arbitrary outlet from the macroscopic

branching-flow system

K diameter ratio for a circular tube, defined as (outside diameter/

inside diameter)

L length of the circular tube

LE entrance length
4 length of the light path in the fluid

M total mass contained within the macroscopic branching-flow

system


xii











N an arbitrary integer

order of an isochromatic band

N1 normal to the surface of the tube at point B

N2 normal to the surface of the tube at point C

n outwardly directed unit vector normal to the surface of the

macroscopic branching-flow system

n, index of refraction of medium 1 (air)

n2 index of refraction of medium 2 (glass or Plexiglas)
n3 irdex of refraction of medium 3 (Milling-Yellow dye)

nx index of refraction for the component vibration parallel to the

optic axis of the fluid

ny index of refraction for the component vibration perpendicular to

the optic axis of the fluid

0 coordinate origin

P pressure of the fluid at any point

P total momentum of the macroscopic branching-flow system

Q volumetric flow rate

QA volumetric flow rate at the injection site A, (1/sec)
Qi volumetric flow rate at the sampling site i, (I/sec)

R inside radius of the circular tube

Re Reynolds number based on apparent viscosity and tube diameter,

(Re = D7poli)

r radial coordinate

? position vector denoting the center of the light path in the fluid


xiii











S denotes the surface of the macroscopic branching-flow system

T period of vibration for the incident light beam

t time

t' defined as t + gT

ts sampling time

V denotes the volume occupied by the macroscopic branching-flow

system

v fluid velocity at any point

Vs velocity of the moving surface of the macroscopic branching-flow
system at any point

v magnitude of the fluid velocity v

Vr radial component of the fluid velocity v

Ve angular component of the fluid velocity v
vz axial component of the fluid velocity v

V cross-sectional average of v

2 cross-sectional average of v2

w subscript denoting the inside wall of the tube

x displacement of the component vibration parallel to the optic axis

of the fluid at any time

arbitrary coordinate perpendicular to the observer's external line

of sight and passing through the center of the tube

y displacement of the component vibration perpendicular to the optic
axis of the fluid at any time

arbitrary coordinate parallel to the observer's external line of

sight and passing through the center of the tube

xiv











y^ limit of integration, (y* = (R2 -a ))
z axial coordinate

Greek Symbols

a fractional displacement of the light path in the fluid from the

center of the tube, (az = a/amax = a/R)

B fractional displacement of the isochromatic band in the flow

photograph, (B = b/bmax = a/R =a)

AP pressure drop over the length L of the circular tube

6 path difference between the interfering vibrations in the fluid

e angle of rotation between the abscissa describing the incident

polarized light and the optic axis of the fluid

0 angular coordinate

01 angle between the light ray in medium 1 and the normal NI

02 angle between the light ray in medium 2 and the normal N1

Q3 angle between the light ray in medium 2 and the normal N2

04 angle between the light ray in medium 3 and the normal N2
A difference operator meaning the sum over all outlet streams minus

the sum over all inlet streams of the macroscopic branching-flow

system

X wavelength of incident light (in vacuo)

\X wavelength of the component vibration parallel to the optic axis

of the fluid

xy wavelength of the component vibration perpendicular to the optic

axis of the fluid











4
pp apparent viscosity of the Milling-Yellow dye (app = PR -
app app 8QL
V fluid viscosity (p = papp for Milling-Yellow dye)

T irrational number, 3.14159...

p fluid density
Summation over either inlet or outlet streams of the macroscopic

branching-flow system

shear-stress tensor

T component of the shear-stress tensor T

Tz component of the shear-stress tensor =

rz component of the shear-stress tensor T
T magnitude of shear stress (at upstream positions, T = prz)

S wall stress, (rT = () ~

D dimensionless average shear stress, (c = T(avg)/T~)

Miscellaneous Symbols

underscore denotes a vector quantity
-- overscore denotes an integral average over a cross section

overscore also denotes a line segment

-- denotes a tensor quantity

II 11 denotes a tensor quantity

(avg) denotes an integral average along the light path

(up) denotes the value at the upstream position

(down) denotes the value at the downstream position


xvi










Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy

STUDIES OF ARTERIAL BRANCHING IN
MODELS USING FLOW BIREFRINGENCE

By


William Joseph Crowe, Jr.

March, 1969

Chairman: Dr. R. W. Fahien
Co-chairman: Dr. L. J. Krovetz
Major Department: Chemical Engineering


The optical technique of flow birefringence (double refraction)

has been used to investigate steady and pulsatile flows through drilled

Plexiglas branches simulating those of the arterial system. This

technique was also used to trace flow disturbances created by inserting

needles and cardiac catheters radially and axially into a laminar flow

through a circular tube.

The birefringent liquid used was a 1.4% aqueous suspension of the

aniline dye Milling Yellow, and birefringence patterns were photographed

on 16 mm motion-picture film. The experiments covered a Reynolds-number

range from 0 to 250.

A scheme has been proposed for calibrating the isochromatic-band

distribution obtained for steady laminar flow through a circular tube

against the average shear stress along the light path through the fluid.

The optical effects arising from curved boundaries and differences in

refractive index have been included in this calibration.


xvii











Analysis of birefringence photographs has revealed that regions

of flow stasis and boundary-layer separation occur along the outer walls

of branches for upstream Reynolds numbers between 20 and 50, depending

on the angle of branch. In addition, it has been shown that regions of

locally increased shear, where stresses exceed the upstream wall stress,

occur along the inner walls of branches if 1. the branching angle is

greater than 30, 2. the upstream Reynolds number is greater than 150

and 3. the downstream arms of the branch are of approximately the same

diameter. Birefringence films for pulsatile flows have shown that these

regions do not move with the incoming pulse but remain centered approxi-

mately one diameter downstream from the point of bifurcation.

It has been suggested that locally increa;ePd shpering stress on

the arterial intima at sites of branching represents that physical

mechanism for initiating wall trauma and atherosclerotic disease which

is most closely supported by pathologic data.

Photographic evidence has been presented for the occurrence of

spiral secondary flows in branches, and the importance of the resultant

mixing effect has been discussed with respect to the Stewart-Hamilton

technique for measuring cardiac output.

The propagation of flow disturbances has been characterized by

defining branch entrance lengths in terms of distorted birefringence

patterns. Entrance-length data and birefringence photographs have shown

that as upstream Reynolds number increases,flow disturbances in branches

are generated and propagated by the following three processes: 1. the

readjustment of a laminar velocity profile, 2. the onset of boundary-


xviii













layer separation and 3. the onset of secondary flow.

Both Milling-Yellow dye and water Lave been used to study the

flow-distribution characteristics of branches for steady flows in the

Reynolds-number range from 0 to 2100. These experimental results have

shown that upstream fluid momentum per unit volume is the principal

dynamic variable affecting branching flow distributions and that area

ratio is a much more significant geometric parameter than is branching

angle.

It has also been shown that wall attachment occurs in branches

to the extent that side-arm flow will exceed that in the main downstream

stem providing that 1. the branching angle is less than 600, 2. the

upstream Reynolds number is below a certain value which is dependent on

the fluid and 3. the diameter of the side arm is approximately equal

to that of the main downstream stem.


xix













SECTION I


INTRODUCTION


During the past twenty years, engineering research has undergone

a transition characterized by continual erosion of once well-defined

boundaries separating individual disciplines. One need look no farther

than the modern electronics industry, with its increasing dependence on

such fields as metallurgy, physical chemistry and chemical engineering,

to appreciate this fact. What might be called the interdisciplinary

revolution has grown both in scope and intensity, affecting not only

engineering disciplines but also creating new interfaces with the life

and social sciences.

Undoubtedly, several factors have contributed to this process

but perhaps two of the most significant have been first, the development

of new materials and devices having broad application (e.g., the high-

speed electric computer and the laser) and second, the refinement of

existing measuring instruments and techniques, coupled with the ever-

present desire of researchers to extend known theoretical and experimen-

tal results to new areas of research.

The field of biomedical engineering has evolved as a direct

consequence of this process and is now enjoying rapid growth both in

this country and internationally. Significantly, this growth has occurred

at both academic and industrial levels and is reflected by new curricula

and publications in the field.












Among those topics receiving attention in the biomedical-

engineering literature, none has received more than the cardiovascular

system. This is not surprising when one considers that diseases of

the circulation are among the most common yet-unsolved problems in

medicine and that the circulatory system, featuring unsteady-state heat,

mass and momentum transport within materials possessing non-Newtonian

rheology and functioning under a closed-loop automatic control system,

involves many of the most interesting and challenging engineering prob-

lems of the day.

The great majority of biomedical-engineering studies dealing with

the cardiovascular system fall into one or both of the following catego-

ries: those presenting experimental data obtained from in vivo, in vitro

or model experiments, and those proposing mathematical models and analogs

to incorporate existing experimental data. The studies of McDonald,

Taylor, Hardung, Womersley, Attinger, Bergel, Noordergraaf and Gessner

are a few examples of these.

However, with some notable exceptions,13141622424748495054

few studies have attempted to correlate either experimental data or

theoretical predictions with abnormal clinical and pathological findings.

That is to say, little attention has been given to providing physical

bases or mechanisms for the formation of observed pathological phenomena,

as opposed to providing models simply to explain observed experimental

data. The recent work of Fry demonstrating that excessive shearing

stress causes wall-tissue erosion is a particularly significant step in













this direction.

The present work represents an attempt to further this approach

by focusing attention on experiments with models simulating arterial

branches and interpreting experimental results in the light of known

findings from the medical literature. The approach here is then in

contrast with an engineering-science approach to the problem, wherein

one would be concerned with solving the equations of change for various

types of fluids and wall conditions.


A. Problem Statement


The problem may be posed generally in two parts: First, in the

flow range common to the arterial system in man, what physical phenomena

may be attributed to the presence of a branch in the flowing stream?

