int. j. engng sci. vol. 34, no. 1, pp. 87-99, 1996...

Pergamon 0020-7225(95)00077-1

Int. J. Engng Sci. Vol. 34, No. 1, pp. 87-99, 1996 Copyright (~ 1996 Elsevier Science Ltd

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F L O W O F E R I N G E N F L U I D ( S I M P L E M I C R O F L U I D ) T H R O U G H A N A R T E R Y W I T H M I L D S T E N O S I S

D. PHILIPt and PEEYUSH CHANDRA~z Department of Mathematics, Indian Institute of Technology, Kanpur, Kanpur-208 016, India

Abstract--This paper presents an analysis for blood flow through a stenosed artery considering the localized effects of the stenosis. The blood vessel is modelled as a rigid tube and the particulate nature of blood is accounted through an Eringen fluid (simple microfluid) model. Further a peripheral layer of Newtonian fluid has been considered in the model to account for the cell free layer. It is observed that the resistance to the flow as well as the wall shear stress increase as the height of the stenosis increases. However, the trend is reversed as the peripheral layer thickness increases.

1. I N T R O D U C T I O N

It is now well known that flow through a stenosed artery can cause serious circulatory disorders. This led to extensive investigations of the characteristics of blood flow through a stenosed artery in recent years. Young [1] was amongst the first ones who attempted to theoretically analyse the interaction of mild stenosis with fluid dynamics of blood flow. Similar problems were subsequently investigated by many other researchers: Forrester and Young [2], Lee and Fung [3], Morgan and Young [4], Shukla et aL [5], Chaturani and Samy [6], etc. However, in all of these studies blood has been characterized as a Newtonian fluid and little attention has been given to its suspension nature. Experimental studies indicate that under certain flow conditions blood exhibits a non-Newtonian behaviour, e.g. a non-parabolic velocity profile and existence of a peripheral layer have been observed for the flow through tubes of small diameter: Bugliarello and Sevilla [7], Cokelet [8], Goldsmith and Skalak [9]. This non-Newtonian behaviour is attributed to the particulate nature of blood: Charm et al. [10], Whitmore [11]. Thus attempts have been made to rheologically describe blood as a suspension of neutrally buoyant deformable particles in viscous fluid using the microcontinuum approach: Cowin [12], Popel et al. [13], Ariman et al. [14, 15], Kang and Eringen [16].

In view of this, Sinha and Singh [17], Srivastava [18] considered a couple stress fluid model for blood flow through stenosed tube to account for the size effects of the suspended particles. Parvathama and Devanathan [19] used a micropolar fluid model to account for particles' spin, while Pralhad and Shultz [20] discussed blood flow through a stenosed artery for different diseases by considering a polar fluid model for blood. However, the effect of cell deformability, which has a pronounced effect on the apparent viscosity of blood, was not considered in these models. Eringen [21, 22] introduced the theory of simple microfluids, also called Eringen fluids, which accounts for the microspin as well as microstretch of the particles in a suspension. Kang and Eringen [16] successfully applied this model to study rheological properties of blood. Therefore, in this paper an analysis is presented for blood flow through a narrow artery with mild stenosis by considering Eringen's simple microfluid model for blood. The model presented here considers only the localized effects of the stenosis and accounts for cell free layer of blood plasma near the tube wall: Bugliarello and Sevilla [7], Shukla et al. [5]. The flow is assumed to be steady, as the pulsatile behaviour of blood flow is not very significant in narrow arteries.

t Present address: AG Technomathematik, Postfatch 3049, Universitat Kaiserslautern, 67653 Kaiserslautern, Germany. To whom all correspondence should be addressed.

87

88 D. PHILIP and P. CHANDRA

Pi

"1 L

Fig. 1. Geometry of arterial stenosis with peripheral layer.

2. F O R M U L A T I O N

In this study, a two-layered model is considered for blood flow in stenosed artery. The model consists of a central layer of Eringen fluid (simple microfluid) in the core region and a peripheral layer of Newtonian fluid (plasma) near the wall of the tube. The configuration of stenosis generally does not have a well defined geometrical shape. As an approximation it is assumed to be axially symmetric in this paper. The radius R (z) of the stenosed tube (Fig. 1) is given as follows (Young [1]):

R(z) = R o - ~ - l + c o s ~ 0 z - d - d < - z < - L o + d .

