Co~npsterR M~th. Applic. VOl. 24, No. 10, pp. 19-31, 1992 0097-4943/92 $5.00 + 0.00 Printed in Great Britain. All rights reserved Copyri~t~) 1992 Pergamon Press Ltd

U N S T E A D Y F L O W OF B L O O D T H R O U G H N A R R O W B L O O D V E S S E L S - -

A M A T H E M A T I C A L A N A L Y S I S

J. C. MlSaA, M. K. PATRA AND B. K. SAHU Department of Mathematics, Indian Institute of Technology

Kharagpur 721302, India

(Received Jane 1991)

Abstract---Of c(mcern in the paper is a study of unsteady flow of blood in a uniform arteriole segment under various bo,,-dA~ conditions at the wall. The blood vessel is modelled as an initially stressed orthotroplc elastic tube filled with a viscous incompressible fluid representing blood which is treated here as a polar fluid. As for the equations governing the motion of the arteriole wall, we have taken a pair of appropriate equations derived in one of our earlier communications, by using suitable constitutive relations and the principle of superimposition of a sinai/additional deformation on a state of known finite deformation. Numerical computations of the derived analytical expressious for velocity distribution, pressure, total angular velocity, wall shear stress, flow rste and resistance to fluid motion have been carried out. Variations of these quantities as functions of frequency and the system parameters have been studied with a view to illustrate the applicability of the mathematical model.

INTRODUCTION

Current problems in biological modelling, fluids, and missile fuel systems have stimulated re- searchers to study the propagation of waves in elastic or viscoelastic tubes containing liquid at rest or filled with a flowing fluid. In the first approximation, the fluids may be regarded as incompressible. Considerable attention has been paid, particularly on the part of physiologists concerned with blood flow and the blood pressure pulse in mammalian arteries. It has been observed by Dintenfass [1], Chien [2], Woodcock [3], Lighthill [4], Caro [5], Schmid-Sch6nbein [6], Fry [7] and many other researchers that the knowledge of rheological and fluid dynamical proper- ties of blood and blood flow, like velocity profiles, velocity gradients, wall shear stress, apparent viscosity of blood, deformability of red cells and the spin boundary condition, could play a vital role in the basic understanding, diagnosis and treatment of many diseases (cardiovascular, renal, diabetic, etc.). Furthermore, the viscosity of blood varies greatly in different areas of circulation and depends on such factors as internal viscosity of red cells, hematocrit, plasma viscosity, ag- gregation of red cells as well as on the luman of the blood vessels, its diameter and on the flow velocity. It has been observed by Chien [2] and Dintenfass [1] that blood viscosity in patients with myocardial infraction is higher than that in a normal individual. Pathogenic investigations of arteriosclerosis show that the incidence of the disease increases with age, is more pronounced and occurs earlier in male humans, is accelerated by hypertension, and prefers certain vessels and locations. Medical evidence indicates that the basic laws of hydraulics determine the behaviour of the flow in the arterial tree after loss of elasticity. Although the human body is adopted to changes in velocity profiles and velocity distributions of blood to a certain extent, when the changes in the velocity are of a very large magnitude and extend over a large period of time, there is evidence that such velocity changes may bring about serious physiological effects, which may sometimes lead to fatality. According to Hiatt el al. [8], prolonged acceleration may cause various physical disorders such as headaches, loss of vision, increase in pulse rate, abdominal pain, venous pooling of blood in the extremities and hemorrhage in the face, neck, eyesockets, lungs and brain.

Typeset by ~4A, Ifl-TEX

19

20 J.C. Miss^ e~ aI.

Since rheological properties and flow behaviour of blood are of importance in the fundamental understanding, treatment and diagnosis of many diseases, and are different in different regions of circulation, many mathematical models have been proposed for blood flow [cf. 9-13].

