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00254
Journal of Mechanics in Medicine and Biology
Vol. 8, No. 4 (2008) 561–576
c World Scientific Publishing Company
A STUDY OF MICROPOLAR FLUID IN AN ANNULAR
TUBE WITH APPLICATION TO BLOOD FLOW
P. MUTHU∗,† , B. V. RATHISH KUMAR and PEEYUSH CHANDRA
Department of Mathematics and Statistics
Indian Institute of Technology, Kanpur
Kanpur-208016, India
∗snklpm@rediffmail.com; [email protected]
Received 23 May 2008
Accepted 26 May 2008
The oscillatory flow of micropolar fluid in an annular region with constriction, provided by variation of the outer tube radius, is investigated. It is assumed that the local
constriction varies slowly over the cross-section of the annular region. The nonlinear
governing equations of the flow are solved using a perturbation method to determine
the flow characteristics. The effect of micropolar fluid parameters on mean flow and
pressure variables is presented.
Keywords: Micropolar fluid; annular tube; oscillatory flow; stenosis.
1. Introduction
The study of flow through an axisymmetric annular tube has been the subject
of many studies, due to its relevance to blood flow in a catheterized artery.
MacDonald1 presented a model of pulsatile flow in an annular tube, and estimated the magnitude of the error in pressure by assuming that the rates of flow
of fluid through the annular and regular tubes are described by the same known
periodic function of time. Back2 estimated the increase in mean flow resistance
during coronary artery catheterization, using models of fluid flow in an annular
tube. This study is based on the observation that, for a pulsatile flow in a tube, the
relation between mean (time-averaged) pressure gradient and flow rate is exactly
the same as its counterpart for steady Poiseuille flow through a rigid tube. Rao
and Padmavathi3 proposed mathematical models by considering fluid flow confined
between two eccentric cylinders in relative motion for the study of catheter movement in blood vessels.
∗ Corresponding
author.
address: Department of Mathematics and Humanities, National Institute of Technology
Warangal, Warangal-506004, Andhra Pradesh, India.
† Present
561
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P. Muthu, B. V. Rathish Kumar & P. Chandra
Investigations have been carried out to observe the influence of the presence/
movement of catheter on blood flow in a stenosed artery, using theoretical models.
Manjula and Devanathan4 proposed a mathematical model to study the effect of
catheter probe in a stenosed artery, and showed that the error in measurements
can be reduced by decreasing the diameter of the catheter. Back et al.5 showed
that mean flow resistance increases in the presence of an annular tube. In a recent
investigation, Sarkar and Jayaraman6 studied (1) the effect of the movement of
the catheter influenced by the pulsatile nature of flow and (2) the contribution
of the steady streaming effect which brings into focus the existence of a non-zero
mean pressure drop in addition to the one predicted by the previous authors, using
Newtonian fluid flow models.
It is well known that many of the physiological fluids behave like suspensions of
deformable or rigid particles in Newtonian fluid. In view of this, some researchers
have used non-Newtonian fluid models for the biofluids. Dash et al.7 estimated
the increased flow resistance in a narrow catheterized artery using the Casson fluid
model. Banerjee et al.8 investigated the changes in flow and mean pressure gradient
across a coronary artery with stenosis in the presence of a translesional catheter
using the finite element method for the Carreau model, a shear-rate-dependent
non-Newtonian fluid model.
Eringen9 proposed the theory of micropolar fluids to study fluids with suspension nature. In this theory, the continuum is regarded as sets of structured particles which contain not only mass and velocity, but also a substructure. That is,
each material volume element contains microvolume elements that can translate
and rotate independently of the motion of macrovolume. In this model, two independent kinematic vector fields are introduced — one representing the translation
velocities of fluid particles; and the other representing angular (spin) velocities of
the particles, called as microrotation vector.10
Ariman11 examined the flow of micropolar fluid in a rigid circular tube, and
observed that it serves as a better model in comparison to the Newtonian one for
the study of blood flow. This is also supported by the investigations of Sawada
et al.12 and Turk et al.13 In view of this, an attempt was made to assess the
magnitude of mean pressure gradient and mean velocity in the flow of micropolar
fluid in an annular region with an elastic outer wall.14 The equations of the fluid
motion supplemented by the response of the wall were solved by the perturbation
technique, assuming small wall diameter variation. The analysis was carried out
for zero mean flow rate and was restricted to small steady streaming Reynolds
number.