R0 otherwise

(1)

Here, R0 is the constant radius of artery in non-stenosed portion, Lo is the length of the stenosis and 6~ is the maximum height of the stenosis.

It is further assumed that: (i) Fluid in the peripheral layer is Newtonian and incompressible whereas the Eringen

fluid in the core region is incompressible as well as micro-incompressible. (ii) Body forces and body moments are absent.

(iii) Flow, which is due to the pressure gradient d p / d z , is one-dimensional and axi-symmetric.

(iv) Stenosis is very mild, which implies that the variation of all the flow characteristics (barring pressure) along the axial direction is negligible.

(v) The length of the tube is large compared to its radius (a subsequent order of magnitude analysis thus implies that the inertial terms can be neglected).

(vi) The peripheral layer is of constant thickness, & Thus the core region is given by 0 <-- r <- R1(z) = R ( z ) - ~ and peripheral region is given by R l ( z ) <- r <- R ( z ) .

Using the above assumption, the equations governing the steady laminar flow in the two regions are given as follows:

Eringen fluid flow through an artery 89

(a) Core region (0 <-r <- R,(z))

As already mentioned the blood is assumed to be an Eringen fluid which flows due to the pressure gradient dp/dz. The field quantities are, velocity vector, ~ = (0, 0, vc), gyration tensor components v~3 and v3~ (other components being zero) and pressure, p. Thus the equation of balance of linear momentum gives, Kang and Eringen [16].

_0 1 dp (t13) + - t,3 = ~ P (2)

Or r dz

while the equations due to balance of first moment stress reduce to

_ 1 0 ( a 1 1 3 ) q- _ ( a l l 3 __ /~223) q- t31 - - $31 ~-~ 0 (3)

Or r

0 1 Or (A131) + -r (A13, - a232 ) q- t13 - $13 = 0 ( 4 )

where tzj, s o, Aijk are given by the constitutive equations of Eringen fluid. The non-zero components can be written as,

t13 = 2/~d~3 + El vo3) + k(v[t31 - - o)13)

t31 = 2/zd31 + ffl v(31) + k(v[311 - 0)31)

S13 = 2/zd13 + 2ff2V(13)

$31 ~-- 2/zd31 + 2~'2V(31)

1 0 ~ 0 A,,3 =~(K1 + K3) orV31 + (K2 + K4) o-Tv,3

A223 = [(K1 + Ks)v31 + (K2 + K4)v13]/2r

1 0v31 1 0V13 A131 = ~ (K, - K3) a---r- + 2 (K2 - K4) a---r-

a232 = [ ( g 1 - g 3 ) v 3 1 n t- ( g 2 - g4)v13]/2r

l ( a v ~ ) = d 3 1 ' 0)13---= 1 0 1 3 c d13 = ~ \ Or / 2 Or - - O)31 )

V13"~ V31 VI3-- V31 VO3)-- 2 -- V(31) and V [ ] 3 l - - 2 - v[131

where/x, ffl, ~2, k, are the viscosity coefficients and have the dimensions of viscosity/z; while Ki's are functions of the viscosity coefficients "Yl to '~15 which have dimensions o f / z L 2, Kang and Eringen [16]. These coefficients satisfy the following restrictions,

/z >--O, 2 ( 2 - ~ >-k >-0, 4/.L (~.2 1 2 - ~ ~'~) - (~' , /2) 2 --- 0

'yl "4- y 2 "-~" • • • -'~- Y15 ~ 0 , "f2 -1-- y l 1 "-~- 'Y14 ~ 0 .

Here v03 ) and vf131 give the symmetric and skew symmetric parts of the v]3. The symmetric part, v(13), represents local stretching and shearing, while the skew symmetric part, V113] , gives local intrinsic angular velocity of the substructure.


(b) Peripheral region (RI(z) <- r <- R(z))

The equation for the Newtonian fluid (plasma) flow with fi = (0, 0, Vp) and constant viscosity /z under pressure gradient dr~/dz is,

I-11 0 (rO_~]] = P .