The quasi-steady pulsatile flow of blood in a capillary vessel for small values of the Womer- sley number has been studied theoretically by Aroesty and Gross [14,15]. The same problem has been studied by Iida and Murata [16] using the Herschel-Bulkley fluid model, the problem being taken to be quasi-steady. One of the characteristics of the pulsatile flow of polar fluids is the flow enhancement (the increase in the mean flow rate due to pulsation). Ariman and his colleagues [17,18] found closed-form solutions of the unsteady flow of blood in conduits using a new spin boundary condition and applied their results to predict the capillary flow behaviour of blood. In order to account for the Fahraeus-Lindquist effect as well as the R.B.C. spin distribu- tion during capillary flow, Chaturani and Upadhya [19] considered a two phase model of blood flow in which a central core fluid is described by the micropolar fluid model and the annular region surrounding the core by a Newtonian fluid. They used the spin boundary condition of Ariman and his colleagues [17,18] at the fluid interface. A similar two phase fluid model has been proposed, but the interface spin is obtained by assuming no couple on the particles at the imme- diate vicinity of the interface, which ensures the continuity of stress across the phase boundary. In two phase models, the Fahraens-Lindquist effect is incorporated through the prior assumption of a cell-free layer which can be determined experimentally as a function of R.B.C. concentration and tube radius.

The network of vessels carrying blood exhibits rather complicated distensibility relations, which in the case of arteries are of great importance in matching the pumping action of the heart to produce an unsteady (due to exercise or hard working) perfusion of the peripheral capillaries. Therefore, the problem that has to be treated is that of unsteady pulsatile flow in an elas- tic/viscoelastic vessel.

In this paper, we have dealt with a continuum model for substructured fluids having prospective applications in haemodynamics of the microcirculatory system. In the model, there are two basic and independent kinematical vector fields--a vector field representing the linear velocities of the fluid particles and an axial vector field representing the angular velocities of the fluid particles. Such a model is usually referred to as a polar fluid model. One of the principal predictions of the theory of polar fluids is an increased effective shear viscosity in capillary tubes. The application of the theory of polar fluids, to the explanation of a number of physical phenomena has been suggested by many researchers. Wolkowa [20] found experimentally that non-polar fluids give the same value of the radius for any given solid boundary, but the radius decreases with increasing polarity of the fluid; the radius was found to be very small for water. Kline and Allen [21-23], Valanis and Sun [24], Ariman [25] and Cowin [26] considered the applicability of the polar fluid theories to the flow of blood, particularly in the microcirculatory system. In view of the above-mentioned facts, blood has been treated here as a polar fluid and its motion is considered to be linear and unsteady, the blood vessels being considered to be flexible and thin-walled. In a state of physiological loading, there is experimental evidence that blood vessels can undergo large deformations. So the governing equations of motion are nonlinear. In an earlier paper by Misra et ai. [27], it has been shown that the equations can be linearized by considering superposition of a small additional deformation on a state of known finite deformations of the blood vessel. The linearized equations of motion have been employed in this paper to account for the wall deformations. The diameter of the blood vessel under study is small and the motion of blood is considered to be harmonic oscillatory; its wave length is large in comparision to the radius of the vessel. Numerical computations are made in order to have an estimate of the velocity components, total angular velocity, pressure, wall shear stress, flow flux and resistance to blood flow at different locations and at different instants of time. The results are compared with the available exact solutions as well as with experimental observations on blood flow.

FORMULATION OF THE PROBLEM

Let us consider an axially symmetric time-dependent laminar flow of blood through a circularly cylindrical arteriole segment of radius a, which is considered as a thin-walled elastic tube of uniform cross-section. Here blood is represented by an incompressible polar fluid. With reference

Unsteady flow of blood 21

to the polar coordinate system (r, 0, z), let u and v be the velocity components of blood at (r, 0, z) in the radial and axial directions respectively, as shown in Figure 1. The circumferential velocity is taken to be negligibly small. The problem is investigated under the following assumptions:

(i) The motion is slow, so that the inertia terms may be neglected, and (ii) body forces, body couples and intrinsic angular momentum are vanishingly small.

/ / / / / / / ~ ) / # / / / , , ~ / ~ / / /

Z

I_.. d _I F "= '~I

Figure I. Flow geometry.

