Biorheology Vol. 15, pp. 119-128 0006-355X/78/0401-0119$02.00/0 © Pergamon Press Ltd. 1978. Printed in Great Britain
VISCOSITY OF POISEUILLE FLOW OF A COUPLE STRESS FLUID WITH APPLICATIONS TO BLOOD FLOW P. Chaturani Department of Mathematics, I.T.T. Bombay-400 076, India
(Received in revised form 10 October 1977) (Accepted by Editors-in-Chief) Abstract - Viscosity of Poiseuille flow of a fluid with couple stress has been studied in this paper. The analytic expression of the viscosity of couple stress fluid has been obtained in the form of modified Bessel's functions of the order of zero and one. Since it is a complicated function of the couple stress parameters a and n, the numerical values of the relative viscosity nr have been computed for various values of a and n. These theoretical results are compared with other theoretical and experimental results on blood flow and suspension flows with 40% concentration. For the quick calculation of nr two simple approximate formulas have been obtained. The values of nr obtained from the exact formula and the approximate formulas are compared and it is found that they are in good agreement (within 6% error). The variation of nr (exact and approximate) with a and n is shown graphically which clearly indicates the existance of a discontinuity in the relative viscosity and the velocity at point (a = 0.0 and n = 1.0). The most important conclusions of this analysis are: (i) the conditions for the existance of Fahraeus-Lindquist effect in a tube flow have been obtained; (ii) Up to this date, only velocity profiles have been used to determine the values of a and n (n chosen arbitrarily); here it is shown that by using the experimental velocity profiles and relative viscosity both the couple stress parameters can be determine quite accurately (n no longer chosen arbitrarily). Finally, some biological implications of this theoretical investigation have been indicated. INTRODUCTION It has been observed that the blood viscosity in patients with myocardial infarction was appreciably higher than in normal individuals (1). The viscosity of blood varies greatly in different areas of the circulation and depends on such factors as the internal viscosity of red cells, hematocrit, plasma viscosity and aggregation of blood cells as well as on the lumen of the blood vessel, its diameter and the flow velocity. We shall study the viscosity of the flow of blood through the tubes of small diameter (of the order of 15-50 ~m diameter). 119
P. Chaturani
120
Poiseuille flow of a couple stress fluid through a circular tube has been studied by Valanis and Sun (2). They have attempted to apply the couple stress theory to the flow of blood through narrow capillaries where non-Newtonian behaviour is pronounced. While comparing their theoretical results with the experimental data of Bugliarello et aZ. (3) on blood flow, they find a very good agreement between the two velocity profiles. It is well known that the viscosity of blood can be a significant factor in the pathogenesis of ischemia and infarction and might play an important role in hypertension. It is also an important factor from the fluid engineering point of view. But for some reasons, Valanis and Sun (2) have made no attempt to study the viscosity of couple stress fluid (blood). Recently, some authors (4,5) have attempted to explain various anomalies associated with blood flow through couple stress theory. Because the viscosity of blood is an important factor in pathogenesis of thrombosis and other cardiovascular diseases and the couple stress theory has been used to study the blood flow (2,4), it seems worthwhile to calculate the viscosity of couple stress fluid flow considered by Valanis and Sun (2). The aim of the paper is to study the viscosity behaviour of the couple stress fluid. VISCOSITY CALCULATION The expression for axial velocity V of Poiseuille flow of a fluid with couple stress has been obtained by Valanis and Sun (2). For the sake of completeness and ready reference that expression (equations (3.22) and (3.22a) of Ref. (2) is reproduced here: 1 -
(---E-)
2
a
- II [ 1 - I
(ar) I a
0
0
(a) ] ,
(I)
where 2 (1-11)
li
a2
(1 +11) a
1 -
.E.
,
11
.:1.'
11
11
V11 0
(2)
Il (a) I (a) 0
a 2K 4
(3)
~
]J
where In is a modified Bessel's function of order n~ II the viscosity of solvent, 11 and 11' the couple stress parameters, K the rate of pressure drop in axial direction, a is tube and r is the radial coordinate. Volumetric flow rate Q for a tube is given by
Q = 2n p fa y V dr
(4)
o
where p i.s density. Substitlltion of equations (I) and (2) into equation (4) gives Q
'rT
P a
2
]J
4
k
1
f
o
~
[1-
~2_li
{I-I (aO/I (a)}] 0
where ~ =
ria .
Now the equation (5) may be written 1n the form
0
d~
,
(5)
Viscosity of Poiseuille Flow Q
P a
'IT
2
y
where
1 4
k ]J
H H +---2 a 2I (a) o
a
f
o
121
Y I (y) dy
(6)
o
= a~
On integration
[1.4-2"H
Q
+
H
a
II (a)
-I-J.
(7)
o(a)
Apparent viscosity na for Poiseuille flow is given by
na
'IT
=
k a4
(8)
8Q
which for the present problem takes the following form (9)
]J
1 -
2H
+
Now the relative viscosity
4H
np of the fluid (suspension or blood) can be written as
[ 1-2ll +
The variation of
~H
(10)
I 1 (a) ] I (a) o
np with a and n is shown through Fig.
I.