Motivated by these studies, the present work attempts to understand the influence of micropolar fluid parameters on the flow rate–pressure gradient relationship in an annular tube with constriction. The nonlinear governing equations for
micropolar fluid flow are solved by using a perturbation scheme in terms of small
values of slope parameter, that is, the ratio of the tube radius to the length of the
spread of constriction. The steady streaming phenomenon, arising primarily from
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563
the interaction of viscous and convective inertia forces due to an oscillating flow
situation, is discussed. The results are presented for mean pressure drop and mean
wall shear stress, and the effects of micropolarity are observed.
2. Formulation of the Problem
Figure 1 shows the schematic diagram of an annular region with a slowly varying
cross-section corresponding to the cylindrical coordinate system (R, Θ, Z), where
Z is the axial coordinate, R is the radial coordinate, and Θ denotes the azimuthal
angle. The geometrical shape of the constricted region is taken as
R0 (1 − f (Z)) if 0 ≤ Z ≤ λ
(1)
Rs (Z) =
otherwise,
R0
where f (Z) describes the geometry of the varying wall and is assumed to have
developed in an axisymmetric manner over the length of the outer tube, and R0
is the radius of the rigid tube in the nonconstricted region. λ is the distance along
the outer tube over which the constriction is spread out. We assume that the inner
tube is located co-axially with the outer tube and that the fluid is flowing in the
annular region kR0 ≤ R ≤ Rs (Z), k < 1.
We consider the oscillatory flow of an incompressible micropolar fluid in the
annular region. Also, the flow is considered to be axisymmetric, fully developed,
and laminar so that the entrance and end effects are neglected. Thus, the governing
equations, in the circular cylindrical coordinates, are given by9
∂W
U
∂U
+
+
= 0,
(2)
∂R
∂Z
R
2
∂U
2µ + κ
∂U
∂U
∂ U
U
∂2U
1 ∂U
∂P
ρ
+U
+W
+
−
+
+
=−
∂τ
∂R
∂Z
∂R
2
∂R2
∂Z 2
R ∂R R2
−κ
R
∂G
,
∂Z
(3)
λ
STENOSIS
Ro
CATHETER
k Ro
STENOSIS
Fig. 1. Geometry of catheterized artery with stenosis.
Z
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P. Muthu, B. V. Rathish Kumar & P. Chandra
2
2µ + κ
∂ W
∂2W
1 ∂W
∂P
+
+
+
∂Z
2
∂R2
∂Z 2
R ∂R
∂G G
+
+κ
,
(4)
∂R R
2
1 ∂G
∂G
∂G
∂ G ∂2G
G
∂G
+
+
+U
+W
= −2κG + γ
−
ρJ
∂τ
∂R
∂Z
∂R2
∂Z 2
R ∂R R2
∂U
∂W
−
+κ
,
(5)
∂Z
∂R
ρ
00254
∂W
∂W
∂W
+U
+W
∂τ
∂R
∂Z
=−
where ρ is the density; µ is the dynamic viscosity coefficient; P is the pressure; τ is
the time; U and W are velocity components in the R- and Z-directions, respectively;
G is the non-vanishing component of the microrotation vector in the Θ direction; J
is the microinertia constant; and κ and γ are the viscosity coefficients for micropolar
fluid, called coefficients of vortex viscosity and gyro-viscosity, respectively.
The oscillatory nature of the fluid will have an influence, however small, on the
instantaneous position of the inner tube, and there may be very little movement
away from its axis. We assume that these effects are very small. However, it is
assumed that the fluid flow has an axisymmetric, periodic motion along the Z-axis
and that it is in phase with the rate of flow with small constant amplitude. In other
words, the wave propagating in the fluid medium is transmitted to the inner tube
as well.
Equations (2)–(5) constitute a nonlinear coupled system of differential equations
governing the oscillatory flow of micropolar fluid in the region bounded by 0 ≤ Z ≤
λ and kR0 ≤ R ≤ Rs (Z). They are to be solved subject to appropriate boundary
conditions. For the problem under study, we apply the following conditions:
U = 0, W = Wc (τ ),
U = 0, W = 0,
G = 0 at R = kR0 ,
G = 0 at R = Rs (Z),
and
where W̃c (τ ) represents the motion of the inner tube. In the absence of movement
of the inner tube, the model corresponds to the oscillatory flow of micropolar fluid
in an annular region between the concentric tubes, one of constant radius and the
other of a cross-section varying along the Z-direction.
