Some of the characteristics of steady and oscillatory blood flow
by Bharat Ochhavlal Shah
A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF
PHILOSOPHY in Chemical Engineering
Montana State University
© Copyright by Bharat Ochhavlal Shah (1978)
Abstract:
An estimate of the difference between the Poiseuillian viscosity and the apparent viscosity at a given
shear rate was made for blood under steady flow conditions. Shear stress-shear rate (viscometric) data
for blood under steady flow conditions were used along with some numerical methods to obtain the
difference between these two viscosities. The difference is significant at low flow rates and high values
of the apparent viscosity.
Experimental work was done on oscillatory flow of blood at different frequencies of oscillation in rigid
circular tubes. An apparatus was built so that oscillatory pressure and flow could be measured. The
apparatus was tested with Newtonian fluids, such as glycerol solution, water, saline and plasma
because the theory of oscillatory flow for a Newtonian fluid is understood very well.
Different red cell suspensions such as red blood cells in plasma, red blood cells in Dextran solutions,
red blood cells in albumin-saline and hardened red blood cells in 0.5% Dextran 40 solution were used
for the experimental work. Flow in two rigid circular glass tubes 400 and 776 microns in diameter was
investigated. The experimental suspension hematocrit was usually about 45%, although 36%
hematocrit was also used.
Experimental results showed that all pressure-time curves were sinusoidal for frequencies of oscillation
of 0.5 hertz through 3 hertz. (The flow-time curves were always sinusoidal because of the inherent
nature of the apparatus.) The pressure-flow data are summarized in this thesis and they are explained in
terms of the known properties of blood and other red cell suspensions. These properties are the
rheological data of blood, the visco-elasticity of blood, the aggregation of red blood cells and the
deformation of red cells.
Red blood cells in high molecular weight Dextran solutions aggregate more strongly than they do in
plasma. Hence, suspensions consisting of red blood cells in such Dextran solutions need higher
pressure gradients to maintain the same flow compared to pressure gradients needed by red cells in
plasma. From the experimental work with red cells in saline solution and hardened red cells in saline, it
was concluded that red blood cell aggregation is significant process in oscillatory blood flow when the
tube diameter is 400 microns. SOME OF THE CHARACTERISTICS OF STEADY
AND OSCILLATORY BLOOD FLOW
by
BHARAT OCHHAVLAL SHAH
A t h e s i s s u b m it t e d i n p a r t i a l f u l f i l l m e n t
o f t h e r e q u ir e m e n t s f o r t h e degree
of
DOCTOR OF PHILOSOPHY
in
Chemical
E n g in e e r in g
A p p ro ve d :
MONTANA STATE UNIVERSITY
Bozeman, Montana
M a rch , 1978
iii
ACKNOWLEDGMENT.
The a u t h o r w is h e s t o th a n k t h e e n t i r e
s t a f f o f t h e D e p artm en t
o f C h e m ic a l E n g in e e r in g o f Montana S t a t e - U n i v e r s i t y f o r t h e i r h e l p .
The a u t h o r i s
e s p e c i a l l y i n d e b t e d t o D r. G .-R . C o k e le t f o r h i s
a d v i c e , a s s i s t a n c e and e nco u ra ge m e n t t h r o u g h o u t t h e c o u r s e o f t h i s
p ro je c t.
S p e c ia l
th a n k s go
t o Mr. G. R. W illia m s o n and Mr. M. C.
Huntsman who h e lp e d i n b u i l d i n g t h e a p p a r a t u s . a n d t r a c k i n g down t h e
v a r i o u s p a r t s and p ie c e s o f t h e e q u ip m e n t d u r i n g t h e w o r k .
Thanks a l s o
go t o M rs. M a ria n E v o n iu k , because she h e lp e d t o p r e p a r e t h e b lo o d
s a m p le s .
The a u t h o r i s
t i o n f o r th e f in a n c ia l
F in a lly ,
th e s is .
v e r y much t h a n k f u l
t o t h e Montana H e a r t A s s o c i a ­
s u p p o rt..
th a n k s go t o M rs. Lee L u n d q u is t f o r t y p i n g t h i s
TABLE OF CONTENTS
.
•■
•.
-
Page
V I T A . ................................
ii
ACKNOWLEDGMENT................... .... ............................................................ .
. . . . . .
iii
LIST. OF TABLES...........................
vi
LIS T OF F IG U R E S ................... .... ..........................................................................
. , .
ix
ABSTRACT........................................................
xi
CHAPTER
I
INTRODUCTION.
...........................................•........................................ ' . . .
I
. SECTION A ( STEADY>FLOW)
II
III
STATEMENT OF THE P R O B L E M .................................
.
REVIEW OF.THE PREVIOUS WORK . . . . . . .
........................
6
................................
3
IV
THEORY AND CALCULATIONAL PROCEDURE..................
, V-,,V..:.. .-V
' '
V
RESULTS AND DISCUSSION. . . ......................
.V I
CONCLUSIONS AND RECOMMENDATIONS
9
.
13
........................'
SECTION B (OSCILLATORY FLOW) . .
V Il
/
STATEMENT OF THE PROBLEM.
. . .
. V I I I , REVIEW OF THE PREVIOUS..WORK .
-.Y
/ %- -y,,
.. < "
\
IX
X
2 2 - ,.
■
. . . . . . .' .........................
.
. ............................ ....
.
24
. . .
28
THEORETICAL BACKGROUND................................................................
APPARATUS AND PROCEDURES.
. . . . . .
.........................................
A.
A p p a ra tu s - G e n e ra l D e s c r i p t i o n . . . . .
................... ....
B . . P re s s u r e Measurement D e v ic e ........................................................
" C.
Flow Measure D e vice .
............................ .
................... ....
P ro ced u re s
D.
C a l i b r a t i o n o f P re s s u r e T r a n s d u c e r ...........................................
E.
C a l i b r a t i o n o f D is p la c e m e n t T r a n s d u c e r . . .........................
F'
Measurement o f t h e C a p i l l a r y - T u b e D ia m e t e r ........................
G.
T e s t i n g o f A p p a r a tu s . . . ....................... ...
H.
P r e p a r a t i o n o f B lo o d S a m p le s ........................ ....
36
'
40.
40
44
44.
46
46?48
5 0'
50
:
V
CHAPTER
X
XI
page
( C o n t in u e d )
I.
O p e r a t i o n ..................................................................................................
J . . R e p r e s e n t a t io n o f D a t a ............................................................. .... .
K.
Use o f R h e o lo g ic a l Data o f C a l c u l a t e
O s c i l l a t o r y B lo o d F lo w ..................................................................
51
53
RESULTS AND D IS C U S S IO N ..............................................................................
56
A.
B.
C.
X II
X III
56
61
85
CONCLUSIONS AND RECOMMENDATIONS ..........................................................
96
A.
C o n c l u s i o n s ..............................................................................................
B. 1 R e c o m m e n d a t io n s .....................................................................................
96
97
APPENDICES.......................................................................
A.
B;
X IV
T e s t i n g o f A p p a ra tu s w i t h N e w to n ia n F l u i d s ............................
Data and D is c u s s io n o f D i f f e r e n t Red C e ll
S u s p e n s io n s U s in g Plasma and D e x tra n S o l u t i o n s . . .
Data and D is c u s s io n o f D i f f e r e n t Red C e ll
S u s p e n s io n s U s in g Plasm a, S a l i n e and
Hardened Red C e l l s ...........................................................................
53
'Computer Program f o r S te a d y F lo w ...............................................
D e t a i l s o f V a r io u s I n s t r u m e n t s . • ...............................................
BIBLIOGRAPHY....................................................................................
99
100
1.01
104
■
Vi
. LIST OF TABLES
Table
'
.
V -I . . .
Page -
COMPARISON OF RESULTS OBTAINED FROM LITERATURE
(1 5 ) AND FROM THIS WORK........................................................................... 19
V -2
FITTING OF RHEOLOGICAL. DATA INTO ANALYTICAL EXPRESSIONS ..
20-
V II-I
VALUES' OF MEAN PRESSURE DROP AT VARIOUS LOCATIONS . . . .
26
V III-
I
V I I I -2
X I-I
X I-4
X I-5
X I-6
'' ■
29 '
SOME OTHER DETAILS ABOUT THE PREVIOUS WORK
DONE ON OSCILLATORY BLOOD F L O W . ................... ................................
30
DATA OF OSCILLATORY BLOOD FLOW and STEADY
■' flow v i s c o m e t r i c d ata
;. H e m a t o c r it - 43.7 % ;
Red C e l l s i n P la sm a ;
Tube D ia m e te r = 776 m ic r o n s .............................. ....
X I-2
'•' X I - 3
•
•
SOME OF THE DETAILS ABOUT THE PREVIOUS WORK .
■ DONE ON OSCILLATORY BLOOD FLOW................... " ..................................
65
DATA.OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
. H e m a t o c r it .= •44.4% ;. Red C e l l s i n 2% D e x tra n 1 50 ;
' Tube D ia m e te r = 776 m i c r o n s ................... .....................................
.
- \ DATA OF-OSCILLATORY BLOOD FLOW and-STEADY '.
FLOW VISCOMETRIC DATA
H e m a t o c r i t ■= 43%;’ Red C e l l s i n 2% D e x tra n 2 5 0 ;
Tube D ia m e te r = 776 m ic r o n s .......................................... ....
66
. .
67
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
. H e m a t o c r it = 4 5.2 % ;
Red C e l l s i n D e x tra n 1 50 ;
Tube D ia m e te r = 776 m ic r o n s . . . . '. . . ........................
68
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
• VsH e m a t o c r it = 45.5% ;
Red C e l l s i n P la sm a ;
Tube D ia m e t e r = 776 m ic r o n s . ...............................................
.
69
■ DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it = 36.1% ;
Red C e l l s i n Plasma
J Tubeli D ia m e te r = 776 m ic r o n s . . . ..........................................
70
vii
Table
X I-7
X I-8
X I-9
X I-IO
X I-Il
X I-I2
X I-I3
X I-1 4
'
Page
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRTC DATA.
H e m a t o c r it = 36.4%;
Red C e l l s i n Plasm a;
Tube D ia m e te r = 400 m i c r o n s ..........................................
73
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it = 46%;
Red C e l l s i n Plasma;
Tube D ia m e te r = 400 m i c r o n s ..........................................
74.
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it = 44%;
Red C e l l s i n Plasm a;
1 Tube D ia m e te r = 400 m i c r o n s ................................. .......................
77
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it = 44%;
Red C e l l s i n D e x tra n 2 5 0 ;
Tube D ia m e te r = 400 m ic r o n s ........................................................
78
DATA OF OSCILLATORY BLOOD FLOW and STEADY
■ FLOW VISCOMETRIC DATA
H e m a t o c r it = 45%;
Red C e l l s i n Plasm a;
Tube D ia m e te r = 400 m i c r o n s ........................................................
83
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
; H e m a t o c r it = 45%;
Red C e l l s i n Plasm a;
Tube D ia m e te r = 400 m i c r o n s ........................................................
84
DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it = 46%;
Red C e l l s i n S a l i n e and 0.5%
A lb u m in ;
Tube D ia m e te r = 400 m i c r o n s .................................
87
DATA OF OSCILLATORY BLOOD FLOW, and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it = 45%;
Red C e l l s i n Plasma;
Tube D ia m e te r. = 400 m i c r o n s ........................................................
88,
v iii
T a b le
Page
X I - I 5 ■ DATA OF OSCILLATORY BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it - 46.5 % ;
Hardened Red C e l l s i n 0.5%
D e x tra n S o l u t i o n ; Tube D ia m e te r = 400 m ic r o n s . . . .
X I-1 6
X I-I7
90
DATA OF OSCILLATORY. BLOOD FLOW and STEADY
FLOW VISCOMETRIC DATA
H e m a t o c r it = 35.6% ;
Hardened Red C e l l s i n 0.5%
D e x tra n S o l u t i o n ; Tube D iam eter. = 400 m ic r o n s . . . .
91
COMPARISON OF VISCOSITY CALCULATED FROM THEORY
AND EXPERIMENTAL ULTIMATE NEWTONIAN VISCOSITY ...................
93
IX
LIS T OF FIGURES
F ig u r e
Page
I-I
APPARENT VISCOSITY VERSUS SHEAR RATE............................ ....
.
3
V -I
SHEAR STRESS-SHEAR RATE DATA UNDER
STEADY CONDITIONS;
HEMATOCRIT = 25.4% ......................................
14
SHEAR. STRESS-SHEAR RATE DATA UNDER
STEADY CONDITIONS;
HEMATOCRIT = 35.8 % ......................................
15
SHEAR STRESS-SHEAR RATE DATA UNDER
STEADY CONDITIONS;
HEMATOCRIT = 4 6 .5 % ......................................
16
V-2
V -3
V-4
V-5
. .
DIFFERENCE BETWEEN THE POISEUILLIAN VISCOSITY AND
APPARENT VISCOSITY FOR DIFFERENT BLOOD
HEMATOCRITS - LOW RANGE . . . . . . .
...........................................
17
DIFFERENCE BETWEEN THE POISEUILLIAN VISCOSITY AMD
APPARENT VISCOSITY FOR DIFFERENT BLOOD
HEMATOCRITS - HIGH RANGE.......................................... ' ..........................
18
V II-I
NATURE OF OSCILLATORY BLOOD FLOW.........................................................
25
IX -I
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE FROM
WOMERSLEY'S THEORY; SMALL TUBE DIAMETER..................................
37
IX -2
X -I
X-2
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE FROM
WOMERSLEY'S THEORY; LARGE TUBE DIAMETER ......................................
SCHEMATIC DIAGRAM OF APPARATUS................................................ ....
. .
38
41
DIAGRAM EXPLAINING THE PRINCIPLE ON .
WHICH SCOTCH-YOKE OPERATES.......................................................................
42
X -3
DIAGRAM OF SCOTCH-YOKE.................................................................................
43
X-4
PRESSURE TRANSDUCER CALIBRATION
47
X -5
CAPILLARY-TUBE DIAMETER C A L IB R A T IO N .............................. • • • •
X I-I
TESTING OF APPARATUS WITH 400 MICRONS CAPILLARY
TUBE - PRESSURE AM PLITUD E.........................................................................
.............................
49
57
X
Figure
X I-2
X I-3
X I-4
X I-5
X I-6
X I-7
X I-8
■■■:
X I-9
X I-IO
X I-Il
XT-12
X I-1 3
Page
TESTING OF. APPARATUS WITH 400 MICRONS
CAPILLARY TUBE - PHASE DIFFERENCE . . ................................. ....
58
TESTING OF APPARATUS WITH 776 MICRONS
- CAPILLARY" TUBE - PRESSURE A M P LITU D E ..........................................
59
TESTING.OF APPARATUS WITH 776 MICRONS '
CAPILLARY TUBE - PHASE DIFFERENCE................... .... .......................
60
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE FOR
DIFFERENT BLOOD SAMPLES.WITH PLASMA AND DEXTRAN;. Tube D ia m e te r = 776 m ic r o n s ...................................... ....
64
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE FOR
.
BLOOD SAMPLES WTTH PLASMA;
. ’ Tube D ia m e te r = 400 m ic r o n s ........................................................
72
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE FOR
„ DIFFERENT BLOOD SAMPLES WITH PLASMA AND
DEXTRAN SOLUTION;
Tube D ia m e te r = 400 m ic r o n s ........................................................
76
FLOW'AMPLITUDE VERSUS PRESSURE AMPLITUDE AT
FREQUENCY = 0 . 5 HERTZ ............................................... ....
79
■ FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE AT
' FREQUENCY = I H E R T Z ............................ .... .........................................
.
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE AT
FREQUENCY = 2 HERTZ .
. .
............................
80
81
' FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE AT
FREQUENCY = 3 HERTZ . . . .
............................ ....
82
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE FOR
RED CELLS.IN PLASMA AND RED CELLS;IN SALINE;
Tube D ia m e te r = 400 m i c r o n s ........................................................
86
. PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE FOR
' HARDENED RED CELLS IN 0.5% DEXTRAN SOLUTION;
Tube D ia m e te r = 400 m ic r o n s
...............................................;
8 9 -;
xi
ABSTRACT
An e s t i m a t e o f t h e d i f f e r e n c e between t h e P o i s e u i l l i a n v i s c o s i t y
and t h e a p p a r e n t v i s c o s i t y a t a g iv e n s h e a r r a t e was made f o r b lo o d
under ste a d y flo w c o n d it io n s .
S he a r s t r e s s - s h e a r r a t e ( v i s c o m e t r i c )
d a ta f o r b lo o d u n d e r s t e a d y f l o w c o n d i t i o n s were used a lo n g w i t h some
n u m e r ic a l methods t o o b t a i n t h e d i f f e r e n c e between t h e s e two v i s c o s i t i e s .
The d i f f e r e n c e i s s i g n i f i c a n t a t lo w f l o w r a t e s and h ig h v a lu e s o f t h e
a p p a re n t v i s c o s i t y .
E x p e r im e n t a l w o rk was done on o s c i l l a t o r y f l o w o f b lo o d a t
d i f f e r e n t fre q u e n c ie s o f o s c i l l a t i o n in r i g i d c i r c u l a r tu b e s .
An appa­
r a t u s w a s . b u i l t so t h a t o s c i l l a t o r y p r e s s u r e and f l o w c o u l d be measured.
The a p p a r a tu s was t e s t e d w i t h N e w to n ia n f l u i d s , such as g l y c e r o l s o l u ­
t i o n , w a t e r , s a l i n e and plasma because t h e t h e o r y o f o s c i l l a t o r y f l o w
f o r a N e w to nia n f l u i d i s u n d e r s to o d v e r y w e l l .
D i f f e r e n t re d c e l l s u s p e n s io n s such as re d b lo o d c e l l s i n
p la s m a ,
re d b lo o d
c e l l s i n D e x tra n
s o l u t i o n s , r e d b lo o d c e l l s i n
a l b u m i n - s a l i n e and harde n ed re d b lo o d c e l l s i n 0.5% D e x tra n 40 s o l u t i o n
were used f o r t h e e x p e r im e n t a l w o r k .
Flow i n two r i g i d c i r c u l a r g la s s
tu b e s 400 and 776 m ic r o n s i n d ia m e t e r was i n v e s t i g a t e d .
The e x p e r i ­
m ental s u s p e n s io n h e m a t o c r i t was u s u a l l y a b o u t 45%, a l t h o u g h 36%
h e m a t o c r i t w a s - a ls o used.
E x p e r im e n t a l r e s u l t s showed t h a t a l l p r e s s u r e - t i m e c u rv e s were
s in u s o id a l f o r fre q u e n c ie s o f o s c i l l a t i o n o f 0 .5 h e r tz th ro u g h 3 h e r t z .
(The f l o w - t i m e c u rv e s were a lw a y s s i n u s o i d a l because o f t h e i n h e r e n t
n a tu re o f th e a p p a ra tu s .)
The p r e s s u r e - f l o w d a ta a r e summarized i n t h i s
t h e s i s and t h e y a r e e x p l a i n e d i n te rm s o f t h e known p r o p e r t i e s o f b lo o d
and o t h e r re d c e l l s u s p e n s io n s .
These p r o p e r t i e s , a r e t h e r h e o l o g i c a l
d a ta o f b l o o d , t h e v i s c o - e l a s t i c i t y o f b l o o d , t h e a g g r e g a t i o n o f re d
b lo o d c e l l s and t h e d e f o r m a t io n o f re d c e l l s .
Red b lo o d c e l l s i n h ig h m o l e c u l a r w e i g h t D e x tra n s o l u t i o n s ag ­
g r e g a t e more s t r o n g l y th a n t h e y . d o i n p la sm a .
Hence, s u s p e n s io n s c o n ­
s i s t i n g o f re d b lo o d c e l l s i n such D e x tra n s o l u t i o n s need h i g h e r p r e s ­
s u r e g r a d i e n t s t o m a i n t a i n t h e same f l o w compared t o p r e s s u r e g r a d i e n t s
needed by re d c e l l s i n p la sm a .
From t h e e x p e r im e n t a l w o rk w i t h re d
c e l l s i n s a l i n e s o l u t i o n and harde n ed re d c e l l s i n s a l i n e . , i t was c o n ­
c lu d e d t h a t re d b lo o d c e l l a g g r e g a t io n i s s i g n i f i c a n t : p ro c e s s i n
o s c i l l a t o r y b lo o d f l o w when t h e tu b e d ia m e t e r i s 400 m ic r o n s . .
3
INTRODUCTION
T h is s t u d y i s an a t t e m p t t o w o rk to w a rd s t h e s o l u t i o n s o f some
o f t h e p ro b le m s i n v o l v e d i n b lo o d f lo w . "
in to
two s e c t i o n s .
The e n t i r e
th e s is
o f tim e , th e flo w is
When t h e p r e s s u r e and t h e f l o w a r e in d e p e n d e n t
te rm e d as s t e a d y f l o w .
When t h e p r e s s u r e and t h e
f l o w a r e some p e r i o d i c f u n c t i o n s o f t i m e , t h e f l o w i s
rig id
d iv id e d
S e c t io n A d e a ls w i t h s t e a d y f l o w and S e c t io n B d e a ls
w ith o s c i l l a t o r y flo w .
Ia t o r y flo w .
is
The w o rk shown i n t h i s
te rm e d as o s c i l -
t h e s i s d e a ls w i t h such f l o w s i n
c i r c u l a r tu b e s .
G eneral
D is c u s s io n
The b a s i c f u n c t i o n o f t h e human c i r c u l a t o r y s y s te m i s
t o p ro v i d e
n o u r is h m e n t t o v a r i o u s p a r t s o f t h e body and t o remove w a s te m a t e r i a l s .
B lo o d , t h e f l u i d
fillin g
t h e c i r c u l a t o r y s y s te m , i s a s u s p e n s io n o f
.. .'.V a r io u s t y p e s o f n o n - s p h e r i c a l , d e f o r m a b le p a r t i c l e s
X - w h ite c e l l s ,
p l a t e l e t s ) , i n an aqueous s o l u t i o n
(p la s m a ).
s u s p e n d in g medium (p la s m a ) has N e w to n ia n r h e o l o g i c a l
'
s u s p e n s io n i s
a n o h fN e w to n ia n f l u i d . ( 5 ) .
g r e a t e s t i n f l u e n c e on t h e s u s p e n s io n
which, i s
'
t h e volume o f re d b lo o d c e l l s
flo w p ro p e rtie s
:
p a th o lo g ic a l
p r o p e r t ie s , th e
is
th e h e m a to c rit
p e r u n i t volume o f w h o le b lo o d .