COUPLE STRESS PARAMETERS DETERMINATION AND VISCOSITY DISCONTINUITY There are two important conclusions which can be drawn from Fig. I. The first this figure and the velocity Profiles (equation (1» the couples stress parameters a and n can be determined quite accurately. It is well known that from the experimental measurements (3) the value of np and V can be determined. Thus for a given relative viscosity* - say np = 31 - we draw a corresponding curve in a-n plane (Fig. 1). Using equation (1), we can also plot the non-dimensional axial velocity V(o)
against a with n as a parameter (Fig. 2). Now for a given axial veloc-
~ -vnV(o). = 0.516 (FLg. 5, of Ref. (2»
ity - say
o
- the line
~(~ )
0.516 is drawn
0
which will intersect many n = constant curves (Fig. 2). These points on intersection individually correspond to a combination of a and n which can give the nondimensional axial velocity = 0.516. A combination of a and n which also lies on the curve np = 81 in Fig. 1. is the proper value of a and n for the given fluid under the given conditions. Thus, it has been shown that the values of a and n can be determined quite accurately. This is in contrast with the statement of Ref. (2) "Because the value of n is not known and cannot even be guessed at". Also, this method has an advantage over Stokes (7) method, where one has to conduct two ex-
* The value of np is not given in Ref. (3). Therefore the value of the viscosity was calculated from the Fig. 15 of the Ref. (6). The computed value of the viscosity is 3.36 centipoise.
P. Chaturani
122
0..8
Exact curve - - 1st approximation . _ . _ . 2nd approximation 0..6
o.A
0..2
-0..2
-o.A
-0.6
-0.8
Fig. 1. periments - one with the flat plate and the other with the cylindrical tube. The second conclusion is: the viscosity is discontinuous at point (a = 0.0 and n = 1.0). It may be seen at a glance from Fig. 1 that all the viscosity curves tend to cluster near the point (a = 0.0, n = 1.0). This shows that the function nr (a,n) has different slopes as we approach the point (0,1). Hence the function nr (a,n) is a discontinuous function of a and n at (0.1). To be more specific, this discontinuity is of irremovable type. Similarly, one can say that the velocity is also a discontinuous function of a and n at this point.
Viscosity of Poiseuille Flow
123
APPROXIMATE EXPRESSIONS OF RELATIVE VISCOSITY
From equations (J), (2) and (10), it is quite clear that V and np are complicated functions of modified Bessel's functions of zero and first order and in turn difficult functions of a and n. It is therefore, going to be quite inconvenient, in particular, for non-professional mathematicians, e.g. physiologists, engineers, etc. to use these results. We shall, therefore, derive two approximate formulae for np and compare it with exact results. First of all, it is assumed that a and n are less than one. With this assumption, the equation (10) reduces to a very simple form np
24(I-n)
=
(II)
a2 7-n)
The values of np calculated from this expression are listed in Table I and are compared with exact values of np (equation (10». It is of interest to note that for a = O. J (concentrated suspensions) the error is only 6%, the error percentage increases with a and n. Looking at the simplicity of the expression (I I) and at the percentage error, it can be said that the equation (I I) is a good approximate formula for calculating np for small values of n and a (small in comparison to one). This formula (equation (II» we shall call the first approximation of np' As expected, this equation gives incorrect np if both n and a are approximately one, and the error is ca. 30-40 % if one of them is approximately one (Table 2). However, if we retain one more term in the numerator of equation (10), then the expression for nr takes the following form:
24(I-n)
+
3 a 2 (3 - n)
(12)
a 2 (7 - n)
which we shall call the second approximation of np. The numerical values of np (for a = 1.0) computed from equations (10), (II) and (12) are shown in Table 2. The improvement in the values of np is significant: in the first approximation when one "7 = I "7 = 0.8 "7 = 0.5 "7 =0.1 "7 = 0.5 "7 = 1.0
·0" "
~o
0.516 0.5
o
~~~--~------~------~
0.45
2
a
Fig. 2.
3
124
P. Chaturani Comparison of first approximation and actual relative viscosities
TABLE I.
n = 0.0
n = -1.0 Actual
nr O. I
0.5 1.0
Approximate
578.7
600.0
25.03
nr
343.0
14.72
6.000
TABLE 2.
nr
336.5
24.00
7.034
Approximate
Actual
nr
n = 0.5
4.438
n
nr
Approximate
183.4
195
Actual
=
Actual
nr
nr
42.43
0.9 Approximate
nr 39.3
13.72
8.335
7.70
2.574
1.572
3.43
3.849
1.95
I. 393
0.393
First and second approximation and actual relative viscosities for a = 1.0
n
First approximation
Second approximation
Actual
0.9
0.393
I. 42
1.393
0.5
I. 95
3.00
2.849
0.0
3.43
4.70
4.438
-1.0
6.00
7.50
7.034
of the parameters is approximately one the error was oa. 30-40%, this in the second approximation reduces to 5-6%; when both a and n are approximately one, then in the first approximation the value of nr was incorrect, but in the second approximation they are within 3% error (Table 2) which is a remarkable improvement. At this stage, it is necessary to mention that while calculating the relative viscosity and/or velocity in the neighbourhood of the point (a = 0.0, n = 1.0) one should be extra careful because there is a discontinuity in the viscosity and velocity at this point. The relative viscosity and velocity in the neighbourhood of this point can be given by the following approximate formulae.