Second, what are the implications of these phenomena with respect to wall

trauma and atherosclerotic disease, known to have a predilection for

formation at such sites in the arterial system? More specifically, the

present study seeks to investigate three major areas of branching dynamics

under conditions of both steady and pulsatile flow by using the optical

technique of flow birefrigence. These three areas are: 1. the distri-

bution of wall shearing stresses in branches insofar as determining

whether local concentrations might occur which could damage the vessel

wall, 2. branching-flow distributions as functions of branching angle

and upstream Reynolds number and 3. convective-mixing effects at

branches and the propagation of disturbances downstream as these relate












to suspended particles in the blood stream and to the Stewart-Hamilton

indicator-dilution technique957 for determining cardiac output.

With the exception of one wye branch, this study has been

restricted to branches of the sngle-side-arm type fabricated from

drilled or milled Plexiglas block. Though model branches of this type

cannot imitate branches of the arterial tree in every detail, particu-

larly their distensibility and wall characteristics, at least some

indications of possible flow behavior in the arterial branches can be

obtained in this way; and for those branches rigidly bound by surrounding

tissue and muscle, such a model represents at least a good first approxi-

mation.

The approach here is primarily experimental and considers only

mechanical interactions between the flowing stream and the branch.

Though biochemical processes do serve a vital function in the cardiovas-

cular system, there is no reason to suppose a priori that biochemical

activity at a bifurcation would be any different from that occurring in

unbranched arterial segments. On the other hand, there is every reason

to suppose that flow behavior at two such sites would be different, per-

haps even to the extent of either altering normal biochemical function

resulting in the development of arterial disease, as Rodbard's work 42

suggests, or simply causing trauma to the vessel wall, such as that

shown in Fry's work.6 This point is further emphasized by the studies

of Texon, Imparato, Helpern and Lord, who have shown experimentally 22












49,50
4950that atherosclerosis can be produced artificially in dogs by

altering blood-vessel geometry.

There are four important reasons why flow birefringence was

chosen as the flow-visualization technique used in the present study:

1. Since flow birefringence is an optical technique, there is no need

to introduce probes or other devices into the flow field thereby causing

external disturbances. 2. It is possible to study flow behavior through-

out the entire branch, not just along a single streamline as with the

technique of dye injection. 3. Prior two-dimensional studies36'37'40,41

have indicated that flow-birefringence patterns can be correlated with the

shearing-stress distribution, so the possibility of obtaining shear-stress

information in branches directly seemed fruitful. 4. It was felt that

flow birefringence would provide a convenient way of tracing flow distur-

bances and entrance effects downstream.

As early as 1964, Wayland had suggested the possibility of using

flow birefringence to study flow disturbances in branches,51 though this

was unknown to the author until recently. Fortunately, the present

study has incorporated most of Wayland's suggestions.


B. Prior Branching Studies

Interest in the problem of branching flows is not new. In fact,

problems of flow distribution have been of engineering concern since

ancient times, and, today, relationships between vessel branching and flow












behavior are receiving considerable attention in connection with the

design of fluid amplifiers and fluidic digital-logic devices.

On the other hand, physiologic interest in branching has arisen

only within the last fifty years and has been sporadic. Recent

studies,47'54 demonstrating that atherosclerosis shows a predilection for

formation at sites of branching and curvature in the arterial system,

have increased the interest in branching. There now seems little doubt

that problems associated with vessel branching on a small scale consti-

tute significant research areas both in engineering and medicine.

Surprisingly little work, either theoretical or experimental,

has been published on the branching problem. Poisson's work,39 concern-

ing reflected and refracted waves at a branch site was among the earliest

theoretical studies of branching. Poisson's results were later repeated

by Lord Rayleigh 46 in his discussion of the propagation of sound through

branched tubes.

The earliest physiologic interest in vessel branching appears to

have been that of Moens34 and Frank,15 who were concerned with the

propagation of pulsewaves through the arterial tree. Though Frank's

study has often been cited, it apparently has never been criticized,

which is surprising in view of the fact that it contains numerous mathe-

matical errors. Frank did discuss the problem of a pulsewave encounter-

ing a bifurcation but only to the extent of repeating Poisson's formula-

tions (incorrectly) as they appeared in Rayleigh's work.












More recent studies of branching have emphasized 'oth

theoretical and experimental approaches. Womersley's study56 represents

the most comprehensive theoretical attack on the problem of mathemati-

cally modeling pressure and flow relationships in the arterial tree.

His proposed model was based on the linearized Navier-Stokes equations

and the equations for an ideally elastic wall. Womersley also considered

wave reflectionsat branches and investigated the frequency dependence

of wave-reflection coefficients and input impedence.

Acrivos, Babcock and Pigford considered the problem of flow

into an idealized manifold consisting of uniformly spaced side ports of

equal area. They proposed a simplified mathematical model based on

friction factors and pressure-recovery coefficients and were able to

show good agreement with their pressure and flow data for both blowing

and sucking manifolds.

Knox24 considered the more general problem of measuring pressure

drops and flow distributions for branches of varying branch angle. He

considered branches of the single-side-arm type and determined pressure

recoveries at bifurcations. He defined a recovery length for branches

as the distance downstream from the branch necessary for the pressure

gradient to regain linearity, and his data showed that recovery lengths

for the side arms were significantly greater than those for the main

stems.

The technique of dye injection was used by Copher and Dick,11

Hahn, Donald and Grier18 and by Barnett and Cochrane5 to study particle

and flow distribution in glass models simulating venous flow. Distinct










8



flow channeling was noted in these experiments.

Helps and McDonald2021 investigated venous-flow patterns both

in vivo and in a glass model and found that at Reynolds numbers less

than 1000 injected dye formed a parabolic profile in the tributary,

but at the junction circulating movement developed. Two sets of secon-

dary flow were set up in each pipe half, and this persisted "for some

distance down the parent trunk until a new parabola was formed across

the single trunk".
33
McDonald33 used the dye-injection method to show that marked

disturbances were present at the aortic bifurcation in a rabbit at

Reynolds numbers less than 1000. He reported that dye could be seen to

impinge on the wall, and eddies were set up near the orifices of the

two branches.

Krovetz25'27'28 conducted similar studies using glass models and

Silastic (silicone rubber) molds of dogs' arteries35 to simulate arterial

branching. He observed and photographed secondary flows which dis-

tributed dye particles across the cross section as if the flow were

turbulent, even though the upstream flow was well within the laminar

range. He found that dye particles were distributed unevenly between

the downstream branches if the incoming dye front was laminar but

were distributed evenly if the incoming front was turbulent. He also

found that the critical upstream Reynolds number for the onset of secon-

dary flow in branches ranged from 58 to 89% of the critical Reynolds












number for the onset of true turbulence in a straight tube.

Another study by Stehbens,45 reporting a series of experiments

on turbulence in glass models simulating arterial bifurcations and the

curvature of the carotid artery, found critical Reynolds numbers rang-

ing from 306 to 1473 "calculated for the lowest rate of flow at which

turbulence could be induced in the models".

Typical experimental results obtained by Krovetz27 are shown

in Figures 1 and 2. Figure 1 shows a laminar dye front encountering

a branch in a glass tube, and Figure 2 is the equivalent experiment

for flow through a plastic mold of a dog's artery. In both cases,

pronounced secondary flow occurs immediately distal to the bifurcation,

and significant mixing occurs in this area. In the glass branch

(Figure 1), this mixing proceeds downstream approximately five diameters

before any dye enters the side arm; and in the mold (Figure 2), no dye

at all enters the upper downstream branch, indicating a definite

channeling effect similar to that observed by Barnett and Cochrane.5

These figures also show that most of the mixing activity occurs near the

inner walls of the branches (the medial-crotch region) and that little,

if any, mixing occurs near the outer walls in the immediate region of the

branches.

Other studies by Attinger3 and by Fox and Hugh14 have confirmed

that the regions along the outer walls in branches represent areas of

boundary-layer separation and local stasis. Attinger used suspended

dust particles to trace flow disturbances in branch models for Reynolds

















REYNOLDS NUMBER


1440


... .... .




2 t
7


A-ISM


Figure 1. Dye-streamline experiment (Evans Blue in water) by
Krovetz for steady flow through a glass branch. Pronounced secon-
dary flow and mixing occur at the branch even though upstream flow
is laminar.



























0 4)
1-I p



0rX

4J -4





N 0


O c
>1



0
>*cri












r-4


co 0
c .





0)
CO










4U
*r4


0 0
C64 0
Zr 0
M





1-1













ri i- ,C
I *U










d) 0
1-1 3 0)













numbers up to 10,000. In addition to observing helical flow, he also

observed cavitation (which he attributed to air dissolved in the

stream) at the highest Reynolds numbers in the regions of boundary-layer

separation along the outer walls. Fox and Hugh considered open-channel

flows and used aluminum powder sprinkled on the flowing surface to

trace disturbances in branched configurations. They postulated that

the boundary-layer separation and stasis along the outer branch walls

are contributing factors for the deposition of atheromatous placque

there.














SECTION II


THEORETICAL CONSIDERATIONS

In this section, we present the basic fluid-dynamic and optical

theory underlying the birefringence technique used and derive the

macroscopic balances of mass and momentum for a "generalized" branching-

flow system. Although the intent here is primarily to develop those

mathematical relations needed later for analyzing the flow photographs,

a brief discussion of flow birefringence, together with a review of

certain previous studies employing this technique, has been included.


A. Fundamental Equations

We start by considering the equations of continuity and motion

in their Lagrangian form8:



-2 = -p V v continuity (1)
Dt



D = t VP -V + pg motion (2)


Here, p is the fluid density, v the fluid velocity, P the

pressure, T the shear-stress tensor and the vector sum of body

forces per unit mass acting on the fluid. D/Dt is the substantial








14




derivative defined as + v V.



It is well known that these equations apply in general for any

continuous medium and in Lagrangian form apply specifically to elements

of fluid moving with the flow.

We wish to consider the set of equations resulting from (1) and

(2) for flow through a uniform cylindrical tube of inside radius R.

Anticipating the experimental flow conditions to be encountered later,

we use the cylindrical coordinates r, 0 and z and impose the following

assumptions upon Equations (1) and (2).


