Med. & Biol. Eng. & Comput., 1983, 21, 438445
Flow of micropolar fluid through a tube with stenosis R. D e v a n a t h a n
S. P a r v a t h a m m a
Department of Applied Mathematics, Indian Institute of Science, Bangalore 560 012, India
Department of Mathematics, University of Agricultural Sciences, Bangalore 560 065, India
mathematical model for the steady flow of non-Newtonian fluid through a stenotic region is presented. The results indicate that the general shape and size of the stenosis together with rheological properties of blood are important in understanding the flow characteristics and the presence of flow separation.
Abstract--A
Keywords--Micromotions, Stenosis
1 Introduction LOCALISEDnarrowing of the blood vessels, commonly referred to as a stenosis, is a frequent result of arterial disease. Such constrictions, in general, disturb the normal blood flow and there is considerable evidence that hydrodynamic factors can play a significant role in the development and progression of this pathological condition. It has been emphasised that several flow characteristics such as wall shear stress, pressure drop and separation in the flow field have direct applications in the clinical aspects of the study (CARo et al., 1971; FRY, 1968). There are various mathematical models of the blood flows through stenosed vessels (TEXON, 1963; FORRESTER and YOUNG, 1970; MORGAN and YOUNG, 1974; McDONALD, 1979; LEE and FUNG, 1970; SCHNECK and OSTRACH, 1975; PADMANABHAN, 1980; PADMANABHAN and DEVANATHAN, 1981). These studies have been very useful in determining the flow characteristics and the presence of flow separation. The other important investigations concerning the steady flow through constricted tubes are by MANTON (1971), CHOW and SODA (1972) and McDONALD (1978), and for non-Newtonian fluids by WILLIAMS and JAVADPOUR (1980) and SHUKLA et al. (1980). In most of the above analyses blood is treated as a homogeneous incompressible fluid. On the other hand, BUGL1ARELLO(1969) and COKELET (1972) have emphasised that under certain flow conditions blood exhibits non-Newtonian behaviour. In particular, blood consists of a suspension of a variety of cells in aqueous solution of organic and inorganic substances. ERINGEN (1964) introduced the concept of simple First received 17th June and in final form 8th October 1982
0140-011 8/83/040438 + 08 $02.00/0 9 IFMBE: 1983
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microfluids to characterise concentrated suspensions of neutrally buoyant deformable particles in a viscous fluid Where individuality of substructures affects the physical outcome of the flow. Such fluid models can be used to rheologically describe polymeric suspensions, normal human blood etc. A subclass of these microfluids is known as micropolar fluids (ERINGEN, 1966), where the fluid microelements are considered to be rigid. Basically, these fluids can support couple stresses and body couples and exhibit microrotational effects and microrotational inertia. With this model ARIMANet al. (1973), studied blood flow and presented an encouraging comparison of the theoretical velocity and cell-rotational velocity profiles with the experimental data of BUGLIARELLO and SEVILLA (1970). Thus the main advantage of using the microcontinuum approach to study blood flows in comparison with the other class of non-Newtonian fluids is that the former model takes care of the rotation of red blood cells by means of an independent kinematic vector called the microrotation vector. In the present analysis the steady flow situation is examined, although the blood flow is pulsatile. The main purpose is to obtain the flow characteristics and impedance and other relevant factors using the microcontinuum approach for the flow through stenosed vessels. Furthermore, the onset of separation and the corresponding critical Reynolds number are determined for a given geometry of the stenosis. 2 Fluid model The constitutive equations for an incompressible micropolar fluid as given by ERINGEN (1966) are Tij = - - p 6 q + ( 2 ~ t + K ) D i j + KOijk(tak--Vk) Mij = fllvi,j+71vj, i .
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(1) . (2)
July 1983
where Tq and Mij are the stress tensor and couple stress tensor, respectively, 6~ is the Kronecker delta , 0~jk an alternating tensor, p the isotropic pressure, r the vorticity vector and v the microrotation vector (a c o m m a after a suffix denotes covariant differentiation).