+~
nr Vn V0
E:
(I
-
(13)
r2
-) 0: 2
E: E:
+ 48
where E:
= a 2,
8
I - n
DISCUSSION Valanis and Sun (2) have compared their theoretically predicted with the experimentally obtained velocity profiles of Bugliarello shown that the two results are in good agreement. While comparing and the experimental velocity profiles, they find that blood ~ith of REC could be represented by a couple Stress fluid with n = 0.5
velocity profiles et aZ. (3) and the theoretical 40% concentration and a = 1.35 (n
Viscosity of Poiseuille Flow
125
is arbitrarily chosen and a determined). For some reasons, they have not calculated the relative voscosity for couple stress fluid. In our opinion, the computation of the relative viscosity is quite important, because it is required along with the velocity profiles to determine the two couple stress parameters a and 11 and also it can be used as a predictive tool in hypertensive and cardiovascular diseases (I). The relative viscosity of couple stress fluid with 11 = 0.5 and a = 1.35 turns out to be 2. IS (using equation (10». This value of 11p appears to be very low for a suspension with 40% concentration (Table 3a). Therefore, an attempt was made, using Fig. I, to obtain the same velocity profiles with higher viscosity (Table 4). It was found that the relative viscosity in the range of 3 to 10 cannot give the required velosity profiles. However, a marginal increase in 11r~ i.e. 11r = 2.5, is possible (Table 4). But this reduction of 11r could be justified by another interpretation which seems to be quite reasonable. From the Table 3(a), it may be observed that the value of 11p obtained for a = 1.35, 11 = 0.5 is almost the half of the other values of 11p' This 50% reduction in the relative viscosity is a characteristic feature of the blood flow through the tubes with small diameter (15-50 ~m diameter) which is known as Fahraeus-Lindquist effect (10). This is due to the existence of a cell-free zone (2-5 ~m wide) next to the tube wall. It is of interest to note that Valanis and Sun (2) have derived their boundary conditions (equation (3.20) of Ref. (2)) by using this idea, i.e. 'existence of a cell-free zone next to tube wall' (p. 8S/2nd para. (2)). Hence, it appears that the analysis of Valanis and Sun (2) is valid for the flows through the small tubes which exhibit Fahraeus-Lindquist effect. It is, therefore, improper to compare the present 11r with 11p of the flows in which the Fahraeus-Lindquist effect is absent. In view of this argument, the reason for the disagreement is obvious. However, if we compute the other relative viscosities of Table 3(a) for the flows with the Fahraeus-Lindquist effect and then compare with the present one, the agreement between th~ two is satisfactory (Table 3(b)). TABLE 3(a).
Comparison of relative viscosities In absence of Fahraeus-Lindquist effect
Investigator
Concentration
Relative viscosity
Cocklet (8)
40 %
4.0
2
Hatschek (9)
40 %
4.0
3
Valanis and Sun (2)
40 %
2.18
4
Woodcock (10)
40 %
5.5
No.
TABLE 3(b).
Comparison of relative viscosltles of flows with Fahraeus-Lindquist effect
Investigator
Concentration
Relative viscosity
Cocklet (8)
40 %
2.0
2
Hatschek (9)
40 %
2.0
3
Valanis and Sun (2)
40 %
2.18
Woodcock (10)
40 %
2.64
No.
4
From Fig. I, it can be seen that for a suspension with given 11r~ a cannot be assigned any arbitrary value between zero to affinity. As a matter of fact, for a given 11p the values of a have a certain upper limit. e.g. 11r = 10.0, 0 < a ~ 0.705; 11r = 5.0, 0 < a ~ 1.2 etc. (Fig. I). One can explain the existence of the upper limit
N 0\
TABLE 4.
Variation of relative viscosity and acial velocity with a and
nr = 10.0 a
=
o. I
0.5
n
=
0.92
0.4
0.705 -1.0
nr = 5.0
nr = 3.0
0.5
1.0
1.2
0.7
-0.3
-1.0
nr = 2.5
0.5
0.9
1.0
1.8
0.88
0.56
0.45 -1.0
1.0
I. 67
0.715 0.0
n
nr 2.45
-1.0
=
I. 53
0.5
2.18 2.94 -1.0
nr = 2.0 0.5
2.0
0.94 -0.2
.'U (")
::r $lJ
rt
r::
Ii
v(O) vn 0
$lJ
0.05
o. II
0.08
0.19
0.24
0.19
0.33
0.36
0.34
0.375 0.42
0.475
0.53
0.516
0.637
0.52
0.54
::l
C'0
Viscosity of Poiseuille Flow
127
of Cl - say Clupper - on the following grounds. We have already shown that the present analysis ~s valid for the flows which exhibit Fahraeus-Lindquist effect. Also we know that this effect is observed in the suspension flows through the small radius tubes (for blood 15-50 ~m diameter), i.e. it is not observed in the flows through the tubes with big radius (for blood flow, diameter >50 ~m). Hence, it is easy to imagine that for a given suspension there exists a critical tube radius aupper such that flow through the tube of radius less than this critical radius will exhibit Fahraeus-Lindquist effect and for the flows through the tubes with radius greater than the critical radius, this effect will not be observed. If there exists a critical tube radius for a given suspension, then there must exist a critical value of Cl - Clupper - because, by definition Cl is the ratio of the tube radius to the material characteristic length and the material characteristic length is fixed for a given suspension. Hence, the value of Clupper corresponds to the critical tube radius for a given suspension (particle size ana concentration). However, it could be interpreted in another way also, i.e. for a fixed tube radius there will be a critical concentration, but in connection with blood which flows through the tubes of different sizes the former interpretation of Clupper is of interest. At this stage it is also important to mention that the present analysis is valid for the suspension flows with Cl ~ Clupper and for the flows with Cl > Clupper Stokes (7) analysis may be