misra2011 A NUMERICAL MODEL FOR THE

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Journal of Mechanics in Medicine and Biology
Vol. 11, No. 3 (2011) 547–562
c World Scientific Publishing Company
DOI: 10.1142/S0219519410003794
A NUMERICAL MODEL FOR THE
MAGNETOHYDRODYNAMIC FLOW
OF BLOOD IN A POROUS CHANNEL
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J. C. MISRA∗,§ , A. SINHA† and G. C. SHIT‡
∗Professor
of Applied Mathematics and Pro Vice-Chancellor (Academic)
Siksha O Anusandhan University, Bhubaneswar, India
†School of Medical Science and Technology
Indian Institute of Technology, Kharagpur, India
‡Department
of Mathematics, Jadavpur University, Kolkata
§misrajc@rediffmail.com
Received 25 November 2009
Revised 9 June 2010
Accepted 30 June 2010
Magnetohydrodynamic (MHD) principles may be used to study the flow of arterial blood
under the action of an applied magnetic field. Such studies are of potential value in the
treatment of cardiovascular disorders that may be associated with accelerated circulation. With an aim to providing a generalized model for studying the flow of blood in
an electromagnetic field environment, a numerical model is developed here, by treating
blood as a non-Newtonian fluid, the motion of which is taken to be governed by Walter’s B-fluid model. The channel flow characteristics of the fluid are studied here, when
the channel is porous and is subjected to an external magnetic field. Using the similarity transformation and boundary layer approximations, the associated nonlinear partial
differential equations of the problem are reduced to nonlinear ordinary differential equations. These are solved numerically by developing a finite difference scheme. The study
provides useful estimates for the influence of Reynolds number Re, Hartmann number
M , and viscoelastic parameter K1 on the flow characteristics. It bears the potential to
explore some important information about the hemodynamical flow of blood in an artery
when it is under the action of an external magnetic field.
Keywords: Walter’s B-fluid; boundary layer flow; viscoelasticity.
1. Introduction
It is known that blood is a suspension of various cells (the major component being
the erythrocytes) in plasma. Since erythrocytes have small negative charge, an
applied magnetic field can influence the movement of erythrocytes and thereby the
flow of blood is likely to be affected. Information available in the literatures regarding the exact mechanism by which the rheological properties of blood are affected
§ Corresponding
author.
547
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J. C. Misra, A. Sinha & G. C. Shit
by a magnetic field is, however, inadequate. Diamagnetic susceptibility of oxidized
hemoglobin and the paramagnetic susceptibility of non-oxidized hemoglobin were
studied by Pauling and Coryell.1 Higashi and his associates2 and Yamagishi3 investigated the orientation of erythrocytes in magnetic fields of strengths 8T and 4T
respectively. They proclaimed that the erythrocytes orient themselves with their
flat side in the direction of the magnetic field. A three-parameter rheological model
was used by Kirkovskii et al.4 in order to study the influence of a variable magnetic
field on the rheological properties of blood of patients suffering from rheumatoid
arthritis. The parameters of the model are related to the hydrodynamic properties
of erythrocyte suspension. They observed that out of the three parameters, only
one has the capacity to affect the rheological properties of blood to an appreciable
extent.