Dij = 1/2(ui,S+us, i),
. (3)
# and K are coefficients of viscosity, and fl~ and 71 are coefficients of gyroviscosity. In the absence of external forces the equations of continuity, m o m e n t u m and angular m o m e n t u m for a micropolar fluid are given by Ui, i = O ,
. . . . . . . .
.
Tji,S = p D u j D t , M j l , j "~ Oij k Tik :
tube. The dimensionless form of these equations can be written as
(1/r)[d/Or(rv)]+~3u/dx = 0 . . . . . . 0u au dp) Re U~x + V~r + ~x
:
\ d r 2 q- r ~rr + ~ - )
Dv I ~ . . . . . . .
(6)
3 Formulation of the problem The specific boundary for the stenosis is considered in the form of a cosine curve as initiated by YOUNG (1968) (See Fig. 1). Thus the shape of the constriction is
/d2v dZv l dr dr 2 + r dr
q-
.
v
~
. (9)
dv
r2 + dx2j-m~xx
'
.(10)
v d2v~ r 2 + dx2J +n
--
.(]i)
-- 2nv = 0
where u and v are the dimensionless velocity components in x, r directions, respectively, and v is the microrotation along the 0 direction. The dimensionless parameters of the problem are Re -
.(7) rt =
= a otherwise where a is the radius of the tube outside the stenotic region, 2Z 0 is the length over which the stenosis extends and 6 represents the maximum protuberance of the stenotic growth into the lumen. The basic equations of motion and the continuity equation are written down in polar coordinates (x, r, 0) with the x-axis coinciding with the axis of the
ld~
= ~072 + r d r
( l +coS n~o ) o ~
~r
(4) . (5)
for-Z
q-m
R f ~v dv dp~ e + dr + /
where p is the density and I is the micromoment of inertia.
RI(X) = a - a / 2
(8)
pu~
#+K
Ka 2 7
_ Reynolds number
, n 1 --
#a 2 , 7
micropolar parameters
m = n/n+n I .
/
(12)
The corresponding boundary conditions are
u=0
}
v=0
at r = rl(x ) = Rl(x)/a
(13)
at r = 0.
(14)
(1/r)(d/dr)(rv) = 0 &/& = 0 U=U (1
1
)
v is finite
0
4 Solution for the velocity field ~------Z 0
Fig. 1
-t
G e o m e t r y o f the stenosed vessel
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It is evident that eqns. 8-11 are highly nonlinear, and, as such, it is not possible to obtain the solutions in a closed form. It is well known that in the absence of
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439
stenosis, the axial velocity profile for the classical case is parabolic, i.e.
Eqns. 18 and 19 are integrated (after multiplying by r) across the tube to obtain the following: rl
u=2~
1-
. . . . . .
(15)
d f ru2d r Re d~ 0
whereas for the micropolar fluid (ERINGEN, 1966), the profile is
~p I r~rrJo 3uqrt+m[rv]~
Re
- - 2 -rzx~xx + I
u = 2fi 1 -
(~) z
~,o(flr/rl)1} +-flzlo(fl) [ io(fl ) 4~
.(20)
1 (16) and
where
drrLr Or (rv) r d r - n c~ -
4ix~fl)
, fl
0
n(2-m)
=
and fi is the mean axial velocity at any given crosssection. We shall simplify eqns. 8-11 by using certain-order analysis used in the works of PADMANABHAN(1980) and FORRESTER and YOUNG (1970) while studying flow through stenosed vessels. In the human arterial system, during the initial stages of the formation of the stenosis, 5/a can be taken in the order of 0"1 (FRY, 1968). T h u s taking a/z o ~ 0 ( 1 ) and 5/a ~ 0 . 1 , corresponding to the mild stenosis cases, we find that ?p/c3r ~ t3p/dx, t3v/3x ~ ~3v/3r. Furthermore, from the equation of continuity (eqn. 8), 3u/3x ~ O(6/a). Also, among the viscous terms in eqn. 9, the term ?2u/3xZ is negligible compared with the other two terms, At this stage it is important to emphasise that, as in the analysis of Forrester and Young, we retain the convective, viscous and micropolar terms in eqn. 9, whereas eqn. 10 yields only one term in view of the above order analysis. Thus, the equations of motion, eqns.9-11, are simplified to a great extent to yield
~p/& = 0 . 3u
~u
Re u S + ~ g + ~
.