* 100
h e m a t o c r i t i s a b o u t 42-46% , b u t i t
fro m normal, v a lu e s i n
W h ile th e
The p a r a m e te r w h ic h has t h e
' * * * * * * • fel g f f w g f f b l V
N o rm a lly ,th e
( r e d b lo o d c e l l s ,
'
s itu a tio n s .
can v a r y c o n s i d e r a b l y
2
In t h e human c i r c u l a t i o n
th ro u g h a r t e r i e s ,
a rte r io le s ., c a p illa r ie s ,
r e t u r n t h e b lo o d t o t h e h e a r t .
a rte r io le s ,
c a p illa rie s
f o r m in g n e tw o rk s
s y s te m , b lo o d i s
v e n u le s and v e i n s , w h ic h
The m i c r o c i r c u l a t i o n c o n s i s t s o f t h e
and v e n u le s .
(5 ,1 9 ).
pumped fr o m t h e h e a r t
The v e s s e ls a r e i n t e r c o n n e c t e d ,
The s i z e o f t h e v e s s e l s t h r o u g h w h ic h b lo o d
f lo w s v a r y f r o m a b o u t 2 -1 0 m ic r o n s t o 2 5 ,0 0 0 m ic r o n s
s u re d ro p a c r o s s t h e
80% o f i t
e x is ts
human c i r c u l a t i o n
across a r t e r i o l e s
th ro u g h o u t th e m ic r o c i r c u la t io n .
The p r e s ­
I n - v i v o e x p e r im e n ts
r e g io n s o f f l o w o s c i l l a t i o n s
Hence o s c i l l a t o r y f l o w e x i s t s
in th e
c i r c u l a t o r y s y s te m .
In - v itro ,
th e r h e o lo g ic a l
s t e a d y f l o w e x p e r im e n t s have been used t o d e te r m in e
p r o p e r t i e s o f b lo o d .
r a t e s , b lo o d a c t s l i k e
a Casson f l u i d
sh ea r-de p en d en t v i s c o s i t y )
(3 ).
te n d to r e v e r s ib ly a g g re g a te .
It
(ro u le a u x ) v a rie s
fo u n d t h a t u n d e r lo w s h e a r
A t lo w s h e a r r a t e s ,
re d b lo o d c e l l s
The l e n g t h o f t h e a g g re g a te s
in v e r s e ly w ith th e shear ra te
(5 ).
T h is k i n d o f
c i r c u l a t o r y syste m and a l s o i n t h e
T h is phenomenon i s
re d b lo o d c e l l s
s t r e s s , and a
The a g g r e g a te s a re fo rm e d o n l y by j o i n ­
phenomenon can o c c u r i n t h e l i v i n g
e x p e r im e n t a l w o r k .
is
(w h ic h has a y i e l d
i n g t o g e t h e r t h e f a c e s o f t h e re d c e l l s .
ra te s
(5 ,1 9 ).
i s a b o u t 100 mm o f Hg b u t a b o u t
and c a p i l l a r i e s .
have shown t h a t t h e r e a r e s u b s t a n t i a l
e n tire
•
re v e rs ib le
(5 ).
a r e d i s p e r s e d and te n d t o d e fo r m .
a g g r e g a t io n a t lo w s h e a r r a t e s and re d b lo o d c e l l
s h e a r r a t e s may e x e r t t h e i r r h e o l o g i c a l
e ffe c ts
A t h ig h s h e a r
Red b lo o d c e l l
d e f o r m a t io n a t h ig h ,
t h r o u g h a common
.
APPARENT VISCOSITY
Red c e l l s i n plasma
H e m a t o c r it = 46.5%
T e m p e ra tu re = 37° C
AGGREGATION
DEFORMATION
SHEAR RATE
FIGURE I - I .
APPARENT VISCOSITY VERSUS SHEAR RATE
4
m echanism, s h e a r - d e p e n d e n t changes i n t h e e f f e c t i v e c e l l
e ffe c tiv e
(7 ).
c e ll,v o lu m e
is
t h e fu n d a m e n ta l
A p p a r e n t v i s c o s i t y o f b lo o d i s
F ig u r e I - I
shows t h e r e l a t i o n
Some i n - v i t r o ,
vo lu m e .
d e t e r m i n a n t o f b lo o d v i s c o s i t y
t h u s a f u n c t i o n o f s h e a r r a t e and
between a p p a r e n t v i s c o s i t y and s h e a r r a t e .
o s c illa to r y
f l o w s t u d i e s have been p e r fo r m e d ,
m o s t ly i n l a r g e d ia m e t e r tu b e s and o s c i l l a t o r y v is c o m e t e r s .
fr o m t h e s e s t u d i e s
The r e s u l t s
have n o t been r e l a t e d t o f l o w mechanisms o f b lo o d ,
and t h e i n t e r p r e t a t i o n
o f t h e d a ta i s o f t e n c lo u d e d by p r i o r a s s u m p tio n s
a b o u t t h e f l o w n a t u r e o f t h e b lo o d ( e . g . , t h e
i n phase w i t h
The
th e flo w d r i v i n g f u n c t io n
is
f l o w re s p o n s e component
assumed t o be p u r e l y v is c o u s
w h i l e t h e component 90° o u t o f phase w i t h t h e d r i v i n g
fu n c tio n
is
assumed t o be p u r e l y e l a s t i c ) .
S te a d y f l o w i s
flo w is
d is c u s s e d i n C h a p te rs I I
d is c u s s e d i n C h a p te rs V I I t h r o u g h X I I .
th ro u g h V I.
O s c illa to ry
,SECTION A
STEADY CONDITIONS
II.
STATEMENT' OF THE PROBLEM
T h is s e c t i o n d e a ls w i t h s t e a d y f l o w .
The r h e o l o g i c a l
d a ta
(th e
r e l a t i o n s h i p between t h e s h e a r s t r e s s and t h e s h e a r r a t e ) . c a n be
o b t a in e d f o r .b lo o d a t d i f f e r e n t h e m a t o c r i t s u s in g v is c o m e t e r s such as
t h e c o n c e n t r i c c y l i n d e r v is c o m e t e r and t h e cone and p l a t e
Such d a ta can be used t o o b t a i n t h e r e l a t i o n s h i p
v is c o m e t e r .
between t h e b u l k a v e r ­
age v e l o c i t y and t h e p r e s s u r e d ro p f o r s t e a d y f l o w o f b lo o d i n r i g i d
c i r c u l a r tu b e s .
I h r e p o r t i n g t h e r e s u l t s o f b lo o d f l o w s t u d i e s , t h e d a ta a r e
sometimes g iv e n i n te rm s o f t h e v i s c o s i t y c a l c u l a t e d f r o m t h e
P o is e u illia n
flu id
e q u a t i o n f o r s t e a d y , u n i f o r m l a m i n a r f l o w o f a N ew tonian
th ro u g h a tu b e .
The v i s c o s i t y c a l c u l a t e d by t h i s
c a lle d th e P o i s e u i l l i a n
e q u a tio n is
v is c o s ity .
Where N^ = t h e P o i s e u i l l i a n
V is c o s ity
R = o u t s i d e r a d i u s o f tu b e
' LI = b u l k a v e ra g e v e l o c i t y
^
= p re s s u re g r a d ie n t p e r u n i t le n g th o f tube
T h is , e q u a t io n , i s
q u e n tly i t s
v a l i d o n l y f o r N ew to nia n f l u i d s
use w i t h b lo o d f l o w d a t a i s
and co n se ­
i n v a l i d , e s p e c i a l l y a t lo w e r
flo w r a te s .
T h is P o i s e u i l l i a n
V is c o s ity is
sometimes compared t o t h e a p p a r ­
e n t v i s c o s i t y o b ta in e d fro m v is c o m e t r ic d a ta .
T h is c o m p a ris o n w i l l
not
7
be v a l i d , even f o r p r a c t i c a l
so t h a t t h e b lo o d i s
q u e s tio n s ,
v e r y n o n -N e w to n ia n .
due t o t h e use o f t h e P o i s e u i l l i a n
if
t h e b lo o d f l o w i s
However, no measure o f e r r o r
V i s c o s i t y has been r e p o r t e d , and t h e
p u rp o s e o f t h i s w o rk was t o p r o v i d e v a lu e s o f t h i s
s te a d y f lo w c o n d it io n s .
e r r o r under v a rio u s
The d i f f e r e n c e between t h e P o i s e u i l l i a n
c o s i t y and t h e a p p a r e n t v i s c o s i t y ,
re fle c ts
lo w enough
f o r a g iv e n s t e a d y b lo o d f l o w ,
t h e m a g n itu d e o f t h e n o n -N e w to n ia n b e h a v io r o f b lo o d .
V is ­
REVIEW OF PREVIOUS WORK
III.
P o i s e u i I l e may have done t h e f i r s t w o rk on b lo o d
s e l f fo u n d t h a t com plex f l u i d s
(5 ).
He h im ­
such as b lo o d do n o t n e c e s s a r i l y obey
t h e P o i s e u i l l ia n e q u a t i o n w h ic h i s
d is c u s s e d on P. 6 .
O t h e r w o rk e rs
a l s o fo u n d t h a t d e v i a t i o n s e x i s t b e t w e e n . t h e p r e s s u r e d ro p c a l c u l a t e d
fr o m t h e P o i s e u i l l i a n
w o rk on b lo o d .
e q u a t i o n and t h a t measured by t h e e x p e r im e n t a l
Fahraeus and L i n d q u i s t p e r fo rm e d some o f t h e c l a s s i c a l
s t u d i e s o f b lo o d f l o w i n t u b e s .
S c o t t - B l a i r and M e r r i l l
w i t h blootd u s in g c a p i l l a r y v i s c o m e t e r s .
w o rks a r e n o t summarized i n t h i s
th e s is .
The d e t a i l s
a l s o worked
o f such i n d i v i d u a l
Fung (1 2 ) g iv e s a r e v ie w o f
t h e w o rk done t h r o u g h ,1969 ( 5 ) .
Work p u b l i s h e d i n r e c e n t y e a r s by Agarw al
(I)
and L ip o w s k y (1 5 )
shows t h a t t h e y used t h e P o i s e u i l l i a n e q u a t i o n f o r c a l c u l a t i n g th e
a p p a r e n t v i s c o s i t y o f b lo o d .
o f b l o o d , t h e b lo o d v e s s e l
L ip o w s k y (1 5 )
in d ic a t e s t h a t th e v e l o c i t y
s i z e and t h e p r e s s u r e d ro p a c r o s s th e b lo o d
v e s s e l were measured and t h e P o i s e u i l l i a n e q u a t i o n was used t o c a l c u l a t e
th e a p p a re n t v i s c o s i t y .
A garw al
(I)
r e p o r ts th e r e l a t i v e
b lo o d a t v a r y i n g h e m a t o c r i t s i n p u lm o n a ry c i r c u l a t i o n .
P o is e u iT lia n e q u a tio n in
o f th e P o is e u ill ia n
h is w o rk.
A d d itio n a l
v is c o s ity o f
He used t h e
exam ples o f such usage
e q u a t i o n can a l s o be fo u n d i n l i t e r a t u r e .
An a t t e m p t has been made h e re t o show t h e m a g n itu d e o f t h e e r r o r
between v i s c o s i t y c a l c u l a t e d fro m t h e P o i s e u i T l i a n e q u a t i o n and th e
a p p a r e n t v i s c o s i t y u n d e r t h e same f l o w c o n d i t i o n s .
IV .
THEORY AND CALCULATIONAL PROCEDURE
F o r a N e w to n ia n f l u i d ,
t h e r e l a t i o n s h i p between t h e p r e s s u r e and
t h e f l o w can be d e r i v e d fr o m t h e la w o f C o n s e r v a t io n o f Momentum f o r
s te a d y , la m in a r f lo w th ro u g h a c i r c u l a r tu b e .
can be. e x p re s s e d by t h e P o i s e u i l l i a n
6.
The r e l a t i o n s h i p o b t a in e d
e q u a tio n .
The v i s c o s i t y d e r i v e d f r o m such e q u a t i o n i s
v is c o s ity .
T h is i s
te rm e d t h e P o i s e u i l l i a n
T h is s e c t i o n d e s c r ib e s t h e c a l c u l a t i o n a l
c u la t in g th e d if fe r e n c e
d is c u s s e d on P.
procedure f o r c a l ­
between t h e P o i s e u i l l i a n v i s c o s i t y and t h e
a p p a re n t v i s c o s i t y .
The r e l a t i o n s h i p
between t h e s h e a r s t r e s s and t h e s h e a r r a t e
(th e v is c o m e tric o r rh e o lo g ic a l
c r its
d a t a ) a t t h r e e d i f f e r e n t b lo o d hemato­
were o b t a i n e d w i t h t h e h e lp o f t h e c o n c e n t r i c c y l i n d e r v is c o m e t e r
and t h e cone a n d ' p l a t e v i s c o m e t e r .
F ig u r e s V - I -th ro u g h V - 3 .
The d a ta a r e p l o t t e d and shown i n
The d e t a i l s
o f t h e v i s c o m e t r i c a p p a r a tu s and
t h e p ro c e d u re s a r e n o t d e s c r ib e d h e r e .
h e m a t o c r i t were d i v i d e d i n t o
several
The d a ta f o r a g iv e n b lo o d
s h e a r r a t e ra n g e s and an a n a l y t i c a l
e x p r e s s io n f o r each ra n ge was o b t a i n e d .
E i t h e r t h e l i n e a r o r t h e n on ­
l i n e a r r e g r e s s i o n a n a l y s i s was g e n e r a l l y used i n t h i s
p ro ce ss:
The
d a ta a r e r e p r e s e n t e d as
G = f (T )
where G = s h e a r r a t e
T = shear s tre s s
or
= f (T )
where u = v e l o c i t y o f f l u i d
(I)
a t ra d iu s r
10
T a b le V-2 summarizes t h e a n a l y t i c a l
e x p r e s s io n s f o r d i f f e r e n t ra n ge s a t
d i f f e r e n t b lo o d h e m a t o c r i t s .
The r e l a t i o n s h i p
between t h e s h e a r s t r e s s and t h e p r e s s u r e d ro p
p e r u n i t l e n g t h o f t h e tu b e f o r s t e a d y , u n i f o r m l a m i n a r f l o w o f any
flu id
t h r o u g h a c i r c u l a r t u b e can be e x p re s s e d as
T =
where
dp
dx
_r
2
( 2)
= p re s s u re
r = ra d ia l
g r a d ie n t p e r u n i t le n g th o f tube
c o o rd in a te
x = lo n g itu d in a l
T h is e q u a t i o n i s
c o o rd in a te
d e r i v e d by s i m p l i f i c a t i o n
o f t h e momentum
e q u a t i o n s f o r such f l o w s i t u a t i o n .
T h e , b u l k a v e ra g e v e l o c i t y U i s
g iv e n by
y _ v o lu m e tric flo w ra te
u
c r o s s - s e c t i o n a l a re a
'
2 iru r d r
2 Trr d r
or
U =
^
ur dr
where R = o u t s i d e r a d i u s o f t u b e
. I n t e g r a t i n g by p a r t s and u s in g t h e b ou n da ry, c o n d i t i o n s ,
11
u = 0
at r = R
u = “ max
at r = 0
U = ^2
/
^a Fi r 2 d r
(3)
U s in g r as an in d e p e n d e n t v a r i a b l e , t h e e q u a t i o n s
(I),
( 2 ) and
( 3 ) were s o lv e d s i m u l t a n e o u s l y by n u m e r ic a l methods and t h e r e l a t i o n ­
s h i p between U and t h e p r e s s u r e d ro p was o b t a i n e d f o r d i f f e r e n t b lo o d
h e m a to c rits .
The c o m p u te r p ro gram o f t h e n u m e r ic a l method i s
shown i n
A p p e n d ix A.
The P o i s e u i l l i a n
e q u a t i o n was used t o c a l c u l a t e t h e P o i s e u i l l i a n
V is c o s ity ,
where N = P o i s e u i l T i a n V i s c o s i t y
P
A p p a r e n t v i s c o s i t y was s i m u l t a n e o u s l y c a l c u l a t e d by t h e d e f i n i ­
tio n ,
V
*
N and N were c a l c u l a t e d a t a g iv e n w a l l
p
a
r..
*
‘
"
s h e a r s t r e s s and t h e
•
d i f f e r e n c e between t h e two was c a l c u l a t e d s i m u l t a n e o u s l y by c o m p u te r.
I t . m a y be rem a rke d t h a t f o r a N e w to nio n f l u i d ,
- S K
T
where u = v i s c o s i t y , a c o n s t a n t .
12
U s in g e q u a t i o n s
( 2 ) and ( 3 ) ,
it
can be shown t h a t
dp _ 8 ViU
"dx
Rr "
The d i f f e r e n c e between
gra ph i s
shown i n
F ig u r e s V -4 and V - 5 .
p a ra m e te rs on t h e g ra p h .
where U = re d u c e d v e l o c i t y
J
and Ng i s
p l o t t e d a g a i n s t Ng and t h e
U and t h e h e m a t o c r i t a re t h e
V.
F i g u r e s . V -I
RESULTS AND DISCUSSION
t h r o u g h V-3 show t h e v i s c o m e t r i c
T a b le V-2 summarizes t h e f i t t i n g
a n a ly tic a l
o f th e r h e o lo g ic a l
( r h e o lo g ic a l) d a ta .
d a ta i n t o t h e
e x p r e s s i o n s . . F ig u r e s V-4 and -V-5 show t h e r e s u l t s
c a lc u la tio n s .
It
is
o f th e
seen t h a t t h e d i f f e r e n c e between t h e P o i s e u i l l i a n
V i s c o s i t y and t h e a p p a r e n t v i s c o s i t y i s
s i g n i f i c a n t a t lo w b u l k a v e ra g e
v e l o c i t y and a t h ig h v a lu e s o f h e m a t o c r i t s .
i s a b o u t 6 c e n t i p o i s e and h e m a t o c r i t i s
When t h e a p p a r e n t v i s c o s i t y
a b o u t 42% and re d u ce d v e l o c i t y
i s a b o u t I sec ^ , t h e d i f f e r e n c e between t h e P o i s e u i l T i a n V i s c o s i t y and
th e a p p a re n t v i s c o s i t y
th e a c tu a l
shows t h e c o m p a ris o n o f r e s u l t s
(1 5 ) and th o s e o b t a i n e d by t h i s w o r k .
re s u lts
(w h ic h i s a b o u t 8% o f
v a lu e o f t h e a p p a r e n t v i s c o s i t y ) .
T a b le V -I
tu re
is a bo u t 0 .5 c e n t ip o is e
o b t a i n e d fro m l i t e r a ­
I t s h o u ld be n o te d t h a t
o b t a in e d by L ip o w s k y (1 5 ) r e p r e s e n t i n - v i v o
and r e s u l t s
shown i n t h i s
th e s is
re p re s e n t i n - v i t r b
e x p e r im e n t a l
d a ta
e x p e r im e n t a l
d a ta .
In T a b le V - I , t h e t u b e h e m a t o c r i t i s e s t i m a t e d fro m t h e r e s u l t s o f
Barbee ( 5 ) .
These t u b e h e m a t o c r i t s a r e n o t t h e e x p e r im e n t a l
o f L ip o w s k y ( 1 5 ) .
re s u lts
The l a s t two columns show t h e c o r r e s p o n d in g r e s u l t s
fr o m t h e g ra p h s shown i n
F ig u r e s V-4 and V - 5 .
b e tw e e n .th e P o i s e u i l l i a n
V i s c o s i t y and t h e a p p a r e n t v i s c o s i t y i s n o t
ve ry s i g n i f i c a n t ,
A lt h o u g h t h e e r r o r
t h e e s t i m a t e d a p p a r e n t v i s c o s i t y i s much lo w e r th a n
t h a t o b t a i n e d by L ip o w s k y ( 1 5 ) .
Some a d d i t i o n a l
exam ples o f t h e use o f t h e P o i s e u i l l i a n
14
H e m a t o c r i t » 2 5 .4 %
T e m p e r a tu r e = 3 7 ° C
ACTUAL DATA
NEWTONIAN F L U ID
(SHEAR R A T E P ,
FIGURE V - I .
SEC
SHEAR STRESS - SHEAR RATE DATA
15
H e m a t o c r it = 35 .8%
T e m p e r a tu r e
= 37° C
ACTUAL DATA
IEWTONIAN FL U ID
(SHEAR R A T E ) \
FIGURE V - 2.
SEC
SHEAR STRESS - SHEAR RATE DATA
16
H e m a t o c r it = 46.5%
T e m p e ra tu re = 37° C
ACTUAL
DATA
NEWTONIAN FLUID
(SHEAR RATE)15, SEC ^5
FIGURE V - 3 .
SHEAR STRESS - SHEAR RATE DATA
17
LOWER RANGE OF V IS C O S IT IE S
— U = 0 .1
sec
— LI = 0 . 5
sec"
----------
HEMATOCRIT = 2 5 .4%
U = I
sec
U = 2 sec
O d Il
U = 5 se c
-U = 20 s e c "
- HEMATOCRIT = 4 6 .5 %
HEMATOCRIT = 3 5 .8%
CENTIPOISE
FIGURE V -4 .
DIFFERENCE BETWEEN Nn AND N3
P
a
18
HIGHER RANGE OF V IS C O S IT IE S
HEMATOCRIT = 2 5 .4%
HEMATOCRIT = 35 .8%
U = 0 .1
sec 1
O d Il
U = 0 .5
U = I
sec
sec
HEMATOCRIT = 4 6 .5 %
C ENTIPO ISE
FIGURE V -5 .
DIFFERENCE BETWEEN Nn AND Na
P
a
19
TABLE V -I
COMPARISON OF RESULTS OBTAINED FROM LITERATURE (1 5 )
V essel
Si ze
(A v g .)