Under the above-mentioned assumptions, the equations of linear and angular momentum of blood (treated as a polar fluid) and the equation of continuity are given by

Ou Op (02u 10u u 02u~ 2tOG, \Ors ,'Or ,.2 Oz~. ) -~z

v - ~ = - ~ + (~ + ~) + + ~ + (2) k o~ + 7 ~ -~z~ ) ~ ' OG [ O~G 10G G O~G ~ Ou

P ~ - =(/~ + 7) k~-~'r~ + r Or r ~ + ~'Yz~ / - 4Gr + 2r ~z ~r ' (3)

and 1 ~ r ( r u ) + ov ; ~ = 0, (4)

where p is the density of blood, r is the rotational viscosity,/~ and 7 are viscosities associated with the gradient of the particle angular velocity, p is the traditional shear viscosity coefficient and G the total angular velocity of a fluid particle at the indicated position.

WALL MOTION

Taking account of the fact that all blood vessels are in a prestressed state and that the wall material is orthotropic and viscoelastic, and incorporating the inertial forces, the surface forces, and the forces of constraint which are, in this case, the reaction of the surrounding tissues, the equations governing the wall motion derived by Misra and Roychoudhury [27], by employing the principle of superimposition of a small additional deformation on a known state of finite deformation of the blood vessel, are as follows:

I~ ~ 0Ill hoPo O~ o~ (ho~,~ hobo 2ho,o~ o~ [ho(~,#-~o) 2~0ho I hopo 02~ 0-7 \ ~ R o + ~ ~---~Ro) + ~ L ~ ; + ~ J + Y = ~---~ 0t=' (6)

in which ~ and ~ are the physical components of the superimposed displacement in the radial and axial directions, Ro and ho denote the mean surface radius and the wall thickness of the segment of the blood vessel, ~1 and ~ are the axial and circumferential stretch ratios and Po is the volume density of the wall tissues. The specific expressions for ¢o, ¢o, ~os, ~e=, ~== and ~=e are given in the Appendix. X and Y are the radial and longitudinal components of the external forces acting on the wall tissues having the following expressions

x = v - 2 o , g - - ~ , ~ + . (7) P ~ a r~---Q

22 J .C. MIsR^ et M.

BOUNDARY CONDITIONS

The same boundary conditions as used in viscous fluid theory for the velocity field can be used in pola~ fluid theory. These conditions--which may be stated as: at the interface between the fluid and the wall, the fluid velocity must be equal to the wall velocity--lead to two kinematic conditions on the boundary. Also we assume that there is a constant spin of the suspended particles (erythrocytes) at the boundary and that the vessel wall is subjected to a periodic acceleration along the axial direction. These may be stated mathematically as:

a~ (i) u - - ~ a t r - - a ,

Or/ (ii) v=~- a r t = a , (S)

1 C9 ( r G ) - O a t r - a , (m) r

-~ = H exp - r =

where H is the amplitude of the applied acceleration, w the circular frequency, and c the wave propagation velocity.

ANALYSIS

Since the vessel wall exhibits periodic acceleration for the velocity distribution, pressure, and the total angulax velocity of the fluid and the displacements of the wall, we seek a solution which is haxmonic in time 't' and the longitudinal coordinate 'z'. Thus, we put

(9)

exp (,__;)], where ~0 and r/0 axe constants and u0(r), vo(r), G0(r), p0(r) axe axbitraxy functions of r alone.

Substitution of expressions (9) into Equations (1)-(4) leads, respectively, to

~pro 1"O2uo l Ouo ( 1 w ' ) ] 2/WrGo ' (11) ip~',,o = - + (~ + ~') [ T ~ + ,. o,- ~ + ~ - uo +

c ~ Or ~ + r 0r c2 v0 + 21" ~.-~'r + ' (12)