In addition to the boundary conditions mentioned above, we assume a constant
volume flux Q at every instant along the annular tube. That is, the flux across a
cross-section of the annular tube is prescribed as
Rs (Z)
2πRW dR = cos(τ ω) for all Z ∈ [0, λ],
(6)
Q(τ, Z) =
kR0
where ω is the frequency of oscillatory flow. This condition is taken to evaluate the
change in pressure gradient due to the presence and movement of the inner tube.
We define the following nondimensional quantities using the characteristic
lengths R0 in the radial direction and λ in the axial direction as well as Ua (typical
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565
axial velocity) as characteristic velocity:
r=
R
,
R0
p=
z=
Z
,
λ
P
,
ρUa2 /δ
t = τ ω,
q=
u=
Q
,
2πR02 Ua
Uλ
,
Ua R0
w=
W
,
Ua
g=
J
,
R02
w̃c =
W̃c
,
Ua
rs =
j=
G
,
Ua /R0
Rs
.
R0
Using these nondimensional quantities in the governing equations and the boundary
conditions, Eqs. (2)–(6) are written in nondimensional form as
∂u ∂w u
+
+ = 0,
(7)
∂r
∂z
r
2
∂ u 1 ∂r
2Re δ 3
∂u
u
2α2 δ 2 ∂u
∂u
2Re ∂p
+
+w
+ δ2
−
+
u
=−
2 + µ1 ∂t
2 + µ1
∂r
∂z
2 + µ1 ∂r
∂r2
r ∂r r2
+ δ4
∂2u
∂g
− 2N 2 δ 2 ,
∂z 2
∂z
(8)
2α2 ∂w
2Re
∂w
∂2w ∂2w
∂w
2Re ∂p
+
+w
+ δ2 2 +
δ u
=−
2 + µ1 ∂t
2 + µ1
∂r
∂z
2 + µ1 ∂z
∂z
∂r2
∂g g
1 ∂w
+ 2N 2
+
+
,
r ∂r
∂r
r
∂g
g
∂2g
∂g
∂ 2 g 1 ∂g
2 ∂g
+ M Re δ u
+w
− 2 + δ2 2
j Mα
= 2 +
∂t
∂r
∂z
∂r
r ∂r
r
∂z
∂u ∂w
−
+ µ1 M δ 2
− 2µ1 M g.
∂z
∂r
(9)
(10)
The boundary conditions become
u = 0, w = w̃c (τ ), g = 0 at r = k,
u = 0, w = 0,
g = 0 at r = rs (z),
rs (z)
q=
rwdr = cos(t),
and
(11)
k
where
α = R0
ωρ
µ
N=
12
,
Re =
µ1
2 + µ1
12
,
ρUa R0
,
µ
M=
δ=
R02
,
l2
R0
,
λ
l2 =
µ1 =
κ
,
µ
γ
.
µ
Here, δ is the geometric parameter, which gives the ratio of the tube radius to
the length of spread of the constriction. The parameters α and Re are the fundamental quantities called Womersley number and Reynolds number, respectively, as
observed in the classical theory governing the oscillatory flow in an annular tube
with constriction.6 The parameters µ1 and M are the nondimensional quantities due
to micropolar fluid flow.14 The nondimensional quantity µ1 characterizes the coupling of Eqs. (8)–(10); and as κ tends to zero, these equations become uncoupled.
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P. Muthu, B. V. Rathish Kumar & P. Chandra
Also, as M tends to infinity, Eqs. (8)–(10) reduce to the classical Navier–Stokes
equations.
Following Kline and Allen,15 Prakash and Sinha,16 and Willson,17 in our study
we assume here that j 1. In view of this, the effect of the microinertia constant
j is neglected and it is taken to be zero in the present study.
The catheter movement w̃c (t) is taken as w̃c (t) = wc cos(t − t0 ), where wc 1
is the maximum velocity amplitude of the moving inner tube and t0 is the phase
lead of the inner tube oscillation over the flow.
3. Analysis
In general, the flow is quite complex and Eqs. (7)–(10) are not amenable to an
analytic solution. We therefore apply a regular perturbation method with δ as a
small parameter. It may be noted that δ 1 corresponds to a slowly varying crosssection of the annular region. Thus, to solve Eqs. (7)–(10), for the velocity field and
the microrotation component, we attempt an approximate solution for u, w, g, p,
and q as a power series in terms of parameter δ:
u = U0 + δU1 + O(δ 2 ).