/ZLr~rr\ 0 r / J (5)

The boundary conditions

(i) No slip condition gives,

Vp=0 at /~=R(z) . (6)

(ii) Symmetry of the flow implies that,

Ovc =0, voa )=0, vp31=0 at r = 0 . (7)

Or

(iii) Continuity of velocity at the interface implies,

vc(R1) = vp(R,). (8)

(iv) Continuity of shear stress at the interface gives

C)pp ~ r r=R, =t'3l~=&" (9)

(v) The determination of proper boundary conditions for vckt) and v[ktl at the interface is rather difficult. In view of this, the conditions for the microstretching and microspin as postulated by Kang and Eringen [16] at the tube wall, are assumed in the present study at the interface. Thus, at r = Rl(z),

Ore OV c v¢13) = A1 Or and V[13l = A2 Or (10)

where A~ and A2 are constants.

3. ANALYSIS

Integrating equation (2) alongwith the boundary condition (7) at r = 0, one obtains,

= ( +_k] 0vc+ rP t13 ~ 2/ Or ~ l V ( 1 3 ) + k v [ 1 3 l = 2 - (11)

which gives the interface condition (9) for stress in the following form:

OVp R1P (12) ~'L~'~- r r=RI = t13 I r = R , - 2

Equation (5) along with the boundary conditions (6) and (12) gives,

- -~-P(R2-r 2) Rl<-r<-R. (13) Up =


On substituting for AI,3, A223, t31, $31, )t131, A232, t13 and s,3 in the equat ions (3) and (4), one obtains, Kang and Eringen [16], Chandra and Agarwal [23],

K , [ V 2 - a2]v31 + K2[V 2 - 0~2]v13 = 0 (14)

krP K3[V 2 - 0~2]v31 -1- Ka[V 2 - 0/42]v13 - - - (15)

(2/~ + k)

where,

K, a 2 = -(st , - 2sr2) = Kza22,

Knot,] = k(2/z - ~',)/(2/z + k)

K3a 2 = -k (2 / z + ~,)/(2/z + k)

0 2 1 0 1 and V 2 = -

- - O r 2 + r Or r 2"

On solving equat ions boundary conditions, (velocity of Er ingen fluid in the core region) are obtained as, Kang and Eringen [16],

(11), (14) and (15) for v¢, Vl3 and v3, along with the appropriate the expressions for vo3)(microstretching), vt~3l(microspin), and v~

where,

- b i e i di a i l l ( a iR l )

b, = [(1 + 2A2)g2 - 2A,h2]/[glh2 -g2hl]

be = [(1 + 2Ae)g, - 2A,h,]/[g, hz - geh,]

(E, - E3) + (El + E3)f (a 2 - ff2)K, g4 + (a 2 - ff2)g2K3 e i - - , fii-- (2 + E3) g l g 3 ( f f 2 _ ff2)

2 ' 2

a~ = o~iRo, a, = a = otRo, a2 = ~ = ¢~Ro, Re = K , / ( ~ R g )

E, = - L'., E2 = 'g~', E3 =--'k , f = r/Ro; k , = k ( z ) - L g = ~/Ro tx tx tx

R = R ( z ) / R o = I - ~ - l + C O S L o e - d - , g ~ = 6 J R 0 .

cl = d /L , £o = Lo/L, g= z /L , Io(x) and I,(x) are modified Bessel functions of order 0 and 1 respectively.

a2 and/32 are obta ined using equations (14) and (15) and are given as the positive roots of the quadrat ic equat ion

x2(KzK3 - K1Ka) + x[ (a 2 + a~)K2K3 - (a2 + ce42)K, Ka] + a2a~K2K3 _ oz20~2K, K4 = 0. (19)

It may be remarked here that 6 = 0, i.e. R~(z) = R(z ) yields a case of no peripheral layer, i.e. the blood is model led as a single layer of Eringen fluid, while R~(z) = 0 gives no core region of Er ingen fluid., i.e. the blood behaves as a Newtonian fluid.