1 o0 ip~vGo = (8 + 7) L'-ff~-r~ + r o t "~ + ~ Go - 4wGo - 2~" Uo + Or ) ' (13)

and ~o = - : - + . ( 1 4 ) \Or Eliminating 'p0' between (11) and (12), we get

ipw Uo =(p,+l") + Uo + + c u o

- - - - U O ~ ' + ~ \ O r c

ro,oo 1ooo (1 ] + 2r L'T~-r2 + + Go • r Or -~ 7~

Unsteady flow of blood 23

Again, eliminating (~_.~..t ar + (iw/e) Uo) between (13) and (15), one obtains

)ooo ) 0r 4 "4- r 0r 3 r'~ + ~ 3 ~ + ~'~ r ~ - - 1. 4 1. 2 '~4 G o - 0 , (16)

where Aa and Aa have expressions as given in the Appendix. The above equation may be rewritten a8 [02 10 (1 )][02 1 0 ( ~ )1 ~ . ~ + - ; ~ - ~+o,~ ~.;-~+-1-~- +o,~ V 0 = 0 (17)

Its solution is of the form Go = ClI1(a11-) + Cs A(a~r), (18)

in which C1 and Cs are arbitrary constants, 11 stands for the modified first kind, first order Beasel's function, and az, as have expressions as shown in the Appendix.

Using (18) and eliminating vo between (13) and (14), we obtain

02uo 10uo (1 o;2) 01---- T 4 f o r ~'2 + ~- u0 = AIl(alr) + BIz(a2r), (19)

the derived expressions for A and B being shown in the Appendix. The complete solution of (19) is given by

Clio; ( 4r+ipo; ~ o;1-) "°= ~ ~+~-~{:rJ~/o)/z1(~11-)+c2~1 (7

(20) ~erCaio; x ( a~4r + + , , + . , -

C2 and C4 being arbitrary constants, K, stands for the modified second kind, first order Bessel's function.

From (14), with the help of (20), one finds

CI{21 ( 4r + ipo; ) o;

C8a2 ( 4r + ipo; +---~--~ ~ + ~ - 4 : ~ ) j ~0(~) + ~ ,~o (~ r).

Using (18) and (21), we have, from (12), the following expression for P0.

i ea,C: (#+r) ( /~+7) a~-~- - 4p r - iPo;(t~ + r + l~ + 7) Po - 2r o;

(21)

ipo;(4r + ipo;)] ,, , , / o; \

.I (22) ,co,.[ + 2r---~--- (p+r)(¢~+7) a~--~- - 4 p r - i p o ; ( p + r + ~ + 7 )

i.o;c4r.~,g _ (o;vo') + ip~)] C o; 1-) + /z0(~21-) + i . c c , go ~ .

Using Equations (10) and (20)-(22), we obtain the following equations from (5) and (6) together with relations (7).

[( ) - C l a n - C a ipc+2Pco; I ° ( ~ a ) + 2 P l t ( c

- C s a n + C4 + iVc Ko , ,~ (2a)

[ hotO 0 o;2 ~1 hol~o O;2 ho ] + ~0 L ,~1,~2 c2~2 Ro 2 (&o - 2¢o) J

/~hoA2 + ~o ~ (~., - ¢ o ) = 0,

N*IO-¢

24 J.C. Mmak et ~L

and -Clala+C22ip'~e I1 (~a)-C3al4+C42ip~c K~ (~a)

(24) , ) + : o

where the expressions of a~i, i - 1, 2, 3, 4 are presented in the Appendix. Further, employing the four boundary conditions given by Equations (8), we get, the following four relations involving the arbitrary constants C~, C2, Cs and C4, respectively.

°,~_ o~-~,/c,)~',(°,") +~,',(-~°1

CI¢~I I 2, ~+7 a~_(w2/e2)jlo(ala)-C'i'o(~ a) C3a2 ( 47". +_ip~o .~ w a) O,

+ ~ ~+~- ~_(~2/c~)/I0(~2~)+C~ig0 ( 7 - i ~ 0 =

C1oqlo(ala) ÷ C3 a21o(a2a) = O,

(26)

(2T)

Clialca (~ 4, +ipw ~ ~o

2--T- ~+~ - ~] - (~2/,2)/ (28)

By solving the Equations (23)-(28), one can calculate uniquely the six unknowns Cj, ~o and ~/0, j - 1,2,3,4. These equations have been solved numerically by the use of Grams elim- instion method on a HP 1000 computer model. Thus, the velocity components, the pressure dmtribution and the total angular velocity of the fluid particle, as well as the wall motion, ~e completely known at any location at any instant of time.