1. = 0
at


steady flow


2. p = constant

Trr

3. v = 0, T = 0
rz


4. z =, = 0
3z 7z -


Trz

0 ,-- 0
a2 -
0 0
zz


5. g = 0


incompressible
flow


angularly-
symmetric flow


fully-developed
flow


no body forces


The symmetry of the stress tensor is recognized in assumption 3, as is

the possible existence of normal stresses (such as would occur in a

Stokesian fluid).












From the continuity equation (1), we have by assumption 2

that V* v = 0. That is, in cylindrical coordinates


1 3(rvr) 1 DvQ
r ar r a



Since the second

and the third term zero

becomes simply


3z
az


term of Equation (3) is zero by assumption 3

by assumption 4, the continuity equation


(rvr) = 0
ar


Integrating Equation (4) directly, we obtain


rvr = G


where the function of integration G is independent of r.

If Equation (5) is solved for vr, then it becomes apparent that

G must be zero on the physical basis that the velocity component vr

cannot be infinite along the axis r = 0. We therefore conclude that

vr itself is zero and that the above assumptions describe a parallel

flow with vz the only remaining velocity component.

The components of the equation of motion (2) in accordance with

assumptions 1 through 5 are












P 1 3(rTrr)
Br r ar


r-component


O-component


aP = _1 3(r-rz)
az r 3z


z-component


Equation (7) shows that the pressure P is independent of 0 and

therefore dependent only on r and z.

If Equation (6) is differentiated partially with respect to z

and the order of differentiation interchanged on the right-hand side,

one has


a aP 1 a a(rTrr)
az ar r ar az


Since the right side of (9) is zero by assumption 4, it follows that


a ap 0
a T 0
Sz 3r


(10)


That is, the radial pressure gradient is independent of axial position.

Because the cross partial derivatives of P are equal, it follows that


a a 0
ar az


(11)


That is, the axial pressure gradient is independent of radial position.


(6)


-- = 0











According to Equation (8), the axial pressure gradient must also be

independent of both 0 and z, since differentiation of that equation

with respect to either 0 or z and interchanging the order of differen-

tiation on the right side renders the right side zero by virtue of

P
either assumption 3 or assumption 4 above. Thus,-- is independent of
3z
all three coordinates, and we have the important result that the axial

pressure gradient is constant.

Equation (8) may now be integrated directly to give



rTrz = (-) + F (12)



where the function F is independent of r.

Solving Equation (12) for I and noting that T cannot be
rz rz
infinite at r = 0 leads to the conclusion that F must be zero, and

therefore



r z= ( ) (13)
rz az 2



That is, the shear stress Trz is a linear function of radial position.

Equation (13) represents the starting point for the isochromatic

shear-stress calibration to be discussed in Section V. There we shall

find it convenient to drop the subscripts on irz and to consider

Equation (13) in terms of the wall stress Twas follows:




wR
W R












where w = ( )



The preceding derivation involved integrating the equations of

continuity and motion over an arbitrary cross section of a circular

tube. We now consider the more general problem of integrating these

equations over an arbitrary volume in space to obtain the so-called

macroscopic balances of mass and momentum.

This problem has been treated by Bird7 for a system consisting

of a "generalized" chemical plant having a single inlet flow and a

single outlet flow. We consider the equivalent problem for a "generalized"

branching-flow system in which no chemical reactions are occurring but

which may have any number of inlet and outlet streams.

The arbitrary region of interest consists of that volume V

bounded laterally by the walls of the branch and bounded at the inlets

and outlets by imaginary control surfaces which are assumed fixed in

space and time and oriented perpendicular to the direction of mean flow

through the inlet or outlet.

It is convenient to start from the equations of continuity and

motion in Eulerian form:



cE =- V. p v continuity (15)
at


S= -V Pvv -Vp-7V* + Pg motion (16)
at











These equations are easily derived from the corresponding Lagrangian

relations (1) and (2), but when written in the above form, they apply

to stationary elements of volume.

Integrating the continuity equation over V we have formally



fff dV = -fff V P v dV (17)
V V


By an extension of the Leibnitz rule for differentiating an

integral,8 the left side of (17) may bo written as



ffJ P dV = fff p dV-ff pv ndS (18)
V V S


where S refers to the surface bounding the branch, ys is the velocity

of this surface at any point and n is the outwardly directed unit vector

normal to the surface. Note that vs is zero for a rigid branch, but

would not be so in the cardiovascular system.

By the Gauss divergence theorem, the right-hand integral of (17)

becomes



ff V. pv dV= ff p v* n dS (19)
V S


where the surface integral is again taken over the bounding surface S,

and v is the velocity of the fluid.

Substituting Equations (18) and (19) into (17) and rearranging

terms gives












d f PdV = .-ff (v -.) n dS (20)
S
The left side of Equation (20) is the time rate of change of the

total mass M within the branch. The integrand on the right side vanishes

on all solid surfaces, and v vanishes on the control surfaces at all

inlets and outlets. If we now assume that the density p does not vary

across the inlet and outlet surfaces, and note that the normal n is

opposite in sense to v for inlet streams but in the same sense as v

for the outlet streams we may carry out the above integration to obtain



dt PiiFi j j j (21)
IN OUT

where the first summation is taken over all inlet streams and the second

over all outlet streams. The subscripts i and j refer to the ith inlet

stream and jth outlet stream respectively. Si and Sj are the cross-

sectional areas of these streams, and the average velocity V is by definition

ff S vdS
I dS

S

where v is the magnitude of v at the cross section of interest.

If we now define the difference operator A as meaning the sum over

all inlet streams minus the sum over all outlet streams, then Equation

(21) may be conveniently rewritten as


d = -A(p V S) (22)
dt










Equation (22) is the macroscopic or over-all unsteady-state

mass balance for the branching-flow system. Under steady-flow

conditions, dM = 0.
IF

Integrating the equation of motion we have


fff 2(-) dV = -fff(V p v v)dV -fff (VP)dV -fff (V7. )dV (23)
V V V V

+ fff (pg)dV
V

The left side of this expression may be transformed as before

to give


fff a(P) dV = d fff (Pv) dV -ff(Pv)v n dS (24)
V t dt V S


The first three integrals on the right side of (23) may be transformed

to surface integrals by modifications of the divergence theorem7 as

follows:



fff (V. p v v) dV = ff (p v)v n dS (25)
V S


Jff (V P)dV = (f P n dS (26)
S


(ff (V- X ) dV = f5 n dS (27)
V S












Substituting Equations (24) through (27) into (23) and

rearranging terms we have



d fff pv dV = -ff pv (v- V) n dS -ff P n dS (28)
dt -
V S S

-ff (c n)dS + fff p dV
S V

The left side of Equation (28) is the time rate of change of

the total momentum P within the branch. Again, the first integrand on

the right side vanishes on all solid surfaces, and ys vanishes at all

inlets and outlets. The remaining contribution from this term points

in the direction of the flow both at the inlets and outlets. The second

term on the right contributes to the influx and efflux of momentum at

the inlets and outlets respectively and to the drag force on all solid

surfaces. The third term represents the viscous contribution to drag on

solid surfaces and gives a usually negligible contribution to momentum

transport at flow inlets and outlets. Finally, the last term on the

right represents the vector sum of all the body forces body acting on

the fluid.

Using the previously defined A notation, we may write the macroscopic

momentum balance for the branching-flow system as




d = A (pvS + PS) -- F + y (29)
dt -drag body











where the vector S has a magnitude equal to the cross-sectional area

of the inlet or outlet of interest, but points in the direction of the

flow. Fdrag and v2 are defined as follows:



-;-rag-
^drag = Pn dS+ffl *.ds
Ssolid Ssolid


ffS 2dS
vfZ fdS
S

dP
Again, for steady-flow conditions --= 0.
dt


B. Flow Birefringence


According to Jerrard,23 "the first published observations on
30
flow birefringence were made in 1873, when Mach30 observed double

refraction in extremely viscous substances such as strong metaphos-

phoric acid and Canada balsam, when poured into a beaker." Soon there-

after, Maxwell31 reported that he had experimented with birefringence

as early as 1866 and had devised a concentric-cylinder apparatus for

studying the phenomenon in Canada balsam. Since that time, flow

birefringence (also called streaming birefringence and flow double

refraction) has been observed in many liquids, and numerous theories

of flow birefringence have been proposed. From the extensive reviews

of these theories presented by Jerrard23 and by Cerf and Scheraga,9

it is evident that no single theory has been successful in predicting













all birefringence phenomena for liquids.

Though most of the early studies of flow birefringence were

directed toward using this as a technique for investigating molecular

size and shape, within the last forty years, many investigators have

reported using birefinngence techniques to study the properties of

flows themselves. Attinger's recent studies3'4 of eddy formation and

flow disturbances both in distensible tubes and in rigid tubes of circu-

lar and elliptical cross section, are typical examples of the qualitative

use of flow birefringence. The most significant quantitative work has

been that by Prados41 and Prados and Peebles,40 who showed that the bire-

fringence technique could be used to determine shear-stress distributions

and velocity profiles for flows through various types of flat channels.

They used aqueous suspensions of Milling-Yellow dye and obtained good

agreement between the flow rates calculated from birefringence measurements

-and the experimentally determined values. Essentially no quantitative

work has been attempted for three-dimensional flow systems, and,apparently,

the first attempt to use flow birefringence for studying unsteady or pul-

satile flow was the 1957 work of Thurston and Hargrove cited by Prados and

Peebles.40 It is probable that the present study represents the first

attempt to use flow birefringence for studying pulsatile flows in branches.

As is the case for photoelastic solids, flow birefringence in

liquids results from the optical anisotropy induced by a shearing field.

The phenomenon is most pronounced in solutions or suspensions containing

asymmetric particles, although it has been observed at high shear rates














in pure liquids (and Newtonian fluids), where the anisotropy

presumably arises because of molecular deformation rather than molecu-

lar reorientation.