PRo [b,(1 + f~)ll(ffr-) + b2(1 + j~)I,(/3r-)] (16) v(,3) = - 4--~-

v031_ ~ o [b,(1 - . / ] ) / , ( f ro + b2(1 -f2)l,(/3F) - Y] (17)

PR~ PR~R, vc = 4/z (~2 _/~2) 4/z [d,{Io(~'k,) - Io(ff~-)} + d2{Io(/3/~,) - Io(/3~-)}] (18)


The flow flux Q, which is defined as,

(R~(z) (R(~) Q =2~Jo rv~ dr + 2~JR,(~) rv,, dr

can be obtained in the following form using equations (13) and (18),

/rR 4 F[~(~), &(~)]

Q 8/z oz

where,

(2 + E3) d, Io(~/~,)

The equation (20) along with the conditions

p = p ~ at z = 0 and

gives pressure drop across the tube as,

P = P o at z = L ,

(20)

21,(/3R,)~] JJ"

81xQL (~ d f P o - P , = 7rR 4 Jo F[/~(~-),/~(~-)] " (21)

It may be noted that in the regions 0<-z - d , and, d + Lo<-z <-L, /~(~-) = 1 a n d / ~ ( f ) = 1 - 8, which imply F[/~(~-), /~1(z-)] is constant, F* (say).

Thus,

8p, QL [ 1 - f~o + [ 3+& dz ] P o - P i - JrR~ F* J,i F[/~(~-),)~,(£)I J (22)

The peripheral resistance A, is given in the non-dimensional form as, Young [1],

A = [1 ' d4~ A=Ao - F?°+£ (23) - - F[/~(~b), ~1(6)1 ]

where, A = (pi - p o ) / Q , Ao = 81a, L/(zR4); 4' = ½ - (z - d - Lo/Z)/Lo. The shearing stress at the wall vw in non-dimensional form is given by

xRg k(e) Zw = 4/~Q Vw = F[/~(~') , /~(f)] " (24)

4. R E S U L T S AND D I S C U S S I O N

The resistance to the flow, h and the wall shear stress, rw are the two important characteristics in the study of blood flow through stenosed artery. The non-dimensional form of these characteristics are given in equations (23) and (24) respectively. The integral appearing in equation (23) is not amenable to analytical solution and so it is integrated numerically using Simpson's rule. Polynomial approximations are taken for the Bessel functions lo(x), I~(x).

Apart from the usual dimensionless parameters, i.e. t]s (the maximum height of the stenosis), L 0 (the length of the stenosis) and 8 (the peripheral layer thickness), one encounters certain additional Eringen fluid parameters (El, E2, E3, K1, K2, K3, K4) in the expressions for ,~ and rw. The parameters E~, E2 and E3 (non-dimensional forms of ~,, ~2, and k respectively) correspond to the deformability and rigidity of the particles, e.g. for a given value of/~, higher values of k represent more rigid structure, while higher values of 1~[ and ]~21 represent more flexible structure. However, the precise relationships of these parameters with deformability or rigidity are not known. Further, these constitutive coefficients not only depend upon the initial concentration of the particles but as well as on the particle shape, deformability, and the fluid

E r i n g e n fluid f low th rough an a r t e ry 93

1.60 -- E 3 = 0.1

= 0 . 0 5 A l [ 0.75 1.50 - _~0 = ~ ,~ o E~ ~ 0.2 _

1.40

1.20

1.1o

1.00 I [ I I f 0.04 0.06 0.08 0. I 0 O. 12

~Ss/R 0 *

Fig. 2. V a r i a t i o n of A wi th ~s/Ro for d i f fe ren t LolL, E I and E 2.

viscosity; again their explicit dependence is not known. In view of this, the flow characteristics have been studied with respect to independent variation of these parameters. Further, the parameters K1, K2, K 3 and g 4 a r e the linear combinations of the viscosity coefficients yl to Y~5. In the present study the non-dimensional parameters Ki (=Ki/I~L 2) for a narrow artery have

1,6

1.5

1.4

1.3

1.2 T 0.04

E I = 0.3

E 2 = 0.3

E3 = 0 . 1 o E 3 = 0.2 ra E3 = 0.3 zx~ = 0 . 0 5

f ~35

I I I 0.06 0.08 0.10

5siR 0 t.

Fig. 3. V a r i a t i o n of A wi th 8JRo for d i f ferent Lo/L and E3.

I 0.12

94 D. PHILIP and P. C H A N D R A

1.3

I ¢~ 1.2

1.1

1.5 - - Lo/L = 0.5

= 0.05

1.4 E2 = 0.3 - - 0 E 3 = 0 . 1 ~ . . . . . . . . . . . . ~ y . ' ' ' - -

a E ~ = 0 . 2 ~ - " - - . . . . , E : : 0 .9 .