Now the wall shear stress component "zr can be calculated by employing the relation,

r,, = (~+ r)~r + 2 r G

r lO (

Also, the total flow rate

(29)

Q- /2~rvdr o

+ G ¢+'~

(30)

Unsteady flow of blood 25

The resistance to flow, ,~ is given by

/ [ eiw(t-(d/¢))] = a o - Poor _ p ° ! ° ) t ' " ' -

Q b2a e iw(t-(zlc))

- ( 1 - b2a

where the expression for b2a is presented in the Appendix.

(31)

A NUMERICAL EXAMPLE

With a view to examining the validity of the mathematical analysis presented in this paper, a specific numerical example will now be considered. The results presented here have been computed by using a HP 1000 computer. In agreement with the observation of Young et al. [28] that the four function theory for the relaxation functions describing the viscoelastic properties of the wall tissues is good enough for most practical applications, the numerical results have been calculated by considering the following four relaxation functions (in units of 102 N/mS):

and

C22(t) = 282 Jr 22 exp(-0.47t°'47),

C2a(t) = 179,

C3~(t) = 270 Jr 16 exp(-1.67t°'2a),

C3a(t) -- 267 Jr 28 exp(--O.51t°'~).

Values of the constant finite strains have been taken as follows:

a0 = 0.02, b0 = 0.005.

Two sets of values have been obtained by taking ~1 " - 1.28, As = 1.55 and '~1 ~- 1.4, )is = 1.61. Our choice of the stretch ratio values of the wall tissues is based upon the experimental observation that the average range for the circumferential stretch ratio is from 1.17 to 1.82, while that of the longitudinal stretch ratio is from 1.28 to 1.61. As for the values of the other physical and material constants, we have taken a = 0.03 mm, p = 1056 kg/m s, P0 = 1100 kg/m 3, h0 = 0.02 mm, Ro = 0.45 mm,/~ + 7 = 12 x 10 -s kg-mm/s, r = 98 x 10 -5 kg/m-s [cf. 29]. Numerical results obtained through the use of the above-mentioned data are exhibited in Figures 2-9.

Numerical values of the mean flow rate are computed for different phase angles ~b and for different sets of values of the stretch ratios )q and )ts. We also simulate two values of the frequency parameter at (= a x / ~ / v , / being the heart rate), viz. 0.02 and 0.06. The results have been displayed in Figure 2. From this figure, it is clearly seen that the phase angle is a dominating factor for the mean flow rate; the extent of variation is more when )q = 1.28, As = 1.55 than when A1 = 1.4, ~s = 1.61. The results are also sensitive to a, with greater influence at higher at. Depending on the phase angle, flow rate could be either higher or lower than the values for the two selected sets of stretch ratio parameters by as much as 10%. It is also interesting to note that the mean flow rate varies almost sinusoidaily with the phase angle and that the mean flow rate is almost independent of diameter variation at two phase angles, differing approximately by 180".

Figure 3 displays characteristics of the mean wall shear stress (WSS) wave forms as a function of ¢. Markedly different behavior is seen at high and low values of at. Wall shear stress is higher when ,~1 = 1.28, ,~s = 1.55 than when ,~1 = 1.4, ,~2 = 1.61 in 0 < ~b <_ 900 and 270* < ~b <_ 360 ° and lower for the above-mentioned sets of values of the parameters in 90 ° < ~b < 2700 for high or low values of at. Greater diameter variation and unsteadiness result in greater sensitivity of mean WSS to phase angle variations. However, WSS is much more semdtive than flow rate to variations in phase angle.

Representative velocity profiles for two different values of at and two different sets of values of the stretch ratio parameters A1 and A2 are shown in Figures 4(a)-4(d). The profiles in Figurm 4(a)

2 6

t 7

u 6

S _

400 I I l I I I I 900 180 ° 270 ° 360"

FiSure S(s). Vm-iation of the mean flow rste with the plume mSle, for a = 0.02.