(12)
Similar expressions can be written for w, g, p, and q. Substituting them in Eqs. (7)–
(11) and collecting the coefficients of like powers of δ on both sides of the equations,
we obtain the following sets of governing equations and boundary conditions for
various orders of flow quantities.
The equations corresponding to the case δ 0 are
∂U0
∂W0
U0
+
+
= 0,
∂r
∂z
r
∂P0
= 0,
∂r
∂G0
∂ 2 W0
2α2 ∂W0
G0
1 ∂W0
2Re ∂P0
2
−
+
2N
+
,
+
=
∂r2
r ∂r
2 + µ1 ∂t
∂r
r
2 + µ1 ∂z
(14)
∂ 2 G0
G0
1 ∂G0
∂W0
− 2 − 2µ1 M G0 − µ1 M
= 0.
+
∂r2
r ∂r
r
∂r
(16)
U0 = 0, W0 = wc cos(t − t0 ),
G0 = 0 at r = k,
U0 = 0, W0 = 0,
G0 = 0 at r = rs (z),
(13)
(15)
and the flux condition becomes
Q0 = cos(t) for all z ∈ [0, 1].
(17)
The equations corresponding to the case δ 1 are
∂W1
U1
∂U1
+
+
= 0,
∂r
∂z
r
(18)
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∂P1
= 0,
∂r
567
(19)
∂G1
∂ 2 W1
2α2 ∂W1
G1
1 ∂W1
2
−
+
2N
+
+
∂r2
r ∂r
2 + µ1 ∂t
∂r
r
∂W0
∂W0
2Re
2Re ∂P1
+
+ W0
=
U0
,
2 + µ1 ∂z
2 + µ1
∂r
∂z
G1
1 ∂G1
∂W1
∂ 2 G1
− 2 − 2µ1 M G1 − µ1 M
= 0.
+
2
∂r
r ∂r
r
∂r
(20)
(21)
U1 = 0, W1 = 0, G1 = 0 at r = k,
U1 = 0, W1 = 0, G1 = 0 at r = rs (z),
and the constant flux condition becomes
Q1 = 0
for all z ∈ [0, 1].
(22)
3.1. Solution for O(1) case
Since the fluid motion is oscillatory with frequency ω, Eqs. (13)–(17) can be satisfied
by a solution of the form
1
(r, z) exp(−it)},
(23)
W0 = {w00 (r, z) exp(it) + w00
2
where the asterisk denotes a complex conjugate. Similar expressions are written for
the quantities U0 , G0 , P0 , and Q0 .
Substituting these equations in Eqs. (13)–(17), we have the following governing
equations for O(1) terms:
∂ 2 w00
∂r2
∂w00
u00
∂u00
+
+
= 0,
∂r
∂z
r
∂p00
= 0,
∂r
∂g00
2α2
g00
1 ∂w00
2Re ∂p00
−i
+
,
+
w00 + 2N 2
=
r ∂r
2 + µ1
∂r
r
2 + µ1 ∂z
∂ 2 g00
g00
1 ∂g00
∂w00
− 2 − 2µ1 M g00 − µ1 M
= 0.
+
2
∂r
r ∂r
r
∂r
(24)
(25)
(26)
(27)
u00 = 0,
w00 = wc exp{−it0 },
g00 = 0
at r = k,
(28)
u00 = 0,
w00 = 0,
g00 = 0
at r = rs (z),
(29)
and the constant flux condition becomes
q00 = 1.0
for all z ∈ [0, 1].
(30)
It may be remarked here that the governing equations for u00 , w00 , g00 , p00 , and
q00 correspond to the case of zeroth order in δ and are proportional to eit . The
solution procedure for finding these quantities is given below.
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On eliminating w00 from Eqs. (26) and (27), we obtain the equation for g00 as
2
2
∂
∂
1 ∂
1
1 ∂
1
2
2
− α1 − 2
− α2 − 2 g00 = 0,
−
−
(31)
∂r2
r ∂r
r
∂r2
r ∂r
r
where
α21 =
A−
√
A2 − 4B
,
2
A = a2 (1 − N 2 ) + iα̃2 ,
α̃2 =
α22 =
A+
√
A2 − 4B
,
2
B = iα̃2 a2 ,
a2 = 2µ1 M,
2α2
.
2 + µ1
The general solution for g00 is
g00 (r, z) = C1 (z)I1 (α1 r) + C2 (z)K1 (α1 r)
+ D1 (z)I1 (α2 r) + D2 (z)K1 (α2 r).