.
.
.
.
.
:(17)
@)
(~2u l~u) =
.
~rT + -rffrr
(c3v ~.) + m ~rr +
(
~u r 2 - - n ~ r --2nv = 0 9 .
rl
Q = r~f~ = f urdr,
. . . . . .
(22)
0
where Q is the volume flow rate. The velocity profile and microrotational profile in a tube of circular cross-section may be assumed to be a polynomial of the type u
- - = ao+alrl+a2rl2+a3rl3+a4rl 4 .
.(23)
Ul V
-- = bo+blrl+b2rl 2
.(24)
V1
r
where r/ = 1 - - - , vxis the centreline cell rotational rl velocity and ao .... , a4 and bo, bl, bz are coefficients to be determined. To evaluate the coefficients in eqns. 23 and 24 completely, additional conditions are obtained from the governing equation, eqn. 9, and the axial-velocity profile eqn. 16, and these are given by
.(18) r~r
+ - r ~rr(rV) a t r = rl(x )
(25)
v)
.(19)
The above system of equations is now solved by the momentum integral method as in FORRESTER and YOUNG (1970). Essentially, the momentum equation is integrated across the stenosed tube and a polynomial type of solution is assumed for the axial velocity profile such that the boundary conditions are satisfied. 440
vrdr = 0(21) 0
where the boundary conditions u = v = 0 at r = r~ have been applied. The integrated form of the continuity equation is
r~r ~2v 1 0v c3r2 + r (~r
r~rrdr-2n 0
~2U
2U
~r z
r~
(1-a) atr =0
. . . . .
(26)
Using the boundary conditions in eqns. 13, 14, 25 and 26 we obtain
a o = bo = b 1
=0, b2 -
nul 2v 1 a 1
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.(27)
July 1983
al +a2+a3+a 4 = 1
_
5 Rer~
14
aa+2a2+3aa+4a 4 = 0 .(28)
a 2 + 3 a 3 + 6 a 4 = ~-- 1
2a2-al
Re = - -Url~ x
2
F(fl) =~-[2I~(fl)-I~(fl)
x
2ul 7
(5+~)
.(33)
where 16~ 2 V
dp
dx
4lo(fl)II(fl)
fl
+~)(8-32)-~I1(3)]
F r o m eqn. 28 we can solve for a~ .... , a4. Thus eqns. 23 and 24 become
The relationship in eqn. 33 suggests that it is possible -uUl
_
10+2e-
( 5 + c t + 3.e r
+
dx !
--Y1
--
9
-
12+8c~+
b/1
q
.
.
.
_
dp
a simple expression for dxx as given below:
dp) q2 r2dx
+ ( 4 + 5 c t + Re Ul r2 d ~ ) r/31
.(29)
dp dx
and N
nu~ tl2(lO+2ct_Rer2dP)
vl -
l )
in eqn. 33, from eqns. 22 and 31, respectively, we obtain
Re
v
d-
hnear functions of dxx" Hence substituting for u and ul
t/1
3Re
dp dx
to evaluate an expression for - - since fi and u t are
14vx
ul
1 dx-x
8Q r~(25 + 3a) [ Q ( 9 7 + llct) Re 1_ 3r I
{1/3+ F(fl)} dr1
.(30)
l
~ x - 10(5+~) J .(34)
We notice that the velocity profiles are functions of
dp 2
Re dxx rl and micropolar parameter ~. Substituting eqn. 29 in eqn. 22 and integrating with respect to r we obtain
Knowing the value of the pressure gradient, it is now possible to evaluate the solution for the axial velocity and v completely 9 Thus eqns. 31 and 34 are substituted in eqn. 29 to obtain the velocity u as a function ofr and X.