A typical flow-birefringence experiment requires that an incident

beam of plane-polarized light (from the polarizer) be passed through

the flowing liquid in a direction perpendicular to the direction of

flow. The observer then "analyzes" the emerging light by viewing this

light through a second polarizer (the analyzer) which is oriented so

that its plane of polarization is crossed 90 with respect to that of

the incident light. It is found that when the liquid is at rest, the

incident beam of polarized light emerges unchanged from the liquid and

is extinguished by the analyzer. Thus, the entire field appears dark

to the observer under zero-flow conditions. However, when the liquid

is set in motion, the observer sees a series of interference bands dis-

tributed through the flow field. These bands appear dark when monochro-

matic light is used and colored when white light is used, and their

presence indicates that only under certain conditions does the incident

beam emerge unchanged from the flowing liquid.

Although the above description pertains to the use of plane-

polarized light, a similar set of results is obtained when circularly

polarized light is used. An incident beam of circularly polarized

light can be produced by inserting a quarter-wave plate between the

polarizer and the liquid such that the optic axis of the quarter-wave













plate makes an angle of 450 with the plane of polarization of the

polarizer. The effect of this quarter-wave plate is to decompose

the incident plane-polarized vibration into two components vibrating

at right angles to one another and 90 out of phase. These are pre-

cisely the properties of circularly polarized light. When circularly

polarized light is used, it is also necessary to insert a quarter-wave

plate between the fluid and the analyzer. This plate is oriented so

that its optic axis is crossed 90* with respect to that of the first

quarter-wave plate. If a light beam emerges unchanged from the fluid

(that is,circularly polarized), the second quarter-wave plate changes

the vibration back to a single plane-polarized beam vibrating perpen-

dicular to the plane of polarization of the analyzer. The beam is then

extinguished. Regardless of the type of polarized light used, it is

true that the observer will see an interference band in the flow field

only at positions where the incident light emerges unchanged from the

liquid.

The effects of passing beams of plane and circularly polarized

light through doubly refracting materials and the conditions under which

such beams will emerge unchanged have been treated analytically by

Rosenberg.43 Basically, if a beam of plane-polarized light enters a

birefringent liquid, this beam will be decomposed into two component

vibrations, one vibrating parallel to the optic axis of the fluid and

the other perpendicular to this axis. Initially, these two vibrations












are in phase and may be represented by



x = a cos (21t) (30)



y = b cos(2nA) (31)
T


where x is the displacement of the component vibration parallel to the

optic axis of the fluid, y the displacement of the component vibration

perpendicular to this axis, a the amplitude of the x-vibration, b the

amplitude of the y-vibration, and T is the period of vibration.

Owing to the optical anisotropy of the flowing liquid, these two

vibrations will in general propagate through the liquid at different

speeds and will emerge out of phase by an amount dependent upon the path

difference (or relative retardation) 6 between them. The emerging

vibrations may therefore be expressed as



x = a cos(2w-t) (32)
T


y = b cos 2ir(- + ) (33)


where X is the wavelength of the incident beam.

If the emerging components are now recombined by eliminating t,

the resulting equation is



+ 2 (22)
x2 + 2xy cos (2i) sin2 4)
ay yZ ab sin ( (34)












Since Equation (34) describes an ellipse, the emerging beam will

in general be elliptically polarized. However, if one compares Equa-

tion (34) with Equations (30) and (31), it is apparent that under the

following conditions the beam will emerge unchanged



6 = NX, N = 0, 1, 2.... (35)



a = 0 or b = 0 (36)



The first of these conditions governs the appearance of the

so-called isochromatic bands (or isochromatics), while the second con-

dition specifies the appearance of the so-called isoclinics. Thus,

when plane-polarized light is used, dark fringes can appear at the analy-

zer from both isochromatics and isoclinics.

Equations (35) and (36) show that the isochromatic bands are

related to the magnitude of the path difference 6, while the isoclinics

appear only when the incident plane-polarized vibration is parallel

(b = 0) or perpendicular (a = 0) to the optic axis of the fluid. It

has been found experimentally that the path difference 6 is dependent

upon the length of the light path and upon the average shearing stress

along this path while the orientation of the optic axis is dependent upon

the direction of shearing stress along the light path. Thus, it is clear

that the isochromatics are related to both the magnitude of the shearing

stress and the length of the light path, while the isoclinics are related












to the direction of shearing stress along the light path.

The analysis is similar for circularly polarized light.

However, since circularly polarized light consists of two mutually

perpendicular vibrations which are of equal amplitude but 90 out of

phase, each of these vibrations will be decomposed into components

vibrating parallel and perpendicular to the optic axis of the fluid.

When the appropriate contributions from the incident beam are added,

the components propagating through the fluid may be represented by



x = a cos 2t' (37)
T


y = a sin 2t' (38)
T


where a is the amplitude of the incident beam, T the period of vibra-

tion and t' = t + y with e being the angle of rotation between the
27r
abscissa describing the incident beam and the optic axis of the fluid.

Again, these components will undergo a relative retardation

resulting in a corresponding phase change. Upon emerging from the fluid

the components can be represented by



x = a cos 2'-T (39)
y T a sin 2

y = a sin 2w( + 6) (40)
T












Equations (39) and (40) can be combined by eliminating t' to

give



x2 + y2 -2xy sin(2i -) = a2 cos2(2ra) (41)
A A


Equation (41) also describes an ellipse, and,hence, when

circularly polarized light is used, the emerging beam will in general

be elliptically polarized. However, comparison of Equation (41) with

Equations (37) and (38) shows that the circularly polarized beam will

emerge unchanged only if 6 = NX. Therefore, when circularly polarized

light is used, only the isochromatic bands can appear in the flow field.

It has been shown that the condition 6 = NA corresponds to con-

structive interference of the emerging vibrations for both plane and

circularly polarized light. Since the two vibrations are known to

travel at different speeds through the fluid, constructive interference

also implies that the difference between the number of waves of each

vibration in the fluid is an integer N. Because the difference in

wave speeds of the two vibrations occurs from changes in wavelength and

not changes in frequency,52 constructive interference requires that



(A/Xx y) = N (42)


where k is the length of the light path, Ax the wavelength associated with

the vibration parallel to the optic axis and y the wavelength associated
y












with the vibration perpendicular to the optic axis.

Since the frequency of vibration f is unchanged in the fluid,

the wave speeds must obey the following equations



Cx fx (43)



Cy = fXy (44)


The wave speed of any light ray in vacuo is given by


c = fX (45)


where c is the speed of light, f the frequency of vibration and X the

wavelength of the vibration in vacuo.

If Equation (45) is now divided by Equations (43) and (44) the

following relations are obtained:


X/Ax = c/cg = nx (46)



X/Xy = c/y = ny (47)


where nx and ny are by definition the indices of refraction for the two

vibrations.

Equation (42) may now be rewritten in terms of the indices of

refraction nx and ny as follows:


(nx -ny)S = N.


(48)












This expression gives the important relation between the so-called

amount of birefringence (nx -n ) and the isochromatic-band order N. The

quantities 2nx and by are know as the optical path lengths for vibra-

tions x and y respectively.52

According to Equation (48) the phase difference in radians between

the two vibrations is given by


A = 27r(nr-ny) (49)



This is the result given without derivation by Jerrard.23

The quantitative application of flow birefringence begins with

Equation (48). The usual assumption employed states that the amount of

--birefringence (nx -ny) is a linear function of the average shear stress

T(avg) along the light path. The results obtained by Prados40 indicate

that this is a good assumption for suspensions of Milling-Yellow dye.

This assumption may be stated mathematically as


(nx ny) = MT(avg) (50)


where the coefficient M is known as the Maxwell constant.

If the assumption given by Equation (50) is substituted into

Equation (48) one has


NX = MAR(avg) (51)

Defining M* as M/X, and rewriting Equation (51) in terms of the








33



isochromatic-band order N we have finally


N = M* T(avg)


(52)


This expression represents the starting point for the analysis

of the flow photographs to be presented in Section V.














SECTION III


EXPERIMENTAL EQUIPMENT


The production of flow-birefringence patterns requires a

relatively simple experimental setup. Basically, the birefringent

liquid must be caused to flow through a transparent test section

located between two crossed polarizing filters (it is common practice

to refer to the incident-light filter as the polarizer and to the second

filter as the analyzer). A light source, which may or may not be mono-

chromatic, must be oriented such that the incident unpolarized light

passes through the polarizer, test section and analyzer respectively.

The observer, looking through the analyzer in the directionof the light

.source, sees optical interference patterns wherever shear stresses are

present and a dark field elsewhere.

The experimental setup used in the present study was designed with

simplicity and ease of operation in mind and is pictured schematically

in Figure 3. For purpose of discussion, this setup may be conveniently

subdivided into three major components: A. The fluid-perfusion system,

B. The flow test sections and C. The optical system. Where pertinent,

the important pieces of peripheral equipment are also discussed.



A. The Fluid-Perfusion System

Since flow birefringence occurs under dynamic conditions,

































C:



24
4<
u 4


z
w
w
U)
u


v,

I3
U.
L


I-
2:
-j
-J


ma




C
a4 4J
0


4
0 frt


o4 0).


o 0
a) 4-4
cU I




04

0
0 OH
co

0 *


X e

0


S4


*rd 0 w
w) ct
1 *c4












00
4*N
CY) C4
143






00n
M 0
Ocd













H 4-) 1
CA D:

60 i <
l-l a)
P^ mf













provision must be made for either recirculating the birefrigent fluid

through a test section or collecting it after a single pass. It was

decided that for branching-flow studies, and in particular for deter-

mining branch flow ratios, the best approach was to provide for a

single pass through the test sections with fluid collection and metering

at the outlets. Direct unobstructed flow metering using graduated cylin-

ders was thus possible, assuring the experimenter that observed flow

ratios were due to branch configuration rather than side effects of

flow metering. The single-pass approach provided the additional advan-

tage of eliminating the need for a constant-temperature bath to remove

viscous heat caused by continuous recirculation. Change in fluid vis-

cosity due to viscous heating was a difficulty reported by Peebles,

Prados and Honeycutt in a previous study.37

General requirements were that the fluid-perfusion system be

-capable of producing both steady and pulsatile flows over the same range

of flow:.ates, that the fluid be maintained at constant temperature and

isolated from outside vibration during experimental runs and that the

flow system be leveled so as to eliminate gravitational effects in the

flow test sections.