(1 5 )
m ic r o n s
E s tim a te d
Tube
H e m a t o c r it
p e r c e n ta g e
Reduced '
V e lo c ity
. U (1 5 )
sec 1
A pparent
V is c o s ity
(1 5 )
c e n t!p o is e
33
• 28
1 0 -2 0
9-4
38
30
5 -2 0
20-8
/I O
HO
OI
0*1
2 -4
9 -2 0
2 5 -1 5
3 0 -1 0
48
32
6 -2 0
58
34
15
39
30
50
3 2 .5
AND FROM THIS WORK
E s t im a te d
A p p a re n t
V is c o s ity
c e n t i p o is e
E s tim a te d
(Np -
N a)
D iffe re n c e
c e n tip o is e
3
0.1
3 .3 ■
0.1
3 .6
3 .3
0 .1 5
0.1
2 0 -7
3 .3
0.1
25
3 .3
8 -2 0
2 5 -7
3 .3
0.1
9 -2 0
1 1 -7
3 .3
0.1
.
.
.
0.1
20
TABLE V-2 .
FITTING OF RHEOLOGICAL DATA INTO ANALYTICAL EXPRESSIONS
T e m p e ra tu re 37° C
H e m a t o c r it
Range
(I)
2 5 .4
O < T <; 0 .0 0 4 6 8 3
0 .0 0 4 6 8 3 < T < 1 . 0
1 .0 < T <. 3 3 .0 6
T > 3 3 .0 6
(2)
3 5 .8
O < T < 0 .0 1 3 0 2
. 0 .0 1 3 0 2 < T < 1 . 0 4
I .04 < T < 3 7 .2
T > 3 7 .2
(3 )
4 6 .5
O < T < .04285
0 .0 4 2 8 5 < T < 1 .4 2
1 .4 2 < T < 4 2 .2 5
T > 4 2 .2 5
A n a ly tic a l
G
G
G
G
=
=
=
=
E x p r e s s io n
O
(6 .2 3 8 T ‘ 5 - .4 2 6 8 )2
3 3 .0 0 5 T 1 - 07332
4 3 .2 9 T
G = O
G = ( 5 . 9 4 4 T • 05 - . 6 7 8 ) 2
G = ( - . 7 0 2 7 + 5 ..9 4 4 T '5 +
.0 4 6 3 T )2
G = 37.174T
G = O .
G = ( 4 . 9 3 1 0 5 T * 5 - 1 .0 2 0 7 4 T ) 2
G = (-1 .3 7 2 7 + 5 .1 3 6 9 T *5 +
.06 5 T )2 .
G = 28.4 09 1 T
V i s c o s i t y can be f o u n d i n t h e l i t e r a t u r e
a t lo w v a lu e s o f re d u ce d v e l o c i t i e s
and t h e e r r o r s a r e s i g n i f i c a n t
and h e m a t o c r i t s .
CONCLUSIONS AND RECOMMENDATIONS
V I.
1.
The d i f f e r e n c e between t h e P o i s e u i l l ia n V i s c o s i t y and t h e
a p p a re n t v i s c o s i t y is
s i g n i f i c a n t a t lo w v a lu e s o f re d u c e d v e l o c i t y
(th e
r a t i o o f b u l k a v e ra g e v e l o c i t y and t u b e d ia m e t e r ) and a t lo w v a lu e s o f
h e m a to c ri ts..
2.
th e a c tu a l
In - v itro
e x p e r im e n t a l
e x p e r im e n t a l w o rk i s
re a s o n s w h ic h a r e s t i l l
3.
If
re s u lts
in -v iv o ,
a r e shown i n t h i s
th e s is .
t h e r e s u l t s may d i f f e r ,
If
fo r
unknown.
th e a c tu a l
e x p e r im e n t a l w o rk i s
in -v itro ,
t h e use o f t h e
P o i s e u i l l i a n e q u a t i o n s h o u ld be made o n l y a t h ig h re d u c e d v e l o c i t i e s
such as 30 s e c "^ o r m ore.
4.
A d d itio n a l
exam ples o f usage o f t h e P o i s e u i l l i a n
may be f o u n d f r o m l i t e r a t u r e
V is c o s ity
s u r v e y and t h e e r r o r s can be e s t i m a t e d .
SECTION B
OSCILLATORY CONDITIONS
V II.
STATEMENT OF THE PROBLEMS
T h is s e c t i o n d e a ls w i t h o s c i l l a t o r y
f l o w o f b lo o d .
g iv e s some id e a o f t h e n a t u r e o f o s c i l l a t o r y
c i r c u l a t o r y s yste m ( 1 8 ) .
F ig u r e V I I - I
f l o w o f b lo o d i n th e human
The r a t e o f h e a r t b e a ts u s u a l l y v a r i e s fr o m
a b o u t 6 0 -7 0 b e a ts p e r m in u t e a t r e s t t o a b o u t 160-170 b e a ts p e r m in u te
a t peak c a r d i a c o u t p u t .
la tio n
is
ty p ic a l
Thus t h e u s e f u l
ra n g e o f f r e q u e n c i e s o f o s c i l ­
a b o u t I t o 3 c y c l e s p e r s e co n d .
T a b le V I I - I
summarizes some
v a lu e s o f mean p r e s s u r e d ro p p e r u n i t l e n g t h o f t h e v e s s e l a t
v a rio u s Id e a tio n s
(2 5 ,2 6 ).
E x p e r im e n t a l w o rk was done t o u n d e r s ta n d some o f t h e c h a r a c t e r ­
is tic s
o f o s c i l l a t o r y b lo o d f l o w i n
r ig id ,
c i r c u l a r g la s s t u b e s .
c h a r a c t e r i s t i c s are th e p re s s u re g r a d ie n t - f lo w r e l a t i o n s h i p
These
under
o s c i l l a t o r y c o n d i t i o n s , t h e v a lu e s o f t h e h e m a t o c r i t o f b l o o d , th e
p o s s ib le e f f e c t s
o f d e f o r m a t io n and a g g r e g a t io n , o f re d b lo o d c e l l s ,
e tc .
I t may a l s o be p o s s i b l e t o c a l c u l a t e t h e p r e s s u r e g r a d i e n t - f l o w
re la tio n
f o r b lo o d u n d e r o s c i l l a t o r y c o n d i t i o n s u s in g t h e momentum
e q u a tio n s
( e q u a t io n s o f m o t io n ) and t h e r h e o l o g i c a l
s te a d y flo w s in
v is c o m e te rs ).
d a ta
( o b t a in e d fro m
B u t such d a ta may p ro v e t o be in a d e q u a te
f o r such c a l c u l a t i o n s , because t h e c h a r a c t e r i s t i c t im e s f o r re d c e l l
a g g r e g a t io n and re d c e l l
o s c illa tio n .
d e f o r m a t io n a r e co m p a ra b le t o t h e tim e s o f f l o w
I n a d d i t i o n , v i s c o e l a s t i c e f f e c t s may be i m p o r t a n t .
Hence t h i s
s e c tio n
in v o lv e s :
( i ) t h e e x p e r im e n t a l w o rk done on o s c i l l a t o r y
flo w .
FEMORAL VEIN
SUBCLAVIAN VEIN
FEMORAL VEIN
FEMORAL ARTERY
SIMULTANEOUS RECORDINGS OF FLOW VELOCITIES
FIGURE V I I - I .
NATURE OF OSCILLATORY BLOOD FLOW
26
.
T a b le .V I I - I
VALUES OF MEAN PRESSURE DROP AT VARIOUS LOCATIONS ( 2 5 , 2 6 )
D im en sio n s
Type o f
B lo o d Vessel
D ia m e te r
m ic r o n s
L e n g th
mm
P re s s u r e Drop
P er U n i t L e n g th
mm Hg/mm
L a rg e A r t e r y
1 0 ,0 0 0
600
-0 .0 1 8 3
T e rm in a l A r t e r y
. I ,600
no
-0 .1 7 3
A rte r io le s
40
2
-1 5 .0
C a p illa rie s
16
I
-1 5 .0
V en u le s
60 .
2
-4 .5
Main V e in
L a rg e V e in
D esce nd in g A o r t a
Venae Cavae
4 ,0 0 0
100
-0 .0 5
2 0 ,0 0 0
600
-0 .0 0 8 3 4
1 6 ,0 0 0 -2 0 ,0 0 0
2 0 ,0 0 0
—
-7 .0 1
x IO "4
— —
-3 .1 6 x IO "4
27
( i i ) th e d is c u s s io n o f th e c a l c u l a t i o n a l
flo w r e la t io n s h ip
m e tric d a ta .
procedure f o r p re s s u r e -
u n d e r o s c i l l a t o r y c o n d i t i o n s u s in g t h e s t e a d y v i s c o ­
V III.
The l i t e r a t u r e
several
REVIEW OF' PREVIOUS WORK
on p u l s a t i l e
b lo o d f l o w has been d e s c r ib e d by
a u t h o r s , such as McDonald ( 1 6 ) , B e rg e l
( 6 ) and A t t i n g e r ( 2 ) .
Womersley (2 7 ) d e r i v e d t h e t h e o r y o f o s c i l l a t o r y
flu id
in a c i r c u l a r tu b e .
m a riz e d on P.
f l o w f o r a N ew tonian
Some o f t h e r e s u l t s o f h i s t h e o r y a re sum­
36.
T a b le s V I I I - I
and V I I I - 2
summarize some o f t h e d e t a i l s o f t h e
e x p e r im e n t a l w o rk done by S e v i l l a - L a r r e a
( 1 9 ) , S in g h (2 0 ) and T h u r s to n
( 2 2 , 2 3 ) . 'T h e s e t a b l e s summarize t h e t u b e d ia m e t e r s , t h e f r e q u e n c i e s
o f o s c illa tio n ,
th e flo w a m p litu d e s , h e m a to c rits , th e p re s s u re a m p li­
t u d e s , e t c . , used by t h e s e w o r k e r s .
" S e v illa -L a rre a
(1 9 ) s t u d i e d p u l s a t i l e
speed m ic r o c in e m a t o g r a p h y .
f l o w o f b lo o d u s in g h ig h
He measured t h e p r e s s u r e w i t h t h e h e lp o f
p r e s s u r e t r a n s d u c e r b u t f o r t h e measurement o f f l o w , he measured t h e re d
b lo o d c e l l
v e lo c ity p r o file s .
He d i d n o t measure t h e f l o w w i t h t h e h e lp
o f a d is p la c e m e n t t r a n s d u c e r o r some o t h e r f l o w measurement d e v i c e .
It
a l s o seems t h a t t h e a p p a r a tu s was n o t t e s t e d w i t h a N e w to n ia n f l u i d ,
i.e .,
flu id s
t h e a p p a r a tu s s h o u ld have been ch ecke d t o show t h a t f o r such
t h e e x p e r im e n t a l
and t h e o r e t i c a l
r e s u l t s a g re e w i t h each o t h e r .
S in g h (2 0 ) used an e l e c t r o m a g n e t i c f l o w m e te r t o measure o s c i l ­
la t o r y flo w .
H is p r e s s u r e m e a s u rin g method was d e s ig n e d so t h a t he
c o u ld e lim in a t e
th e e n tra n c e e f f e c t s
t a p s i n h i s tu b e s
(1 6 6 0 y , 3 2 4 0 y ) .
o f th e tu b e .
He had two p r e s s u r e
These t a p s were p la c e d s u f f i c i e n t l y
f a r fr o m t h e ends o f t h e tu b e so t h a t end e f f e c t s were n e g l i g i b l e .
H is
29
TABLE V I I I - I
SOME OF THE DETAILS ABOUT THE PREVIOUS WORK
DONE ON OSCILLATORY BLOOD FLOW
W o rke r
S e v illa -L a rre a
(1 9 )
S in g h
(2 0 )
C a p illa ry Tube
D ia m e te r
cm
L en g th
o f Tube
cm
Number o f
Tube's i n
P a ra lle l
N
H e m a t o c r it
p e r c e n ta g e
0 .0 0 4
6 .5
I
0 .0 0 7
,1 0 .0
I
0 .0 6 0 8
0 .1 6 6
0 .3 2 4
7 .5
7 0 .0
1 0 0 .0
I
I
I
40
40
40
1 1 .1 4
6
46
. 0 .6 5 4
0 .9 8 4
1 .2 3 6
1.401
1 .9 9 3
2 .8 6
5 .8 6
1 3 .1 6
50
50
32
12'
8
5.
2
I
46
46
46
46
46
46
46
46
T h u rs to n
(2 2 )
0 .0 8 6
T h u r s to n
(2 3 )
0 .0 4 3
0 .0 6 1 0
0 .0 7 8 8
0 .0 9 4
0 .1 3 2
0 .1 9 1 8
0 .3 9 8
0 .7 0
T e m p e ra tu re
°C
• 5 ,1 0
2 0 ,4 0
5 ,1 0
2 0 ,4 0
25
25
23
23
23
'
2 1 .5
-
24
24
24
24 .
24
24
24
24
TABLE V H I - 2
■SOME OTHER DETAILS ABOUT THE PREVIOUS WORK DONE ON OSCILLATORY BLOOD FLOW
Some O t h e r D e t a i l s
W orker
S e v illa -L a rre a
(1 9 )
S in gh
(2 0 )
T h u r s to n
(2 2 )
T h u r s to n
(2 3 )
F requency
.. o f
O s c illa tio n
c .p .s .
'
0 .0
3 .6
8 .4
0 .6
th ro u g h
10
.
2
0 .2
th ro u g h
200
Flow
A m pli t u d e
cm3/ s e c
D ia m e te r
o f Tube .
cm
P r e s s u r e Drop
A m p lit u d e
d y n e s /c m 3
N a tu re o f Flow
0 .0 0 4
6 x IO " 6 to
20 x I O " 5
8 x IO 3 to
79 x I O 3
0 .0 0 7
19 x 1 0 " b_ to
60 x 1 0 " 6 .
4 x IO 3 to
36 X l O 3
0 .0 8
I O 5 t o I .64
x IO 5
0 .1 6 6
0 .8
2200 t o 4720
0 .3 2 4
1 .6
360 t o 1800
0 .0 8 6
M O "5 to IO " 3
I to 2 x IO 3
S te a d y ,
o s c i l l a t o r y and
p u ls a tile
0 .0 4 3
to
0 .7 0
1.0"6 t o 1 0 " i
I O " 1 t o TOO
P u r e ly
o s c illa to r y
0 .0 6 0 8
.
C o m b in a tio n o f
s te a d y and
o s c illa to r y or
p u ls a tile .
P u r e ly
o s c illa to ry
31
a p p a r a tu s was t e s t e d f o r a N e w to nia n f l u i d .
a m p lit u d e t o f l o w a m p lit u d e i s
re s u lts
f o r b l o o d , he f i r s t
im pe de n ce.
The r a t i o
o f p ressure
'T o a n a ly z e h i s e x p e r im e n t a l
p l o t t e d t h e impedence v e rs u s f r e q u e n c y f o r
N e w to nia n f l u i d s w i t h d i f f e r e n t v i s c o s i t i e s . .
The e x p e r im e n t a l
impedence
f o r b lo o d a t a g iv e n f r e q u e n c y was measured and th e n p l o t t e d on t h e same
g ra p h f o r d i f f e r e n t f r e q u e n c i e s o f o s c i l l a t i o n .
A t a g iv e n f r e q u e n c y ,
t h e v i s c o s i t y o f a N e w to n ia n f l u i d , w h ic h g iv e s an impedence e qual t o
t h a t o f b l o o d , was c o n s i d e r e d as t h e b l o o d 's , " e q u i v a l e n t v i s c o s i t y . "
The r e s u l t s
d e riv e d w it h t h i s
e q u i v a l e n t v i s c o s i t y a r e o n l y good f o r
b lo o d a t t h a t f r e q u e n c y o f o s c i l l a t i o n .
B lo o d i s n o n -N e w to n ia n and t h e
p r e d i c t i o n s a b o u t b lo o d fr o m a N e w to n ia n f l u i d
re tic a lly
so und.
t h e o r y may n o t be t h e o ­
Hence, t h e a n a l y s i s o f h i s e x p e r im e n t a l
re s u lts
needs
f u r t h e r s tu d y ..
T h u r s to n
( 2 2 , 2 3 ) p e r fo rm e d b o th t h q t h e o r e t i c a l
w o rk on o s c i l l a t o r y f l o w o f b lo o d .
tic
flu id .
A v is c o u s m a t e r i a l
is
He c o n s id e r e d b lo o d as a v i s c o e l a s ­
one w h ic h f l o w s , when s t r e s s i s
a p p lie d .
An e l a s t i c m a t e r i a l
a p p lie d .
T h u r s to n c o n s i d e r e d b lo o d t o show c h a r a c t e r i s t i c s o f b o th
t h e v is c o u s and t h e e l a s t i c
is
and e x p e r im e n t a l
one w h ic h d e fo r m s , when s t r e s s i s
s ta te s .
He d e r i v e d t h e t h e o r y o f o s c i l ­
la t o r y flo w f o r a lin e a r v is c o - e la s t ic
(2 3 ).
The t u b e i s
assumed t o be f i l l e d
e f f ic ie n t o f v is c o s ity
n* = n'
- in "
flu id
in a r i g i d
w ith a f l u i d
c i r c u l a r tu b e
o f com p lex
co­
32
n' = re a l
p a rt
n " = im a g in a r y p a r t
U s in g t h e N a v ie r - S t o k e s e q u a t i o n s w i t h a co m p lex v i s c o s i t y s u b ­
s titu te d
f o r th e usual
co m p lex f l o w .
v is c o u s v i s c o s i t y ,
A N e w to n ia n f l u i d
a v is c o -e la s tic f l u i d
he d e r i v e d e q u a t i o n s f o r t h e
does n o t show any e l a s t i c
shows b o th v i s c o u s and e l a s t i c e f f e c t s
a re r e p r e s e n t e d by t h e r e a l and i m a g in a r y te rm s i n
such d i v i s i o n
o f v is c o s ity in to
v is c o u s f l u i d
a ls o .
two. p a r t s
is
H is e x p e r im e n t a l w o rk on o s c i l l a t o r y
t r a n s d u c e r t o measure t h e o s c i l l a t o r y
s tir r e r
e f f e c t , w h ile
flo w .
th is
and t h e y
th e o ry .
But
p o s s ib le f o r a p u re ly
flo w in v o lv e s a v e lo c it y
B u t he d i d n o t have a
i n h i s a p p a r a tu s so t h a t t h e re d b lo o d c e l l s may s e t t l e and t h e
h e m a t o c r i t o f b lo o d may n o t re m a in u n i f o r m .
He a l s o had a membrane i n
t h e fe e d r e s e r v o i r w h ic h s e p a r a t e d t h e b lo o d fr o m w a t e r .
The p r e s s u r e
was measured i n t h a t s e c t i o n o f f e e d r e s e r v o i r w h ic h c o n t a in e d w a t e r .
H is e x p e r im e n t a l
re s u lts
show t h a t t h e r e i s a l i n e a r r e l a t i o n s h i p
between t h e p r e s s u r e and t h e f l o w a t lo w e r s h e a r . r a t e s
lo w e r f l o w r a t e s ) .
T h is o b s e r v a t i o n
t i o n s o f many o t h e r w o r k e r s
is
( i .e .,
in c o n t r a d ic t io n w it h o bse rva ­
( 2 , 3 , 1 3 ) , because b lo o d i s
f o u n d t o be
n o n -N e w to n ia n a t lo w s h e a r r a t e s and t h e r e e x i s t s a f i n i t e
Hence a t lo w e r s h e a r r a t e s , t h e r e l a t i o n s h i p
t h e f l o w s h o u ld be n o n - l i n e a r .
at
y ie ld
s tre s s . •
between t h e p r e s s u r e and
F o r t h e s e and o t h e r r e a s o n s , T h u r s t o n 's
33
e x p e r im e n t a l w o rk needs t o be r e p e a t e d .
K lin e
(1 3 ) c o n s id e r e d b lo o d as a p o l a r f l u i d .
In p o l a r f l u i d
t h e o r y , b lo o d i s c o n s i d e r e d as a c o l l e c t i o n o f p a r t i c l e s
te n s o r a n a ly s is
is
B u t h i s w o rk i s
p u re ly th e o r e t ic a l
flu id
used t o u n d e r s ta n d t h e o s c i l l a t o r y
.
f l o w o f b lo o d .
and Cowin (1 0 ) a rg u e d t h a t p o l a r
t h e o r y does n o t a d e q u a t e l y model b lo o d f l o w i n m i c r o c i r c u l a t i o n .
A ls o , i t
is
p o s itio n
in a tu b e , b u t p o la r f l u i d
tio n
and s t r e s s -
o b s e rv e d t h a t t h e h e m a t o c r i t o f b lo o d changes w i t h r a d i a l
t h e o r y does n o t t a k e i n t o
such a v a r i a b i l i t y .
It
•
has been m e n tio n e d on P. 24 t h a t o s c i l l a t o r y
f l o w o f b lo o d
m ig h t be p r e d i c t a b l e f r o m s t e a d y f l o w s h e a r s t r e s s - s h e a r r a t e
m e tric )
d a ta .
c o n s id e ra -
(v is c o ­
B u t such d a ta may p ro v e t o be in a d e q u a te , because o f t h e
v i s c o - e l a s t i c i t y o f . b l o o d a n d / o r because t h e k i n e t i c s o f re d c e l l
d e f o r m a t io n a re s lo w compared t o t h e o s c i l l a t o r y
tim e .
t h e id e a s r e g a r d i n g v i s c o - e l a s t i c i t y and t h e r e d c e l l
d e f o r m a t io n a r e d is c u s s e d i n t h i s
L e s s n e r and c o - w o r k e r s
Hence, some o f
a g g r e g a t io n and
c h a p te r.
(1 4 ) s t u d i e d t h e v i s c o - e l a s t i c p r o p e r t i e s
o f b lo o d u s in g a v i s c o - e l a s t o m e r u n d e r o s c i l l a t o r y c o n d i t i o n s .
d e t e r m in e d t h e r e l a x a t i o n
between 0 . 3 t o 0 . 8 s e c o n d .
They
t im e s p e c tru m f o r b lo o d o v e r a t im e range
They c o n c lu d e d t h a t re d c e l l
a g g r e g a t io n
p ro c e s s e s have c h a r a c t e r i s t i c t im e c o n s t a n t s o f 0 . 8 second o r l e s s .