Nomenclature
η
σ
ρ
ν
B0
k0
H
(u, v)
Re
M
K1
:
:
:
:
:
:
:
:
:
:
:
Non-dimensional distance
Electrical conductivity
Density of blood
Kinematic viscosity of blood
Strength of the applied magnetic field
Coefficient of viscoelasticity
Channel width
Velocity components along x- and y-directions respectively
Reynolds number
Magnetic parameter
Viscoelastic parameter
The study of blood flow under different situations was carried out by several
authors. A mathematical model was developed by Misra and Shit5 for studying
blood flow through a stenosed arterial segment by taking into account the slip
velocity at the wall of the artery. Later on, Misra et al.6 theoretically analyzed
the flow of blood obeying Casson’s fluid model through an artery having multiple
stenoses. Fung and Sobin7 studied the flow of blood between two endothelial layers,
using the sheet model. The mass transfer problem in an alveolar sheet was treated
by Tang and Fung.8 McCracken9 used vortex-grid methods to investigate blood
flow through the heart valves. The morphometric basis for sheet flow was studied
by Sobin et al.10 Peskin11 used numerical techniques to examine the flow of blood
in the heart. In recent years, the subject hydromagnetics has attracted the attention of many researchers, due not only to its inherent importance, but also to its
multi-fold applications to problems having physiological/geophysical/astrophysical
significance. The primary contributors to the magnetohydrodynamic (MHD)12–14
blood flow studied characteristics of blood flow in lungs, where blood can be visualized as flowing between the opposing layers of capillary endothelium, held apart by
endothelium-covered “posts” made of special tissue. The capillary endothelium is,
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Magnetohydrodynamic Flow of Blood
549
in turn, covered by a thin layer (interstitial space) lining the alveoli, which has been
treated as a porous medium. The application of magneto-hydrodynamic principles
is one method for effecting the flow field in a desired direction by altering the structure of the boundary layer. In fact, the application of MHD principles to biology
and medicine has been receiving growing interest of physiologists, medical practitioners, and fluid dynamicists, owing to its importance in biomedical engineering
as well as in the treatment of various pathological conditions.
Vardanyan15 explored the potential use of MHD principles in prevention and
rational therapy of arterial hypertension. He reported that a magnetic field applied
in a direction transverse to an artery bears the potential to alter the flow rate of
blood. While investigating the effect of a magnetic field on the apparent viscosity of human blood, Haik et al.16,17 observed that blood viscosity is significantly
affected by the action of a strong magnetic field. This can be attributed to the
variation in the orientation of the erythrocytes due to the action of the magnetic
field.
Mathematical modeling of blood flow in a channel with stretching walls was
performed by Misra et al.18,19 by treating blood as a non-Newtonian viscoelastic
fluid. In a separate communication, Misra et al.20 presented a mathematical analysis for the electrical and magnetic fields in osseous tissues induced due to the
propagation of torsional waves. The analysis was carried out by paying due attention to the piezoelectric behavior of osseous tissues, in conformity to experimental
reports. Misra et al.21,22 investigated the steady flow of an electrically conducting
fluid in a parallel plate channel in the presence of a uniform transverse magnetic
field. A numerical model was formulated and analyzed by Misra and Shit23 with
the motivation of studying the flow of blood through arteries under the action of a
transverse magnetic field. They examined the cases of arteries of various sizes and
paid due attention to the elastic response of arteries.
From the discussion made above, it turns out that the hydrodynamic changes
in the flow of blood can take place due to certain external factors. It is known
from the study of magnetohydrodynamics that when a transverse magnetic field is
applied externally to an electrically conducting fluid, electric currents are induced
in the fluid. The interaction between these induced currents and the applied magnetic field produces a body force (Lorentz force) which tends to retard the movement of the fluid. Since blood is an electrically conducting fluid, the motion of
blood is expected to be controlled by applying a magnetic field. Thus studies concerned with the effects of a magnetic field on the flow characteristics of blood are
of profuse interest to physiologists and clinicians. A theoretical analysis for the
flow of blood in ascending and descending aortic segments in the presence of a
magnetic field was performed by Kinouchi et al.24 It was observed by Fukuda and
Kaibara25 as well as by Stoltz and Lucius26 that, under certain conditions, blood
exhibits viscoelastic behavior which may be attributed to the viscoelastic properties
of the individual erythrocytes and the internal structures formed by cellular
interactions.
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J. C. Misra, A. Sinha & G. C. Shit
The present study is motivated toward a theoretical investigation of the MHD
boundary layer flow of blood in the aortic arch. The non-Newtonian character of
the fluid is considered to obey Walter’s liquid-B model. This makes the equation
of motion highly nonlinear. It is a partial differential equation of order three. The
form of the equation being complex, it is not possible to find its exact solution. The
method of perturbation has been used, and then a finite difference scheme has been
developed to achieve the solution.