Steady flows were generated using the flow system illustrated

in Figure 3. This system consisted of a glass tank approximately 30 cm

high which could be sealed with a rubber stopper and pressurized using

an external compressed-air supply. Though a larger constant-head tank













was available in the laboratory, accurate flow control over a much

wider range of flow rates was possible using the pressurized system.

A pressure gauge permitted tank air pressure to be monitored to better

than 1 mm of mercury over the range 0 to 200 mm of mercury.

The fluid tank was equipped with a tapering outlet nozzle

converging to an outside diameter of 9 mm, just sufficient to match

the outside diameter of the glass connecting tube joining the fluid

tank to the flow test section. This provided for a smooth transition

flow from tank to connecting tube. The connecting tube used for all

experiments was a 3.00 mm diameter section of precision-bore glass

tubing, 96 cm in length. This length was sufficient to insure that

fully developed flow entered the test sections.

A tight-fitting rubber sleeve over the nozzle and connecting

tube served both as a seal against leaks and a structural support, thus

assuring a secure flush connection between nozzle and connecting tube.

In addition, this sleeve provided a simple means for adapting the flow

system from steady-to pulsatile-flow operation. To do this, it was

simply necessary to slide the rubber sleeve off the connecting tube

while pinching it to arrest any flow due to fluid in the tank. The

glass tube could then be connected directly to the pulsatile-flow source

using a section of rubber tubing.

Pulsatile flows simulating heart action were generated using a

positive-displacement rotary bellows pump (Model 1000, manufactured by

Research Appliance Company, Pittsburgh, Pennsylvania). This pump

employed a rotating eccentric cam to displace fluid through a ball valve.













The resulting flow wave was measured independently using a square-wave

electromagnetic flowmeter (Model 301, manufactured by Carolina Medical

Electronics Company, Winston Salem, North Carolina) and is shown in

Figure 4.

Whether operating under steady-or pulsatile-flow conditions, the

entire fluid-perfusion system was mounted on sturdy laboratory tables

and maintained in an air-conditioned laboratory at a constant ambient

temperature of 250C. External vibration was small but could be mini-

mized by using felt pads for supports and by turning off the air-condi-

tioning system during experimental runs.


B. The Flow Test Sections


There are several desirable properties for flow test sections

to be used in birefrigence work. Among these are that they should be

as nearly transparent as possible and of reasonable cost. They should

also be easily fabricated from a material in which residual stresses

can be removed. If the last requirement is not met, the resulting defor-

mation of crystal planes inthe material may render it optically active

with respect to polarized light, hence interfering with experimental

results.

While there may be no ideal material in every respect, previous
40 41
studies441 have reported success using Plexiglas test sections. This

material can be fabricated with relative ease and is reasonably priced,
















0


*1-
4J1
0l


VI 0
- a-



LEl
'0 -

o ix 0
pr 10


*O 0


o *0 M



2 z
V IL 0
u- V)










a- A



0 I-'
oo fa
-~ 0



WCu









( >












although the cost of labor for fabrication may be considerable.

Plexiglas is optically active in polarized light when residual stress

is present, but this can be removed by proper temperature curing. Such

test sections can be polished both inside and out to a high degree of

transparency, rendering them ideal from the optical point of view..

The flow test sections used in the present study have been

listed in Table 1. With the exception of sections 1, 2 and 7, all test

sections were constructed of drilled Plexiglas block and were fabricated

in various branched configurations in an attempt to simulate typical

branches found in the cardiovascular system.

Fabrication cost imposed a hard limit on the number and variety

of branches which could be investigated.

Figure 5 is a photograph of the branched test sections listed in

Table 1. Over-all size of these branches was limited by the lengths of

the drills available. Test section 7 was milled to a 3 mm wide by 2 mm

deep rectangular cross section so that an experimental comparison could

be made regarding the effects of internal geometry upon optical and flow-

ratio behavior.

Great care was required in drilling the Plexiglas block, as

eccentric motion of the drill caused internal burring which could not

be removed by later polishing. Care was also necessary to insure that

branches met at the proper angles and lay in a plane paralleling that of

the broad face of the Plexiglas block.

The upstream faces of all blocks were drilled to a depth of








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approximately 1 cm on center with the upstream branch stem to

accommodate the glass connecting tube with a snug and flush internal

fit. Duco cement was used as a leak seal around the connecting tube

on the upstream block faces. Figure 5 shows the connecting tube joined

to test section 6. Branch outlets were mounted in a similar fashion and

very satisfactory results were obtained using Plexiglas tubing for these.


C. The Optical System


There are three basic requirements to be considered in designing

the optical system. First, this system must be oriented such that the

light path is perpendicular to the direction of flow being studied.

Secondly, it should be possible to produce incident circularly polarized

light as well as plane-polarized light of arbitrary angular orientation.

Finally, it should be possible to photograph the resulting birefrigence

patterns. Figure 3 illustrates the optical scheme adopted.

The light-diffusing screen, consisting of a 60 cm square plate

of smoked glass, was mounted horizontally on runners between the two

laboratory tables. Mounted in this fashion, the screen provided both a

uniformly lighted background and a rigid level support for the polarizer

and test sections.

Two Sylvania flood lamps (type R-32, color temperature 32000K,

375 watt), constituting a white.light source, were mounted on ring stands

which rested on the floor. These lamps could be easily raised to compen-

sate for intensity loss due to aging, and heat could be dissipated by a












fan. A white-light source was chosen, because it was felt that the

resulting colored bands would be easier to distinguish in the final

photographs than the light and dark bands obtained with monochromatic

light.

As Figure 3 shows, the polarizer was mounted directly on the

light-diffusing screen. Plane-polarized light of various orientations

was produced using the edge of the screen as an azimuthal angular

reference. A second polarizing filter mounted directly on the camera

lens served as the analyzer.

Circularly polarized light could be produced by inserting

quarter-wave plates (not shown) between the polarizer and test section

and between the test section and analyzer respectively. Again, the edge

of the diffusing screen was used as an angular reference for properly

inserting these.

The camera used was a Bolex Paillard H-16 (16 mm) Reflex motion-

picture camera equipped with a 25 mm lens (1 : 1.4 Switar Rx). This lens

in conjunction with 5 mm and 10 mm extension tubes permitted photography

from a distance of a few centimeters. The film used was Ektachrome (ER

type B) and a Weston light meter (Model 745) was used to check light

intensity at the camera. The camera was mounted on a tripod taped securely

to the floor. This arrangement permitted easy focusing using the tripod

crank. Once focused, the camera could be locked in position and the

reflex viewing used to accurately check on proper analyzer orientation

by light extinction under zero-flow conditions.













SECTION IV


EXPERIMENTAL TECHNIQUE


The experimental work was composed of four parts: A.

Preparation and storage of the dye suspension, B. Determination of

fluid physical properties, C. Flow regulation and collection and D.

Photography.


A. Preparation and Storage of the Dye Suspension


The doubly refracting liquid used was a 1.4% aqueous suspension

of the aniline dye Milling Yellow (obtained from National Aniline

Division, Allied Chemical Company). The suspension was prepared directly

from the commercial product, a yellow powder, using the method suggested
36
by Peebles, Garber and Jury.3

Distilled water, about 50% in excess of that desired for the final

volume, is heating to boiling. To this is added a thick paste of dye

powder and distilled water, sufficient to make a 1% suspension by weight.

Boiling and agitation are allowed to proceed until excess water has evapo-

rated. Since birefringent sensitivity is markedly affected by dye concen-

tration, the final suspension should contain 1.3 to 1.6% dye for best

results. The double-refraction property may be tested during evaporation

by withdrawing a small sample into a test tube, cooling to room temperature

and observing whether or not interference patterns are produced when the












tube is agitated between illuminated crossed polarizing plates.

After the dye suspension was prepared in sufficient quantity,

it was stored in sealed polyethylene tanks until needed for use. It

was found that suspensions stored in this fashion were very stable with

respect to particle agglutination over a period of two weeks, after

which some coagulation occurred.


B. Determination of Fluid Physical Properties


To date, the most extensive physical-property studies of Milling-

Yellow suspensions are those of Peebles, Prados and Honeycutt37 estab-

lishing the birefringent and theological behavior. Their studies have

shown that viscosity data for Milling-Yellow suspensions are accurately

correlated by the Powell-Eyring equation for pseudo-plastic materials.

Little work has been done to establish other physical properties for

such suspensions.

The physical properties required for the present study were

viscosity, density and static refractive index of a 1.4% dye suspension

at 250C.

Viscosity data were obtained by using the 96 cm length of

precision-bore 3 mm glass tubing as a capillary viscometer. The dye

suspension was permitted to flow through the tube under steady-flow con-

ditions for several different but known pressure gradients. The corres-

ponding flow rates were then measured and the Hagen-Poiseuille equation

used to calculate apparent viscosity (that is, the viscosity which a New-












tonian fluid would have flowing through the tube at the same flow

rate under the same pressure gradient). Typical experimental results

are listed in Table 3, Appendix B, and Figure 6 shows apparent viscosi-

ties and the corresponding Reynolds numbers plotted as functions of

flow rate Q in the 3 mm diameter tube. These data are in strong agree-

ment with the data reported by Peebles, Prados and Honeycutt.37

Fluid density was determined by weighing 25 ml aliquots of the
3
-dye suspension at 25"C. The density was found to be 1.00 gm/cm3

Staticrefractiveindex measurements were made using a Bausch

and Lomb refractometer. The static refractive index at 250C was found

to be 1.338 for the 1.4% suspension. Static refractive index, however,

should not be confused with the indices of refraction associated with

the ordinary and extraordinary rays when the fluid is in motion. It is

not clear just what relationship the static index bears to these two,

though Peterlin and Stuart38 have taken it to be the mean value of the

dynamic indices.


C. Flow Regulation and Collection


The procedure for steady-flow experiments was as follows: Prior

to filling the pressure tank with fluid, short rubber tubes were fitted

over each flow outlet and clamped to prevent flow. During the experi-

ments, these tubes were unclamped and served to divert outlet flows to

large collecting beakers without altering flow distributions in the

test sections.
