2o'~5

2.00 g-

~ 135

"O

1.50

1.25 ,.N

\ .~,=~.2a /

J.C. MmRA 8t =l.

18{

1,6

12

~ 1 - ' i . 2 8 ~ =1 .55

'J l '~ ~ I" t I I I t I

90" 180 ° "270" 360 °

Figure 3(b). V~, is t ion of t he m e a n flow ra te wi th the plume A-~e , for r, _- 0.06.

t - - 6 N

E U

r. , > .

~2

E

~-0

• A t - - I . 2 0 , / A ~ = 1.55

1 . 0 0 t I I I I I . I o2 I .I I I . i , i 0 11"/2 1T 3'11"/2 21T 0 Tr/2 TT 311"/2 211'

,:1:,--.-~ ¢ - - - - ~

Figm~ 3(~) Dis t r ibu t ion ~ w~dl she~r stress, for Figure 3(b). Dis tn 'but ion ~ wall s]Im~ stress, a = 0.02. for ~ --- 0.06.

and 4(b) correspond to the phase angle of 180 o while the profiles in Figures 4(c) and 4(d) depict the situation when the phase angle is 0 °. For the lower value of a, very little negative velocity is observed. For the higher value of a, however, considerable velocity reversal is seen in the near wail region with smociated steep velocity grandients near the wall (high WSS) during the reversed velocity portion of the cycle.

From Figure 5, one may note that when AI - 1.4 and As = 1.61, the fluid resistance diminishes sharply for values of a < 0.03 and the variation is of rectangular hyperbolic nature for both ~b = 0 ° and ~b = 180 °. When AI = 1.28 and As = 1.55, the resistance increases steadily for

< 0.01 and then decreases gradually for higher values of a. It is interesting to note that, in all csses, the resistance is smaller for lower phsse angles.

Real parts of the non-dimansional axial velocity profiles are s~sin presented in Figure 6, for different values of a and for different palm of values of the stretch ratios A] = 1.4, A2 = 1.61. It is ob6erved that the velocity profiles are strongly dependent on frequency. However, for both the values of the frequency psrsmetem studied here, the greatest velocity corresponds to wt = 0. Further computational results (not presented here) indicate that, for higher values of a, the velocity profiles are not &ppreciably influenced by the initial stress.

Figures 7(a) and 7(b) illustrate the variation of the axial velocity, Re(v), ss well am its amplitude with the applied periodic sccvkrstion at dil~mnt,instants of time for two different values of a and the presssigued valims Of the stretch ratios. These figures indicate that the magnitude of the

Unsteady flow of blood 27 40 !-ii

. . . . t : 0.7.5

- 1 0 I , I I I I I I I 0.0 0.2 0./-, 0.6 0.8 1.0

1.4, , ~ = 1.61 and ~ ~ l e 180 °.

z.O

t 3 0

~zo

-J 10 <~ uJ n~

- 1 0 1 I I I I I I I I 0.0 0.2 0.4 0.6 0.8 tO

1-~-1 =

Figure 4(c). Axial velodty at dii~-~-ent t i m e

fractjctm for ¢t = 0.02, ~1 ----- 1.4, )t2 ---- 1.61 and phue anSle 00.

t ~

F i ~ r e 5.

4O

t ~ 3 0

E ~J

- - 2 0

. J

w 10

I I I I I , I I L -5 0.0 0.2 0/, 0.6 0.8 1.0

Figure 4(b). Ax/al velocity distr ibution a t different instants of time, when ~ -- 0.06, ,~z -- 1.28, ~2 = 1.55 and phase angle 180 °.

40

30

E 20 t J

. 3 1 0

W

0

- t = 0 . 2 5

- I 0 ' t t J_ I I I ,, , J 0.0 0.2 0/-, 0.6 0.8 1.0

(-~-) - - -~

Fis~"e 4(d). A ~ v e l ~ t y ~ diffe--~mt time ~ " t ~ u s for a --. 0.06,/~z = 1.28, ~2 = 1.55 and p h a s e angle 0 ° .