The influence of various parameters, such as Reynolds number, Hartmann number, and the viscoelastic parameter, has been examined for the cases of both
suction and injection. The results are presented in graphical form. The graphs
highlight the velocity distribution of blood flowing in the aortic arch under various situations and lead to several important predictions. The results indicate very
clearly the threshold value of the magnitude of the magnetic field strength that
should be exposed to the human body. The results of the study are expected to
be of profound importance to medical surgeons in regulating blood flow during
surgery.
2. Mathematical Formulation of the Problem
The circulatory system mainly consists of three-dimensional (3D) vessels. However,
in some cases, such as in micro-vessels of the lungs, motion of blood is approximately
2D and it can be considered as channel flow. We consider steady flow in a channel
having porous boundaries bounded by two thin parallel plates, under the action of
a transverse magnetic field. A physical sketch of the geometry is shown in Fig. 1.
The x-axis is taken along the centre line of the channel, parallel to the channel
surface and y-axis in the transverse direction. The flow is taken to be symmetric
about x-axis. The porous walls of the channel can be represented as y = H/2 and
y = −H/2 (H being the channel width). The fluid injection/extraction is supposed
to take place through the porous walls with velocity V /2. Here, V > 0 corresponds
to suction and V < 0 to injection.
y
B0
B0
y=H/2
y=0
x
B0
Fig. 1. Schematic representation of the model geometry.
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Magnetohydrodynamic Flow of Blood
551
Let u and v be the velocity components along x-axis and y-axis respectively
and B0 be the applied magnetic field. For small magnetic Reynolds number, the
induced fields are negligibly small. In the absence of pressure gradient, the equation
for MHD boundary layer flow of an incompressible fluid (considered here as Walter’s
liquid-B fluid) is
∂u
∂ 2u
∂ 3 u ∂u ∂ 2 u ∂u ∂ 2 u
∂3u
∂u
+v
= ν 2 − k0 u
+
v
+
−
u
∂x
∂y
∂y
∂x∂y 2
∂y 3
∂x ∂y 2
∂y ∂x∂y
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−(σB02 /ρ)u,
(1)
while the continuity equation is
∂u ∂v
+
= 0,
∂x ∂y
(2)
in which ρ, ν, B0 , σ, and k0 are respectively the density, kinematic viscosity, strength
of the applied magnetic field, electrical conductivity, and the coefficient of viscoelasticity of the fluid. Assuming that the flow is symmetric about the central line y = 0
of the channel, we focus our attention to the flow in the region 0 ≤ y ≤ H2 only.
Then the boundary conditions to be taken care of are as follows:
∂u
= v = 0 at y = 0
∂y
(3)
and
u = 0, v =
V
2
at y =
H
.
2
(4)
We now introduce the non-dimensional quantities defined by
ξ=
x
y
, η = , u = V ξf (η)
H
H
and v = −V f (η).
(5)
It may be noted that the continuity equation (2) is automatically satisfied.
In terms of the non-dimensional variables, Eq. (1) reads
f − M f + Re(f f − f 2 ) = K1 [2f f − f f iv − f 2 ]
(6)
σB 2 H 2
0
in which Re = VνH is the Reynolds number, M = Ha2 =
is the square of
ν
V k0
the Hartman number and K1 = Hν is the viscoelastic parameter.
With the use of transformation (5), the boundary conditions (3) and (4) become
f (η) = f (η) = 0 at η = 0
(7)
and
f (η) = 0 and f (η) = −
1
2
at η =
1
.
2
(8)
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J. C. Misra, A. Sinha & G. C. Shit
3. Perturbation Analysis
Considering the viscoelastic parameter K1 to be small, we apply a perturbation
expansion approach by writing
f = f0 (η) + K1 f1 (η) + K12 f2 (η) + · · ·
(9)
Substituting this into Eq. (6), equating coefficients of like power of K1 and ignoring
the quadratic and higher powers of K1 , we obtain
f0 − M f0 + Re(f0 f0 − f02 ) = 0
(10)
f1 − M f1 + Re(f0 f1 + f1 f0 − 2f0 f1 ) = 2f0 f0 − f0 f0iv − f02 .