T h u s , fro m t h e r e l a x a t i o n
q u e n c ie s o f o s c i l l a t i o n
it
t im e s p e c t r u m , i t
seems t h a t a t lo w e r f r e ­
may be p o s s i b l e t o p r e d i c t o s c i l l a t o r y f l o w
34
b e h a v i o r o f b lo o d fr o m t h e s t e a d y v i s c o m e t r i c d a t a .
B u t , t h e re d b lo o d c e l l s
a r e a l s o d e f o r m a b le .
Work done by
Evans (1 1 ) and Waugh (2 4 ) d e s c r ib e s t h e d e f o r m a t io n o f re d b lo o d c e l l s .
They c o n s id e r e d t h e re d b lo o d c e l l
lo w v i s c o s i t y N e w to nia n f l u i d
as a membrane e n v e lo p e c o n t a i n i n g a
(an aqueous" s o l u t i o n o f h e m o g lo b in ) .
For
t h e membrane, t h e y f o r m u l a t e d and e x p e r i m e n t a l l y d e t e r m in e d f o u r
m a te ria l
c o n s ta n ts .
e la s tic ity ,
I
These c o n s t a n t s were te rm e d t h e s h e a r modulus o f
th e v i s c o s i t y
" s h e a r , " and a p l a s t i c
(in
t h e v i s c o - e l a s t i c d o m a in ) , t h e y i e l d
v is c o s ity .
Below a y i e l d c o n d i t i o n , t h e membrane
is
r e v e r s i b l y d e fo r m a b le
( v i s c o - e l a s t i c ) , b u t above t h e y i e l d
it
b e g in s t o f l o w i n a v i s c o u s , p l a s t i c m anner.
c o n d itio n
The r e d b lo o d c e l l
has
t h e shape o f a. b i- c o n c a v e d i s k w i t h a mean d ia m e t e r o f a b o u t e i g h t
m ic r o n s and t h i c k n e s s o f a b o u t one t o two m ic r o n s
because i t
(5 ).
It
is f le x ib le ,
i s e s s e n t i a l l y a t h i n membrane c o n t a i n e r i n c o m p l e t e l y f i l l e d
w ith a f l u i d .
It
is
a liv in g
c e ll.
It
is
c a p a b le o f r e s p o n d in g t o
o s m o tic p r e s s u r e changes i n t h e s u s p e n d in g m e d ia.
Waugh (2 4 ) and Evans
(1 1 ) f o u n d t h a t t h e t im e t o r e c o v e r t h e d e f o r m a t io n o c c u r r e d w i t h t h e
re d b lo o d c e l l s may be o f t h e o r d e r o f 0 . 3 se co n d s.
lo w e r f r e q u e n c y o f o s c i l l a t i o n , o s c i l l a t o r y
s t u d i e d fr o m t h e s t e a d y v i s c o m e t r i c d a t a .
; o s c illa tio n ,
it
Hence, a g a i n , a t
f l o w o f b lo o d can be
B ut a t h i g h e r f r e q u e n c y o f
may n o t be p o s s i b l e t o do s o .
W o rk.done by s e v e r a l w o rk e rs has been d e s c r ib e d b r i e f l y
c h a p te r.
From t h i s
in t h i s
d i s c u s s i o n , t h e p r e s e n t s t u d y was made c o n s i d e r i n g
35
th e f o l lo w in g a s p e c ts .
1.
b lo o d c e l l s
A m a g n e tic s t i r r e r
(h e m a to c rit)
is
used so t h a t t h e c o n c e n t r a t i o n o f re d
re m a in s u n i f o r m
( i .e .,
c e lls
do n o t s e t t l e )
in
t h e a p p a r a tu s r e s e r v o i r s .
2.
A p o s s i b l e in a d e q u a c y o f t h e use o f an e l e c t r o m a g n e t i c f l o w ­
m e te r can be a v o id e d .
Such f lo w m e t e r s may r e c o r d t h e same r e s u l t s f o r
two d i f f e r e n t o s c i l l a t o r y
3.
flo w s .
P re s s u r e s a r e measured by d i r e c t c o n t a c t o f t h e b lo o d w i t h
p re ssu re tra n s d u c e rs .
4.
The e x p e r im e n t a l
known p r o p e r t i e s o f b lo o d .
•
d a ta s h o u ld be e x p l a i n e d i n te rm s o f t h e
IX .
THEORETICAL BACKGROUND
The t h e o r y o f . o s c i l l a t o r y f l o w f o r a N e w to nia n f l u i d
c u l a r tu b e was s t u d i e d by Womersley ( 2 7 ) .
as f o l l o w s .
4k
When t h e p r e s s u r e d ro p i s
"
Pm cos
in a c i r ­
H is r e s u l t s may be summarized
g iv e n by
(w t)
where Pffl = a m p lit u d e o f p r e s s u r e d ro p
a) = a n g u l a r f r e q u e n c y
= 2irf
f
= l i n e a r fre qu e ncy
t = i n s t a n t a n e o u s t im e
p = p re ssu re
x = a x ia l
th e flo w is
d is ta n c e
g iv e n by
d4
Q= ^
'
^lO
Pm
si n (uit + <j>)
where R = r a d i u s o f t h e tu b e
y = v is c o s ity o f f lu id
i
Mio
The q u a n t i t i e s
and * have been d e r i v e d as a f u n c t i o n o f a , w h ic h i s
g iv e n by
a = (B !m .)0.'5
= f r e q u e n c y p a r a m e te r
where p = d e n s i t y o f f l u i d
-
•
37
5 °
S
g
tr>
I
°
0 . 00!
—I
3000
6000
9000
12000
PRESSURE A M P LITUD E, DYNES/CM3
FIGURE I X - I .
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE FROM THEORY
( P o i n t s shown i n t h i s f i g u r e a re c a l c u l a t e d fro m
th e o ry .)
38
Tube R adius = 0 .0 4 cm
V i s c o s i t y = 0 .0 3 5 p o is e
D e n s i t y = I gm/cm3
FLOW AMPLITUDE, CM3/SEC
LARGE TUBE RADIUS
0
3000
6000
9000
PRESSURE AMPLITUDE, DYNES/CM3
FIGURE I X - 2 .
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE FROM THEORY
39
The q u a n t i t y M10 r e p r e s e n t s modulus of" t h e s i n u s o i d a l o s c i l l a ­
t i o n o f volume f l o w .
The q u a n t i t y <f> r e p r e s e n t s t h e phase d i f f e r e n c e
between p r e s s u r e wave and f l o w wave.
B oth
and <f> can be o b t a in e d as
a f u n c t i o n o f. a fr o m a t a b l e w h ic h was c o n s t r u c t e d by Womersley ( 2 7 ) .
F ig u r e s
IX -I
and I X - 2 g iv e some id e a a b o u t how t h e f l o w a m p lit u d e v a r i e s
w i t h t h e p r e s s u r e a m p lit u d e f o r d i f f e r e n t v a lu e s o f a , a c c o r d in g t o
th is
th e o ry .
Flow a m p lit u d e =
r
u
P
m az
= V
P re s s u r e a m p lit u d e = Pm
m
'
X.
APPARATUS AND PROCEDURES
E x p e r im e n t a l Work
A.
A p p a ra tu s
The s c h e m a tic d ia g ra m o f t h e a p p a r a tu s i s
The a p p a r a tu s i s
o f a s c o tc h -y o k e .
The s i n u s o i d a l
F ig u r e
o p e ra te s .
(w h ic h i s
s ig n a l
X-2 and F ig u r e
B a s ic a lly i t
g e n e ra te d .
th e r e c ip r o c a t in g
s c o tc h -y o k e .
m o tio n i s
when a p a r t i c l e
As shown i n
d r ille d
F ig u r e
in a
p r o j e c t e d on t h e d ia m e t e r o f t h e
X -3,
i n p a r t A.
in th e h o r iz o n ta l
a p in
d ia m e t e r i s
c a l l e d s im p le
(p a rtic le )
is
in
between t h e g u id e s
Thus, th e s in u s o id a l
th ro u g h a g ear box.
t a c h o m e t e r and i s
( p a r t C ).
-
im p a r t s t h e o s c i l l a t o r y m o tio n t o t h e f l u i d
The s c o t c h - y o k e i s
in
im p a r t e d t o t h e p l u n g e r o f a s y r i n g e
..
c a p i l l a r y tu b e .
fitte d
As p a r t A r o t a t e s , t h e p a r t B
d ire c tio n
The o s c i l l a t o r y m o tio n o f p a r t B i s
w h ic h , i n t u r n ,
ro ta te s
T h is same a n a lo g y can be e x a c t l y a p p l i e d t o th e
a g ro o v e w h ic h i s
o s c illa te s
( o s c il­
i n o t h e r w o r d s , s im p le h a rm o n ic m o tio n
c i r c u l a t o r y p a t h , t h e p r o j e c t e d m o t io n on i t s
h a rm o n ic m o t io n .
g e n e r a te d w i t h t h e h e lp
X - 3 e x p l a i n t h e p r i n c i p l e on
in to
As shown i n F ig u r e X - 2 ,
c i r c u l a t o r y m o tio n and i t s
is
c o n v e r t s t h e c i r c u l a r m o tio n o f a. s h a f t
d r i v e n by an e l e c t r i c m o t o r )
l a t i n g ) m o tio n o f a p i s t o n , o r ,
is
X -I.
d e s c r ib e d by c o n s i d e r i n g one p a r t a f t e r a n o t h e r .
S c o tch -Y o ke .
w h ic h i t
shown" i n F ig u r e
s ig n a l
is
in th e
g e n e r a te d .
d r i v e n by a v a r i a b l e speed e l e c t r i c m o t o r ,
The m o to r speed i s m o n it o r e d by an e l e c t r o n i c
r e g u l a t e d by a f e e d - b a c k c i r c u i t .
Speed R e g u la t o r
O s c illo g ra p h ic
R e c o rd e r
Ampli f i e r
Gear Box
P re s s u re
T ra n s d u c e r
E l e c t r i c M o to r
D is p la c e m e n t
T ra n s d u c e r
Feed R e s e r v o i r
S y r in g e
D is c h a rg e
R e s e rv o ir
C a p illa ry
S c o tc h -y o k e
M a g n e tic
S tirre r
FIGURE X - I .
SCHEMATIC DIAGRAM OF APPARATUS
Stand
PLATE
PARTICLE
ROTATES
RECIPROCATES
FIGURE X -2 .
DIAGRAM EXPLAINING PRINCIPLE-OF SCOTCH YOKE
P art A
P art C
RECIPROCATES
ROTATES
P art B
FIGURE X -3 .
SCOTCH-YOKE
GOES TO
SYRINGE
44
S y r in g e
(A p p e n d ix B ) .
to th e s c o tc h -y o k e
(see F ig u r e
The p l u n g e r o f t h e s y r i n g e i s c o n n e c te d
X - I).
o u t o f t h e c a p i l l a r y t u b e w h ic h i s
T h e ' s y r i n g e pumps b lo o d i n t o and
fix e d
i n between t h e two r e s e r v o i r s .
R e s e r v o i r s . a n d Tube (A p p e n d ix B ) .
The r e s e r v o i r s a re c o n n e c te d
t o t h e t u b e w i t h t h e h e lp o f s q u a re p l a t e s and s c re w s .
g lu e d a t t h e ends o f t h e t u b e .
g a s k e t"
The p l a t e s a re
To p r e v e n t le a k a g e , s i l i c o n e
(a r u b b e r base a d h e s iv e s e a l a n t )
is
"fo rm -a -
a p p l i e d i n between th e
s q u a re p l a t e s and t h e r e s e r v o i r s . .
The b lo o d i n t h e fe e d r e s e r v o i r can be s t i r r e d
s tir r e r
B.
(A p p e n d ix B) t o m a i n t a i n t h e h e m a t o c r i t o f b lo o d u n i f o r m .
P re s s u r e Measurement D e v ic e (A p p e n d ix B)
P r e s s u r e a t u p s tre a m s i d e
(fe e d s id e )
P23Dd p r e s s u r e t r a n s d u c e r w h ic h i s
p u t o f th e a m p l i f i e r i s
re co rd e r.
C.
by a m a g n e tic
i s measured b y 'a S tath am
c o n n e c te d t o an a m p l i f i e r .
The o u t ­
f e d t o a. H e w le t t - P a c k a r d o s c i l l o g r a p h i c
The p r e s s u r e i n t h e d i s c h a r g e r e s e r v o i r i s
a t m o s p h e r ic .
Flow Measurement D e v ic e (A p p e n d ix B)
The d is p la c e m e n t o f t h e p l u n g e r o f t h e s y r i n g e i s
m o n it o r e d by
a d is p la c e m e n t t r a n s d u c e r whose o u t p u t is . f e d t o an a m p l i f i e r , w h ic h i s
c o n n e c te d t o t h e r e c o r d e r .
The d is p la c e m e n t t r a n s d u c e r measures t h e
d is p la c e m e n t o f t h e p lu n g e r b u t fr o m t h e d is p la c e m e n t s i g n a l
s ib le
to c a lc u la t e th e flo w s ig n a l.
If
t h e d is p la c e m e n t wave i s
g iv e n b y .
it
is
pos­
45
S .= Sm s i n tot .
, m
where S = d is p la c e m e n t a t t im e
t = i n s t a n t a n e o u s t im e
w = 2 Trf
= a n g u la r fre q u e n c y
f
S =
m
= l i n e a r fre q u e n c y
a m p lit u d e o f t h e wave
th e n t h e p l u n g e r speed o f movement i s
g iv e n b y ,
3 ! = Smco cos cot
The f l u i d
v o l u m e t r i c f l o w r a t e fr o m ( o r i n t o )
th e s y rin g e is
g iv e n b y .
w here Q = i n s t a n t a n e o u s f l o w
ds- = d ia m e t e r o f s y r i n g e p lu n g e r
C o m b in in g t h e l a s t two e q u a t i o n s ,
Q = I ds V
cos ut
T h is can a l s o be w r i t t e n a s ,
'
Q = Qm cos cot
"here Qm = £ ^
Thus, i t
is
p o s s ib le to c a lc u la t e
flu id
f l o w a m p lit u d e fro m
s y r i n g e p lu n g e r d is p la c e m e n t a m p l i t u d e .
R e c o rd e r (A p p e n d ix B ) .
The H e w le t t - P a c k a r d m o d e l o s c i l l o g r a p h i c
46
r e c o r d e r has two c h a n n e ls .
It
and on t h e o t h e r ch a n n e l
can s i m u l t a n e o u s l y r e c o r d t h e p r e s s u r e wave.
it
can r e c o r d t h e p lu n g e r d is p la c e m e n t wave,
P ro c e d u re s .
D.
C a l i b r a t i o n o f t h e P re s s u r e T r a n s d u c e r
. The p r e s s u r e t r a n s d u c e r measures t h e p r e s s u r e i n , t e r m s o f an
o u tp u t v o lta g e .
d uce r.
Hence, i t
The r e l a t i o n s h i p
is
nece ssa ry to c a l i b r a t e
between t h e p r e s s u r e measured i n te rm s o f v o l t s
and t h e p r e s s u r e measured i n te rm s o f mm o f Hg i s
o f th e p re ssu re tra n s d u c e r.
is
te rm e d t h e c a l i b r a t i o n
T h is c a l i b r a t i o n was o b t a in e d w i t h t h e h e lp
o f a p r e c i s e m e rc u ry manometer (A p p e n d ix B ) .
t o be l i n e a r and i t
th e p re ssu re t r a n s ­
shown i n
F ig u r e X -4 .
The r e l a t i o n s h i p was fo u n d
A l i n e a r re g re s s io n a n a ly ­
s i s was used t o o b t a i n an e x p r e s s io n w h ic h r e l a t e s t h e p r e s s u r e i n v o l t s
t o t h e p r e s s u r e i n mm o f Hg.
P = 1 1 4 .5 V + I .7
where V = p r e s s u r e i n v o l t s
P = p r e s s u r e i n mm o f Hg.
E.
C a l i b r a t i o n o f t h e D is p la c e m e n t T r a n s d u c e r
T h is t r a n s d u c e r a l s o measures t h e . d is p la c e m e n t o f t h e p lu n g e r o f
t h e s y r i n g e i n te rm s o f an o u t p u t v o l t a g e .
o b ta in th e r e la t io n s h ip
Hence, i t
is
n e c e s s a ry t o
between the. d is p la c e m e n t measured i n term s o f
v o l t s and t h e d is p la c e m e n t measured i n te rm s o f c e n t i m e t e r s .
b r a t i o n was o b t a i n e d u s in g a p r e c i s i o n m ic r o m e t e r .
T h is c a l i ­
The r e l a t i o n s h i p
47
V , VOLTS
FIGURE X -4 .
PRESSURE TRANSDUCER CALIBRATION
48
used i s
g iv e n b y ,
I
r e c o r d e r c h a r t d i v i s i o n = 0 . 0 6 8 5 ’ em
where I d i v i s i o n
c o rre s p o n d s t o t h e s e n s i t i v i t y o f 10 m v / v / f u l l
s c a le .
F.
Measurement o f t h e C a p i l l a r y - T u b e D ia m e te r
T h is was done by p e r f o r m i n g an e x p e r im e n t i n w h ic h w a t e r (a
N e w to nia n f l u i d )
la ry -tu b e w ith
was pumped u n d e r s t e a d y c o n d i t i o n s t h r o u g h t h e c a p i l ­
t h e h e l p o f a s y r i n g e pump ( A p p e n d i x . B ) . . The p r e s s u r e
d ro p a c r o s s t h e tu b e a t v a r i o u s v o l u m e t r i c r a t e s was measured w i t h t h e
h e lp o f t h e p r e s s u r e t r a n s d u c e r , w h ic h i s
The P o i s e u i l l ia n e q u a t i o n
f i x e d i n t h e fe e d r e s e r v o i r .
(good f o r a N e w to n ia n f l u i d
such as w a t e r ) was
used t o o b t a i n t h e d ia m e t e r o f t h e t u b e .
r4
= _Q_ 8yL
AP
TT
■
:
where Q =. v o l u m e t r i c f l o w
Ap = pressure drop across the tube
u = viscosity of water
L = length o f the tube
R = radius of the tube
T h u s , t h e r a d i u s o f t h e t u b e can be c a l c u l a t e d .
shows t h e gra ph o f t h e v o l u m e t r i c f l o w r a t e a g a i n s t
F ig u r e X-5
th e p re ssu re d ro p .
From t h e s lo p e o f t h e g r a p h , t h e r a d i u s o f t h e t u b e can be c a l c u l a t e d .
F lu id :
W ater
T e m p e r a t u r e : 25° C
Tube L e n g th :
3 9 .5 cm
Tube R a d iu s :
200 m ic ro n s
PRESSURE DROP, MM OF HG
FIGURE X -5 .
CAPILLARY TUBE CALIBRATION
50
6.
T e s t i n g o f t h e A p p a ra tu s
'A s i t
is
d is c u s s e d on P.
36, t h e r e e x i s t s a t h e o r e t i c a l
t i o n s h i p between t h e measured p r e s s u r e and t h e measured f l o w ,
o s c i l l a t o r y c o n d itio n s
( f o r a N e w to n ia n f l u i d ) .
were p e r fo r m e d .
(New­
The a g ree m e n t between t h e t h e o r y and
t h e e x p e r im e n t a l w o rk was good o v e r a c e r t a i n
H.
s o lu tio n s
S im u lta n e o u s r e c o r d i n g s o f t h e p r e s s u r e -
wave and t h e f lo w - w a v e were made.
tu b e d ia m e t e r s .
und e r
Hence, t o ju d g e t h e
adequacy o f t h e a p p a r a tu s some e x p e r im e n t s on g l y c e r o l
to n ia n f l u i d s )
re la ­
ra n g e o f f r e q u e n c y and
The r e s u l t s a r e d is c u s s e d on P. 56.
. P r e p a r a t i o n o f B lo o d Samples
I.
450 ml human b lo o d was c o l l e c t e d by v e n e p u n c tu r e i n t o a
b lo o d pack c o n t a i n i n g an aqueous s o l u t i o n o f 63 ml c i t r a t e
d e x tro s e a n tic o a g u la n t.
206 mg c i t r i c
p h o sp h a te
Each 63 ml o f CPD c o n t a i n s :
a c id
( h y d r o u s ) , USP
1 .6 6 g sodium c i t r a t e
( h y d r o u s ) , USP
■
140 mg so dium b i p h o s p h a te , USP
1.61
'
2.
g d e x tro s e
( h y d r o u s ) , USP
The b lo o d was th e n c e n t r i f u g e d a t 3500 RPM f o r 10 m i n u t e s .
U s in g a P a s t e u r c a p i l l a r y p i p e t t e ,
t h e plasma was w it h d r a w n and s a v e d ,
w h i l e t h e b u f f y c o a t ( p r i m a r i l y w h i t e b lo o d c e l l s
and p l a t e l e t s ) was
d is c a rd e d :
3.
The RB C s were suspended w i t h
.5% b o v in e a lb u m in i n i s o t o n i c
51
s a l i n e , and c e n t r i f u g e d a g a i n ; t h e r a t i o
o f RBCs t o
.5% a l b u m i n / s a l i n e
b e in g 1 : 4 .
4.
The s u p e r n a t a n t p lu s any r e m a in in g b u f f y c o a t was d e ca n te d
a f t e r c e n t r i f u g i n g , t h e n t h e RBCs w ere a g a in suspended w i t h
.5%
a l b u m i n / s a l i n e and c e n t r i f u g e d .
5.
The s u p e r n a t a n t was d e c a n te d and t h e r e m a in in g RBCs were
suspended w i t h e i t h e r i t s
own p la s m a , 2% D e x tra n 150 o r . 2% D e x tra n 250
to d e s ire d h e m a to c rit.
6.
Two sam ples o f w e l l - m i x e d RBC s u s p e n s io n were w ith d r a w n
i n t o n o n - h e p a r i n i z e d m i c r o h e m a t o c r i t tu b e s and s e a le d on one end w i t h
c rito s e a l.
The tu b e s were c e n t r i f u g e d i n a m i c r o h e m a t o c r i t c e n t r i f u g e
f o r 4 m in u te s a t 1 1 ,5 0 0 RPM.
7.