1.0

0.9 I I I I 0.04 0.06 0.08 0.10 0.12

8s/R 0 b

Fig. 4. Variation of ~. with ,SJRo for different E~ and E>

been chosen as R1 = 1.1 / ~ 2 = 1.0, /~3 = - 0 . 9 , I ~ 4 = 0.6, Kang and Er ingen [16] and results are e l abo ra t ed th rough Figs 2 -11 and Tab les 1 and 2.

T h e resul ts for zero microsp in and zero micros t re tch at the interface, i.e. A1 = A2 = 0 are p re sen ted in Tab les 1 and 2 whereas Figs 2-11 depict results for A~ = A2 = 0.25. It is ev ident f rom Tab les 1 and 2 tha t bo th X and rw (calculated at the m a x i m u m height of the stenosis)

1 . 2 4

1.20

1.16

1.12

1 .08

1.04 0.04

Lo/L = 0.5

E I = 0.6

E 2 = 0.3 J

I I I I 0.06 0.08 0.10 0.12

6s/R o

Fig. 5. Variation of A with 6JRo for different ~ (peripheral layer width).


1.55

1.45

E I = 0.3

E 3 = 0.1

o E 2 = 0.3

t, E 2 = 0.4

~. 0.5

1.25

1.15 ~ ' ' ' - l I I I 0.04 0,06 0.08 0.10 0.12

8 / R 0

Fig. 6. Variation of h with ~JRo for different E2, LolL and 6.

increase as the stenosis height ~s increases, but decrease as the peripheral layer thickness increases. Similar behaviour was observed by Shukla et al. [5] who considered the fluid to be Newtonian in both the regions but with different viscosities. The resistance to the flow gets enhanced as E] and E3 increase and E2 decreases.

Figures 2-4 show the variation of X vs ~s when ~ (peripheral layer thickness)= 0.05. It is

1.4

£S 34:14

1.3

T 1.2

I.I

1.0 0.04

E 2 = 0.40 ~b.9

_ _E 3 = 0.30 ~ ~

L 0 = 0.75 5 o E o = 0 . 5 0 % ~ . 0 ~

• L0 = 0.05 ~ E , ~" 0"~. - - ' ~

•

~ .05 ~

I I I I 0.06 0.08 0.10 0.12

8siR 0

Fig. 7. Variation of X with 8 J R o for different E I , Lo /L and 6.


i,,.,~

1.34

1.30

1.26

1.22

1.18

1.14

1.10

E I = 0 . 6

E 2 = 0 . 3

/x L 0 = 0 . 7 5

o t ' o = 0 . 5 0

..

J %

1.06 ] 0 .04 0 .06 0.08 0 .10 0.12

8 / R o ),

F i g . 8. V a r i a t i o n o f ~, with 65/R o for different LolL, 6 a n d E s,

observed that in the case of Eringen fluid also, the flow resistance becomes larger as 6s (height of the stenosis) or Lo (length of the stenosis) increases.

Figure 5 shows the effect of the increase in peripheral layer thickness for a given Eringen fluid. It is clear that increase in peripheral layer thickness decreases A. Further A takes higher

1.6 - -

1.5

1.4 -

1.3

1.2

1.1

L o l L = 0 . 5

= 0 . 0 5

E3 = 0 . 1

o E 2 = 0 . 2

A E 2 = 0 . 3

[] E 2 = 0 . 4 E I = 0 . 6

E l = 0 . 3

1.0

0.9 i 0.04

I 1 I 0,06 0.08 0 .10

8 s / R 0 ~-

F i g . 9. V a r i a t i o n o f ~w with 8 s i r o for different E 1 and E z.

I 0.12


1.9

1.7

V L0/L = 0.5

= 0.05 E 2 = 0.3

o E3 = 0.1

A E 3 = 0 .2

l 1.5

1.3

1.1

E l = 0.3

0.9 J 0.04 0.06 0.08 0. I 0 0.12

8s/R o

Fig. 10. Variation of rw with ~JR o for different E I and E 3.

values when the tube is filled with Eringen fluid only in comparison to the case of Newtonian fluid. Thus, a peripheral layer facilitates the blood flow in diseased cases.