420 { (97+llc~r2x Q +
u~-
Rer2dpt 210 dxx
"
.(31)
u
2
fi
(25+3ct)r 2
Again, substitution of the expression for u in the 1.h.s. of eqn. 20, taken from eqn. 16, and substitution ofu and v from eqns. 29 and 30 in the r.h.s, of eqn. 20 yield
2Re dr 1 x [~r~ ~x{1/3+F(fl)}{llq-43q2+45~3
rl
Re ddx
f 4fi2
- 1504) + 0~(0- 5r/2 + 703 - 304)}
Ii_(r/rl)2 + ~4c~I~
5 {485(2q-02) (97+ 11~)
0
x ~I~
lt]2rdr
(Io(3)
-
5 14
Rer 2
dp dx
2u 1
7
(5+~).
+ ~(3040 - 15202 - 38803 + 29104)
j
.(32)
Evaluating the integral in the 1.h.s. of eqn. 32 and using suitable recurrence relations for Bessel functions, the above relationship simplifies to 2Re
dx{r2 ~z [1/3 + F(fl)] }
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+ l l ~ 2 ( 2 q - q 2 - 4 0 3 +304)}1
.(35)
Incidentally, in the limiting case the solution in eqn. 35 coincides with the solution given by FORRESTER and YOUNG (1970) when c~ --. 0, for a Newtonian fluid. It is evident that the axial velocity is very much influenced owing to the presence of microstructure in the fluid.
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441
5 Resistive impedance and shear stress The resistance to flow, A, is defined as (McDoNALD, 1960)
reattachment condition from eqn. 39 by equating the wall shear stress to zero, which gives
1 ) dr 1 Re
AP A = -Q
d~-
.(36) 15(485 + 152ct + l l e 2)
where AP is the pressure drop across the stenosis l between the cross-section x = +_ 2" The pressure drop can be calculated by integrating the expression for dp given in eqn. 34 and we get dx
8Q
AP
Re ( l ~ dr~
3
;, rl
~r~lf dxx = 20'4
i'l
l/2
xRe
Evidently this condition depends mainly on Re and ~, and enables us to determine a critical value for the Re causing separation. Incidentally, eqn. 40 takes a very neat form in Newtonian fluids, as for ct --* 0 we have
[Q(97 + llot) {13 + F(fl) 1
(25 + 3ct) Re
1/2
r~xxdX-lO(5+~)
r~dX
-l/2
.(37)
-1/2
It is of great interest to determine where the diverging portion separation will begin. To locate this point 1
---
dr x
r x dx
If we specify the shape of the stenosis, it is possible to integrate eqn. 37 and to get an expression for the pressure drop across the stenosis, and hence impedance can be obtained. To highlight the role stenosis plays, the impedance is calculated in the absence of stenosis, i.e., when r~ = 1 to get the impedance AN as (AN = resistance for no stenosis) AN -
80(5+~) Re(25+3a)
l.
.
.
.
.
.
.
.(39)
where Tland T2 are given by T, -
=
(5,rl)
(d2r,) (,'r,)2 r' dx2;-',dxJ
dx \r 1 dx J =
r2
8(ll+e) {1/3 +F(fl)} 3(25+3~) 40(485 + 152e+ 11c~2)
Thus, the location of the initial separation point is given by X -
-
= 1/Tzsec-1 (2a/6-1)
. . . . .
(42)
It should be mentioned that the point of separation is independent of Re and c~. Table 1 gives the critical Reynolds number a t different cross-sections for various values of n.
6 Discussion of the results A mathematical model for the steady flow of nonNewtonian fluid through a stenotic region has been studied. Some of the physiologically relevant factors have been evaluated in the analysis. For the numerical scheme, we have selected the form of the stenosis to be a cosine curve as given in eqn. 7. Fig. 2 shows the velocity profiles for some particular
Table 1. Critical Reynolds number for different n and at different cross-sections x/z o
(25+3e)(97 + l l e )
n
With the help of the shear stress distribution on the wall it is possible to study the phenomenon of separation in the flow field. The separation and reattachment points occur when the wall shear stress vanishes. Thus we can calculate the separation and
442
O1)
= 0
Z0
T2 1 Re r~
.