H e m a t o c r it v a lu e s were o b t a i n e d as f o l l o w s ;
H e ig h t o f RBCs ( c e n t i m e t e r s ) . ______
T o t a l h e i g h t o f sample ( c e n t i m e t e r s )
in n
Each tu b e was measured s e p a r a t e l y , and t h e a v e ra g e t a k e n f o r f i n a l
h e m a t o c r i t v a lu e i n %.
I.
O p e r a t io n
F irs t o f a l l ,
b o ile d g ly c e rin e )
the
t h e a p p a r a tu s i s . f i l l e d
and t h e same i s
w ith b o ile d w a te r (o r
passed t h r o u g h t h e a p p a r a tu s so t h a t
b u b b le s w h ic h may be p r e s e n t i n t h e system a r e d i s s o l v e d .
p o i n t , t h e syste m can be t e s t e d , as i t
is
At th is
d is c u s s e d on P-. 50.
To ru n t h e a p p a r a t u s , t h e s w i t c h o f t h e speed, r e g u l a t o r f o r t h e
52
e l e c t r i c m o t o r .c a n be s w it c h e d o n .
The s c o t c h - y o k e pumps w a t e r i n t o and
fro m t h e c a p i l l a r y t u b e
between two r e s e r v o i r s )
(w h ic h l i e s
s y r i n g e and t h e c o n n e c t i n g t u b e .
and d is p la c e m e n t i s
v i a th e
S im u lta n e o u s r e c o r d i n g o f p r e s s u r e
done by s w i t c h i n g on t h e a m p l i f i e r s o f t h e p r e s s u r e
t r a n s d u c e r (w h ic h i s
fix e d
i n t h e f e e d r e s e r v o i r ) and d is p la c e m e n t
t r a n s d u c e r and t h e o s c i l l o g r a p h i c r e c o r d e r . .
The f r e q u e n c y o f o s c i l l a t i o n
speed o f t h e e l e c t r i c m o t o r .
'I
can be changed by c h a n g in g th e
The a m p lit u d e o f o s c i l l a t i o n
changed by c h a n g in g t h e s t r o k e o f t h e o s c i l l a t i o n .
o s c illa tio n
is
fitte d
can be
The s t r o k e o f
can be changed by c h a n g in g t h e p o s i t i o n o f t h e p i n w h ic h
in s id e th e s c o tc h -y o k e
A f t e r t h e a p p a r a tu s i s
(F ig u re
X -3 ).
te s te d , s a lin e s o lu tio n
is
passed t h r o u g h
t h e a p p a r a tu s so t h a t no a i r b u b b le escapes i n s i d e t h e a p p a r a t u s .
in e s o lu t i o n
S a l­
d r i v e s o u t t h e w a t e r o r g l y c e r i n e s o l u t i o n w i t h w h ic h t h e
a p p a r a tu s was i n i t i a l l y
fille d .
60 cc o f s a l i n e s o l u t i o n
t h e a p p a r a tu s i s
fille d
is
I t was f o u n d e x p e r i m e n t a l l y t h a t a b o u t
re q u ire d to d r iv e o u t th e g ly c e r in e .
w ith th e s a lin e
s o l u t i o n , plasma i s
t h r o u g h t h e a p p a r a tu s so t h a t no a i r b u b b le e s c a p e s .
A fte r
passed
Plasma g iv e s a
c o a t i n g t o t h e i n s i d e s u r f a c e o f t h e c a p i l l a r y tu b e so t h a t t h e re d
c e lls
do n o t s t i c k
fille d
A f t e r w a r d s , t h e a p p a ra tu s i s
w i t h b lo o d sample so t h a t no a i r b u b b le escapes i n s i d e th e
a p p a ra tu s .
p la s m a .
t o t h e g la s s s u r f a c e .
A b o u t 60 cc o f b lo o d sample i s
The a p p a r a tu s i s
n e c e s s a ry .to d r iv e o u t th e
now re a d y t o t a k e t h e p r e s s u r e - f l o w d a ta o f
53
t h e b lo o d sa m p le.
If
it
is
d e s ire d to f i l l
t h e a p p a r a tu s w i t h some
o t h e r b lo o d s a m p le , t h e a p p a r a tu s can be f i l l e d
w i t h t h e o t h e r b lo o d ,
sample i m m e d i a t e l y . a f t e r t h e e x p e r im e n t a l w o rk w i t h t h e p r e v io u s b lo od sample i s
fin is h e d .
A b o u t 60 cc o f t h e o t h e r b lo o d sample i s
A f t e r t h e a p p a r a tu s i s
o f b lo o d sample i s
fille d
re q u ire d .
w i t h a g iv e n b lo o d s a m p le , t h e h e m a t o c r i t
checked t o e n s u re t h a t t h e a p p a r a tu s c o n t a i n s b lo o d
o f th e d e s ire d h e m a to c rit.
Two d i f f e r e n t tu b e s were used f o r e x p e r im e n t a l w o rk (see
A p p e n d ix B f o r t h e d im e n s io n s o f t u b e s ) .
J.
R e p r e s e n t a t io n o f Data
The p r e s s u r e f l o w d a ta o f d i f f e r e n t b lo o d sam ples a r e t a b u l a t e d
in th e n e x t c h a p te r.
Some o f t h e d a ta a r e r e p r e s e n t e d g r a p h i c a l l y .
The t a b l e s show t h e p r e s s u r e a m p lit u d e and f l o w a m p lit u d e as a f u n c t i o n
o f th e fre q u e n c y o f o s c i l l a t i o n .
The g ra p h s a ls o show t h e p r e s s u r e
a m p lit u d e s as a f u n c t i o n o f t h e f l o w a m p l i t u d e s .
The e x p l a n a t i o n o f d a ta i s
d is c u s s e d i n t h e n e x t c h a p t e r i n
te rm s o f t h e known p r o p e r t i e s o f b lo o d .
K. - Use o f , R h e o l o g i c a l
Data t o C a l c u l a t e O s c i l l a t o r y B lo o d Flow
The s t e a d y f l o w r h e o l o g i c a l
shear ra te r e la t io n )
can be w r i t t e n
T = f (-f^ )
where
t
= shear s tre s s
d a ta f o r b lo o d ( t h e s h e a r s t r e s s as
0)
54
-g p = shear ra te
The l i n e a r i z e d N a v ie r - S t o k e s e q u a t i o n f o r t h e c y l i n d r i c a l
d in a te s
(in
s i m p l i f i e d f o r m ) can be w r i t t e n
p n
=
+ 7 J F
x = a x ia l
as
(rx )
where u = v e l o c i t y o f f l u i d
r = ra d ia l
co o r­
(2 )
a t ra d iu s r
c o o rd in a te
c o o rd in a te
t = i n s t a n t a n e o u s t im e
Assum ing t h e p r e s s u r e d ro p o f t h e fo r m
= A cos tot
(3 )
w here A = p r e s s u r e - w a v e a m p lit u d e
p = in s ta n ta n e o u s p re s s u re
S o lv in g e q u a tio n s
s ib le
(I),
( 2 ) and ( 3 ) s i m u l t a n e o u s l y , i t
is
pos­
to fin d
u
=u ( r , t )
(4 )
In te g ra tio n s
. Q =
can be p e r fo rm e d t o c a l c u l a t e t h e f l o w ,
Z tru rd r
(5 )
where R = r a d i u s o f t h e t u b e
Q = in s ta n ta n e o u s flo w
Thus, i f
t h e p re s s u r e -w a v e i s
known, t h e f lo w - w a v e can be
o b ta in e d .
B u t , as i s m e n tio n e d on P... 2 4, such a method o f c a l c u l a t i n g t h e
55
p re s s u re -flo w r e la t io n s h ip
u n d e r o s c i l l a t o r y c o n d i t i o n s may p ro v e t o be
in a d e q u a t e because o f p o s s i b l e v i s c o - e l a s t i c i t y o f b lo o d a n d / o r th e
d e f o r m a t io n o f t h e re d b lo o d c e l l s .
X I.
A.
T e s t i n g o f A p p a ra tu s W ith N e w to n ia n F l u i d s
As i t
is
d is c u s s e d on P. 5 0 , t h e a p p a r a tu s was t e s t e d by p e r ­
f o r m in g e x p e r im e n t s w i t h
it
RESULTS AND DISCUSSION
is
g ly c e ro l
s o lu tio n s .
From t h e t h e o r y
(P .
36),
seen t h a t f o u r f a c t o r s can be c h e c k e d — t h e f r e q u e n c y , t h e p r e s s u r e
a m p l i t u d e , t h e phase d i f f e r e n c e , and t h e shape o f t h e f lo w - w a v e and
th e p re ssu re -w a ve .
These f a c t o r s
ta k e ca re o f th e r e l a t io n s h ip
t h e p re s s u r e -w a v e and t h e f lo w - w a v e .
between
F re q ue n cy and shape o f t h e two
waves were a lw a y s fo u n d t o be i n a g re e m e n t w i t h t h e t h e o r y .
F ig u re s
X I - I t h r o u g h X I - 4 show t h e a g ree m e n t between t h e t h e o r y and e x p e r im e n ts
f o r t h e p r e s s u r e a m p lit u d e and t h e phase d i f f e r e n c e a t v a r i o u s f r e ­
q u e n c ie s .
F ig u r e s X I - I
d ia m e t e r o f 400 m ic r o n s .
about 5 J
and X I - 2 show t h e r e s u l t s
V is c o s it y o f th e g ly c e r o l
is
s o lu tio n
used was
c e n t i p o i s e . F ig u r e s show t h e ch eck between t h e t h e o r y and*
e x p e r im e n t a l w o rk a t two d i f f e r e n t s t r o k e s
It
f o r a tu b e w i t h a
( 0 . 1 9 cm and 0 .1 3 9 7 cm).
seen t h a t a t t h e f r e q u e n c y o f 3 c y c l e s p e r s e c o n d , t h e e x p e r i ­
m e n ta l
th e o ry .
p r e s s u r e a m p lit u d e i s
a b o u t 4-5% l e s s th a n t h a t p r e d i c t e d by t h e
T h is may be because o f t h e s l i g h t c o m p lia n c e o f t h e r e s e r v o i r
and c o n n e c t i n g l i n e s ,
th e c o m p r e s s ib i lit y o f g ly c e r o l
e f f e c t o f th e f r e e s u rfa c e o f th e g ly c e r o l
re s e rv o ir.
s o lu tio n
s o l u t i o n and t h e
i n t h e d is c h a r g e
Hence, a c o r r e c t i o n f a c t o r s h o u ld be a p p l i e d .
done by r e d u c in g t h e a c t u a l
f l o w i n s i d e t h e tu b e by a b o u t 5%.
F ig u r e s X I - 3 and X I - 4 show t h e r e s u l t s
t e r c a p illa ry -tu b e .
T h is may be
Two d i f f e r e n t s t r o k e s
f o r a 776 m ic r o n s d ia m e ­
( 0 . 2 8 8 cm and 0 .3 8 6 cm) were
57
EXPERIMENTAL
THEORETICAL
Tube D ia m e t e r = 400 m ic r o n s
Tube L e n g th = 5 cm
G ly c e r o l S o l u t i o n
V i s c o s i t y = .05 7 p o is e
T e m p e ra tu re = 2 5 . 5 ° C
PRESSURE AMPLITUDE,DYNES/CM3
60000
45000
S t r o k e = .19 cm
30000
S t r o k e = .1397 cm
15000
FREQUENCY OF OSCILLATION,CYCLES PER SECOND
F ig u r e X I - I .
TESTING OF APPARATUS
58
PHASE DIFFERENCE
EXPERIMENTAL
THEORETICAL
S t r o k e = 0 .1 9 cm
Tube D ia m e te r = 400 m ic r o n s
G ly c e r o l S o l u t i o n
V i s c o s i t y = 0 .0 5 7 p o is e
PHASE DIFFERENCE
EXPERIMENTAL
THEORETICAL
S t r o k e = 0 .1 3 9 7 cm
Tube D ia m e te r = 400 m ic r o n s
G ly c e r o l S o l u t i o n
V i s c o s i t y - 0 .0 5 7 p o is e
0 .5
I
2
FREQUENCY OF OSCILLATION,CYCLES PER SECOND
FIGURE X I - 2
TESTING OF APPARATUS
3
59
Tube D ia m e te r = 776
Tube L e n g th = 3 0 .0 5
G ly c e ro l S o lu tio n
V i s c o s i t y = .0465
T e m p e ra tu re = 2 5 . 1 °
m ic r o n s
cm
p o is e
C
PRESSURE AMPLITUDE, DYNES/CM3
EXPERIMENTAL
THEORETICAL
S t r o k e = 0 .3 8 5 7 cm
S t r o k e - 0 .2 8 3 cm
FREQUENCY OF OSCILLATION,CYCLES PER SECOND
FIGURE X I - 3 .
TESTING OF APPARATUS
.
PHASE DIFFERENCE
60
Tube D ia m e te r = 776 m ic r o n s
G ly c e ro l S o lu tio n
EXPERIMENTAL
THEORETICAL
PHASE DIFFERENCE
S t r o k e = 0 .2 8 8 cm
S t r o k e = 0 .3 8 5 7 cm
EXPERIMENTAL
THEORETICAL
FREQUENCY OF OSCILLATION,CYCLES PER SECOND
FIGURE X I - 4 .
TESTING OF APPARATUS
61
used.
A g a in i t
is
t h e e x p e r im e n t a l
by t h e t h e o r y
seen t h a t a t t h e f r e q u e n c y o f 3 c y c l e s p e r s e co n d ,
p r e s s u r e a m p lit u d e i s
somewhat l e s s th a n t h a t p r e d i c t e d
(when t h e s t r o k e o f o s c i l l a t i o n
o f o s c illa tio n
is
h ig h , th e r e s u lt s
a r e good.
c a p illa ry -tu b e
is
c u t down, t h e r e s u l t s
is
lo w ).
When th e s t r o k e
I f t h e l e n g t h o f th e
can be im p ro v e d .
F o r t h e 776
m ic r o n s d ia m e t e r c a p i l l a r y . t u b e , no c o r r e c t i o n s were a p p l i e d f o r o s c i l ­
l a t o r y b lo o d f l o w .
W ith t h e e x c e p t i o n o f t h e r e s u l t s . a t t h e f r e q u e n c y o f o s c i l l a t i o n . a t 3 c y c l e s p e r s e c o n d , t h e a g ree m e n t between t h e t h e o r y and t h e
e x p e r im e n t a l w o rk i s
e x p e r im e n t a l
B.
good.
The a v e ra g e d e v i a t i o n between t h e t h e o r y and
r e s u l t s was l e s s th a n +2% f o r w a t e r , s a l i n e and plasm a.
Data and D is c u s s io n f o r D i f f e r e n t Red C e ll
and D e x tra n S o l u t i o n s
ta b le s X I- I
S u s p e n sio n s U sin g Plasma
t h r o u g h X I - 6 summarize t h e r e s u l t s o f e x p e r im e n t a l
w o rk done f o r b lo o d w i t h a 776 m ic r o n d ia m e t e r c a p i l l a r y t u b e .
T a b le s
show t h e p r e s s u r e a m p lit u d e and t h e f l o w a m p lit u d e as a f u n c t i o n o f
fre q u e n c y o f o s c i l l a t i o n
fo r o s c illa to r y
b lo o d f l o w .
The p r e s s u r e
waves and t h e d is p la c e m e n t waves ( f r o m w h ic h t h e f l o w waves can be
d e r i v e d ) were fo u n d t o be s i n u s o i d a l
f o r m ost o f t h e e x p e r im e n t a l w o r k .
S l i g h t n o n - s i n u s o i d a l c h a r a c t e r i s t i c s were o b s e rv e d a t v e r y lo w f r e ­
quency o f o s c i l l a t i o n
such as 0 .2 5 h e r t z .
fre q u e n c y are re p o r te d in t h i s
o id a l.
A ll
th e s is !
No e x p e r im e n t a l
O th e rw is e , a l l
d a ta o f t h i s
waves were sinus-=..
I
t h e b lo o d s a m p le s , w h ic h were used i n a 776 m ic r o n d ia m e t e r
I
62
c a p i l l a r y t u b e were ru n i n t o s t e a d y f l o w v is c o m e t e r s
d e r v is c o m e t e r and c o n e .a n d p l a t e v i s c o m e t e r ) .
f l o w v i s c o m e t r i c d a ta
(c o n c e n tric c y l i n ­
T h is g iv e s t h e s t e a d y
( o r t h e s h e a r s t r e s s - s h e a r r a t e d a t a ) f o r t h e same
b lo o d sam ples w h ic h were ru n i n t h e a p p a r a tu s
fo r o s c illa to r y
flo w .
T a b le s a t t h e b o tto m o f each page show t h e c o r r e s p o n d in g v i s c o m e t r i c
d a ta .
As i t
is
i n d i c a t e d on P. 2 4 , t h e s e d a ta can be used t o p r e d i c t
o s c i l l a t o r y b lo o d f l o w .
is
In t h i s
d is c u s s e d , (se e P. 5 3 ) .
th e s is ,
t h e method o f such c a l c u l a t i o n s
No c a l c u l a t i o n a l
r e s u l t s a r e shown.
I f t h e v i s c o m e t r i c d a t a a r e . p l o t t e d so t h a t t h e o r d i n a t e i s
( s h e a r s t r e s s ) 2 and t h e a b s c is s a i s
w o u ld be s i m i l a r t o t h a t shown i n
is
th e r a t i o
if
a p p a re n t v i s c o s i t y
is
flu id
.
F ig u r e V - I .
is
p lo t
The a p p a r e n t v i s c o s i t y
o f t h e s h e a r s t r e s s and t h e s h e a r r a t e .
From th e s e d a t a ,
p l o t t e d as a f u n c t i o n o f s h e a r r a t e , t h e gra ph
w o u ld be s i m i l a r t o t h a t shown i n
it
( s h e a r r a t e ) 2, a t y p i c a l
F ig u r e
I-I.
From F ig u r e V - I and I - I ,
seen t h a t u n d e r s t e a d y c o n d i t i o n s b lo o d behaves as a N ew tonian
a t v e r y h ig h s h e a r r a t e s .
It
is
a l s o seen t h a t at. lo w s h e a r r a t e s
•• / \
th e a p p a re n t v i s c o s i t y
shear ra te .
is
v e r y h ig h and a y i e l d s t r e s s e x i s t s a t z e ro
Hence, b lo o d i s
n a t u r e o f F ig u r e I - I
o f o s c illa to r y
n o n -N e w to n ia n a t lo w s h e a r r a t e s .
and F ig u r e V - I
b lo o d f l o w .
can be used t o e x p l a i n t h e r e s u l t s
In a d d i t i o n
t o th e s e p r o p e r t i e s , th e p o s ­
s ib le e f f e c t s o f v i s c o - e l a s t i c i t y o f b lo o d ,
and re d b lo o d c e l l
The
re d b lo o d c e l l
d e f o r m a t io n a re a l s o c o n s id e r e d i n t h i s
a g g r e g a t io n
th e s is .
63
T a b le s X I - I , X I - 5 and X1-6 show r e s u l t s
fro m re d c e l l s
in
p la sm a .
f o r b lo o d samples, made
T a b le X I - 5 shows t h e r e s u l t s
h e m a to c rit equal
t o 45.5% and T a b le X I - 6 shows r e s u l t s
h e m a t o c r i t e qual
t o 36.1% .
It
is
is
,
f o r a b lo o d
seen t h a t f o r t h e same s t r o k e , h i g h e r
p r e s s u r e s a r e n e c e s s a ry a t h e m a t o c r i t e q u a l t o 45.5% .
th is
f o r a b lo o d
As i s e x p e c t e d ,
because b lo o d i s more v is c o u s a t t h e h e m a t o c r i t e q u a l t o
45.5%.
T a b le X I - I
shows t h e r e s u l t s
43.1% .
The r e s u l t s a r e p l o t t e d i n
tu d e a g a i n s t t h e f l o w a m p l i t u d e .
s i m i l a r f o r th e r e s u lt s
The n a t u r e o f the. g ra p h w i l l
d is c u s s e d on P.
‘
■
6 2.
be
T h is graph can
S in c e b lo o d i s n on -
■
•
N e w to n ia n a t lo w s h e a r f a t e s and t h e a p p a r e n t v i s c o s i t y i s
a t lo w f l o w r a t e s ,
to
F ig u r e X I - 5 as t h e p r e s s u r e a m p l i ­
shown i n T a b le s X I - 5 and X I - 6 .
be e x p l a i n e d fr o m w h a t i s
C ' -
f o r b lo o d h e m a t o c r i t equal
h ig h p r e s s u r e s a r e needed.
v e r y h ig h
B u t s i n c e b lo o d i s
N ew to nia n a t h ig h s h e a r r a t e s , a t h ig h f l o w r a t e s t h e p r e s s u r e s f a l l
on
a s tra ig h t lin e .
D e x tra n i s a h ig h m o l e c u l a r w e i g h t n e u t r a l
h e lp s re d c e l l s
to a g g re g a te .
p o ly s a c c h a rid e .
It
D e x tra n 250 has an a v e ra g e m o le c u l a r
w e i g h t o f a b o u t 2 5 0 ,0 0 0 w h i l e t h a t o f D e x tra n 1 50 i s
c o n c e n t r a t i o n ra n g e used h e r d , D e x tra n 250 s o l u t i o n
c a p a b i l i t y o f a g g r e g a t i n g re d b lo o d c e l l s
15 0 , 0 0 0 .
In t h e
has a g r e a t e r
th a n D e x tra n 150 s o l u t i o n .
T a b le s X I - 2 , X I - 3 and X I - 4 show t h e r e s u l t s o f b lo o d sam ples u s in g
D e x tra n s o l u t i o n s - .
s a m p le s .
F ig u r e X I - 5 compares t h e r e s u l t s o f d i f f e r e n t b ip o d
S in c e D e x tra n te n d s t o a g g r e g a t e t h e re d b lo o d c e l l s , h i g h e r
64
Red C e l l s i n D e x tra n 250
H e m a t o c r it = 43%
V 1
Tube D ia m e te r = 776 m ic r o n s
/ R e d C e lls in
^ D e x t r a n 150
H e m a t o c r it = 44.4%
Q
6000
Red C e l l s i n Plasma
H e m a t o c r it = 43.7%
0 .0 2 4
048
0 .0 9 6
FLOW AMPLITUDE, CM3ZSEC
FIGURE X I - 5 .