The effect of the Eringen fluid parameters E~ and E2 on X is found to be opposite to each other. It is clear from Figs 2, 6 and 7 that an increase in E~ enhances the resistance to the flow while that in E2 reduces it. Unlike the case of A1 = A2 = 0 where the effects of E1 and E2 are

T

1.6

1.5

1.4

1.3

1.2

E I = 0 .6

E 2 = 0.3

E 3 = 0.1

%

I.I 0.04 0.06 0.08 0.10 0.12

6s/R o D

Fig. 11. Variation of fw with 8s/Ro for different ~ (peripheral layer width).

98 D. PHILIP and P. C H A N D R A

Table 1. Peripherallayerthickness(~)= 0.1

6~ El E2 E3 A rw

1 0.04 0.5 0.3 0.07 1.1386791 1.2284889 2 0.04 0.5 0.3 0.(/9 1.1704005 1.2609274 3 0.04 0.5 0.5 0.07 1.386752 1.2284844 4 0.04 0.5 0.5 0.09 1.1703953 1.2609215 5 0.04 1.0 0.3 0.07 1.1388472 1.2286735 6 0.04 1.0 0.3 0.09 1.1706287 1.2611773 7 0.04 1.0 0.5 0.07 1.1388398 1.2286652 8 0.04 1.0 (1.5 0.09 1.706185 1.2611660 9 0.08 0.5 0.3 0.07 1.1917793 1.3878380

10 0.08 0.5 0.3 0.09 1.2242435 1.4217867 11 0.08 0.5 0.5 0.07 1.1917751 1.3878330 12 0.08 0.5 0.5 0.09 1.2242379 1.4217803 13 0.08 1.0 0.3 0.07 1.1919561 1.3880483 14 0.08 1.0 0.3 0.09 1.2244831 1.4220702 15 0.08 1.0 0.5 0.07 1.1919482 1.3880389 16 0.08 1.0 0.5 0.09 1.2244724 1.4220572

Table 2. Peripherallayerthickness(6)= 0.2

~ El E2 E~ ~ rw

1 0.04 0.5 0.3 0.07 1.0936822 1.1811299 2 0.04 0.5 0.3 0.09 1.1095525 1.1970510 3 0.04 0.5 0.5 0.07 1.0936797 1.1811270 4 0.04 0.5 0.5 0.09 1.1095492 1.1970474 5 0.04 1.0 0.3 0.07 1.0937872 1.1812405 6 0.04 1.0 0.3 0.09 1.1096913 1.1971972 7 0.04 1.0 0.3 0.07 1.0937823 1.1812353 8 0.04 1.0 0.5 0.09 1.1096849 1.1971902 9 0.08 0.5 0.3 0.07 1.1451915 1.3362423

10 0.08 0.5 0.3 0.09 1.1613095 1.3524337 11 0.08 0.5 0.5 0.07 1.1451889 1.3362394 12 0.08 0.5 0.5 0.09 1.1613060 1.3524296 13 0.08 1.0 0.3 0.07 1.1453001 1.3363623 14 0.08 1.0 0.3 0.(/9 1.1614529 1.3525913 15 0.08 1.0 0.5 0.07 1.1452950 1.3363567 16 0.08 1.0 0.5 0.09 1.1614462 1.3525839

not significant, ~ shows appreciable variations with El and E2 when Am = A2 = 0.25 (Figs 6 and 7). The effect of the parameter E3, in this case, is to reduce ~, as E3 varies between 0.1 and 0.2, however X increases for higher values of E3 (Figs 3 and 4). The combined effects of/20 and 6~ with the Eringen fluid parameters are further elaborated in Figs 6-8.

In Figs 9-11 the shear stress rw at the maximum height of the stenosis, is plotted for various values of El, E2 and E3. The qualitative effect of these parameters on ~,~ is similar to that observed in the case of X. It is noted from Fig. 11 that ~,~ increases as ~ increases and this effect is further enhanced in the case of Eringen fluid.

5. C O N C L U S I O N

The present study investigates the flow of blood, which has been modelled by a simple microfluid in the core region with a Newtonian fluid peripheral layer, in a tube in the presence of very mild stenosis. It is observed that resistance to the flow (A) and wall shear stress (rw) increase when 6~ increases. These characteristics are further enhanced by certain combinations of Eringen fluid parameters.


Acknowledgement--The authors are grateful to Dr Prawal Sinha for his helpful suggestions in the preparation of the final version of the manuscript.

R E F E R E N C E S

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(Revision received 28 March 1995; accepted 26 July 1995)

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