used. Further, intuitively one can say that there must exist Cll ower for a given suspension because this analysis will fail when the tube diameter is of the order of one or two particle diameter (for blood microcirculation). But we are unable to determine the value of Cll ower from this analysis, perhaps because of the discontinuity at the point (Cl = 0.0, n = 1.0). At present, the physics associated with the parameter n is not well understood, perhaps it is related with the angular velocity of the red blood cells. CONCLUSIONS An analytic expression (equation (10» for the relative viscosity nr of Poiseuille flow of the fluid with couple stress has been obtained. The variation of relative viscosity with the couple stress parameters Cl and n is shown in Table I. It is observed that while computing the numerical values of nr for small values of Cl (0. I or less) extra care has to be taken, otherwise one might get wrong results. It is shown that nr is a complicated function of Cl and n. Therefore, for quick calculations of nr, two approximate (6 % error) but very simple formulas (equations (II) and (12» have been obtained. The numerical values of nr computed from these approximate formulas are compared with the actual values of nr for various values of Cl and n (Table 2 and Fig. I). It is found that these approximate formulas are quite satisfactory. Further, these approximate formulas might be used for checking the actual computational results fornr. The relative viscosity of couple stress fluid has been computed for one particular case (Cl = 1.35, n = 0.5 which corresponds to 40 % concentration) of Ref. (2). It turns out to be 2.18. For the flows through the tubes of small diamete.r (15-50 ~m diameter), the relative viscosity of blood with 40% RBC concentration lies in the range 1.7 to 2.12 (10). Thus, from the relative viscosity point of view, the results of reference (2) and the other theoretical and the experimental results (8,9) are in reasonably good agreement. However, at this stage it must be mentioned that the theory developed by Valanis and Sun (2) is satisfactory only for the flows through the small diameter tubes (IS-50 ~m diameter). One of the most important conclusions of this analysis is: both the couple stress parameters Cl and n can be determined from the experimental values of velocity profiles and relative viscosity. This is in contrast with the statement in Ref. (2)
[I,R.Y.
!
128
P. Chaturani
"Because the value of II is not known and cannot even be guessed at" (p. 91, (2)). We could have determined the exact values of a and II for the suspension used by Bugliarello et al. (3), had they supplied the value of the relative viscosity along with the velocity profiles. Another important conclusion is: the velocity and relative viscosity are discontinuous at point (a = 0.0, n = 1.0), therefore, their values have to be computed carefully in the neighbourhood of this point. A simple approximation formula (13) has been obtained for the computation of llr in this neighbourhood. At this stage, the physical significance associated with this discontinuity is not clear. It is of interest to note that by this analysis one can de.termine the maximum tube radius 'a'upper for a given suspension (particle size, shape, concentration and relative viscosity given i.e. material characteristic length given). The flow through the tubes with radius 'a' < aupper will exhibit Fahraeus-Lindquist effect. A similar argument could be given about the material characteristic length (particle size and concentration). The important point is a, the ratio of the tube radius to material characteristic length, for the flow under discussion should be less than the. corresponding aupper (Fig. I) then only Fahraeus-Lindquist effect will be observed. The.re will be al ower also, but due to discontinuity at a = 0.0, n = 1.0, we are unable to analyse it. In conclusion, it may be said that this mathematical approach may be useful for predicting the flow profiles in blood vessels of small size (15-50 ~m diameter) which are useful for measuring the volume flow in a blood vessel (10). The information regarding the relative viscosity could be used a pathogenisis tool for predicting many cardiovascular diseases (I). In view of the importance of these theoretical investigations to the diagnosis of cardiovascular diseases, it is hoped that experiments would be conducted in this direction which might lead to the better understanding of the physics associated with the couple stress parameters a and n. Acknowledgement - The author is thankful to Mr. V.S. Upadhya for carrying the computa"tional work of Table 2 anc checking the other results. It is also a pleasure to record the gratefulness of the two Referees for their constructive comments on the earlier version of this paper. Thanks are also due to the Institute authorities for providing the facilities for research. REFERENCES I.
2. 3. 4. 5. 6. 7. 8. 9. 10.
Dintenfass, L. Cardiovasc. Med. 2, 337, 1977. Valanis, K.C. and Sun, C.T. Biorheology 6, 85, 1969. Bugliarello, G., Kapur, C. and Hsiao, G.-Symposium on Biorheology (edited by A.L. Copley) 351, Wiley-Interscience, New York, 1965. Popel, A.S., Regirer. S.A. and Usick, P.I. Biorheology~, 427, 1974. Chaturani, P. Biorheology 13, 243, 1976. Bugliarello, G. and Sevilla,- J. Biorheology 7, 85, 1970. Stokes, V.K. Phys. Fluids 9, 1709, 1966. Cokelet, G.R., Merril, E.W~, Gilliland, E.R., Shin, M., Britten, A. and Wells, R.E. Trans. Soc. Rheol. 7, 303, 1963. Matschek, E. The Viscosity of LIquids. Bell, London, 1928. Woodcock, J.P. Rep. Prog. Phys. 39, 65, 1976.