(32)
Substituting Eq. (32) in Eq. (27) and using Eq. (26), one gets
w00 (r, z) = ã1 (C1 (z)I0 (α1 r) − C2 (z)K0 (α1 r))
+ ã2 (D1 (z)I0 (α2 r) − D2 (z)K0 (α2 r)) −
R̃e ∂p00
,
β ∂z
(33)
where
ã1 =
1
{a1 (α21 − β) + 2N 2 α1 } + a1 ,
β
a1 =
2
α1
−
,
µ1 M
α1
a2 =
ã2 =
2
α2
−
,
µ1 M
α2
1
{a2 (α22 − β) + 2N 2 α2 } + a2 ,
β
β = iα2 ,
and R̃e =
2Re
.
2 + µ1
From the continuity equation (24), we get u00 (r, z) as
ã1 (1)
(1)
u00 (r, z) = −
(C (z)I1 (α1 r) + C2 (z)K1 (α1 r))
α1 1
ã2 (1)
(1)
(D (z)I1 (α2 r) + D2 (z)K1 (α2 r))
α2 1
(1) D̃1 (z)
R̃e r ∂p00
,
+
−
2β
∂z
r
+
(34)
where the superscripts with parentheses indicate derivatives with respect to z. D̃1 (z)
is obtained by using the boundary condition for u00 at r = k, and is given by
ã1 (1)
(1)
D̃1 = k
(C (z)I1 (α1 k) + C2 (z)K1 (α1 k))
α1 1
(1) R̃e k ∂p00
ã2 (1)
(1)
+ (D1 (z)I1 (α2 k) + D2 (z)K1 (α2 k)) −
.
(35)
α2
2β
∂z
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569
00
The other unknowns, namely, C1 (z), C2 (z), D1 (z), D2 (z), and ∂p
∂z , are obtained
by the boundary conditions (28)–(30), and are given by the following system of
equations:
C1 (z)I1 (α1 k) + C2 (z)K1 (α1 k) + D1 (z)I1 (α2 k) + D2 (z)K1 (α2 k) = 0,
(36)
C1 (z)I1 (α1 rs (z)) + C2 (z)K1 (α1 rs (z)) + D1 (z)I1 (α2 rs (z))
+ D2 (z)K1 (α2 rs (z)) = 0,
(37)
ã1 (C1 (z)I0 (α1 k) − C2 (z)K0 (α1 k)) + ã2 (D1 (z)I0 (α2 k)
− D2 (z)K0 (α2 k)) −
R̃e ∂p00
= wc exp{−it0 },
β ∂z
(38)
ã1 (C1 (z)I0 (α1 rs (z)) − C2 (z)K0 (α1 rs (z))) + ã2 (D1 (z)I0 (α2 rs (z))
R̃e ∂p00
= 0.
β ∂z
Furthermore, the constant flux condition implies
ã1
C1 (z) (rs (z)I1 (α1 rs (z)) − kI1 (α1 k))
α1
ã1
+ C2 (z) (rs (z)K1 (α1 rs (z)) − kK1 (α1 k))
α1
ã2
+ D1 (z) (rs (z)I1 (α2 rs (z)) − kI1 (α2 k))
α2
ã2
+ D2 (z) (rs (z)K1 (α2 rs (z)) − kK1 (α2 k))
α2
∂p00
R̃e 2
2
(r (z) − k )
−
= 1.
2β s
∂z
− D2 (z)K0 (α2 rs (z))) −
(39)
(40)
Since it may not be possible to write an explicit analytical expression for C1 (z),
00
C2 (z), D1 (z), D2 (z), and ∂p
∂z , we solve them numerically and calculate the velocity
profiles and microrotation fields corresponding to the case of zeroth-order approximation of δ.
3.2. Solution for O(δ) case — steady streaming analysis
In this case, we refer to Eqs. (18)–(22). Equation (20) suggests the following form
for W1 :
(r, z)e−2it ].
W1 = ws (r, z) + [w11 (r, z)e2it + w11
(41)
Similar expressions are taken for U1 , G1 , and P1 .
In the following, we are interested in the time mean quantities. The time mean
velocity w̄ is defined as
2π
1
w(r, z, t)dt.
(42)
w̄ =
2π 0
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In view of Eq. (12), the form of flow variable quantities, and using Eqs. (23) and
(41), we get
w̄ = δws (r, z) + O(δ 2 ).