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE
0 .1 4 4
65
TABLE X I - I
DATA OF OSCILLATORY BLOOD FLOW
F re q u e n cy o f O s c i l l a t i o n
c .p .-s .
■
:
:
o .s
1 .0
2 .0
3 .0
.
,
...
P re s s u r e A m p lit u d e
Dynes/cm 3.
Flow A m p litu d e
cm3/ s e c
1 6 7 5 .7
3 1 9 9 .8
6 2 3 5 .4
9 1 9 4 .7
0 .0 3 4 6
0 .0 6 9 3
0 .1 3 8 6
0 .2 0 7 9
■■
•
•
Red c e l l s i n plasma
Tube D ia m e t e r :
0 .0 7 7 6 cm.
Tube L e n g th :
3 0 .0 5 cm
H e m a t o c r i t : .43.7%
T e m p e r a tu re :
24° C
STEADY FLOW VISCOMETRIC DATA
■ S hear Rate
sec-1 '
0 .6 3 3 6
1 .2 4 5 7
. 3 .0 1 5 4
5 .9 4 0 4
. 1 1 .8 2 4 0
2 9.6 2 7 5
5 9.8 9 8 9
7 5 .0 0 0 0
1 50 .00 0 0
3 0 0 .0 0 0
7 5 0 .0 0 0 .
1 5 0 0 .0 0 0
Red c e l l s in p la s m a .
H e m a to c r it: 43.7%
T e m p e ra tu re : 2 4 .8 ° C
0 .1 4 3 7
0 .2 0 1 4
0.3 8 04 .
0 .6 3 2 2
I .0746
2 .1 6 1 6
3 .8 1 0 3
' 4 .6 3 5 . •
8 .7 4 5 '
1 6 .0 2 0
•36.675
6 0 .3 0 0 ■
.
A pp a re nt V is c o s it y
p o is e
Shear. S t r e s s
d y n e s /c m 2
*
'
'
0 .2 2 6 9
0 .1 6 1 6
0.1261
0 .1 0 6 4
0 .0 9 0 8
0 .0 7 2 9
0 .0 6 3 6
0 .0 6 1 8
■ 0.0583
0 .0 5 3 4
0 .0 4 8 9
0 .0 4 0 2
- .
•
.
.
66
TABLE X I- 2
DATA OF OSCILLATORY BLOOD FLOW
F re q u e n cy o f O s c i l l a t i o n
c .p .s .
P re s s u r e A m p lit u d e
d y n e s /c m 3
Flow A m p litu d e
cm3/ s e c
1 7 6 4 .6
3 5 0 4 .7
6 4 7 6 .7
9 7 5 3 .6
0 .0 3 46 5
0 .0 6 9 3
0 .1 3 8 6
0 .2 0 7 9
0 .5
1 .0
2 .0
3 .0
Red c e l l s i n D e x tra n 150
Tube D ia m e t e r :
.0776 cm
Tube L e n g th :
3 0 .0 5 cm
H e m a to c rit:
44.4%
T e m p e r a tu re :
2 4 .5 ° C
STEADY FLOW VISCOMETRIC DATA
S he a r Rate
s e c -1
3 .1 4 4 7
6 .1 8 3 8
1 2.1 8 5 4
2 9 .6 4 7 4
5 7.8 4 1 6
7 5 .0 0
1 5 0 .0 0
3 0 0 .0 0
7 5 0 .0 0
■1500.00
Red c e l l s in D e x tra n 150
H e m a to c r it: 44.4%
T e m p e ra tu re : 2 4 .8 ° C
S he a r S t r e s s
d y n e s /c m 2
0 .5 0 2 0
0 .7 5 5 3
1.1641
2.2601
4 .0 1 4 7
5 .2 9 5
1 0 .2 0
1 7 .9 7
40.-65
6 4 .8
A pp a re nt V is c o s it y
p o is e
.1596
0.1221
0 .0 9 5 5
0.0.762
0 .0 6 9 4
0 .0 7 0 6
0 .0 6 8
0 .0 5 9 9
0 :0 5 4 2
0 .0 4 3 2
67
TABLE X I - 3
-
DATA OF OSCILLATORY BLOOD FLOW.
Fre q u e n cy o f O s c i l l a t i o n
c .p .s .
P r e s s u r e A m p lit u d e
d y n e s /c m 3
0 .5
1 ,0
2 .0
3 .0
1 7 9 0 .0
3530.1
6 9 0 8 .6
1 00 0 7.6
Flow A m p litu d e
cm3/ s e c
0 .0 3 46 5
0 .0 6 9 3
0 .1 3 8 6
0 .2 0 7 9
Red c e l l s i n D e x tra n 250
Tube D ia m e t e r :
0 .0 7 7 6 cm
Tube L e n g th :
3 0 .0 5 cm
H e m a to c rit:
43%
T e m p e r a tu re :
25° C
STEADY FLOW VISCOMETRIC DATA
S hear Rate
s e c -1
1 .3 3 4 2
3 .2 3 7 9
6.3121
1 2 .3 9 1 3
2 9 .8 0 4 3
5 7 .6 3 6 8
7 5 .0 0 0
1 5 0 .0 0 0
3 0 0 .0 0 0
7 5 0 .0 0 0
1 50 0 .0 0 0
Red c e l l s in D e x tra n 250
H e m a to c r it: 43%
T e m p e ra tu r e : 2 4 .8 ° C .
S hear S t r e s s
d y n e s /c m 2
0 .4 3 3 4
0 .6 5 2 7
0 .9 7 6 5
I .3838
2 .6 4 0 9
4 .6 4 2 5
5 .0 7 7 5
1 0 .4 4
1 8 .4 5
4 1 .8 5
6 4 .5
A pparent V is c o s it y
p o is e
0 .3 2 4 8
0 .2 0 1 6
0 .1 5 4 7 t
0.1116.
0 .0 8 8 6
0 .0 8 0 5
’ 0 .0 6 7 7
0 .0 6 9 6
0 .0 6 1 5
0 .0 5 5 8
0 .0 4 3
'
68
TABLE X I -4
DATA OF OSCILLATORY BLOOD FLOW
Fre q u e n cy o f O s c i l l a t i o n
. c .p .s .
P r e s s u r e A m p lit u d e
d y n e s /c m 3
0 .5
1 .0
2 .0
3 .0
Flow A m p lit u d e
cm3/sec=
1 2 1 8 .5
2 6 1 5 .6
5 1 5 5 .8
7 6 9 6 .0
Red c e l l s i n D e x tra n 150
Tube D ia m e t e r :
.0776 cm
Tube L e n g th :
3 0 .0 5 cm
H e m a to c rit:
45.2%
T e m p e r a tu re :
25° C
0 .0 2 4
0 .0 4 8
0 .0 9 6
0 .1 4 4
'
STEADY FLOW VISCOMETRIC DATA
S he a r Rate
sec*"1
0 .6 8 4 6
1 .3 4 2 0
3 .2 2 2 6
6 .2 6 5 9
. 1 2 .2 8 0 5
2 9 .7 0 1 6
5 8 .1 8 2 2
7 5 .0 0 0
1 5 0 .0 0 0 ■
3 0 0 .0 0 0
7 5 0 .0 0 0
1 5 0 0 .0 0
Red c e l l s in D e x tra n 150
H e m a to c r it: 45.2%
T e m p e ra tu re : 2 5 ° C
S hear S t r e s s
d y n e s /c m 2 .
0 .2 8 1 2
0 .3 6 2 9
0 .5 7 1 0
0 .8 4 3 3
1 .2 1 9 7
2.3671 ■
4 .1 8 5 6
5 .5 5 7 5
9 .6 6
1 8 .3 0
4 0 .8
7 5 .6
A pp a re nt V is c o s it y
p o is e
0 .4 1 0 8
0 .2 7 0 4
0 .1 7 7 2
0 .1 3 4 5
0 .0 9 9 3
0 .0 7 9 6
0 .0 7 1 9
0.0741
0 .0 6 4 4
0 .0 6 1 0
0 .0 5 4 4
0 .0 5 0 4
’
69
TABLE. X I- 5
DATA OF.OSCILLATORY BLOOD FLOW .
P r e s s u r e A m p lit u d e
d y n e s /c m 3
Fre q u e n cy o f O s c i l l a t i o n
c .p .s .
0 .5
1 .0
2 .0
3 .0
Flow A m p lit u d e
cm3/ s e c
S tro k e I
S t r o k e 2- ‘ ■ S t r o k e I
2 3 6 1 .6
4 4 6 9 .9
8966.1
1 3 6 6 5 .5
. 1 0 1 5 .3
1 9 6 7 .8
3 7 5 8 .7
5 6 1 3 .0 8
0 .0 5 8 5
0 .1 1 7
0 .2 3 4 •
0.351
S tro k e 2
0 .0 2 3 8 3
0 .0 4 7 6 6
0 .0 9 5 3 2
0 .1 4 2 9 8
Red c e l l s i n plasma
Tube D ia m e t e r :
0 .0 7 7 6 cm
Tube L e n g th :
3 0 .0 5 cm
H e m a to c rit:
45.5%
T e m p e r a tu re :
2 5 .2 ° C
\
STEADY FLOW VISCOMETRIC DATA
S hear Rate
sec 1
0 .3 2 2 6
0 .6 3 0 6
1 .2 3 8 2
3 .0 0 0 5
5 .9 2 9 8
1 1.8 51 2
2 9 .8 5
6 0 .6 2 9 7
7 5 .0 0 0
1 5 0 .0 0 0
3 00 .00 0
7 5 0 .0 0 0
1 50 0 .0 0 0
Red c e l l s in p la sm a
H e m a to c r it: 45.5%
T e m p e ra tu re : 2 3 .3 ° C
S hear S t r e s s
d y n e s /c m 2
0 .0 9 3 6
0 .1 3 1 3
0 .1 9 0 6
0 .3 7 5 6
0 .6 2 7 4
1 .0 3 5 4
2 .0 3 6 2
3 .5 6 5 6
4 .5 9
8 .6 5 5
1 5 .1 2
3 6.2 25
7 0 .2
.
A pp a re nt V is c o s it y
p o is e
0.2901
0 .2 0 8 2
0 .1 5 3 9
0.1251
0 .1 0 5 8
0 .0 8 7 3
0 .0 6 8 2
0 .0 5 8 8
0 .0 6 1 2
0 .0 5 7 7
0 .0 5 0 4
0 .0 4 8 3
0 .0 4 6 8
•
70
TABLE X I - 6 .
DATA OF OSCILLATORY BLOOD FLOW
P re s s u r e A m p lit u d e
d y n e s /c m 3
F re q u e n cy o f O s c i l l a t i o n
c .p .s .
-•
0 .5
1 .0
2 .0
3 .0
Flow A m p lit u d e
cm3/ s e c
S tro k e I
S tro k e 2
S tro k e I
S tro k e 2
9 5 1 .8
1 6 1 2 .2
3 3 1 4 .2
5 0 2 8 .8
1 9 4 2 .4
3862.2.
7 6 1 9 .8
1 1 5 0 6 .3
0 .0 2 3 8 6
0.04761 :
0 .0 9 5 3 2
0 .1 4 2 9 8
0 .0 5 6 8
0 .1 1 3 6
0 .2 2 7 2
0 .3 4 0 7
Red c e l l s i n plasma
Tube D ia m e t e r :
0 .0 7 7 6 cm
Tube L e n g th :
3 0 .0 5 cm
H e m a to c rit:
36.1%
T e m p e r a tu re :
25° C
STEADY FLOW VISCOMETRIC DATA
S he a r Rate
s e c -1
0 .6 1 6 6
1 .2152
2 .9 6 7 5
5 .8 6 2 7
. 1 1 .6 8 0 3
. 2 9 .T 9 1 3
'5 8 .6 9 8 1
7 5 .0 0 0
1 5 0 .0 0 0
3 00 .00 0
7 5 0 .0 0 0
1 50 0 .0 0 0
R e d ic e lls in p la sm a
H e m a to c r it: 36.1%
T e m p e ra tu re : 2 3 .3 ° C
Shear S t r e s s
d y n e s /c m 2
0 :0 7 4 6
0 .1 1 5 6
. 0 .2 2 1 9
0 .3 9 3 5
0 .7 0 3 5
I .5248
2 .8 1 5 8
3 .9 0
7 .4 4
1 3 .2
3 0 .2 2 5
5 4 .3
A pp a re nt V is c o s it y
. p o is e
0.1209.
0.0951
0 .0 7 4 7
0.0671
0 .0 6 0 2
0 .0 5 2 2
0 .0 4 7 9
0 .0 5 2
‘ 0 .0 4 9 6
0 .0 4 4
0 .0 4 0 3
0 .0 3 6 2
71
p r e s s u r e s a r e r e q u i r e d t o m a i n t a i n t h e same f l o w th a n t h o s e r e q u i r e d
by p la s m a .
D e x tra n 250 needs t h e maximum p r e s s u r e .
T h is i s a ls o
because D e x tra n in c r e a s e s t h e v i s c o s i t y o f c o n t in u o u s phase.- .
T a b le s X I - 7 and X I - 8 show t h e r e s u l t s o f b lo o d sam ples w i t h
h e m a t o c r i t s o f 36.4% and 46% and tu b e d ia m e t e r o f 4d0 m ic r o n s .
r e s u lt s are p lo t t e d
h e m a to c rit,
it
it
in
F ig u r e X I - 6 .
Same
S in c e b lo o d i s more v is c o u s a t 46%
needs h i g h e r p r e s s u r e s t o m a i n t a i n t h e same f l o w .
i s m e n tio n e d t h a t a l l
t o be s i n u s o i d a l .
A g a in ,
p r e s s u r e -w a v e s and flo w - w a v e s were o b se rv e d
T a b le s a t t h e b o tto m o f each page show t h e s te a d y
f l o w v i s c o m e t r i c d a ta f o r t h e b lo o d sample w h ic h has a h e m a t o c r i t c lo s e
t o t h a t w h ic h was ru n i n t h e a p p a r a t u s .
B lo o d w h ic h was ru n i n th e
a p p a r a tu s was n o t ru n i n t h e c o n c e n t r i c c y l i n d e r ( lo w s h e a r r a t e ) v i s c o ­
m e t e r , w h i l e w o r k in g w i t h 400 m ic r o n s c a p i l l a r y - t u b e
same b lo o d was ru n i n cone and p l a t e
T h u s , f o r a sample i n t h e o s c i l l a t o r y
d ia m e t e r .
(h ig h shear r a t e )
B ut
v is c o m e t e r .
f l o w a p p a r a t u s , h ig h s h e a r r a t e s
d a ta were o b t a in e d and t h e y a r e shown i n an a d j a c e n t c o lu m n .
The n a t u r e o f t h e gra ph i n
te rm s o f r h e o l o g i c a l
v i s c o s i t y o f b lo o d i s
h ig h s h e a r r a t e s ,
it
F ig u r e X I - 6 can be e x p l a i n e d i n
p r o p e r t i e s o f b lo o d .
A t lo w s h e a r r a t e s , t h e
h ig h and hence h ig h p r e s s u r e i s
is
n eeded.
N ew to nia n and hence p r e s s u r e i s
But a t
p ro p o rtio n a l
to flo w .
It
s h o u ld be m e n tio n e d t h a t t h e f l o w a m p lit u d e i s
2.5% a t f r e q u e n c y o f o s c i l l a t i o n
re d uce d by
o f 2 c y c l e s p e r second and 5% a t 3
72
Tube D ia m e te r = 400 m ic ro n s
PRESSURE AMPLITUDE, DYNES/CM3
5 0 ,0 0 0
4 0 ,0 0 0
3 0 ,0 0 0
Red C e l l s
in
Plasma
H e m a t o c r it s
36.4%
20,000
10,000
.01176
.02352
.04705
FLOW AMPLITUDE, CM3ZSEC
FIGURE X I - 6 .
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE
.07057
73
TABLE X I - 7
DATA OF OSCILLATORY BLOOD FLOW
F re q u e n cy o f O s c i l l a t i o n
c .p .s .
0 .5
1 .0
2 .0
3 .0
P r e s s u r e A m p lit u d e
d y n e s /c m 3
7 3 2 3 .3
1 3 9 6 4 .3
2 7 3 2 2 .6
4 0 9 1 0 .0
F lo w A m p lit u d e
cm 3/ s e c
.01176
.02352
.04705
.0 4 5 8 7 *
.07057
.06 7 04
Red c e l l s i n plasma
Tube D ia m e te r = 0 .0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it = 36.4%
T e m p e ra tu re = 2 5 . 5 ° C
STEADY FLOW VISCOMETRIC DATA
S he a r Rate
sec 1
0 .6 1 6 6
1 .2 1 5 2
2 .9 6 7 5
5 .8 6 2 7
1 1 .6 8 0 3
2 9 .1 9 1 3
58.6981
75
150
300
750
1500 .
S hear S t r e s s
d y n e s /c m 2
0 .0 7 4 6
0 .1 1 5 6
0 ,2 2 1 9
0 .3 9 3 5
0 .7 0 3 5
1 .5 2 4 8
2 .8 1 5 8
4 .4 8 5
3 .9
8 .1 6
7 .4 4
1 3 .2
1 3 .5
'
3 0 .2 2 5
3 1 .6 5
5 7 .1 5 .
5 4 .3
Red c e l l s i n plasma
H e m a t o c r it •= 36.1%
T e m p e ra tu re = 2 3 . 3 ° C
* T h i s column shows t h e c o r r e c t e d v a l u e s . .
A pp a re nt V is c o s it y
p o is e
0 .1 2 0 9
0,0951
0 .0 7 4 7
0.0671
0 .0 6 0 2
0 .0 5 2 2
0 .0 4 7 9
.0 .0 5 2
0 .0 4 9 6
0 .0 4 4
0 .0 4 0 3
0 .0 3 6 2
■
.0598
.0544
.045
.0422
.0381
74
TABLE X I - 8
DATA OF OSCILLATORY BLOOD FLOW
F re q u e n cy o f O s c i l l a t i o n
c .p .s .
.
0 .5
1 .0
2 .0
3 .0
P re s s u r e A m p lit u d e
d y n e s /c m 3
.
Flow A m p lit u d e
cm3/ s e c
.01176
.02352
.04705
.04587
.07057
.06704
8 4 6 8 .3
1 6 4 8 3 .3
3 2 1 3 1 .6
4 6 2 5 3 .3
Red c e l l s i n plasma
Tube D ia m e te r = 0 .0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it = 46%
T e m p e ra tu re = 25° C
- STEADY FLOW VISCOMETRIC DATA
Shear, Rate
. sec 1 ;
.6846
1 .3 4 2 0
3 .2 2 2 6
6 .2 6 5 9
1 2 .2 8 0 5
. 2 9 .7 0 1 6
5 8 :1 8 2 2
75
150
300
750
1500
:
Red c e l l s in p la sm a
H e m a to c r it = 45.2%
T e m p e ra tu re . = 2 5 .0 ° C
S hear S t r e s s
d y n e s /c m 2
0 .2 8 1 2
0 .3 6 2 9
. 0 .5 7 1 0
0 .8 4 3 3
1 .2 1 9 7
2.3671
4 .1 8 5 6
5 .5 5 7 5
5 .5 0 5
9 .6 6
1 0 .0 2
1 8 .3
17.01
4 0 .8
4 0 .2 7 5 .
7 5 .6
7 2 .0
A pp a re nt V is c o s ity . '
p o is e
■ 0 .4 1 0 8
0 .2 7 0 4
0 .1 7 7 2
0 .1 3 4 5
0 .0 9 9 3
0 .0 7 9 6
0 .0 7 1 9
0.0741
. .0734
0 .0 6 4 4
.0668
0.061
.0567
0 .0 5 4 4
.0537
0 .0 5 0 4
.048 •
75 -■■■
c y c l e s ' p e r s e co n d .
The . c o r r e c t e d " f l o w a m p l i t u d e s , a re shown i n an
a d j a c e n t .'column o f v a r i o u s t a b l e s .
These c o r r e c t i o n
f a c t o r s a re a p p l i e d
because o f s l i g h t e x p a n s io n o f fe e d r e s e r v o i r and c o n n e c t i n g tu b e
tween s y r i n g e and f e e d r e s e r v o i r )
s u rfa c e .
Such c o r r e c t i o n
and a l s o because o f e f f e c t o f f r e e
f a c t o r s were fo u n d t o be n e c e s s a ry t o o b t a i n
t h e a g ree m e n t between t h e o r y and e x p e r im e n t a l r e s u l t s
tio n
(a N e w to nia n f l u i d
s t o o d , P. 3 6 ) .
(b e ­
f o r g ly c e ro l s o lu ­
f o r w h ic h t h e o r y o f o s c i l l a t o r y . f l o w
Such c o r r e c t i o n f a c t o r s a r e a p p l i e d f o r a l l
a m p lit u d e s a t f r e q u e n c y o f o s c i l l a t i o n
is
u n d e r­
flo w
o f . 2 c y c l e s p e r second and 3
c y c l e s p e r second w h i l e w o r k in g w i t h 400 m ic r o n s d ia m e t e r c a p i l l a r y
tu b e .
T a b le s X I - 9 and X I - 1 0 summarize t h e r e s u l t s o b t a i n e d w i t h r e d
c e ll
s u s p e n s io n s u s in g plasma and D e x tra n 250 s o l u t i o n as c o n t in u o u s
p hase.
The re d c e l l s were o b t a in e d f r o m t h e same b lo o d .
o b t a i n e d w i t h a 400 m ic r o n s d ia m e t e r c a p i l l a r y - t u b e .
th e r e s u lt s
g r a p h ic a lly .
The d a ta were
F ig u r e X I - 7 shows
S in c e D e x tra n h e lp s re d b lo o d c e l l s t o a g g r e ­
g a t e , h i g h e r p r e s s u r e s a r e r e q u i r e d t o m a i n t a i n t h e same f l o w compared
t o those, r e q u i r e d by p la sm a .
T h is i s
a l s o because D e x tra n in c r e a s e s t h e
v i s c o s i t y o f c o n t in u o u s p hase. .
T a b le s .X I - I l
and X I - 1 2 show t h e p r e s s u r e a m p l i t u d e - f l o w a m p l i -
t u d e ; d a ta a t f o u r d i f f e r e n t s t r o k e s f o r t h e same b lo o d sa m p le .