VISCOSITY OF POISEUILLE FLOW OF A COUPLE STRESS FLUID WITH APPLICATIONS TO BLOOD FLOW P. Chaturani Department of Mathematics, I.T.T. Bombay-400 076, India
(Received in revised form 10 October 1977) (Accepted by Editors-in-Chief) Abstract - Viscosity of Poiseuille flow of a fluid with couple stress has been studied in this paper. The analytic expression of the viscosity of couple stress fluid has been obtained in the form of modified Bessel's functions of the order of zero and one. Since it is a complicated function of the couple stress parameters a and n, the numerical values of the relative viscosity nr have been computed for various values of a and n. These theoretical results are compared with other theoretical and experimental results on blood flow and suspension flows with 40% concentration. For the quick calculation of nr two simple approximate formulas have been obtained. The values of nr obtained from the exact formula and the approximate formulas are compared and it is found that they are in good agreement (within 6% error). The variation of nr (exact and approximate) with a and n is shown graphically which clearly indicates the existance of a discontinuity in the relative viscosity and the velocity at point (a = 0.0 and n = 1.0). The most important conclusions of this analysis are: (i) the conditions for the existance of Fahraeus-Lindquist effect in a tube flow have been obtained; (ii) Up to this date, only velocity profiles have been used to determine the values of a and n (n chosen arbitrarily); here it is shown that by using the experimental velocity profiles and relative viscosity both the couple stress parameters can be determine quite accurately (n no longer chosen arbitrarily). Finally, some biological implications of this theoretical investigation have been indicated. INTRODUCTION It has been observed that the blood viscosity in patients with myocardial infarction was appreciably higher than in normal individuals (1). The viscosity of blood varies greatly in different areas of the circulation and depends on such factors as the internal viscosity of red cells, hematocrit, plasma viscosity and aggregation of blood cells as well as on the lumen of the blood vessel, its diameter and the flow velocity. We shall study the viscosity of the flow of blood through the tubes of small diameter (of the order of 15-50 ~m diameter). 119
P. Chaturani
120
Poiseuille flow of a couple stress fluid through a circular tube has been studied by Valanis and Sun (2). They have attempted to apply the couple stress theory to the flow of blood through narrow capillaries where non-Newtonian behaviour is pronounced. While comparing their theoretical results with the experimental data of Bugliarello et aZ. (3) on blood flow, they find a very good agreement between the two velocity profiles. It is well known that the viscosity of blood can be a significant factor in the pathogenesis of ischemia and infarction and might play an important role in hypertension. It is also an important factor from the fluid engineering point of view. But for some reasons, Valanis and Sun (2) have made no attempt to study the viscosity of couple stress fluid (blood). Recently, some authors (4,5) have attempted to explain various anomalies associated with blood flow through couple stress theory. Because the viscosity of blood is an important factor in pathogenesis of thrombosis and other cardiovascular diseases and the couple stress theory has been used to study the blood flow (2,4), it seems worthwhile to calculate the viscosity of couple stress fluid flow considered by Valanis and Sun (2). The aim of the paper is to study the viscosity behaviour of the couple stress fluid. VISCOSITY CALCULATION The expression for axial velocity V of Poiseuille flow of a fluid with couple stress has been obtained by Valanis and Sun (2). For the sake of completeness and ready reference that expression (equations (3.22) and (3.22a) of Ref. (2) is reproduced here: 1 -
(---E-)
2
a
- II [ 1 - I
(ar) I a
0
0
(a) ] ,
(I)
where 2 (1-11)
li
a2
(1 +11) a
1 -
.E.
,
11
.:1.'
11
11
V11 0
(2)
Il (a) I (a) 0
a 2K 4
(3)
~
]J
where In is a modified Bessel's function of order n~ II the viscosity of solvent, 11 and 11' the couple stress parameters, K the rate of pressure drop in axial direction, a is tube and r is the radial coordinate. Volumetric flow rate Q for a tube is given by
Q = 2n p fa y V dr
(4)
o
where p i.s density. Substitlltion of equations (I) and (2) into equation (4) gives Q
'rT
P a
2
]J
4
k
1
f
o
~
[1-
~2_li
{I-I (aO/I (a)}] 0
where ~ =
ria .
Now the equation (5) may be written 1n the form
0
d~
,
(5)
Viscosity of Poiseuille Flow Q
P a
'IT
2
y
where
1 4
k ]J
H H +---2 a 2I (a) o
a
f
o
121
Y I (y) dy
(6)
o
= a~
On integration
[1.4-2"H
Q
+
H
a
II (a)
-I-J.
(7)
o(a)
Apparent viscosity na for Poiseuille flow is given by
na
'IT
=
k a4
(8)
8Q
which for the present problem takes the following form (9)
]J
1 -
2H
+
Now the relative viscosity
4H
np of the fluid (suspension or blood) can be written as
[ 1-2ll +
The variation of
~H
(10)
I 1 (a) ] I (a) o
np with a and n is shown through Fig.
I.