(43)
Similarly, time mean pressure is obtained as p̄ = δps (z) + O(δ 2 ). It may be noted
that this steady component arises from the interaction of viscous and convective
inertia forces, which exist in an oscillating flow situation due to the nonlinear term
on the left-hand side of Eq. (20). This is described in the literature as a secondary
“streaming effect”.18
In this analysis, we consider the time average of velocity components over a
period of oscillation of the flow up to O(δ). It follows from Eqs. (12), (23), and (41)
that ws is the only nonvanishing quantity which represents the mean flow in the
axial direction z. Similarly, us , gs , ps , and qs represent the mean radial velocity,
mean microrotational velocity, mean pressure, and mean flow rate, respectively.
Thus, we have from Eqs. (18)–(22)
∂ws
us
∂us
+
+
= 0,
∂r
∂z
r
∂ps
= 0,
∂r
∂ 2 ws
∂gs
gs
1 ∂ws
2
+
2N
+
+
∂r2
r ∂r
∂r
r
∂ps
∂w
∂w00
∂w
∂w00
+ R̃e u00 00 + u00
+ w00 00 + w00
= R̃e
,
∂z
∂r
∂r
∂z
∂z
gs
1 ∂gs
∂ws
∂ 2 gs
− 2 − 2µ1 M gs − µ1 M
= 0.
+
∂r2
r ∂r
r
∂r
(44)
(45)
(46)
(47)
us = 0,
ws = 0,
gs = 0
at r = k,
(48)
us = 0,
ws = 0,
gs = 0
at r = rs (z),
(49)
and qs = 0 for all z ∈ [0, 1].
(50)
Using Eqs. (46) and (47), we get the governing equation for gs as
∂ 2 gs
gs
a2 R̃e
1 ∂gs
a2 R̃e ∂ps
a2
2
−
r
+
F
C3 (z),
+
−
α
g
=
(r,
z)
+
s
1
3
∂r2
r ∂r
r2
4
∂z
2r
2r
where
F1 (r, z) =
r
k
(51)
∂w00
∂w00
∂w00
∂w00
+ u00
+ w00
+ w00
r u00
dr,
∂r
∂r
∂z
∂z
(52)
α23 = a2 (1 − N 2 ), and C3 (z) is the integration constant (to be determined).
The solution for gs (r, z) is written as
gs (r, z) = D3 (z)I1 (α3 r) + D4 (z)K1 (α3 r) −
+
R̃e r
∂ps
2
4(1 − N ) ∂z
1
a2 R̃e
(K1 (α3 r)V1 + I1 (α3 r)V2 ) −
C3 (z),
8
2(1 − N 2 )r
(53)
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where
V1 (r, z) = −
r
k
I1 (α3 r)F1 (r, z)dr
and V2 (r, z) =
571
r
k
K1 (α3 r)F1 (r, z)dr.
Using Eqs. (46), we can calculate ws (r, z) as
r
1
r2 R̃e ∂ps
+ Re
F1 (r, z)dr + C3 (z) log(r)
ws (r, z) =
4 ∂z
r
kr
+ C4 (z) − 2N 2
gs (r, z)dr,
(54)
k
where C3 (z) and C4 (z) are functions of z only (to be determined).
s
Thus, C3 (z), C4 (z), D3 (z), D4 (z), and ∂p
∂z have to be determined from the
boundary conditions for gs and ws , i.e. Eqs. (48)–(50). They give the following
system of equations:
D3 (z)I1 (α3 k) + D4 (z)K1 (α3 k) −
+
a2 R̃e
1
(K1 (α3 k)V1 + I1 (α3 k)V2 ) −
C3 (z) = 0,
8
2(1 − N 2 )k
D3 (z)I1 (α3 rs (z)) + D4 (z)K1 (α3 rs (z)) −
+
R̃e k
∂ps
4(1 − N 2 ) ∂z
R̃e rs (z) ∂ps
4(1 − N 2 ) ∂z
a2 R̃e
1
(K1 (α3 rs (z))V1 + I1 (α3 rs (z))V2 ) −
C3 (z) = 0,
8
2(1 − N 2 )rs (z)
k 2 R̃e ∂ps
+ C3 (z) log(k) + C4 (z) = 0.