The
s t r o k e o f t h e s y r i n g e c h a n g e d . a t a g iv e n f r e q u e n c y o f o s c i l l a t i o n
( P.
52),
The
U s in g th e s e d a t a . F ig u r e s X I - 8 t h r o u g h X I - I I w ere p l o t t e d .
76
Tube D ia m e te r = 400 m ic r o n s
H e m a t o c r it = 44%
PRESSURE AMPLITUDE, DYMES/CM3
5 0 ,0 0 0
4 0 ,0 0 0
3 0 ,0 0 0
20,000
LEGEND
Red C e l l s
Plasma
in
10,000
Red C e l l s i n
D e x tra n 250
.02352
.04705
.07057
FLOW AMPLITUDE, CM3/SEC
FIGURE X I - 7 .
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE
77
TABLE X I- 9
DATA.OF OSCILLATORY BLOOD FLOW
Fre q u e n cy o f O s c i l l a t i o n
c .p .s .
.
0 .5
1 .0
2 .0
3 .0
P r e s s u r e A m p lit u d e
d y n e s /c m 3
8 0 1 0 .3
1 4 2 6 9 .6
2 8 6 9 6 .6
4 1 6 7 3 .3
Flow A m p litu d e
cm3/ s e c
.01176
0 .0 2 3 5
0 .0 4 7 0 5
.04587
0 .0 7 0 5 7
.06704
Red C e l l s i n Plasma
Tube D ia m a te r = 0 . 0 4 cm .
Tube L e n g th = 5 cm
H e m a t o c r it = 44%
T e m p e ra tu re = 2 4 . 8 ° C
STEADY FLOW VISCOMETRIC DATA
. S he a r Rate
sec™1
0 .6 3 3 6
1 .2 4 5 7
. 3 .0 1 5 4
5 .9 4 0 4
1 1 .8 2 4 .
2 9 .6 2 7 5
5 9 .8 9 8 9
: 75 :
150
300750
1500
.
Red C e lls in P lasm a
H e m a to c r it = 43.7%
T e m p e ra tu re = 2 4 . S0C
S hear S t r e s s
d y n e s /c m 2
0 .1 4 3 7
0 .2 0 1 4
0 .3 8 0 4
0 .6 3 2 2
1 .0 7 4 6
2 .1 6 1 6
3 .8 1 0 3
4 .6 3 5
3 .4 5 6
8 .7 4 5
7 .4 5 5
1 6 .0 2 V :
1 3 .0 5
3 6 .6 7 5
32.1 .
6 0 .3
6 0 .4 5
A p p a re n t V i s c o s i t y
p o is e
0 .2 2 6 9
0 .1 6 1 6
0 .1 2 6 0
0 .1 0 6 4
0 .0 9 0 8
0 .0 7 2 9
0 .0 6 3 6
0 .0 6 1 8
.0462
0 .0 5 8 3
.0497
0 .0 5 3 4
' .0435
0 .0 4 8 9
.0428
0 .0 4 0 2
.0403
78
TABLE X I- IO
DATA OF OSCILLATORY BLOOD FLOW
Fre q u e n cy o f O s c i l l a t i o n
c .p .s .
0 .5
1 :0 .
2 .0
3 .0
P r e s s u r e A m p lit u d e
d y n e s /c m 3
83.15.6
1 7 0 1 7 .6
3 3 9 6 3 .6
4 8 8 4 8 .6
F lo w A m p lit u d e
cm3/ s e c
0 .0 1 1 7 6
0 .0 2 3 5
0 .0 4 7 0 5
.04587
0 .0 7 0 4 7
.06704
Red c e l l s i n D e x tra n 250 (2% s o l u t i o n )
Tube D ia m e te r = 0 .0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it .= 44%
T e m p e ra tu re = 2 4 . 9 ° C
• STEADY FLOW VISCOMETRIC DATA
Shear1 Rate
sec 1 .
.1 .3 3 42 .
3 .2 3 7 9 '
6.3121
1 2 :3 9 1 3
2 9 .8 0 4 3
.5 7 .6 3 6 8
,75.
• 150
300
750
1500
S hear S t r e s s
d y n e s /c m 2
0 .4 3 3 4
0 .6 5 2 7
0 ,9 7 6 5
• 1 .3 8 3 8
. 2 .6 4 0 9
4 .6 4 2 5
4 .0 7 7 5
4 .6 8 7 5
1 0 .4 4
8 .8 5
1 8 .4 5
,. 1 5 .6
4 1 .8 5
3 7 .0 5
6 4 .5
6 6 .1 5
Red c e l l s in D e x tra n 250 (2% s o lu t i o n )
H e m a to c r it = .43%
T e m p e ra tu re = 2 4 .8 ° C
A p p a re n t V i s c o s i t y
p o is e
•- •
.
0 .3 2 4 8
0 .2 0 1 6
0 .1 5 4 7
0 .1 1 1 6
0 .0 8 8 6 .
0 .0 8 0 5
.0677
.0625
.0696
.059
.0615
.052 ,
.05 5 8
.0 4 9 4 .
.043
.0441
79
2000
3000
FLOW AMPLITUDE, CM3ZSEC
IOOO
4000
Q Tube D ia m e t e r —
= 0 .0 7 7 6 cm
A
F re q ue n cy o f O s c i l l a t i o n
iOOO
10,000
Tube D ia m e te r
= 0 .0 4 cm
= 0 . 5 C y c le Per
Second
14,000
PRESSURE AMPLITUDE, DYNES/CM3
FIGURE X I- 8 .
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE
80
8000
12000
16000
FLOW AMPLITUDE, CM3ZSEC
4000
20000
O Tube D ia m e te r
= .0776 cm
A
= .04 cm
0.005
0.003
F re q ue n cy o f O s c i l l a t i o n
= I C y c le Per Second
5000
7000
PRESSURE AMPLITUDE, DYNES/CM3
FIGURE X I - 9 .
Tube D ia m e te r
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE
81
27000
FLOW AMPLITUDE, CM3ZSEC
17000
37000
47000
F re q ue n cy o f O s c i l l a t i o n
= 2 C y c le s P er Second
O Tube D ia m e te r = 0 .0 7 7 6 cm
^
Tube D ia m e te r = 0 .0 4 cm
0
7600
9600
PRESSURE AMPLITUDE, DYNES/CM3
FIGURE X I- 1 0 .
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE
11600
82
1 0 ,0 0 0
3 0 ,0 0 0
5 0 ,0 0 0
7 0 ,0 0 0
Tube D ia m e te r = 776 m ic r o n s
A
Tube D ia m e te r = 400 m ic r o n s
FLOW AMPLITUDE, CM3ZSEC
O
9 0 ,0 0 0
F re q ue n cy o f O s c i l l a t i o n
= 3 C y c le s Per Second
80
8 ,3 8 0
1 0 ,3 8 0
PRESSURE AMPLITUDE,DYNES/CM3
F ig u r e X I - I l .
FLOW AMPLITUDE VERSUS PRESSURE AMPLITUDE
1 2,3 80
83
TABLE X I - I l
DATA OF OSCILLATORY BLOOD FLOW
P r e s s u r e A m p lit u d e
d y n e s /c m 3
F re q u e n cy o f O s c i l l a t i o n
c .p .s .
0 .5
1 .0
2 .0
3 .0
F lo w A m p lit u d e
cm3/ s e c
S tro k e I
S tro k e 2
S tro k e I
S tro ke 2
2 3 7 5 .9
4 2 7 0 .0
7 7 8 1 .3
1 0 9 8 7 .3
5 0 3 3 .3
9 3 8 4 .3
1 8 1 6 2 .6
2 6 4 0 6 .6
0 .0 0 2 7 8 8
0 .0 0 5 5 7 6
0 .0 1 1 1 5
0 .0 1 6 7 2 7
0.0 0 72 3 13
0 .0 1 44 6 2
0 .0 28925
0 .0 4 33 8
Red c e l l s i n plasma
Tube D ia m e te r = 0 . 0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it = 45%
T e m p e ra tu re = 2 5 . 5 ° C
STEADY FLOW VISCOMETRIC DATA
S he a r Rate
sec 1
0 .6 3 3 6
1 .2 4 5 7
3 .0 1 5 4
5 .9 4 0 4
1 1 .8 2 4
2 9 .6 2 7 5
5 9 .8 9 8 9
75
150
300
750
1500
Red c e l l s in p la sm a
H e m a to c r it = 43.7%
T e m p e ra tu re = 2 4 .8 ° C
S hear S t r e s s .
d y n e s /c m 2
0 .1 4 3 7
0 .2 0 1 4
0 .3 8 0 4
0 .6 3 2 2
1 .0 7 4 6
2 .1 6 1 6
3 .8 1 0 3
4 .6 3 5
3 .6 6 7 5
8 .7 4 5
7 .6 9 5
1 6 .0 2
1 3 .7 4 '
3 6 .6 7 5
3 3 .3 7 5
6 0 .3
5 8 .5
A pp a re nt V is c o s it y
p o is e
0 .2 2 6 9
0 .1 6 1 6
0.1261
0 .1 0 6 4
0 .0 9 0 8
0 .0 7 2 9
0 .0 6 3 6
0 .0 6 1 8
.0489
0 .0 5 8 3
0 .0 5 1 3
0 .0 5 3 4
0 .0 4 5 8
0 .0 4 8 9
0 .0 4 45
0 .0 4 0 2
0 .0 3 9
84
TABLE X I - 1 2
DATA;OF OSCILLATORY BLOOD FLOW
Fre q u e n cy o f O s c i l l a t i o n
c .p .s .
.
0 .5
1 .0
2 .0
3 .0
P r e s s u r e A m p lit u d e
d y n e s /c m 3
Flow A m p litu d e
cm-3/ s e c
S tro k e 3
S tro k e 4
S tro k e 3
S tro k e 4
8 0 8 6 .6
1 5 8 7 2 .6
3 2 0 5 5 .3
46406 .
1 2 6 6 6 .6
2 4 5 7 4 .6
4 9 3 0 6 .6
7 22 0 6.6
0 .0 1 2 6 3
0 .0 2 5 2 6
0 .0 5 0 5 3 2
0 .0 7 5 7 9 .
0 .0 1 9 7
0 .0 3 95
0.0791
0 .1 1 8 6
R e d , c e l l s , i n plasma
,
Tube D ia m e te r = 0 .0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it = 45%
T e m p e ra tu re = 2 5 . 5 ° C
STEADY FLOW VISCOMETRIC DATA
S he a r Rate
sec” 1
0 .6 3 3 6
1 .2 4 5 7
3 .0 1 5 4
5 .9 4 0 4
1 1 .8 2 4
2 9 .6 2 7 5
5 9 .8 9 8 9
75
150
. 300
750
■ 1500 ■
Red c e l l s in p la sm a
H e m a to c r it = 43.7%
T e m p e ra tu re = 2 4 .8 ° C
S he a r S t r e s s
d yn e s /c m 2
0 .1 4 3 7
0 .2 0 1 4
0 .3 8 0 4
.
0 .6 3 2 2
. 1 .0 7 4 6
2 .1 6 1 6
. 3 .8 1 0 3
4 .6 3 5
3 .6 6 7 5
8 .7 4 5
7 .6 9 5
1 6 .0 2
1 3 .7 4
3 6 .6 7 5
.3 3 .3 7 5
6 0 .3
5 8 .5
A p p a re n t V i s c o s i t y
p o is e
0 .2 2 6 9
0 .1 6 1 6
0.1261
. ■ 0 .1 0 6 4
0 .0 9 0 8
0 .0 7 2 9
0 .0 6 3 6
0 .0 6 1 8
0 .0 5 8 3
0 .0 5 3 4
0 .0 4 8 9
0 .0 4 0 2
.0489
0 .0 5 1 3
0 .0 4 5 8
0 .0 4 4 5
0 .0 3 9
.
85
lo w e r c u r v e s i n
F ig u r e s X I - 8 t h r o u g h X I - I l
a r e f o r a tu b e o f 400 m ic r o n s
d ia m e t e r and t h e u p p e r c u r v e s a re f o r a tu b e o f 776 m ic r o n s d ia m e t e r .
I t s h o u ld be n o te d t h a t t h e u p p e r c u r v e s c o n s i s t o f t h r e e p o i n t s ;
th e
i n t e r m e d i a t e p o i n t r e p r e s e n t s one b lo o d sample and t h e o t h e r two p o i n t s
r e p r e s e n t s a d i f f e r e n t b lo o d sa m p le .
o b t a in e d w i t h one b lo o d sa m p le .
The lo w e r c u r v e s show t h e d a ta
Hence, t h e t r e n d shown by t h e two
c u rv e s do n o t seem t o be t h e same.
B u t t h e n a t u r e o f lo w e r c u rv e s can
be e x p l a i n e d i n te rm s o f t h e r h e o l o g i c a l
s h e a r r a t e s , b lo o d i s
Hence, r e l a t i v e l y
b lo o d a c t s l i k e
p o rtio n a l
C.
p r o p e r t i e s o f b lo o d .
n o n -N e w to n ia n and a p p a r e n t v i s c o s i t y
h ig h e r pre ssures a re nece ssa ry.
N e w to n ia n f l u i d
is
A t lo w
h ig h .
A t h ig h e r shear ra te s
and hence t h e p r e s s u r e s s h o u ld be p r o ­
t o th e flo w s .
Data and D is c u s s io n f o r D i f f e r e n t Red B lo o d C e l l s S usp e n sio n U sin g
P lasm a, S a l i n e S o l u t i o n and Hardened Red C e l l s
T a b le X I - 1 3 shows t h e r e s u l t s
a lb u m in - s a lin e s o lu tio n
tio n
f o r 400 m ic r o n s d ia m e t e r t u b e .
o f t h e re d b lo o d c e l l s was 46%.
t h e same re d b lo o d c e l l s
o f re d b lo o d c e l l s
i n p la s m a .
The b l o o d . h e m a t o c r i t was 45%.
n o t a g g r e g a t e i n 0.5% a l b u m i n - s a l i n e s o l u t i o n .
re d c e l l s ,
c e lls
Red b lo o d c e l l s
e x p e r im e n t was p e r fo r m e d .
do
B u t t h e y can a g g re g a te
Hence t o u n d e r s ta n d t h e p o s s i b l e e f f e c t s
th is
The c o n c e n t r a ­
T a b le X I - 1 4 shows t h e r e s u l t s o f
F ig u r e X I - I 2 shows t h e same r e s u l t s , g r a p h i c a l l y .
i n p la s m a .
i n 0.5%
o f a g g r e g a t io n o f
B lo od sample made o u t o f r e d
i n plasma needs a h i g h e r p r e s s u r e t o m a i n t a i n t h e same f l o w
86
T u be D i a m e t e r = 4 0 0 m i c r o n s
H e m a t o c r i t = 46%
25000
20000
15000
10000
A
O
0.0 1 4 4 6
0 .0 2 8 9
Red C e l l s
P lasm a
in
Red C e l l s
S a lin e
in
0 .0 4 3 3
FLOW AMPLITUD E, CM3ZSEC
FIGURE X I - I 2 .
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE
87
TABLE XI -1 3
DATA.OF OSCILLATORY BLOOD FLOW
F re q u e n c y o f O s c i l l a t i o n
c .p .s .
P re s s u r e A m p lit u d e
d y n e s /c m 3 .
0 .5
1.0
2.0
4 1 5 5 .5
7 8 5 7 .6
1 4 7 2 7 .6
2 1 5 9 7 .6
.
3 .0
F lo w Ampli t u d e
cm3/ s e c
0.0 0 72 3 13
. 0 .0 14462
0 .0 2 8 9 2 5 . .0282
0 .0 4 3 3 8 8
.04122
Red c e l l s i n S a l i n e and 0.5% A lb u m in
Tube D ia m e te r = 0 .0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it = 46%.
T e m p e ra tu re = 2 4 . 8 ° C
STEADY FLOW VISCOMETRIC DATA
S he a r Rate
se c"1 . '
1 .1 4 6 8
2 .8 9 0 3
5 .8 0 9 7
1 1 .6 8 3 4
2 9 .2 0 1 8
. 5 8 .1 5 0 4
75
150
' 300
750
1500
S hear S t r e s s
d y n e s /c m 2
0 .0 9 6 6
0 .2 2 9 2
0 .4 2 8 2
0 .7 8 3 7 :
1 .7 2 4 4
.
3 .1 5 6 8
3 .4 6 5
3 .2 6 2 5
6 .9 9
6 .9 9
1 2 .3 6
1 2 .5 7
2 8 .4 2 5 •
2 9 .6 2 5 .
5 1 .1 5
5 2 .0 5
Red c e l l s in S a lin e and 0.5% A lb u m in
H e m a to c r it = 45%
T e m p e ra tu re = 2 4 .5 ° C
A pp a re nt V is c o s it y
p o is e 0 .0 8 4 2
0 .0 7 9 3
0 .0 7 3 7
0 .0 6 7 0
0 .0 5 9 0
0 .0 5 4 2
0 .0 4 6 2
0 .0 4 3 5
0 .0 4 6 6
0 .0 4 6 6
0 .0 4 1 2
0 .0 4 1 9
0 .0 3 9 5
0 .0 3 7 9
0 .0 3 4 7
0.0341
88
TABLE X I - M
DATA OP OSCILLATORY BLOOD FLOW
Fre q u e n cy o f O s c i l l a t i o n
. c .p .s .
0 .5
P r e s s u r e A m p lit u d e
d y n e s /c m 3
5 0 3 3 .3
9 3 8 4 .3
1 8 1 6 2 .6
2 6 4 0 6 .6
1.0
2.0
3 .0
F l ow Ampli t u d e
cm3/s e c .
0.007231
0 .0 1 44 6 2
0 .0 2 8 9 2 5
.0282
0 .0 4 3 3 8 8
.0412186
Red C e l l s i n plasma
Tube D ia m e te r = 0 .0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it = 45%
T e m p e ra tu re = 2 5 . 6 ° C
STEADY FLOW VISCOMETRIC DATA
S hear Rate
sec” 1 .
0 .6 3 3 6
1 .2 4 5 7 1
3 .0 1 5 4
5 .9 4 0 4 '
1 1 .8 2 4 0
2 9 .6 2 7 5
5 9 .8 9 8 9
75
150
300
750
1500
Red c e l l s in p lasm a
H e m a to c r it = 43.7%
T e m p e ra tu re = 2 4 .8 ° C
S hear S t r e s s
d y n e s /c m 2
0 .1 4 3 7
0 .2 0 1 4
0 .3 8 0 4
0 .6 3 2 2
1 .0 7 4 6
2 .1 6 1 6
3 .8 1 0 3
4 .6 3 5
3 .6 6 7 5
8 .7 4 5
7 :6 9 5
1 6 .0 2
1 3 .7 4
3 6 .6 7 5
3 3 .3 7 5 •
6 0 .3 0 0
5 8 .5
A pp a re nt V is c o s it y
p o is e
0 .2 2 6 9
0 .1 6 1 6
0.1261
0 .1 0 6 4 ■
0 .0 9 0 8
0 .0 7 2 9
0 .0 6 3 6
0 .0 6 1 8
0 .0 4 89
0 .0 5 8 3
0 .0 5 1 3
0 .0 5 3 4
0 .0 4 5 8
0 .0 4 8 9
0 .0 4 45
0 .0 4 0 2
0 .0 3 9
89
Tube D ia m e te r = 400 m ic r o n s
Hardened Red C e l l s
40000
30000
20000
LEGEND
O H e m a t o c r it
10000
A
.02352
.04705
H e m a t o c r i t = 35.6%
.07057
FLOW AMPLITUDE, CM3ZSEC
FIGURE X I - 1 3.
4 6 .5 2 L
PRESSURE AMPLITUDE VERSUS FLOW AMPLITUDE
90
TABLE X I - 1 5
DATA OF OSCILLATORY BLOOD FLOW
F re q u e n cy o f O s c i l l a t i o n
c .p .s .
.
P r e s s u r e A m p lit u d e
d y n e s /c m 3
0 .5
6 6 3 6 .3
13888
28086
4 1 6 7 3 .3
1.0
2.0
3 .0
Flow A m p litu d e
cm3/ s e c
0 .0 1 1 7 6
. 0 .0 2 3 5
0 .0 4 7 0 5
.04587
0 ,0 7 0 5 7 . .06704
Hardened re d c e l l s i n 0.5% D e x tra n s o l u t i o n
Tube D ia m e te r = 0 .0 4 cm
' Tube L e n g th = 5 cm
H e m a t o c r it = 46.5%
T e m p e ra tu re = 2 5 . 5 ° C
STEADY FLOW VISCOMETRIC DATA
Shea.r_.Rate
s e c " 1.
S hear S t r e s s
d yn e s /c m 2
75
510
300
750
1500
3 .2 6 2 5
7 .2 3
1 3 .2 9
3 4 .8 7 5
6 5 .2 5
Hardened r e d c e l l s i n 0.5% D e x tra n s o l u t i o n
H e m a t o c r it = 46.5%
T e m p e ra tu re = 2 4 . 5 ° C
A pp a re nt V is c o s it y
p o is e
0 .0 4 3 5
. 0 .0 4 8 6
0 .0 4 4 3
0 .0 4 6 5
0 .0 4 3 5
91
TABLE ..X I-1 6
DATA OF OSCILLATORY BLOOD FLOW
F re q u e n cy o f O s c i l l a t i o n
c .p .s .
0 .5
P r e s s u r e A m p lit u d e
d y n e s /c m 3
■
Flow A m p lit u d e
cm3/ s e c .
.01176
.02352
.0470.5
.04587
.07057
.06704
4 9 5 7 .0
9 23 1 .6 7
1 8 7 7 3 .3 3
2 7 8 9 5 .1 6
1.0
2.0
3 .0
Hardened re d c e l l s i n 0.5% D e x tra n S o l u t i o n
Tube D ia m e te r = 0 .0 4 cm
Tube L e n g th = 5 cm
H e m a t o c r it = 3 5 . 6 %
T e m p e ra tu re = 24° C
STEADY FLOW VISCOMETRIC DATA
S hear Rate
s e c " 1.
.