COUPLE STRESS PARAMETERS DETERMINATION AND VISCOSITY DISCONTINUITY There are two important conclusions which can be drawn from Fig. I. The first this figure and the velocity Profiles (equation (1» the couples stress parameters a and n can be determined quite accurately. It is well known that from the experimental measurements (3) the value of np and V can be determined. Thus for a given relative viscosity* - say np = 31 - we draw a corresponding curve in a-n plane (Fig. 1). Using equation (1), we can also plot the non-dimensional axial velocity V(o)
against a with n as a parameter (Fig. 2). Now for a given axial veloc-
~ -vnV(o). = 0.516 (FLg. 5, of Ref. (2»
ity - say
o
- the line
~(~ )
0.516 is drawn
0
which will intersect many n = constant curves (Fig. 2). These points on intersection individually correspond to a combination of a and n which can give the nondimensional axial velocity = 0.516. A combination of a and n which also lies on the curve np = 81 in Fig. 1. is the proper value of a and n for the given fluid under the given conditions. Thus, it has been shown that the values of a and n can be determined quite accurately. This is in contrast with the statement of Ref. (2) "Because the value of n is not known and cannot even be guessed at". Also, this method has an advantage over Stokes (7) method, where one has to conduct two ex-
* The value of np is not given in Ref. (3). Therefore the value of the viscosity was calculated from the Fig. 15 of the Ref. (6). The computed value of the viscosity is 3.36 centipoise.
P. Chaturani
122
0..8
Exact curve - - 1st approximation . _ . _ . 2nd approximation 0..6
o.A
0..2
-0..2
-o.A
-0.6
-0.8
Fig. 1. periments - one with the flat plate and the other with the cylindrical tube. The second conclusion is: the viscosity is discontinuous at point (a = 0.0 and n = 1.0). It may be seen at a glance from Fig. 1 that all the viscosity curves tend to cluster near the point (a = 0.0, n = 1.0). This shows that the function nr (a,n) has different slopes as we approach the point (0,1). Hence the function nr (a,n) is a discontinuous function of a and n at (0.1). To be more specific, this discontinuity is of irremovable type. Similarly, one can say that the velocity is also a discontinuous function of a and n at this point.
Viscosity of Poiseuille Flow
123
APPROXIMATE EXPRESSIONS OF RELATIVE VISCOSITY
From equations (J), (2) and (10), it is quite clear that V and np are complicated functions of modified Bessel's functions of zero and first order and in turn difficult functions of a and n. It is therefore, going to be quite inconvenient, in particular, for non-professional mathematicians, e.g. physiologists, engineers, etc. to use these results. We shall, therefore, derive two approximate formulae for np and compare it with exact results. First of all, it is assumed that a and n are less than one. With this assumption, the equation (10) reduces to a very simple form np
24(I-n)
=
(II)
a2 7-n)
The values of np calculated from this expression are listed in Table I and are compared with exact values of np (equation (10». It is of interest to note that for a = O. J (concentrated suspensions) the error is only 6%, the error percentage increases with a and n. Looking at the simplicity of the expression (I I) and at the percentage error, it can be said that the equation (I I) is a good approximate formula for calculating np for small values of n and a (small in comparison to one). This formula (equation (II» we shall call the first approximation of np' As expected, this equation gives incorrect np if both n and a are approximately one, and the error is ca. 30-40 % if one of them is approximately one (Table 2). However, if we retain one more term in the numerator of equation (10), then the expression for nr takes the following form:
24(I-n)
+
3 a 2 (3 - n)
(12)
a 2 (7 - n)
which we shall call the second approximation of np. The numerical values of np (for a = 1.0) computed from equations (10), (II) and (12) are shown in Table 2. The improvement in the values of np is significant: in the first approximation when one "7 = I "7 = 0.8 "7 = 0.5 "7 =0.1 "7 = 0.5 "7 = 1.0
·0" "
~o
0.516 0.5
o
~~~--~------~------~
0.45
2
a
Fig. 2.
3
124
P. Chaturani Comparison of first approximation and actual relative viscosities
TABLE I.
n = 0.0
n = -1.0 Actual
nr O. I
0.5 1.0
Approximate
578.7
600.0
25.03
nr
343.0
14.72
6.000
TABLE 2.
nr
336.5
24.00
7.034
Approximate
Actual
nr
n = 0.5
4.438
n
nr
Approximate
183.4
195
Actual
=
Actual
nr
nr
42.43
0.9 Approximate
nr 39.3
13.72
8.335
7.70
2.574
1.572
3.43
3.849
1.95
I. 393
0.393
First and second approximation and actual relative viscosities for a = 1.0
n
First approximation
Second approximation
Actual
0.9
0.393
I. 42
1.393
0.5
I. 95
3.00
2.849
0.0
3.43
4.70
4.438
-1.0
6.00
7.50
7.034
of the parameters is approximately one the error was oa. 30-40%, this in the second approximation reduces to 5-6%; when both a and n are approximately one, then in the first approximation the value of nr was incorrect, but in the second approximation they are within 3% error (Table 2) which is a remarkable improvement. At this stage, it is necessary to mention that while calculating the relative viscosity and/or velocity in the neighbourhood of the point (a = 0.0, n = 1.0) one should be extra careful because there is a discontinuity in the viscosity and velocity at this point. The relative viscosity and velocity in the neighbourhood of this point can be given by the following approximate formulae.