4 ∂z
Since
rs (z)
gs (r, z)dr =
k
(55)
(56)
(57)
D3 (z)
(I0 (α3 rs (z)) − I0 (α3 k))
α3
−
R̃e [rs2 (z) − k 2 ] ∂ps
D4 (z)
(K0 (α3 rs (z)) − K0 (α3 k)) −
α3
8(1 − N 2 ) ∂z
R̃e
1
[log(rs (z)) − log(k)]C3 (z) +
2
2(1 − N )
8(1 − N 2 )
rs (z)
F1 (r, z)
dr ,
× α3 (I0 (α3 rs (z))V1 − K0 (α3 rs (z))V2 ) −
r
k
−
(58)
the following can be obtained from the boundary condition of ws at rs (z):
−
2N 2 D4 (z)
2N 2 D3 (z)
{I0 (α3 rs (z)) − I0 (α3 k)} +
{K0 (α3 rs (z)) − K0 (α3 k)}
α3
α3
R̃e
N 2 [rs2 (z) − k 2 ] ∂ps
+
rs2 (z) +
4
(1 − N 2 )
∂z
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P. Muthu, B. V. Rathish Kumar & P. Chandra
N2
[log(r
(z))
−
log(k)]
C3 (z) + C4 (z)
s
(1 − N 2 )
R̃e
N2
= − F2 (rs (z), z) 1 +
4
(1 − N 2 )
+
+
log(rs (z)) +
R̃e N 2 α3
{I0 (α3 rs (z))V1 − K0 (α3 rs (z))V2 }.
4(1 − N 2 )
(59)
Now, the constant flux condition (50) implies
r4
16
rs (z)
rs (z)
R̃e rs (z)
∂ps
R̃e
+
rF2 (r, z)dr + C3 (z)
r log(r)dr
∂z
4 k
k
k
2 rs (z)
rs (z)
r
+
C4 (z) − 2N 2
rF3 (r, z)dr = 0,
2 k
k
where
F2 (r, z) =
k
r
1
F1 (r, z)dr
r
and F3 (r, z) =
k
(60)
r
gs (r, z)dr.
The above system of linear equations for the five unknowns, viz., C3 (z), C4 (z),
s
D3 (z), D4 (z), and ∂p
∂z , is solved numerically at each axial location z. Solving the
s
above equations, we get the induced pressure gradient ∂p
∂z along with other flow
quantities. This mean pressure gradient is the first correction to the pressure gradient calculated in the zeroth-order approximation of δ. Using these values, ws can
be determined from Eq. (54).
4. Results and Discussion
The objective of this analysis is to study the steady streaming concept involved
in the oscillatory flow of micropolar fluid in an annular tube with constriction
provided by the variation of the outer tube radius, and the resulting mean quantities
that characterize the flow situation. The numerical calculations have been made by
choosing a bell-shaped constricted geometry rs (z) = 1 − exp{−[π(z − 0.5)]2 } for
0 ≤ z ≤ 1. Here, is the nondimensional length of maximum protuberance towards
the catheter and it is located axisymmetrically at z = 0.5. In all of our numerical
calculations, is fixed at 0.15 for mild constriction. Furthermore, we have used the
polynomial expressions for the Bessel functions19 in our numerical calculations.
It may be recalled that µ1 and M characterize the coefficient of vortex viscosity κ
and the coefficient of gyro-viscosity γ of micropolar fluids, respectively. An increase
in κ is reflected as an increase in the parameter µ1 , while an increase in γ results in
decreasing values of M . It may be mentioned here that in the context of blood flow
studies, the viscosity ratio µ1 represents a polar effect which occurs between blood
corpuscles and fluid.12 The microstructure size effect parameter M means the ratio
of a corpuscle to the radius of annular region.
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573
4.1. Mean pressure drop
z s
The mean pressure drop is defined as ∆p¯s = δ∆ps (z), where ∆ps (z) = 0 ∂p
∂z dz.
Since δ is constant, we discuss about ∆ps . To obtain ∆ps , the calculated values
s
of ∂p
∂z are integrated numerically using the trapezoidal rule. It may be noted that
∆ps is the correction to the pressure drop value corresponding to O(1) results.
Figure 2 shows the variation of ∆ps along the axial direction z for various
values of catheter radius k and for micropolar parameters. For fixed values of µ1
and M , there is an increase in the mean pressure drop value as k ranges from
0.3 to 0.5, when α = 5.0 and wc = 0.0. It is known that, for steady flow in a
constricted tube, the pressure drop increases with an increase in the protuberance
of the constriction when the flow rate is constant. From our analysis, we observe
that the mean pressure drop also increases due to the size of catheter for micropolar
fluid flow. The maximum value is indicated at the peak of the stenosis.