is maximised with respect to x, giving the
relation for the initial separation point as
(38)
It is interesting to note that the impedance is inversely proportional to Reynolds number. The shearing stress in the dimensionless form on the wall is simplified to
T1 drx zw - r~ dx
.(40)
(97 + 1 le)(11 + cQ{1/3 + F(fl)}
0
1
5
10
X/Z o
0.435 0-635 0.835
127.50 154.6069 308.5143
2 2 6 - 9 2 300.2794 307.5070 268.2562 354.9788 363.5230 535.2986 708.3514 725-4012
Medical & Biological Engineering & Computing
July 1983
choice of the parameters. It is clearly seen that there is a deviation from the parabolic profile for the Reynolds
o 1OL
0
-1.0
1"0
x - -
Fig. 2
Axial velocity profiles for micropolar fluid, Re = 150
number range of 150; the profiles downstream from the throat show some departure from the parabolic form, but only a slight flattening of the profile is detected upstream. For larger Reynolds numbers the flattening of the upstream profiles has become much more pronounced producing a high shearing stress at the wall (Fig. 3). From the computation it is found that peak values of the profile decrease with increasing values of the non-Newtonian parameter. A similar trend in the velocity profiles has been observed by WILLIAMS and JAVADPOUR (1980) for elasticoviscous fluid through slowly varying cross-sections.
A typical plot showing the distribution of shearing stress on the wall is given in Fig.4 for different Reynolds numbers. It is seen that for any Reynolds number the shearing stress reaches a maximum immediately preceding the throat and then rapidly decreases in the diverging section. When separation occurs, it becomes negative over some downsteam length of the tube, as illustrated in Fig. 4. It is also clear that as the Reynolds number increases, the point of maximum shear stress moves upsteam owing to the slope of the tube wall, Such a result has been noticed by MANTON (1971) for Newtonian fluids. For a comparative study the variation of shear-stress distribution "Cs/'~N at different cross-sections is
0.~l
3-16
o.E~
0.12
O.z
0-08 1
0;
0.0~
x
0
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1,7 1.8 ~'SIrN
Fig. 5
-1"0 Fig. 3
6 x --
Re=250 //
4O
Re=150 ./ /
/: b.
in Fig. 5, where Zs and zN are, respectively, the 1-0 provided w for stenosed walls and nonstenosed walls. It is clear
Axial velocity profiles for micropolar fluid, Re = 250
5O
t 3O
sTeparation streamline
-../
Re=130~i"
that the wall shear stress increases with the height of the stenosis, this increase is less owing to the micropolar behaviour of the fluid. The flow separation is said to take place when Zw = 0 on the wall. Consequently, flow separation and reattachment can be studied. Using our solutions obtained earlier, a phenomenon of separation in axisymmetric stenosis is given in Fig. 6 with localised regions of recirculating flow (a schematic diagram). As
seporcttion point~ ~--separQted region
///////~//
////S}/ //~// i'LL__
20
!~ /Re:O
0 -5
~tion
i!~\/
10
110
0
1.'0
amline
2-()
xlz 0
Fig. 4
Variation of shear stress Zs/zNfor different 3/a and for different cross-section x when n = 1
Fig. 6
Shear stress distribution for micropolar fluid (n = l)
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July 1983
Schematic diagram of axisymmetric stenosis
the
separation
in
OR
443
the Reynolds number is increased gradually, a critical value is reached at which laminar separation takes place. When the flow is increased beyond this limiting value, eddies containing recirculating fluid are likely to appear: The flow is still laminar both outside and within the separated region. To understand the phenomenon in more detail, Table 1 gives the values of critical Reynolds number for different values of n at different cross-sections. It is interesting to note from this Table that the critical Reynolds number increases with increasing value of n and X/Zo, which in turn suggests that, for rheologically complex fluids, the onset of separation takes place at a relatively larger Reynolds number compared with Newtonian fluids. Incidentally, CHow and SODA (1972) have compared their critical Reynolds numbers with those of FORRESTER and YOUNG (1970) for Newtonian fluids. It is obvious that critical Reynolds numbers when n = 0 (in Table 1) will coincide with their value. Another factor of great importance is the impedance. This can be calculated by integrating eqn. 34 if we fix particular form for r~(x). In fact, by taking ra(x ) to be a cosine curve, we get
Reynolds number and wall shear stress are all modified due to the rheologically complex fluids.