+~
nr Vn V0
E:
(I
-
(13)
r2
-) 0: 2
E: E:
+ 48
where E:
= a 2,
8
I - n
DISCUSSION Valanis and Sun (2) have compared their theoretically predicted with the experimentally obtained velocity profiles of Bugliarello shown that the two results are in good agreement. While comparing and the experimental velocity profiles, they find that blood ~ith of REC could be represented by a couple Stress fluid with n = 0.5
velocity profiles et aZ. (3) and the theoretical 40% concentration and a = 1.35 (n
Viscosity of Poiseuille Flow
125
is arbitrarily chosen and a determined). For some reasons, they have not calculated the relative voscosity for couple stress fluid. In our opinion, the computation of the relative viscosity is quite important, because it is required along with the velocity profiles to determine the two couple stress parameters a and 11 and also it can be used as a predictive tool in hypertensive and cardiovascular diseases (I). The relative viscosity of couple stress fluid with 11 = 0.5 and a = 1.35 turns out to be 2. IS (using equation (10». This value of 11p appears to be very low for a suspension with 40% concentration (Table 3a). Therefore, an attempt was made, using Fig. I, to obtain the same velocity profiles with higher viscosity (Table 4). It was found that the relative viscosity in the range of 3 to 10 cannot give the required velosity profiles. However, a marginal increase in 11r~ i.e. 11r = 2.5, is possible (Table 4). But this reduction of 11r could be justified by another interpretation which seems to be quite reasonable. From the Table 3(a), it may be observed that the value of 11p obtained for a = 1.35, 11 = 0.5 is almost the half of the other values of 11p' This 50% reduction in the relative viscosity is a characteristic feature of the blood flow through the tubes with small diameter (15-50 ~m diameter) which is known as Fahraeus-Lindquist effect (10). This is due to the existence of a cell-free zone (2-5 ~m wide) next to the tube wall. It is of interest to note that Valanis and Sun (2) have derived their boundary conditions (equation (3.20) of Ref. (2)) by using this idea, i.e. 'existence of a cell-free zone next to tube wall' (p. 8S/2nd para. (2)). Hence, it appears that the analysis of Valanis and Sun (2) is valid for the flows through the small tubes which exhibit Fahraeus-Lindquist effect. It is, therefore, improper to compare the present 11r with 11p of the flows in which the Fahraeus-Lindquist effect is absent. In view of this argument, the reason for the disagreement is obvious. However, if we compute the other relative viscosities of Table 3(a) for the flows with the Fahraeus-Lindquist effect and then compare with the present one, the agreement between th~ two is satisfactory (Table 3(b)). TABLE 3(a).
Comparison of relative viscosities In absence of Fahraeus-Lindquist effect
Investigator
Concentration
Relative viscosity
Cocklet (8)
40 %
4.0
2
Hatschek (9)
40 %
4.0
3
Valanis and Sun (2)
40 %
2.18
4
Woodcock (10)
40 %
5.5
No.
TABLE 3(b).
Comparison of relative viscosltles of flows with Fahraeus-Lindquist effect
Investigator
Concentration
Relative viscosity
Cocklet (8)
40 %
2.0
2
Hatschek (9)
40 %
2.0
3
Valanis and Sun (2)
40 %
2.18
Woodcock (10)
40 %
2.64
No.
4
From Fig. I, it can be seen that for a suspension with given 11r~ a cannot be assigned any arbitrary value between zero to affinity. As a matter of fact, for a given 11p the values of a have a certain upper limit. e.g. 11r = 10.0, 0 < a ~ 0.705; 11r = 5.0, 0 < a ~ 1.2 etc. (Fig. I). One can explain the existence of the upper limit
N 0\
TABLE 4.
Variation of relative viscosity and acial velocity with a and
nr = 10.0 a
=
o. I
0.5
n
=
0.92
0.4
0.705 -1.0
nr = 5.0
nr = 3.0
0.5
1.0
1.2
0.7
-0.3
-1.0
nr = 2.5
0.5
0.9
1.0
1.8
0.88
0.56
0.45 -1.0
1.0
I. 67
0.715 0.0
n
nr 2.45
-1.0
=
I. 53
0.5
2.18 2.94 -1.0
nr = 2.0 0.5
2.0
0.94 -0.2
.'U (")
::r $lJ
rt
r::
Ii
v(O) vn 0
$lJ
0.05
o. II
0.08
0.19
0.24
0.19
0.33
0.36
0.34
0.375 0.42
0.475
0.53
0.516
0.637
0.52
0.54
::l
C'0
Viscosity of Poiseuille Flow
127
of Cl - say Clupper - on the following grounds. We have already shown that the present analysis ~s valid for the flows which exhibit Fahraeus-Lindquist effect. Also we know that this effect is observed in the suspension flows through the small radius tubes (for blood 15-50 ~m diameter), i.e. it is not observed in the flows through the tubes with big radius (for blood flow, diameter >50 ~m). Hence, it is easy to imagine that for a given suspension there exists a critical tube radius aupper such that flow through the tube of radius less than this critical radius will exhibit Fahraeus-Lindquist effect and for the flows through the tubes with radius greater than the critical radius, this effect will not be observed. If there exists a critical tube radius for a given suspension, then there must exist a critical value of Cl - Clupper - because, by definition Cl is the ratio of the tube radius to the material characteristic length and the material characteristic length is fixed for a given suspension. Hence, the value of Clupper corresponds to the critical tube radius for a given suspension (particle size ana concentration). However, it could be interpreted in another way also, i.e. for a fixed tube radius there will be a critical concentration, but in connection with blood which flows through the tubes of different sizes the former interpretation of Clupper is of interest. At this stage it is also important to mention that the present analysis is valid for the suspension flows with Cl ~ Clupper and for the flows with Cl > Clupper Stokes (7) analysis may be