(11)
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and
Using (9) in (7) and (8), the boundary conditions for f0 and f1 become
1
1
−1
, f0
f0 (0) = f0 (0) = 0 and f0
=
=0
2
2
2
(12)
and
f1 (0) = f1 (0) = f1 (1/2) = f1 (1/2) = 0.
(13)
In the next section, we develop a numerical scheme for solving the nonlinear equations (10) and (11) subject to the boundary conditions (12) and (13).
4. Numerical Methods
The system of coupled nonlinear ordinary differential equations (10) and (11) along
with the boundary conditions (12) and (13) has been solved numerically by employing a finite difference scheme. We use Newton’s linearization method to linearize
the discretized equations. The essential feature of this technique is that it is based
on a finite difference scheme, which has better stability characteristics and is quite
efficient. It is found to yield sufficiently accurate results. The finite difference technique leads to a system which is tri-diagonal and therefore has speedy convergence.
Moreover, it brings about economical memory space to store the coefficients. The
computational work has been carried out by taking δη = 0.0125. We have examined
that further reduction in δη does not produce any significant change. This ensures
the stability of our numerical scheme. The finite difference scheme that we have
developed is briefly described below.
Substituting f0 = F in Eqs. (10) and (12), we get
F − M F + Re(f0 F − F 2 ) = 0
(14)
and
f0 (0) = F (0) = 0,
F (1/2) = 0,
f0 (1/2) = −1/2.
(15)
Similarly, substituting f1 = G in Eqs. (11) and (13), we find
G − M G + Re(f0 G + f1 f0 − 2f0 G) = 2f0 f0 − f0 f0iv − f02
(16)
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Magnetohydrodynamic Flow of Blood
553
and
f1 (0) = G (0) = G(1/2) = f1 (1/2) = 0.
(17)
Using the central difference scheme for derivatives with respect to η, we can write
Ti+1 − Ti−1
+ O((δη)2 )
2δη
(18)
Ti+1 − 2Ti + Ti−1
+ O((δη)2 ),
(δη)2
(19)
(T )i =
and
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(T )i =
where T stands for F or G, i is the grid index in η-direction with ηi = i ∗ δη,
i = 0, 1, . . . , m and δη is the increment along the η-axis. Newton’s linearization
method has been applied to linearize the discretized equations as indicated below.
When the values of the dependent variables at the nth iteration are known, the
corresponding values of these variables at the next iteration are obtained by using
the equation
Tin+1 = Tin + (∆Ti )n
(20)
in which (∆Ti )n represents the error at the nth iteration, i = 0, 1, 2, . . . , n.
5. Results and Discussion
The perturbation method and the numerical procedure outlined in the two preceding sections have been employed to solve the differential equations (10) and (11)
subject to the boundary conditions (12) and (13). Our endeavor here has been confined to obtaining numerical estimates of the flow parameters in the case of blood
flow in the aortic arch. Let us consider that the aortic arch is under the influence
of a strong magnetic field of strength B0 = 8T (Tesla). This was the magnetic field
strength used by Higashi et al.2 in their experimental study on the orientation of
erythrocytes in blood subjected to a strong magnetic field. Further, in finding the
estimates, we have taken blood density, ρ = 1050 kg/m3 and electrical conductivity
of blood, σ = 0.8 s/m. The numerical results estimated by us have been presented
graphically in Figs. 2–9. These figures illustrate the variation of the axial and transverse velocities in the channel flow of blood, with change in Reynolds number Re,
Hartmann number M , and viscoelastic parameter K1 in the cases of both suction
(Re > 0) and injection (Re < 0). As in Refs. 11, 14 and 27, we have examined
different cases taking Re = −10, −5, 5 and 10; K1 = 0.0, 0.005, 0.01 and 0.05, while
different magnetic field strengths examined are given by M = 0, 2, 4, 6, 8 and 10 (cf.
Ref. 28).
Figures 2 and 3 give the distribution of the transverse velocity and the axial
velocity respectively for different values of the magnetic parameter M . Figure 2
shows that f decreases with the magnetic parameter M , in the case of suction.
This implies that the velocity of blood in the transverse direction can be reduced
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J. C. Misra, A. Sinha & G. C. Shit
0
M=2.0
M=4.0
M=6.0
M=8.0
-0.1
f
-0.2
-0.4
-0.5
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 2. Distribution of f for different values of M (when Re = 5.0 and K1 = 0.005).