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The pressure tank and test section were positioned for

photography and the tank filled to a depth of 20 cm with the Milling-

Yellow-dye suspension. The laboratory compressed-air supply was then

connected to the tank via a three-way valve opened initially to the

atmosphere. The pressure gauge in series between the valve and tank

permitted tank air pressure to be monitored directly.

For an experimental run,the outlet flows were opened and the

valve partially closed until the desired driving pressure was registered

on the pressure gauge. Successive experimental runs were made by incre-

menting the driving pressure in steps of 10 mm of mercury. Before and

after each run, the three-way valve was opened to the atmosphere, thus

reducing excess tank pressure and lowering the outlet flows. This also

permitted the liquid level in the tank to be returned to the 20 cm

level. Depending upon the flow rate, experimental runs lasted for 30

seconds or 1 minute. However, the tank was of sufficient capacitance

that the liquid level did not drop more than a few millimeters during any

experimental run. Outlet flows were measured with graduated cylinders.

For the pulsatile-flow experiments, the pressure tank was replaced

by the bellows pump described previously. Since the flow output of this

pump was limited to the flow wave already shown in Figure 4, no flow

regulation was possible here.


D. Photography


Prior to actually photographing the experimental birefringence

patterns it was necessary to standardize several important photographic












variables such as light intensity, framing speeds, exposures, depth of

field, the film used and processing.

Several preliminary runs were made to test these factors using

a color standard made from strips of Kodak red, green and yellow Wratten

filters. It was found that Ektachrome film (ER type B) with a correspon-

ding light source having a color temperature of 3200K gave very good

color reproduction when proper exposures were used. All films were pro-

cessed by the same company so as to minimize any differences due to

processing.

Experimental birefringence patterns were photographed with the

camera shooting down at the test section mounted in the horizontal plane.

After the test section was positioned, the desired exposure and depth of

field were set and the lights turned on. The camera was then focused

and locked into position. The reflex viewing was used to assure that

polarizer and analyzer were crossed by checking for maximum light extinc-

tion under zero-flow conditions. Light intensity was checked at the

camera using a Weston light meter (Model 745).

The flow was started, and after the desired flow conditions were

achieved, the resulting birefringence pattern was photographed. Steady

flows were photographed at 16 frames/sec and pulsatile flows at 64

frames/sec.

The pertinent photographic data for the flow photographs appearing

in the next section have been listed in Table 4, Appendix C.















SECTION V


THE ANALYSIS OF FLOW-PATTERN PHOTOGRAPHS
AND EXPERIMENTAL RESULTS


In this section, selected frame enlargements of the 16 mm

film are presented together with experimental data obtained from flow

measurements and frame-by-frame analysis of the film.

The section is divided into three parts. In Part A, the

isochromatic-band pattern obtained for a fully developed, angularly

symmetric laminar flow through a tube of circular cross section is

considered, and a method is proposed for calibrating this pattern

against shearing stress, including the effects of curved boundaries

and differences in refractive index in the analysis. In Part B, steady-

flow data and isochromatic photographs are presented, and birefringence

theory together with the calibration of Part A is applied to estimate

shear-stress distributions for flows around a cardiac catheter and

through a 90 branch. Experimental results are then compared with

results from the medical literature concerning wall trauma and athero-

sclerotic disease at sites of branching in arteries. In Part C, branch-

entrance-length data for the pulsatile-flow experiments are presented,

and discussion is focused on the areas of stasis, boundary-layer sepa-

ration and increased shear readily observed in the films.











A. Isochromatic Shear-Stress Calibration
for Laminar Tube Flow

The problem of isochromatic calibration in flow-birefringence

work is that of associating an appropriate value of shear stress with

any given isochromatic band in the flow field. In general, this

requires that a birefringence experiment be run under conditions of

known geometry, shear-stress distribution and isochromatic-band distri-

bution. A correspondence or calibration may then be drawn between the

shear field and isochromatic field and this calibration used to determine

the shear field in geometries where only the isochromatic field is known.

It will be recalled from Section II that previous studies have

demonstrated experimentally that fringe order N is directly proportional

to the product of T(avg), the average shear stress along the light path,

and X, the length of this path in the fluid. According to Equation (52)

that relationship was given as N = M*ZT(avg) where M* is a characteristic

of the particular fluid. Although M* is not constant for all fluids, the

data of Prados41 show that this is the case for Milling-Yellow suspensions.

It is evident from Equation (52) that for a three-dimensional

flow system in which the light-path length (fluid depth) associated

with a given band N is constant, the shear stress T(avg) associated

with this band will also be constant. However, if this light-path

length varies along the band, then the corresponding shear stress will

vary inversely as the path length, and there will no longer be only one

appropriate shear stress associated with the given band. A reasonable













approach for this situation would be to establish some reference point

along the band (perhaps upstream from the position of interest), and

to refer the stress at other positions on the band to the value at this

point. This, of course, requires that the light-path length be known

at the downstream position.

Previous experimentalists utilizing flow birefringence were able

to avoid this difficulty associated with test sections of varying depth

by using essentially two-dimensional systems (such as high-aspect-ratio

channels).

As will be seen in Part B, the isochromatic bands obtained in

the present study represent a combination of the above two cases, with

the undisturbed upstream state corresponding to constant light-path

length in the fluid and the distorted downstream state to variable

length. For this reason,we shall establish the upstream-flow condition

as a reference state for branching flows and turn now to the problem of

calibrating the isochromatic-band distribution obtained for laminar tube

flow.

We consider a steady, incompressible, fully developed laminar

flow through a cylindrical tube. A typical isochromatic-band distribu-

tion observed for such a flow is that shown in Figure 7 for test section

1. The flow here is from left to right at a Reynolds number of 50

(based on apparent viscosity and inside tube diameter), and the incident

light is plane polarized perpendicular to the axis of the tube. It is











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seen that the bands parallel the direction of flow and are not displaced

from their original radial position as they pass downstream.

In seeking to associate a characteristic value of shear stress

with each band, the observer must remember two things: First, successive

bands seen in the flow photograph are the net result of viewing the

flow through a curved wall, and,hence, through varying liquid depths in

the radial direction. Second, since curved boundaries are involved, it

is necessary to consider the possibility of optical distortion arising

in the flow photograph due to differences in refractive index between

the different media involved.

A cross-sectional representation of the over-all situation is

shown in Figure 8. For convenience, the coordinate origin 0 has been

chosen at the center of the tube with lines O-N1 and 0-N2 representing

normals to the tube surface at points B and C respectively. Each of

the three media involved is characterized optically by a static index

of refraction (nl, n2, n3). The path of a typical light ray as it

enters the fluid from below and passes to the observer is indicated by

line DCBA, and it will be noted that with the exception of the ray

passing through the center of the tube, all rays seen by the observer

pass obliquely from one medium to the next, thereby undergoing refraction.

This refraction is characterized by the angles 01, 02, 03 and 04 which

are predictable from geometrical considerations and Snell's law of

refraction. These relationships will be considered presently. The











Y
OBSERVER


LIGHT- DIFFUSING SCREEN


Figure 8. Schematic cross section of a circular tube, illustrating
the refraction of a typical light ray passing from the fluid to the
observer. Light path DCBA is characterized by angles 01,02, 03 and
04. These angles are determined by the parameter K and Snell's law
of refraction. nl, n2 and n3 are indices of refraction. N1 and N2
are normals.












observer (camera) viewing from above sees only those light rays

emerging parallel to the y-axis, and this implies that chords such as

DC which are the light paths in the fluid corresponding to isochro-

matic bands seen the the flow photograph, will not in general be parallel

to the y-axis.

For the present analysis, we shall choose chord DC as being a

typical chord associated with an isochromatic band seen at point A in

the photograph and shall identify this chord by its perpendicular dis-

placement from the origin. The lateral displacement from the y-axis

of the band at point A is, of course, dependent on the chord's displace-

ment from the origin, and this dependence will now be established by

considering the above-mentioned relationships.

It should be mentioned that there is some question as to the

justification for using Snell's law of refraction for a doubly refrac-

ting material, since it is known that the ordinary ray obeys this law

but the extraordinary ray does not. In addition, each of these rays

has its own dynamic index of refraction, so the use of the static index

must also be justified. Actually, these questions pose no difficulty

here, because both rays travel the same path, and, as Peterlin and

Stuart have pointed out,38 the difference between the three indices is

very small. Therefore, for the purpose of determining the light path,

the static index and Snell's law may be used.

Referring again to Figure 8, one may write the following relations

between angles 01, 02, 03 and 04;












n1 sin 01 = n2 sin 02 by Snell's law (53)

n2 sin 03 = n3 sin 04 by Snell's law (54)

R cos(03-02) + BC cos 02 = KR Geometry (55)

R sin(03-02) = BC- sin 02 Geometry (56)

If 81 is considered the independent variable (this can be done because

of the principle of optical reversibility52), then these equations may

be solved simultaneously to give


-1 n1
02 = sin (nH sin 01) (57)
Sn1
83 = sin-('-a2-sin 1) (58)
-1 n,
94 = sin (K-.- sin 01) (59)


-where the parameter K is simply (outside diameter)/(inside diameter).

For the special case of a hole drilled parallel to a flat face of a

Plexiglas block, these equations still apply providing K is given a

numerical value of unity and nI is replaced by n2. This is equivalent

to immersing the observer in the Plexiglas, since perpendicular viewing

would create no refraction at the air-block interface.

One notes from Equation (59) that the angle 04 is completely

independent of medium 2 (that is n2). This is a well-known result in

optics. Another interesting and important observation is that the

diameter ratio K plays a central role in determining how much of the

flow field is actually visible to the observer. This fact may be

appreciated by considering the following relationships, which may be












deduced from equations (57) through

study n2>n3>nl.