~ 3 ~ ~, , ,r l~e,>,2_-~.ss, ,_-~eo o -%

~ . ~ . ~ s = 1 . 2 8 ~ , 2 = 1.55~ ~ = 0 ° t -

f2

,. 1 ~1,"=0 ° " ~ , . . . , . . ~ ~

0 f i f I t

0 . 0 0 5 0 .01 0 . 0 2 0 . 0 3 0 . 0 4 0 . 0 5 0 . 0 6

Flow resistance versus frequency for diHe~ent s tretch ratios and phase

28

1.0

0,8

~ 0.4

- - 0 . .J'

UJ

-0.4

-0,8

:1 .0

~ z,O

-E2o

J .C. MISRA ef =i.

0 -0.5 0 0.5 1.0

:e

-3 <~ u.I

-0./,

1.0 =

0.8

0.4

0

0 -0.5 0 0.5 tO

(=) (b)

Figure 6. PredicMed velocity profile as ~ funct ion of radial distance, for A1 = 1.4, A2 = 1.61 and (s) a = 0.02, (b) ~ = 0.06 at d/fferent t ime fractions.

0~0/,//=~,~,~,~,,,,~,,., t = 0.001

~ - ~ " =....~. t =0.25 ~ t =0.50

t =0.75 I I I I I I I

~.o so ioo H (m/s z)

t= 0.0ol 1 ~ 16

t= 0.25

E t= o.so ~ e

t=035 ~" 0

0 -2 '

0 40 80 100 H (m/s z)

(-) (b)

Filp~e 7. Vmis~ion o~ real part m~d mnplitude of axial velocity with pefiodlc accel- or=taon ~ ~ t ~ o~ tim, (a) = = 0.02, (b) a = 0.06:

• A1 = 1.4, A2 = 1.61 " " " A t = 1 . 2 8 , A2 = 1 . 5 5

velocity increases with the increase in the m~nitude of the appled acceleration. It is further seen that while the real part of the axial velocity is significantly influenced by the values of the stretch ratio parameters, the amplitude of the velocity is almost independent of these factors. Irrespective of the value of ~, the velocity is higher when ~I = 1.4 and A2 = 1.61 than when ~I = 1.28 and ~2 = 1.55.

In a similar way, the variation of Re(u) with H has been presented throush Figures 8(a) and 8(b) at different instants of time. It may be noted that the radial velocity increases with time and that at any particular instant of time, the radial velocity is directly proportional to the ms6nitude of the applied acceleration. One may observe further that the magnitude of the radial velocity decreeses with the increase in frequency.

The variation of the total angular velocity, G, at different instants of time with the applied acceleration has also been investigated and the computed results have been presented in F~- urea 9(a) and 9(b). It is observed that the total angular velocity ¢hsnses a little when the values of (~ and the stretch ratic~ AI and ~2 are changed. It may also he noted that at a particular

E

tY

0.02

0.01

-0.01

-0.02

Unsteady flow of blood

t = 0.001 t = 0.25

t=0.5

t = 0.75

I I ~ I I t

E

G rY

0.005 0.004

0.002

-0.002,

-0.004

I I i I i -o.o3 4o 8o "°'°°So 4o 8o H ( m/s z) ~ H ( m/s z)

(a) (b) Figure 8. Variation of radial velocity with periodic acceleration at ditferent instants

0,8,

I 0.6 0

~9

~ 0.4

0.2

t = 0.001

t =0.25

t =0.5

t =0.75 I I

of time Arid stretch ~ e t e r s ; (a) a = 0.02, (b) ¢f --- 0.06: ~1 = 1.4, '~2 ---- 1.61

- - - "~Z ---- 1 . 2 , ,~2 = 1.55

(X =0.02 ~'1 ='1"2 8 / Az=1.55 /

/ / / / t = l . o / / / t =0.75

~ o ~ / / / , ~ t =0 .50

t = o 2 5

40 80 100 H(m/5 z )