0
M=2.0
M=4.0
M=6.0
M=8.0
-0.2
-0.4
-0.6
-0.8
f’
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-0.3
-1
-1.2
-1.4
-1.6
-1.8
-2
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 3. Distribution of f for different values of M (Re = 5.0 and K1 = 0.005).
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Magnetohydrodynamic Flow of Blood
0
555
Re=-10.0
Re=-5.0
Re=5.0
Re=10.0
-0.1
f
-0.2
-0.4
-0.5
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 4. Distribution of f for different values of Reynolds number Re (M = 4.0 and K1 = 0.005).
0
Re=-10.0
Re=-5.0
Re=5.0
Re=10.0
-0.5
-1
-1.5
f’
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-0.3
-2
-2.5
-3
-3.5
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 5. Distribution of f for different values of Reynolds number (M = 4.0 and K1 = 0.005).
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J. C. Misra, A. Sinha & G. C. Shit
0
Re=-10.0
Re=-5.0
Re=5.0
Re=10.0
-0.1
f
-0.2
-0.4
-0.5
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 6. Distribution of f for different values of Reynolds number (M = 0.0 and K1 = 0.005).
0
Re=-10.0
Re=-5.0
Re=5.0
Re=10.0
-0.5
-1
-1.5
f’
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-0.3
-2
-2.5
-3
-3.5
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 7. Distribution of f for different values of Reynolds number (M = 0.0 and K1 = 0.005).
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Magnetohydrodynamic Flow of Blood
0
557
K1=0.0
K1=0.01
K1=0.05
-0.1
f
-0.2
-0.4
-0.5
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 8. Variation of f with vertical distance for different values of viscoelastic parameter K1 (M =
10.0 and Re = 5.0).
0
-0.2
-0.4
K1=0.0
K1=0.01
K1=0.05
-0.6
-0.8
f’
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-0.3
-1
-1.2
-1.4
-1.6
-1.8
-2
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 9. Variation of f i in the vertical direction for different values of viscoelastic parameter
K1 (M = 10.0 and Re = 5.0).
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J. C. Misra, A. Sinha & G. C. Shit
by increasing the strength of magnetic field. Figure 3 indicates that the axial velocity decreases when the value of magnetic parameter M increases upto the point
η = 0.16, beyond which a reverse trend is observed. Thus we can make a conjecture that the (resultant) velocity of blood can be controlled by suitably adjusting
(increasing/reducing) the strength of magnetic field.
Figures 4 and 5 depict the effect of the magnitude of the Reynolds number on
the variations of f and f in the cases of both suction and injection. Figure 4 shows
that as in the case of suction, the transverse velocity increases as Reynolds number
reduces; however, in the case of injection, it increases with an increase in the value
of Reynolds number. During suction, blood velocity in the axial direction increases
with a rise in the Reynolds number (cf. Fig. 5) upto a certain height of the channel,
beyond which it decreases with an increase in Reynolds number. It may, however,
be noted that during injection, there is a reverse trend.
Nature of the transverse and axial velocity distributions in the absence of any
external magnetic field has been illustrated in Figs. 6 and 7. A comparison between
Figs. 4 and 6 reveals that the variation of transverse velocity in the presence/absence
of the magnetic field is of similar nature, but the quantitative estimates are different.
A similar observation is made for the axial velocity by comparing Figs. 5 and 7.
Figures 8 and 9 give the distribution of both the velocity components during
suction, corresponding to different values of the viscoelastic parameter, when the
system is subjected to a sufficiently strong magnetic field. Figure 8 shows that the
transverse velocity is enhanced when the viscoelasticity of blood increases. One
0
-0.2
-0.4
Present study
Result of T. Hayat et al. (2006)
-0.6
-0.8
f’
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558
8,
-1
-1.2
-1.4
-1.6
-1.8
-2
0
0.05
0.1
0.15
0.2
0.25
η
0.3
0.35
0.4
0.45
0.5
Fig. 10. Axial velocity distribution for Re = 3.0, M = 6.0 and K1 = 0.005. (Comparison of the
present numerical solution with the analytical solution of Hayat et al.)