(59) and the fact that in this


THEN


K = 1.0 and n1 = n2
(Plexiglas-block case)
n3
1 nl

n3
K n
n1


"3 n2
n3 < K < --
n, n1


n1

n2
K > --
n
"1


04 > 01 3 = 02


01 > 04 > 03 > 02


e4 = 81 > 83 > e2
04 = 01 > 3 > 02



04 > 01 > 03 > 02


04 > 01 = 03 > 02


04 > 03 > E1 > 82


These relationships show that with the exception of the second case (61),

04 is always greater than or equal to 01. This is important, because as

shown in Figure 8, the magnitude of 04 must range from 0 to 90 if the

observer is to see the entire flow field. At the same time, it will be

noted that 01 physically can be no greater than 90. If case (61) holds,

that is, if K falls between 1.0 and n3/n1, then 04 will be less than 01,

and the observer will not be able to see the entire flow field but only
-1 i -1 n1
the part of it for which 4 ranges from sin (K--n-) to + sin (K-

as dictated by Equation (59). Thus, an experimenter conducting


(60)


(61)


(62)



(63)


(64)


(65)











flow-birefringence experiments employing circular tubes and wishing

to visualize the entire flow field must consider the geometrical range

1 < K < n3/n1 a forbidden zone when choosing experimental test

sections.

For the flow pictured in Figure 7, the appropriate numerical

values are K = 3.0, nI = 1.00, n2 = 1.5 and n3 = 1.338. Since

this value of K is greater than n3/nl, one is assured that the entire

flow field is visible in that photograph. In fact, the entire flow

field is visible in all photographs appearing in this study. For the

branching-flow figures the appropriate numerical values are K = 1.0,

n1 = n2 = 1.55 and n3 = 1.338.

It can be seen from Figure 8 that the perpendicular displacement

of the typical chord DC from the origin is given by R sin 04, and that

the lateral displacement of the corresponding band at point A is KR

sin 01. Defining a distortion ratio as (band displacement)/(chord dis-
sin 01
placement), one obtains a distortion ratio of K From
sin 04
Equation (59) the value of sin 01/ sinO4 is n3/Kn1. Substituting this

above gives a distortion ratio of n3/n1. Thus, our analysis shows that

for circular geometry the distortion ratio is constant and independent

of both the dimensions and the material of medium 2 (the tube).

Furthermore, this distortion represents a magnification effect with the

distortion ratio n3/n1 as the magnification factor. In the present study,

this factor has the value 1.338 for the glass tubes; but since nI must be

replaced by n2 for the Plexiglas-block studies, the factor becomes 0.865












for them. In either case, the only optical distortion arising from

curvature is magnification or shrinking of the original image by a con-

stant known factor. We can use this information to obtain one more

important result. Let the band displacement at point A be denoted by

"b" and the corresponding chord displacement by "a". Then, according

to the above discussion, b = an3/n1. The displacement b will have its

maximum value b(max) when a has its maximum value a(max) = R. Thus,

b(max) = Rn3/nl. If both displacements are now made dimensionless by

normalizing them to their maximum values and the fractional displacements

"8" and "a" defined as 8 = b/b(max and a = a/a(max), we obtain

the important result that B = (an3/nl)/(Rn3/nl) = a/R = a. This

means that the fractional displacement dimensionlesss radius) of an

isochromatic band measured in the flow photograph is precisely equal to

that of the corresponding chord (light path) measured inside the tube.

Now, having related the geometrical properties of the band in the

flow photograph to those of the corresponding light path in the fluid,

we are in a position to derive the expression relating the fractional

displacement a (or 8) to the average shear stress along the chord DC.

Up to this point, no mention has been made of the nature of the

shear~stress distribution in the fluid. From here on, however, it will

be assumed that the flow field in the tube is angularly symmetric, which

implies that the stress field is also angularly symmetric. The assumption

of angular symmetry was verified experimentally for the band distribution












shown in Figure 7.

In Section II it was shown that for the steady, incompressible,

angularly symmetric and fully developed flow being considered here

the equations of motion could be integrated to give the shear-stress

field as a linear function of radial position r. That function was

given by


T r (66)


aP R
where the wall stress Tw is (--) --


The problem of determining the average value of T over the light

path DC can be considerably simplified by taking advantage of the

--angular symmetry.

-Consider the enlarged view of the flow cross section shown

schematically in Figure 9. Though the light path DC might in general

occupy a position such as that indicated by the position vector r, the

assumption of angular symmetry permits this chord to be rotated about

the origin to an arbitrary new position without in any way altering

the shear-stress distribution along the chord. In particular, if one

chooses to rotate chord DC to the position D'C' parallel to the y-axis,

then the average stress from D' to C' is the same as the average stress

from y = 0 to y = y*, owing to symmetry with respect to reflection

in the x-axis. Thus, the average shear stress along the chord can be

obtained by integrating from y = 0 toy = y* = (R2 2
obtained by integrating T from y = 0 to y = y* = (R -a ) where























Y*


O a



r











Figure 9. Schematic fiow cross section, illustrating a typical light
path along which the average shear stress is required. For angularly
symmetric flow, the average stress along light path DC is equal to
that along path D'C'.











the parameter a is the perpendicular displacement of the chord from

the origin.

Along the path D'C',Equation (66) takes the form



(a2+ y2 )
T = Tw (67)
T= R


and the average shear stress T(avg) is given by


fTdy
(avg) =, y* = (R -a (68)

I dy


Substituting d from Equation (67) into Equation (68) and

performing the indicated integration gives


2 2 2 +4
w a (R2-a ) +R
r(avg) = --- 1 + ] In I[ a
R(R2 a)


If we now define the dimensionless variable ( as T(avg) /Tw

and rewrite Equation (69) in terms of the fractional displacement a,

we obtain finally

1 2 (1-i_2) + 1i
T(avg) = 1+ a In (1- (70)
W 2 (1-a2)5 a


This equation gives the desired average shear stress along the

light path as a fraction of the wall stress in the tube. On the basis

of physical intuitionone would expect D to converge to the values of













0.5 and 1.0 for a = 0 and a = 1 respectively. Although inspection

of Equation (70) shows that P is indeterminate at both endpoints, a

single application of L'Hospital's rule is sufficient to demonstrate

that 4 does indeed converge to the expected values.

The length Z of the chord D'C' (or DC) may be written in terms

of a by considering the geometry of Figure 9. The relationship

between A and a is


I /2R = (1 -a ) (71)


The right side of Equation (71) thus gives the light-path length in the

fluid as a fraction of the inside tube diameter, and, as expected, this

length has a value of one diameter for a = 0 and zero for a = 1.

Numerical results from Equations (70) and (71) have been

summarized in Table 2. The first column in the table represents either

the fractional displacement of the light path from the center of the

tube or the fractional displacement of an isochromatic band as measured

in the flow photograph, The second column gives the length of the corres-

ponding light path in the fluid in diameters, and the third column gives

the average shear stress along the light path as a fraction of the wall

stress in the tube.

As an example of the use of this table, consider the green band

which was seen in Figure 7. In the photograph, the center of this band

is displaced 10.4 mm from the center of the tube. The edge of the

birefringence pattern is displaced 13.0 mm from the center. The fractional











Table 2

LIGHT-PATH LENGTH AND AVERAGE SHEAR STRESS
ALONG THIS PATH AS DIMENSIONLESS FUNCTIONS
OF FRACTIONAL DISPLACEMENT IN THE FLUID MEDIUM*


Fractional
Displacement, (a or B)


0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0.450
0.500
0.550
0.600
0.650
0.700
0.750
0.800
0.850
0.900
0.950
1.000


Dimensionless
Path Length, (/2R)

1.000
0.999
0.995
0.989
0.980
0.968
0.954
0.937
0.917
0.893
0.866
0.835
0.800
0.760
0.714
0.661
0.600
0.527
0.436
0.312
0.000


Dimensionless
Average Shear Stress, D


0.500
0.505
0.515
0.529
0.547
0.567
0.588
0.612
0.637
0.663
0.690
0.718
0.747
0.777
0.807
0.838
0.870
0.902
0.934
0.967
1.000


*NOTES: a. This calibration applies only for fully developed
angularly symmetric laminar flow in a circular tube.
b. Fractional displacement a measures the perpendicular
displacement between the center of the tube and the
chord representing the light path. Fractional dis-
placement B measured in the flow photograph for the
corresponding isochromatic band is exactly equal to a.
c. A/2R is calculated from Equation (71).
d. i is calculated from Equation (70).











displacement of this band is therefore 8 = 10.4/13.0 = 0.8. Consulting

Table 2, one finds that the light-path length in the fluid corresponding

to this displacement is 0.600 diameters. Since the tube is known to

have an inside diameter of 3.00 mm, the light path is 1.80 mm long.

Finally, from Table 2, the average shear stress associated with this

band is 0.870 times the know wall stress of 90 dynes/cm2. Therefore,

the shear stress associated with the green band is 78.3 dynes/cm2.

It should be emphasized that the calibration given by Table 2

applies only to isochromatic bands obtained for tubes of circular-

cylindrical geometry under the steady laminar-flow conditions described

earlier. These are the upstream conditions prevailing in the steady-flow

experiments of this study.

--The problem of determining the shear stress at downstream

entrance-flow positions in terms of the upstream conditions will be

considered in Part B with numerical examples for flows around a cardiac

catheter and through a 900 branch.


B. Steady-Flow Experiments

To facilitate discussion of the following figures, it would be

appropriate to discuss first the conventions which have been followed

regarding the orientation of the flow photographs and general terminology.

The photographs appearing in this section show typical isochroma-

tic-band patterns for flows through both straight and branched test

sections. The observer should keep in mind that for each photograph












the flow is steady and from left to right and for branched test

sections this flow is such as to impinge on the bifurcation (that is,

all branched flows are of the diverging type).

The convention followed in labeling branches is that side arms

have been designated "B" and downstream main stems "A", with the term

"upstream" referring to the pre-branch segment.

All Reynolds numbers have been based on apparent viscosity and

the pertinent tube diameter as a characteristic length (for the single

branch of rectangular cross section, this length has been taken as

four times the hydraulic radius, defined as the cross-sectional area

of the stream divided by the wetted perimeter).

The term "flow ratio" applies only to branched test sections and

is defined as the ratio of the volumetric flow rate in the side arm to

that for the downstream main stem (i.e., QB/QA). Similarly, the term

"area ratio" is defined as the ratio of the cross-sectional area of

the stream in the side arm to that for the downstream main stem (i.e.,

AB/AA) .