75
150
300
750
1500
S hear S t r e s s
d yn e s /c m 2
1 .8 3 7 5
4 .2
7 .6 8
21.0
4 1 .5 5
Hardened r e d c e l l s i n 0.5% D e x tra n S o l u t i o n
H e m a t o c r it = 35.6%
T e m p e ra tu re = 2 4 . 5 ° C ■
A pp a re nt V is c o s it y
p o is e .
•
.0245
.02 8
.0256
.02 8
.0277
92
compared t o t h e re d c e l l
a lb u m in - s a lin e s o lu t io n .
t i n u o u s phase ( s a l i n e ) .
s u s p e n s io n made o u t o f re d c e l l s
in 0 . 5%
0.5% a lb u m in in c r e a s e s t h e v i s c o s i t y o f c o n ­
I t s h o u ld be m e n tio n e d t h a t re d b lo o d c e l l s
•
can d e fo rm i n s a l i n e and a l s o i n p la sm a .
T a b le s X I - I 5 and X I - I 6 show t h e r e s u l t s f o r re d c e l l s
s io n s c o n s i s t i n g o f h a rde n ed re d c e l l s
F ig u r e X I - I 3 shows t h e same r e s u l t s
i n 0.5% D e x tra n 40 s o l u t i o n .
g r a p h ic a lly .
harde n ed re d c e l l s were 46.5% and 35.6%.
C o n c e n tra tio n s o f
Hardened re d c e l l s
by a t r e a t m e n t w i t h an i s o t o n i c g l u t a r a l d e h y d e s o l u t i o n .
a g g r e g a te and t h e y do n o t d e fo rm .
c e lls
su spe n ­
a re made
They do n o t
Hence a s u s p e n s io n o f hardened re d
i n s a l i n e has a c o n s t a n t v i s c o s i t y .
b lo o d samples made o u t o f re d c e l l s
It
is
seen f r o m P s . 73-74
i n plasma w i t h h e m a t o c r i t s o f a b o u t
36% and 46% need h i g h e r p r e s s u r e s compared t o s i m i l a r sam ples w i t h
h ardened re d c e l l s .
F in a lly ,
b lo o d c e l l
a n a ly s is
a g g r e g a t io n and re d b lo o d c e l l
is
p e r fo r m e d .
N e w to nia n f l u i d
.
t o g e t some id e a a b o u t t h e p o s s i b l e e f f e c t s o f re d
-
-
From t h e t h e o r y o f o s c i l l a t o r y
(C h a p te r I X ) ,
S
it
is
i
where R = r a d i u s o f tu b e
Pfn = p r e s s u r e a m p lit u d e
Q
d e fo rm a tio n , th e fo llo w in g
= f l o w a m p lit u d e
flo w f o r a
seen t h a t t h e v i s c o s i t y i s
g iv e n b y ,
93
F o r t h e e x p e r im e n t a l w o rk done w i t h re d c e l l
s u s p e n s io n s u s in g
p la s m a , a l b u m i n - s a l i n e s o l u t i o n and harde n ed re d c e l l s ,
th e v a lu e , o f a
2
is
u s u a l l y le s s t h a n I .
The q u a n t i t y
can be a p p r o x im a te d t o
a b o u t 0 .1 2 5 f o r lo w va lu e s ' o f a (.27).
From F ig u r e s X I - 6 , X I - 1 2 and X I - 1 3 ,
o f Pm v e r s u s Qm i s
n e a rly a s t r a i g h t l i n e
it
is
seen t h a t t h e graph
a t h ig h e r flo w r a te s .
t h e s lo p e o f t h e gra ph t h e v a lu e o f PmZQm can be o b t a i n e d .
From
Knowing
M1
1O
t h e v a lu e o f PmZQm, r a d i u s o f t h e t u b e and
e qual
to 0 .1 2 5 , th e
v i s c o s i t y o f a g iv e n re d c e l T s u s p e n s io n can be c a l c u l a t e d fro m th e
above e q u a t i o n .
shear ra te s
I t was a l s o m e n tio n e d t h a t b lo o d i s
(h ig h e r flo w r a t e s ) .
N e w to n ia n a t h ig h
The v i s c o s i t y o f re d b lo o d c e l l
p e n s io n s a t h ig h s h e a r r a t e was d e t e r m in e d e x p e r i m e n t a l l y .
th is
e x p e r im e n t a l
sus­
Hence,
v i s c o s i t y can be compared w i t h t h e v i s c o s i t y c a l ­
c u l a t e d f r o m t h e above e q u a t i o n , and t h e y a r e compared i n t h e f o l l o w i n g
ta b le .
TABLE X I - I 7
F ig u r e
Number
X I- 1 2
X I-1 3
X I-1 2
X I-6
X I-6
Red C e ll
S usp e n sio n
Red c e l l s i n a l b u m i n - s a l i n e
Hardened re d c e l l s
Red c e l l s i n plasma
Red c e l l s i n plasma •
Red c e l l s i n plasma
H e m a t o c r it
p e r c e n ta g e
. 46
3 5 .6
• 45
3 6 .4
. 46
V is c o s ity
From Above
E q u a tio n
p o is e
0 .0 3 6 2 3
0 .0 2 5 9
. 0 .0 4 3
0 .0 5 0 3 7
0 .0 5 8 2
E x p e rim e n ta l
V is c o s ity
p o is e
0.0341
0 .0 2 6 7
0 .0 3 9
0.0381
0 .0 4 8
94
From T a b le X I - I 7 , i t
is
seen t h a t t h e two v i s c o s i t y v a l u e s , as
d e t e r m in e d
fr o m t h e o s c i l l a t o r y d a ta and fr o m t h e v i s c o m e t r i c m easure­
m e n ts , a r e
i n a g re e m e n t f o r t h e s u s p e n s io n o f re d c e l l s
s a l i n e and
f o r t h e h ardened re d c e l l s
re d c e l l s
s u s p e n s io n .
i n a lb u m in -
F o r s u s p e n s io n s o f
i n p la s m a , t h e v i s c o s i t y c a l c u l a t e d fro m t h e o s c i l l a t o r y f l o w
d a ta a r e l a r g e r t h a n t h e v i s c o s i t y d e t e r m in e d by v i s c o m e t r i c means.
S in c e h a rde n ed re d c e l l s
n e i t h e r d e fo rm n o r a g g r e g a t e (and
hence fo r m N ew to nia n S u s p e n s io n s ) , i t
v i s c o s i t y v a lu e s a g re e f o r . t h i s
is
n o t s u r p r i s i n g t h a t t h e two
s u s p e n s io n .
W h ile norm al
re d c e l l s
suspended i n a l b u m i n - s a l i n e a l s o do n o t a g g r e g a t e , t h e y a r e , how eve r,
d e fo rm a b le a n d , i n s t e a d y f l o w , such s u s p e n s io n s a re n o n -N e w to n ia n
(p s e u d o p la s tic ).
I t w o u ld t h e r e f o r e be e x p e c te d t h a t t h e v i s c o s i t y
d e t e r m in e d f r o m o s c i l l a t o r y
d a ta w o u ld be l a r g e r th a n t h a t d e te rm in e d
fro m h ig h s h e a r r a t e v i s c o m e t r i c d a t a .
The f a c t t h a t t h e two v i s c o s i t y
v a lu e s a g re e s u p p o r t s t h e id e a t h a t t h e k i n e t i c s
a r e n o t s y m m e t r ic a l
( t h e d e f o r m a t io n o f a c e l l
a s tre s s e d c o n d itio n
is
o f c e ll
d e f o r m a t io n
fro m a r e l a x e d shape t o
f a s t e r t h a n t h e r e l a x a t i o n p r o c e s s ) and t h a t
t h e s lo w e r r e l a x a t i o n p ro c e s s has a c h a r a c t e r i s t i c t im e c o n s t a n t w h ic h
is
l o n g compared t o t h e t im e c o n s t a n t o f t h e o s c i l l a t o r y
flo w s s tu d ie d .
The f a c t t h a t t h e v i s c o s i t y c a l c u l a t e d fro m o s c i l l a t o r y f l o w
d a ta f o r re d c e l l s
suspended i n plasma i s
r a t e v i s c o m e t r i c v a lu e i n d i c a t e s
f a s t p ro ce ss.
Because o f c e l l
th a t c e ll
l a r g e r t h a n t h e h ig h s h e a r
a g g r e g a t io n i s
a r e la tiv e ly
a g g r e g a t i o n d u r i n g lo w s h e a r s t r e s s
95
p o r tio n s o f th e o s c i l l a t o r y
f l o w , t h e a p p a r e n t v i s c o s i t y o f. t h e s u s ­
p e n s io n w i l l
be h i g h e r t h a n t h e h ig h s h e a r f a t e . v i s c o m e t r i c v a l u e .
a d d itio n ,
must be t h a t t h e c e l l
it
o s c illa to r y
f lo w s i s
shape a t a lo w s h e a r s t r e s s i n t h e '
d i f f e r e n t fr o m t h e c e l l
m e t r i c f l o w s a t t h e same lo w s h e a r s t r e s s ;
s t e a d y v i s c o m e t r i c d a ta w i l l
In
shape d u r i n g s t e a d y v i s c o hence t h e lo w s h e a r s t r e s s
n o t a p p ly t o o s c i l l a t o r y
flo w s .
X II.
A.
C o n c lu s io n s
1.
w e ll
CONCLUSIONS AND RECOMMENDATIONS
A p p a ra tu s b u i l t f o r o s c i l l a t o r y
f o r N e w to n ia n f l u i d s
5 to 6 c e n t ip o is e ) .
d i a m e t e r , s m a ll
o s c illa tio n
such as g l y c e r o l
f l o w o f b lo o d w orks v e r y
s o lu tio n s
( v is c o s it y about
B u t w h i l e . w o r k i n g w i t h t h e t u b e , - o f 400 m ic r o n s i n
c o rre c tio n
f a c t o r s a r e n e c e s s a ry a t f r e q u e n c i e s o f
o f 2 c y c l e s p e r second and 3 c y c l e s p e r s e c o n d .
These
c o r r e c t i o n f a c t o r s ' a r e a p p l i e d because o f s l i g h t e x p a n s io n o f fe e d
r e s e r v o i r s , arid c o n n e c t i n g t u b e and c o m p r e s s i b i l i t y o f g l y c e r o l
s o lu tio n s .
2.
The p r e s s u r e -w a v e s were f o u n d t o be s i n u s o i d a l
q u e n c ie s o f o s c i l l a t i o n s
s e co n d .
o f 0 . 5 c y c l e s p e r second t h r o u g h 3 c y c le s p e r
The f lo w - w a v e s were a lw a y s s i n u s o i d a l
i s d e r i v e d f r o m t h e d is p la c e m e n t wave w h ic h i s
3.
m ic r o n s
because t h e f l o w wave
a lw a ys s i n u s o i d a l .
The e n t r a n c e e f f e c t s o f tu b e s w i t h d ia m e t e r s o f a b o u t 400
( l e n g t h 5 cm) and 776 m ic r o n s
be s i g n i f i c a n t .
4.
fo r fre ­
( l e n g t h 3 0 .0 5 cm) do n o t seem t o
.
The s t i r r e r
i n s i d e t h e fe e d r e s e r v o i r i s
f o u n d t o be n e c e s ­
s a r y t o m a i n t a i n t h e c o n c e n t r a t i o n o f re d b lo o d c e l l S. u n ifo rm ,.
5.
The n a t u r e o f t h e g ra p h s o f p r e s s u r e a m p lit u d e s v e rs u s f l o w
a m p lit u d e s can be e x p l a i n e d q u a l i t a t i v e l y
i n te rm s o f t h e r h e o l o g i c a l
p r o p e r t i e s o f b lo o d .
6.
In t h e p re s e n c e o f h ig h m o l e c u l a r w e i g h t D e x t r a n , re d b lo o d
97
c e lls
a g g re g a te .
Hence h i g h e r p r e s s u r e s a r e needed t o m a in t a in , same
f l o w s f o r re d c e l l s
re d c e l l s
7.
suspended i n such D e x tra n s o l u t i o n s
compared t o
i n p la sm a.
Red c e l l s
do n o t a g g r e g a t e i n . s a l i n e
d e fo rm i n s a l i n e s o l u t i o n .
t h e y do n o t d e fg rm .
o c c u r i n p la s m a .
re d b l o o d . c e l l
Hardened r e d c e l l s
B u t re d b lo o d c e l l
From t h e e x p e r im e n t a l
s o l u t i o n b u t t h e y can
do n o t a g g r e g a t e and
a g g r e g a t io n and d e f o r m a t io n can
re s u lts
it
is
c o n c lu d e d t h a t
a g g r e g a t io n i s an i m p o r t a n t p ro c e s s i n o s c i l l a t o r y b lo o d
f l o w , when t h e tu b e d ia m e t e r i s 400 m i c r o n s .
8.
Lower p r e s s u r e i s
b lo o d h e m a t o c r i t ( r e d c e l l s
needed t o m a i n t a i n t h e same f l o w f o r 36%
i n p la sm a ) compared t o t h e p r e s s u r e needed
f o r 46% b lo o d h e m a t o c r i t ( r e d c e l l s
i n p la s m a ) .
T h is
is
because b lo o d
i s more v is c o u s a t 46% b lo o d h e m a t o c r i t .
•
•
. B . ~Recommendations
1.
To o b t a i n a . p e r f e c t a g re e m e n t between t h e t h e o r y and e x p e r i ­
m en ta l w o rk a t h i g h e r f r e q u e n c i e s . o f o s c i l l a t i o n
(3 c y c l e s p e r second o r
h i g h e r ) , t h e s i z e o f t h e fe e d r e s e r v o i r can be re d u c e d and tu b e l e n g t h
can be r e d u c e d .
T h is s u g g e s t io n may be u s e f u l w h i l e w o r k in g w i t h s t i l l
s m a l l e r d ia m e t e r tu b e s
2.
to escape.
(s a y a b o u t 200 m i c r o n s ) .
W h ile f i l l i n g
. Hence, i t
is
t h e a p p a r a tu s w i t h . b l o o d , b u b b le s a re l i k e l y
recommended t h a t a b u b b le t r a p may be p la c e d
between t h e o u t s i d e s o u r c e fr o m w h ic h re d b lo o d c e l l
s u s p e n s io n i s
98
o b t a i n e d and t h e i n l e t fr o m w h ic h t h e a p p a r a tu s i s
3.
A lt h o u g h t h e n a t u r e o f t h e e x p e r im e n t a l
q u a lita tiv e ly
s ib le .
in t h i s
fille d .
re s u lts
is e x p la in e d
t h e s i s , a q u a n t i t a t i v e e x p l a n a t i o n may be p o s -
T h is may be done by p r o c e d u r e K. shown on P. 53 a lo n g w i t h t h e
p r o p e r n u m e r ic a l methods..
The q u a n t i t a t i v e e f f e c t s o f d e f o r m a t io n and
a g g r e g a t i o n o f re d b lo o d c e l l s
can a l s o be s t u d i e d w i t h t h e h e lp o f
s i m i l a r n u m e r ic a l m e th o d s .
4.
As a f u r t h e r s t u d y i n t h i s
and o s c i l l a t o r y
5.
flo w
( p u ls a tile
flo w )
fie ld ,
t h e c o m b in a t io n o f s t e a d y .
can be s t u d i e d .
F o r o s c i l l a t o r y b lo o d f l o w , t h e phase d i f f e r e n c e s between
p r e s s u r e waves and f l o w waves a r e n o t r e p o r t e d i n t h i s
th e s is .
But i t
s h o u ld be done i n f u t u r e s t u d i e s .
6.
When tu b e d ia m e t e r i s h i g h e r t h a n 300 m i c r o n s , t h e tu b e
b lo o d h e m a t o c r i t i s
Hence i n
e x p e c te d t o be t h e same as fe e d b lo o d h e m a t o c r i t .
t h i s w o r k , t u b e h e m a t o c r i t was n o t measured d u r i n g e x p e r i ­
m e n ta l w o r k .
B u t, i f
t u b e o f s m a l l e r d ia m e t e r ( l e s s th a n 300 m ic r o n s )
i s u s e d , t u b e h e m a t o c r i t s h o u ld be .m easured.
APPENDICES
TOO
APPENDIX A
COMPUTER PROGRAM FOR CALCULATING THE DIFFERENCE
BETWEEN THE POISEUILLIAN AND APPARENT VISCOSITY
5
8
1
INPUT DP,N
.
.
Rl=O.;R0=.05;DR=.001;U=O.;G=O.;SUM=O.;KOUNT=O.
H=.05/N
Fl=G=iRl5i;i2/R0;;:t2
R2=R1+H
T =DP5iR2/2 .
IFCT.GE.O..AND.T.LE..0130234)00 ' T0 I
IFCT.GT..0130234.AND.T.LE.I.04)60 T0 2
IFCT.GT.1 .04.AND.T .LE.37•2)G0 T0 3
IF(T. GT .'37.2)60 T0 9
G=O . .
G0
2
3
G0
9
11
4-
T0
11
G=C 5 • 9 4 4 2 5 " CT 555i. 5 ) - . 6 7 8 3 6 ) " ” 2
G0
T0
11
G=C - . 7 0 2 7 + 5 . 9 4 4 }{C t 5i5i. 5 ) + • 0 4 6 3 5 6 5;T ) 5i5i2
T0
11
G = 3 7 . 1 7 4 7 2 5iT
CONTINUE
F 2=G5iR 2 555;2/R05{5{2
FAVG-C'Fl+F2) / 2 .
SUM=SUM+FAVG
H e m a t o c r it
F1=F2 ‘
R1=R2
KOUNT=KOUNT+!
■ IFCK0UNT.EQ.N)
GO
TO
4 .
XNA=TZG
GO 'T0
8
TOTAL = H 5iSUM
'UBAR=T0TAL/C 2 .'55RO)
CNP = CRO5i:i2)5iDP/C8. "TOTAL)
. D I F FERENCE=CNP-XNA
OUTPUT UBAR,XNA,CNP,DIFFERENCE
GO
TO
5
END
35.8%
101
APPENDIX B
T h is a p p e n d ix d e s c r ib e s some o f t h e d e t a i l s
about t h e . in s t r u ­
ments used i n t h e a p p a r a t u s .
I .
E l e c t r i c M o to r and A c c e s s o r i e s :
A.
.
E l e c t r i c M o to r
M o to m a tic R M o to r G e n e r a t o r
Made by E l e c t r o - C r a f t C o r p o r a t i o n , H o p k in s , M in n .
B.
Gear Box:
Type M;
R e d u c to r R a t io m e t e r
S iz e 1 0 9 ;
M o to r C a t.
No:
In p u t H .P .:
F la n g e d R e d u c to r
R a t io 5 ;
AASD
0 .0 3 5
O u tp u t T o rq u e :
4
In p u t R .P .M .:
1750
"
O u tp u t R .P .M .:
U .S . P a t e n t :
350
.28 68031
Made by B osto n G ea r, Q u in c y , Mass.
C.
Speed R e g u la t o r .
Model TR 9020
M i n a r i k E l e c t r i c C o . , L . A.
102
II.
P re s s u r e Measurement D e v ic e s : .
A.
P re s s u r e T r a n s d u c e r . ...
P23Dd S tath am
Made by G o u ld -S ta th a m I n s t r u m e n t s ,
R ic o
B.
I n c . , Hato R ey, P u e rto
00919
A m p lifie r
.
.
U n iv e r s a l T r a n s d u c e r ;
.
Readout M o d e l. SC 1001;
S tath am P23Dd
R a n g e ,is 0 t o +300 mmHg.
C.
Low-Gain DC P r e a m p l i f i e r
17402A
Made by H e w le t t - P a c k a r d
D.
Manometer
P re s s u r e No. 1425 Hook C-age
Made by Dwyer I n s t r u m e n t s ,
III.
I n c . , M ic h ig a n C i t y ,
I n d . , U .S .A .
Flow Measurement D e v ic e s :
A.
D is p la c e m e n t T r a n s d u c e r
Type 1000 M-HR;
S/N 115
Made by S c h a e v it z E n g i n e e r i n g , Camden, Ni J .
B.
AC C a r r i e r P r e a m p l i f i e r
P l u g - I n 17403A
Made by H e w l e t t Packard
08101
'
.
103
C.
S y r in g e
( i n s i d e d ia m e t e r = .05 7 cm)
M i c r o ! i t e r S y r in g e # 71 0 ;
P a t e n t No. 2933037
Made by H a m ilto n Company, Reno, Nevada
IV .
- O s c illo g r a p h ic R e co rd e rs:
A.
O s c i l l o g r a p h i c R e c o rd e r
7402A
■
. Made by H e w le t t - P a c k a r d
B.
AC C a r r i e r P r e a m p l i f i e r
P l u g - I n I 7403A
Made by H e w le t t P ackard
V.
' M is c e ll a n e o u s
A.
Ite m s :
S y r in g e Pump ( f o r s t e a d y f l o w )
. I n f u s i o n W ith d ra w a l
Pump
H a rv a rd A p p a r a t u s , 150 D over Road, M i l l i s ,
B.
MA
02054
R e s e r v o i r s and Tubes
I)
R e s e rv o irs
Made o u t o f I n c i t e .
C a p a c it y o f f e e d r e s e r v o i r :
12-13 ml
2) C a p i l l a r y Tubes
Made o u t o f g l a s s .
I n s i d e D ia m e te r
400 m ic r o n s
776- m ic r o n s
O u t s id e D ia m e te r
7 mm.
7 mm
L ength
5 . 0 cm
3 0 .0 5 cm.
3) M a g n e tic S t i r r e r .
S p i n b a r , a B e l - A r t P r o d u c t ; . VWR S c i e n t i f i c .
m olded H n ic o -V
O n e -p ie c e
BIBLIOGRAPHY
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2 3 ..
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p p. 1 9 1 -1 9 9 , 1976.
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Waugh, R . , and E. A. E vans, " V i s c o e l a s t i c P r o p e r t i e s o f E r y t h r o c y t e
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B lo o d F lo w , " M i c r o -
MONTANA STATE UNIVERSITY LIBRARIES
CO
III I111111IlIllilll
D3T8
Shl4
cop.2
7132
10C II 1 2 2 5 7
Shah, Bharat O
Some of the characteristics of steady and
oscillatory blood flow
!TNTElttTJTR*ARY
2 V*i—^ ; i ^ITERUBRARY LO