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Magnetohydrodynamic Flow of Blood
559
may note from this figure that the transverse velocity decreases in the vertical
direction as blood moves from the axis of the vessel towards the wall and it attains
its minimum at the wall. Figure 9 indicates that during suction, with an increase
in blood viscosity, axial velocity increases upto the point η = 0.3, beyond which it
decreases gradually.
With an aim to validate our numerical model, we have compared our results
for the axial velocity distribution with those reported recently by Hayat et al.29
who carried out a similar study under some simplifying assumptions and obtained
analytical solution by using the homotopy analysis method (HAM). We notice that
the results of our numerical model are in excellent agreement with those reported
by Hayat et al.29 (cf. Fig. 10).
6. Concluding Remarks
The magnetohydrodynamic flow of blood has been the concern in this study. The
solution to the nonlinear equations that govern the flow is obtained by using a
perturbation analysis and by developing a suitable numerical scheme. The analytical
solutions for the hydrodynamic case, disregarding inertia effects and fluid viscosity
can be obtained as a limiting case by choosing M = 0, Re = 0, and K1 = 0.
Magnetic/electromagnetic field therapies have gained sufficient popularity in
recent years. They are particularly used to relieve pain and also to accelerate
fracture healing in bones. These therapeutic procedures have also been being
employed to accelerate the flow of blood and also to treat various health disorders
like arthritis in humans. This study confirms that application of magnetic field is
also very useful to regulate blood flow, particularly during surgery.
Voltage generated due to MHD effect on blood flow in the body may be observed
on electrocardiogram (ECG) during cardiac magnetic resonance imaging (MRI).
The largest MHD voltage magnitude is observed in the direction perpendicular to
the magnetic field lines and the direction of the fluid flow (cf. Refs. 30 and 31).
It has been observed that MHD voltages distort the ECG and result in triggering
problems for MR image acquisition.32 These voltages are also related to the flow
behavior of blood. So an examination of the MHD effect is expected to provide useful
information about the blood flow. The spatial distributions of electromagnetic fields
in human body by switched magnetic field gradients in MRI have been studied
numerically by Bencsik et al.33
The present study is particularly motivated toward examining the flow of blood
in the aortic arch due to multi-fold reasons. The aorta is the largest blood vessel
in the human body, and the velocity of blood is greatest in the aorta. Also, blood
flow through the aortic arch takes place almost perpendicularly to the magnetic
field lines. The present study bears the potential to examine the change that takes
place in blood velocity, as the magnetic field strength (M ), blood viscoelasticity
(measured by K1 ), and Reynolds number (Re) change.
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J. C. Misra, A. Sinha & G. C. Shit
Moreover, although modeling of blood flow in arteries may appear to be more
intuitive by considering the tube flow, it is worthwhile to mention that flow in an
axisymmetric tube resembles the flow behavior in a channel. This has been the
opinion of several previous investigators.34–36 It is with this rationale the problem
has been formulated by considering 2D channel flow. The results presented here are
applicable to the case of blood vessels in several situations mentioned earlier. Such
a consideration is more suitable in the case of vessels having radius ≤ 0.25 mm in
the microcirculatory system where the non-Newtonian behavior of blood is very
prominent and the erythrocyte concentration is quite high. Owing to the high concentration of erythrocytes, even in the normal physiological state, the vessel geometry in many cases takes a 2D form and the flow of blood well approximates to a
channel flow.
The investigation reveals very clearly that blood velocity can be controlled by
suitably adjusting (increasing/decreasing) the magnetic field strength. The results
presented should be of sufficient interest to surgeons who usually want to keep the
blood flow at a desired level during the entire surgical procedure.
Acknowledgment
The authors are thankful to both the reviewers for their kind words of appreciation in respect of the present research as well as for their nice comments on the
applicability of the results presented. The original manuscript has been revised on
the basis of the reviewers’ suggestions. One of the authors (A. Sinha) is grateful to
CSIR, New Delhi for granting SRF to him.
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