Unless otherwise noted, the values of wall stress and Reynolds

number given in the figures refer to the upstream (pre-branch) values

of these quantities. In each case, upstream wall stress has been

experimentally determined as the negative of the pressure gradient mul"

tiplied by half the inside radius at the upstream position.

Finally, all entrance-length measurements have been expressed in












dimensionless form by normalizing them to the diameter of the branch

for which they apply (i.e., side-arm entrance lengths have been

normalized with respect to side-arm diameter and main-stem entrance

lengths with respect to main-stem diameter).

Prior to the actual branching studies, it was decided to employ

the birefringence technique to obtain isochromatic patterns for steady

flows around various needles and cardiac catheters inserted radially

and axially into a circular tube. Since the birefringence technique

had not been used previously for this purpose, it was hoped that useful

information would be obtained regarding the creation and propagation

of flow disturbances caused by these devices.

Figure 10 shows an isochromatic-band distribution for steady

flow around a catheter (O.D. = 1.67 mm) inserted axially into a cylin-

drical tube (I.D. = 3.00 mm). Here, plane-polarized light has been

used with the plane of polarization perpendicular to the axis of the

tube. The upstream Reynolds number for the experiment was 10, and the

wall stress was 75 dynes/cm2

In the upstream region, the bands are distributed much as they

were for the fully developed flow in Figure 7, but as they encounter the

catheter, these bands become noticeably displaced toward the tube wall.

The flow-interference front caused by the catheter extends approximately

one diameter upstream, and the region between the catheter and the tube

wall is recognized as one of accelerated flow owing to the restriction of

flow area. The band crowding seen in this region is typical of the











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patterns seen for regions of accelerated flow and will also be seen

in photographs of branching flows. This crowding of bands provides a

clue to areas where one would expect to find increased velocity

gradients and shearing stress.

It is apparent from Figure 10 that the flow in the entrance

region surrounding the catheter does not correspond to the fully

developed flow discussed in Part A. This means that the shear-stress

calibration given by Equation (70) in Part A cannot be applied directly

to calculate the stress distribution for the entrance region. In fact,

since the light-path length varies along the bands in this region, there

is no unique stress which can be associated with any band. However, it

was intimated in Part A that the fundamental Equation N = M*jT(avg)

could be used to relate the stress distribution in the entrance region

to that in the upstream region of fully developed flow, where the cali-

bration of Part A does apply. In particular, since the coefficient M*

has been found to be independent of shear rate for Milling-Yellow sus-

pensions, one can apply the above equation to an upstream and downstream

point on the same band (or on two different bands of the same order N)

to obtain


T(avg, down) = k (up) T(avg, up) (72)
k (down)


From Equation (72) it is evident that the average shear stress

along the light path at the downstream position can be calculated from












the upstream value for the given band providing that the light-path

length at each of the two points is known.

It was shown in Part A that for flow through tubes the light-path

length k is related directly to the observed dimensionless displacement

3 of the isochromatic band at the point in question by Equation (71),

1/2R = (1-82) where we have used the fact that B = a. Since this

equation was derived solely from geometrical considerations, it may be

applied accurately to both upstream and downstream positions and

Equation (72) written as


(i- 2(up))%
T(avg, down) = 2 T(avg, up) (73)
(1-8 (down))

-Here- (up) and 8(down) are the dimensionless displacements of the

isochromatic band measured in the flow photograph at the upstream and

downstream points respectively.

Rewriting Equation (73) in terms of the upstream wall stress

T,(up) and the dimensionless average shear stress D(up) one has

2 u
T(avg, down) = (1- (up) (p) p) (74)
(1-_ (down))

All the quantities on the right side of Equation (74) are known,

either from the birefringence experiment or from the shear-stress cali-

bration of Part A. The quantities D(up), (l-2 (up)) and (l- 2(down))

can be obtained either from Table 2 in Part A for the appropriate value

of 8 or by direct calculation.












As an example, consider the dark band appearing upstream in

the center of the tube shown in Figure 10. Encountering the catheter,

this band is displaced from its original position, 0 = 0, through

intermediate values of 8 to a final position, B = 0.5, between the tip

of the catheter and the tube wall. Choosing this point as the downstream

point of interest and consulting Table 2, one finds the following:


8(up) = 0, (l-82(up))4 = 1.000 and D(up) = 0.500


s(down) = 0.5, (l- 2(down))2 = 0.866


The known wall stress for this experiment is given in Figure 10 as

75 dynes/cm2. Inserting these values in Equation (74) gives


t(avg, down) = (1.000/0.866) (0.500) (75) = 43.3 dynes/cm2.


For comparison, the average shear stress corresponding to this band in

the upstream position is


2
t(avg, up) = 4(up) Tw(Up) = (0.500) (75) = 37.5 dynes/cm2


As expected, the region of accelerated flow is also one of higher shear.

It should be emphasized that the above calculation gives only

the average shear stress along the light path at a single downstream

position and applies only when the reference (upstream) flow is fully

developed as described earlier. By repeating this calculation for bands












falling close to the wall of the tube one can obtain an estimate of

the shear stress acting on the wall.

Figure 11 shows another experiment in which a beveled needle

(shank diameter = 0.92 mm) has been inserted through the wall of the

cylindrical tube (I.D. = 3.00 mm) and oriented with the bevel facing

upstream. The Reynolds number for this experiment was 40, and the

upstream wall stress was 80 dynes/cm2. Again, plane-polarized light

has been used with the plane of polarization perpendicular to the

axis of the tube.

The photograph has been underexposed to emphasize the two

distinct vortex patterns seen trailing behind the needle. These

patterns persist for approximately two diameters downstream where the

disturbance becomes damped out because of fluid viscosity. The region

of disturbed flow extends less than one diameter upstream.

Though the vortex patterns are similar in size and shape in

this photograph, motion pictures reveal that when the needle is rota-

ted 900, the lower vortex disappears almost entirely while the upper

one remains essentially unchanged. This is evidence that the edge of

the bevel creates far less disturbance than does the face. When the

needle is rotated further until the bevel faces downstream, the lower

vortex reappears but is not as prominent as the upper one. The same

experiment using a cylindrical needle having no bevel produces only a

single vortex pattern which appears in the center of the tube and is

much more streamlined than those in Figure 11. This disturbance also


















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persists downstream about two diameters.

The most significant finding from these experiments was that

when the needles were withdrawn to a position flush with the wall of

the tube no disturbances could be seen in the birefringence patterns.

This strongly suggests that lateral pressure measurements could be

made in small-diameter models as well as in vivo without appreciably

disturbing the flow.

Figure 12 shows an isochromatic-band distribution obtained

using circularly polarized light for the first of the branching-flow

experiments. The branch shown (test section 6) has a branching angle

of 300 and an area ratio of unity (the side arm and main downstream

stem both having an inside diameter of 3.17 mm). The upstream

Reynolds number for this experiment was 116 with a corresponding wall

stress of 110 dynes/cm2.

Although the bands are parallel to the axis of the tube in the

upstream segment, they tend to converge toward the inner walls and

distribute themselves concavely about the bifurcation point in the

entrance-flow region. This region of disturbed flow extends downstream

for a few diameters in both arms until the bands return to the parallel

disturbution indicative of laminar flow. The fact that the bands tend

to be distributed concavely about a stagnation point, such as the

point of bifurcation, is helpful in locating zones of stasis at other

positions in a flow test section. This will be more apparent in later

photographs. The displacement of bands toward the inner walls of the
























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branch indicates that the regions directly adjacent to the bifurcation

represent areas of accelerated flow and increased shearing stress,

while the absence of bands toward the outer walls indicates boundary-

layer separation in those areas. Aside from the regions immediately

adjacent to the bifurcation, there is no evidence for increased stress

at any other site in this branch.

Figure 13 gives a comparison of the flow-distribution

characteristics for three different 30" branches. Here, flow ratio

(QB/QA) has been plotted against upstream Reynolds number with area

ratio (AB/AA) as a parameter. The upper curve applies to the branch

just discussed, and the lower two apply to branches having side-arm

diameters of 2 mm and 1 mm respectively.

TFrom the over-all decrease of flow ratio with increasing

Reynolds number shown by all three branches, it is evident that the

upstream flow increasingly favors a straight path into the main down-

stream stem A as the upstream fluid momentum per unit volume increases.

The marked separation between successive pairs of curves in Figure 13

simply verifies that reducing the side-arm diameter is an effective

means of controlling side-arm flow over the entire range of upstream

Reynolds numbers. It will be seen later that this control is much better

than that achieved by increasing the branching angle. The disproportionate

increase of flow ratio which occurs below the Reynolds number of 25 is

undoubtedly a manifestation of wall attachment by the flow, similar to

that observed by Fox and Hugh14 in their experiments with open-channel
















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flows. The data of Figure 13 indicate that for 30 branches wall

attachment can cause the side-arm-flow to greatly exceed the main-

stem flow, but only if the branch area ratio approaches unity and

only below a certain upstream Reynolds number.

The phenomenon of wall attachment (sometimes called the Coanda

effect) is known to arise because of a low-pressure region generated

along the outer side-arm wall owing to fluid moving along curved

streamlines concave toward this wall. In branches, this low-pressure

region tends to divert a greater fraction of the incoming flow into

the side arm until the upstream momentum forces can overcome the pressure

forces causing the attachment. At still higher flow rates, the boundary

layer separates from the outer wall and a vortex is generated there.

The data of Knox24 indicate that there is a sharp drop in pressure

associated with this vortex and a corresponding rise of pressure in the

main stem due to the exchange of incoming fluid kinetic energy for pres-

sure energy there.

It has been found that very reliable fluid amplifiers and fluidic

digital-logic devices can be produced by carefully designing branches

with control ports at the vortex regions. Fluid can be introduced

through these ports to destroy a given low-pressure zone, thus causing

the main flow to switch from one arm of the branch to the other. Unfor-

tunately, very little has been published concerning the branching-flow

characteristics of fluidics devices, since most of this information has

been considered proprietary.