0 0 0 0

:oo • I - A t = 1.4 .To 0.6}-'xz 161 /

/ t ---LO /,2//' ~ / ~ ~'t'=0"75

c~ 0 . 4 ( , 3 / , ~ / . ,. / t = 0 . 5 0

Q:: / v / ~ / / O ~ , ~ , ~ t =0.25

0.2 ~ / , ~ , . t = 0.01

, I I I I 4 0 80 100

H ( m/s z}

2 9

(a) (b)

Figure 9. Vari,*tlon of non-dimensional total angu/ar velocity with applied accelera- tion at different instants of time.

i n s t an t of t ime , t o t a l angu la r veloci ty varies l inear ly wi th appl ied acce le ra t ion and increases wi th t h e increase in the app l i ed acce le ra t ion as well as wi th the passage of t ime. T h e a m p l i t u d e o f t he t o t a l angu la r veloci ty is found to be changed a l i t t le (in m a g n i t u d e only) when the values o f a , ~1 and ,~2 are changed.

REFERENCES

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A P P E N D I X

Expressions of d i t r ~ t quantities appearing in Equations (5) and (6):

~o = (1 + 2.o) [~oC22(oo) + boCs2(oo)], ~0 = (1 + 2bo) [a0C23(oo) + boC33(oo)],

~oo = (1 + 2ao) (~o. + ~°c). ~o. = (1 + 2bo) ( ~ + ~b.), ~.e = (1 + 2bo) ( # ~ + ~b.), ~.z = (1 + 2ao) (~°c + ~° . ) ,

w ~ r e

c22(t) = 282 + 22 exp(-o.47t°-'T),

c3~ (t) = 27o + 16 e x p ( - 1.67t °'22),

c33 0 ) = 267 + 28 exp(-o .51t ° ' ' ' ) ,

c23(t) = 179,

~oc = (1 + 2ao) c22(0),

Unsteady flow of blood 31

~=. = (1 + l,.ol / dC~(,.__._)) _ , , , , d,. o

¢'b~ = (1 + / = o ) Cs~(o), 4'b, = O, 0,== = (1 + ~bo)Css(O),

/ ~C~3(oo) e_i~ , ~. , , ~ = (1 + ;~bo)Css(O), ~ ° , = (1 + ~bo) d~

o

. f d033(8) e_iw , ~b, = (1 + 2~0) T d.,

o

where a0 and bo represent the constant finite strains and i = V/'~'~. Ex3pre~ons for ~1 and ~2 appearing in Equation (18):

~l = , and ~2 = 7 4At)ill" ~

with

20 "I 41~1"+ipw(#+'y+/~+~') ~,s : - ~ - + O, + ~)(~ + "~)

wt w241~T+ipol(#.i.,.f+l~+1. ) , ==1 x~ = c-T + c'T O, + ~)(# + '~)

ipw(4~ + ipoo) t" (~, + ~)(# + .~)

Expresekms for A and B in Equation (19):

2c'r (#+'3') cq ~ - --(4~ +ipw) , B = iwC32c~. +., (o==- g)-<,. +,.+]. E x p r e u i ~ for =ii, =12, als and "<14 appearing in Equations (23) and (24):

a l l :

a12 -~-

~,i,.,~. #+~ ,q_(=,2/ca)) ~,oSo@,~=)--*~(~,~,=)=

[ ( c i~ i l~-,,., lo(~,r') O' + - -

,iwc~. #+'* a ~ Z ~ 7 - O ) ) ,~2/o(~,")--1~(,~2")c,

• ipw(47 + ipw)] , c2J'~i 'O(=i")[(iv, "1" "r)(~'-t" 3') (=~ -- - .~-) - 4#'r-- ipi( l l -F'r . i - [J +'7)+ " ~ 2 ~ J

( ~ ) ( 4'r-l-ipw ) ¢113 --

~' c'= = + # +'~ 4 (~,2=).

Expression for b2s in Equation (31):

[c,( Csa(~ .

4 ' r+ ip~ / . • ,