used. Further, intuitively one can say that there must exist Cll ower for a given suspension because this analysis will fail when the tube diameter is of the order of one or two particle diameter (for blood microcirculation). But we are unable to determine the value of Cll ower from this analysis, perhaps because of the discontinuity at the point (Cl = 0.0, n = 1.0). At present, the physics associated with the parameter n is not well understood, perhaps it is related with the angular velocity of the red blood cells. CONCLUSIONS An analytic expression (equation (10» for the relative viscosity nr of Poiseuille flow of the fluid with couple stress has been obtained. The variation of relative viscosity with the couple stress parameters Cl and n is shown in Table I. It is observed that while computing the numerical values of nr for small values of Cl (0. I or less) extra care has to be taken, otherwise one might get wrong results. It is shown that nr is a complicated function of Cl and n. Therefore, for quick calculations of nr, two approximate (6 % error) but very simple formulas (equations (II) and (12» have been obtained. The numerical values of nr computed from these approximate formulas are compared with the actual values of nr for various values of Cl and n (Table 2 and Fig. I). It is found that these approximate formulas are quite satisfactory. Further, these approximate formulas might be used for checking the actual computational results fornr. The relative viscosity of couple stress fluid has been computed for one particular case (Cl = 1.35, n = 0.5 which corresponds to 40 % concentration) of Ref. (2). It turns out to be 2.18. For the flows through the tubes of small diamete.r (15-50 ~m diameter), the relative viscosity of blood with 40% RBC concentration lies in the range 1.7 to 2.12 (10). Thus, from the relative viscosity point of view, the results of reference (2) and the other theoretical and the experimental results (8,9) are in reasonably good agreement. However, at this stage it must be mentioned that the theory developed by Valanis and Sun (2) is satisfactory only for the flows through the small diameter tubes (IS-50 ~m diameter). One of the most important conclusions of this analysis is: both the couple stress parameters Cl and n can be determined from the experimental values of velocity profiles and relative viscosity. This is in contrast with the statement in Ref. (2)
[I,R.Y.
!
128
P. Chaturani
"Because the value of II is not known and cannot even be guessed at" (p. 91, (2)). We could have determined the exact values of a and II for the suspension used by Bugliarello et al. (3), had they supplied the value of the relative viscosity along with the velocity profiles. Another important conclusion is: the velocity and relative viscosity are discontinuous at point (a = 0.0, n = 1.0), therefore, their values have to be computed carefully in the neighbourhood of this point. A simple approximation formula (13) has been obtained for the computation of llr in this neighbourhood. At this stage, the physical significance associated with this discontinuity is not clear. It is of interest to note that by this analysis one can de.termine the maximum tube radius 'a'upper for a given suspension (particle size, shape, concentration and relative viscosity given i.e. material characteristic length given). The flow through the tubes with radius 'a' < aupper will exhibit Fahraeus-Lindquist effect. A similar argument could be given about the material characteristic length (particle size and concentration). The important point is a, the ratio of the tube radius to material characteristic length, for the flow under discussion should be less than the. corresponding aupper (Fig. I) then only Fahraeus-Lindquist effect will be observed. The.re will be al ower also, but due to discontinuity at a = 0.0, n = 1.0, we are unable to analyse it. In conclusion, it may be said that this mathematical approach may be useful for predicting the flow profiles in blood vessels of small size (15-50 ~m diameter) which are useful for measuring the volume flow in a blood vessel (10). The information regarding the relative viscosity could be used a pathogenisis tool for predicting many cardiovascular diseases (I). In view of the importance of these theoretical investigations to the diagnosis of cardiovascular diseases, it is hoped that experiments would be conducted in this direction which might lead to the better understanding of the physics associated with the couple stress parameters a and n. Acknowledgement - The author is thankful to Mr. V.S. Upadhya for carrying the computa"tional work of Table 2 anc checking the other results. It is also a pleasure to record the gratefulness of the two Referees for their constructive comments on the earlier version of this paper. Thanks are also due to the Institute authorities for providing the facilities for research. REFERENCES I.
2. 3. 4. 5. 6. 7. 8. 9. 10.
Dintenfass, L. Cardiovasc. Med. 2, 337, 1977. Valanis, K.C. and Sun, C.T. Biorheology 6, 85, 1969. Bugliarello, G., Kapur, C. and Hsiao, G.-Symposium on Biorheology (edited by A.L. Copley) 351, Wiley-Interscience, New York, 1965. Popel, A.S., Regirer. S.A. and Usick, P.I. Biorheology~, 427, 1974. Chaturani, P. Biorheology 13, 243, 1976. Bugliarello, G. and Sevilla,- J. Biorheology 7, 85, 1970. Stokes, V.K. Phys. Fluids 9, 1709, 1966. Cokelet, G.R., Merril, E.W~, Gilliland, E.R., Shin, M., Britten, A. and Wells, R.E. Trans. Soc. Rheol. 7, 303, 1963. Matschek, E. The Viscosity of LIquids. Bell, London, 1928. Woodcock, J.P. Rep. Prog. Phys. 39, 65, 1976.