1"9V
16
9
a (1
Fig. 7 AN
la2(a-- 6) 2 x (1-12c+30c2-20c3)}]
n=O
Distribution of impedance Jbr di[]i~rent values of micropolar fluid
10-0 M
9.c
+ l-Re (97 + llcQ{i/3 + F(fl)}ZoC
k
(1 - 10c +
6(5 + ct)a2(a - ~)2
30C 2 -- 35C 3
+ 14C4)]
8'(] .(43)
.~7"C ~6'C
where
~ 5"C
c= ~1/2+ !1-2a/6)1) } ( 4x/a/6(a/6 -
M
.E ~.c
Fig. 7 represents the impedance for different n. A comparison of the graph with the results obtained by SHUKLA et al. (1980) is also given. Fig. 8 depicts the impedance for different values of micropolar parameter. It is interesting to conclude from these Figures that impedance decreases for increasing values of the non-Newtonian parameter, whether the nonNewtonian character is described by micropolar or power-law model or Casson's model. In conclusion, it should be mentioned that the results of MANTON (1971), CHOW and SODA (1972) and WILLIAMSand JAVADPOUR(1980) are mainly for low and moderate Reynolds number, whereas in the present case the analysis is exclusively for large Reynolds numbers. It is seen that the presence of separation in the flow field, the critical range of 444
//
3<
2"C 1"r
o
0"04 0"08 0"12 0"16 0"20 0"24 0"28 0"32 0
Fig. 8
Comparative study of impedance for different classes of non-Newtonian fluids. M--micropolar fluid; P--power-law model; C-Casson's model
Acknowledgment--The authors are grateful to the referees for their valuable comments which improved the original manuscript.
Medical & Biological Engineering & Computing
July 1983
References
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MANTON, M.J. (1971) Low Reynolds number flow in slowly varying axisymmetric tubes. J. Fluid Mech., 49, 451-459. MORGAN, B. E. and YOUNG, D. F. (1974). An integral method for the analysis of flow in arterial stenoses. Bull. Math. Biol., 36, 39-53. McDONALD, D.A. (1960) Bloodflow in arteries. Williams & Wilkins, Baltimore. McDONALD, D.A. (1978) Steady flow in tubes of slowly varying cross section. J. Appl. Mech., 45, 475-480. McDONALD, D.A. (1979) On steady flow through modelled vascular stenoses. J. Biomech., 12, 13-20. PADMANABHAN, N. (1980) Mathematical model of arterial stenosis. Med. & Biol. Eng. & Comput., 18, 281-286. PADMANABHAN, N. and DEVANATHAN, R. (1981) Mathematical model of an arterial stenosis allowing for tethering, ibid., 19, 385 390. SCHNECK, D. J. and OSTRACH, S. (1975) Pulsatile blood flow in a channel of small exponential divergence--I. The linear approximation for low Reynolds number. J. Fluid Eng., 16, 353 360. SHUKLA, J. B., PARIHAR,R. S. and RAO, B. R. P. (1980) Effects of stenosis on non-Newtonian flow of blood in an artery. Bull. Math. Biol., 42, 283-294. TEXON, M. (1963) The role of vascular dynamics in the development of atherosclerosis. In Atherosclerosis and its origin. SANDLER, M. and BOURNE, G. H. (Eds.), Academic Press, New York. WILLIAMS, E. W. and JAVADPOUR,S. H. (1980) The flow of an elastico-viscous liquid in an axisymmetric pipe of slowly varying cross section. J. Non-Newtonian Fluid Mech., 7, 171-188. YOUNG, D. F. (1968) Effects of a time dependent stenosis on flow through a tube. J. Eng. for Industry, 90, 248 254.
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