Figures 2(a) and 2(b) show the effect of µ1 on ∆ps for M = 1.0. An increase in
µ1 implies an increase in the mean pressure drop. Thus, an increase in the viscosity
of fluid influences the mean pressure drop, and shows that the size of catheter as
well as the fluid viscosity change the flow characteristics.
2.5
0.18
0.16
µ 1 = 1.0
k = 0.5
0.14
0.12
k = 0.5
2
M = 1.0
k = 0.4
∆ ps
∆ ps
M = 1.0
1.5
k = 0.3
0.1
k = 0.3
0.08
µ 1 = 10.0
k = 0.4
1
0.06
0.5
0.04
0.02
0
0
−0.02
0
0.2
0.4
0.6
0.8
−0.5
1
0
0.2
0.4
0.6
0.8
1
Axial distance (z)
Axial distance (z)
(a)
(b)
0.2
0.2
0
0
−0.2
k = 0.5
−0.6
−0.8
k = 0.3
k = 0.4
µ 1 = 10.0
−1
k = 0.3
M = 10.0
−1.2
M = 10.0
−1
−1.2
−0.6
−0.8
µ 1 = 1.0
k = 0.4
s
−0.4
−0.4
∆p
s
∆p
−0.2
k = 0.5
−1.4
0
0.2
0.4
0.6
Axial distance (z)
(c)
0.8
1
−1.6
0
0.2
0.4
0.6
0.8
1
Axial distance (z)
(d)
Fig. 2. Distribution of mean pressure drop ∆ps along the stenotic length z for different catheter
size k when α = 5.0 and wc = 0.0.
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P. Muthu, B. V. Rathish Kumar & P. Chandra
574
For M = 10.0 [Figs. 2(c) and 2(d)], ∆ps becomes negative for all values of
k = 0.3 to 0.5. This indicates that, for some values of M , there may be zero
correction to pressure drop value corresponding to the O(1) result, thus indicating
a strong dependence of ∆ps on the micropolar parameter M . For M = 10, variation
of µ1 implies a quantitative difference in the absolute value of ∆ps .
4.2. Mean wall shear stress
The mean wall shear stress (WSS) due to steady streaming is given by
∂ws
.
γ1 (z) =
∂r r=rs (z)
Figure 3 depicts graphs of γ1 , with respect to the axial coordinate z, to observe
the influence of catheter size in the presence of micropolar effects. In all cases, it
is seen that γ1 reaches a maximum immediately preceding the throat and then
rapidly decreases to zero at the throat of constriction; in the diverging section of
15
6
k = 0.5
k = 0.5
10
k = 0.4
5
4
M = 1.0
2
0
γ1
γ1
k = 0.3
µ 1 = 1.0
−2
−10
−4
−15
0
0.2
0.4
0.6
0.8
−6
1
M = 1.0
k = 0.3
0
−5
µ 1 = 10.0
k = 0.4
0
0.2
(a)
0.8
1
4
k = 0.5
8
µ 1 = 1.0
3
k = 0.5
µ 1 = 10.0
2
k = 0.4
M = 10.0
k = 0.4
6
M = 10.0
4
1
0
γ1
2
γ1
0.6
(b)
10
k = 0.3
−2
0
k = 0.3
−1
−4
−2
−6
−3
−8
−10
0.4
Axial distance (z)
Axial distance (z)
0
0.2
0.4
0.6
Axial distance (z)
(c)
0.8
1
−4
0
0.2
0.4
0.6
0.8
1
Axial distance (z)
(d)
Fig. 3. Distribution of mean shear stress γ1 along the stenotic length z for different catheter size
k when α = 5.0 and wc = 0.0.
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575
the constriction, γ1 reaches a minimum and approaches to zero. A similar profile
has been predicted for γ1 in the Newtonian fluid flow study.6
From Figs. 3(a) to 3(d), it is seen that the peak value of the profile increases with
increasing values of catheter size k. An increase in the viscosity ratio parameter µ1
decreases these values. Thus, an enhanced viscous effect implies smaller values of
mean WSS at the constricted wall. These values are further reduced by increasing
the size parameter M .
5. Conclusions
The main contribution of the present paper is to see the micropolar nature of blood
flow in a catheterized artery. The analysis was carried out for micropolar fluid flow
in an annular tube with constriction provided by the variation of outer tube radius.
It was found that, although the mean flow rate is zero, steady streaming analysis
predicts a nonvanishing mean pressure drop. The effects of catheter size on wall
shear stress have also been discussed.
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