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International Journal of Pure and Applied Mathematical Sciences. ISSN 0972-9828 Volume 9, Number 1 (2016), pp. 75-88 © Research India Publications http://www.ripublication.com
Mathematical modeling of blood flow in an inclined tapered artery under MHD effect through porous
medium
Ajay Kumar1, R. S. Chandel2, Rajesh Shrivastava3, Keerty Shrivastava4 & Sanjeet Kumar5
1:SCOPE College of Engineering, Bhopal (M. P)-462026, India,
E-mail: ajaykumarrgpv@gmail. com 2:Government Geetanjali Girls College, Bhopal (M. P)-462001 India,
E-mail: rschandel2009@yahoo. co. in 3:Government Science and Commerce College, Benazir Bhopal (M. P)-462016, India,
E-mail: rajeshraju0101@rediffmail. com 4:Government Post Graduate Bhel College, Bhopal (M. P)-462021, India,
E-mail: keertyshrivastava@yahoo. in 5:Lakshmi Narain College of Technology, Bhopal (M. P)-462021, India,
E-mail: sanjeet_418@yahoo. com
Abstract
The purpose of this work is to study the effect of blood flow in an inclined tapered artery under MHD effect through porous medium. Blood is considered as an electrically conducting Newtonian fluid. In this model, the analytical expressions for the volumetric flow rate, pressure gradient, wall shearing stress, and velocity profile have been derived. The problem is described by the usual MHD equations along with suitable boundary conditions. There is also a noticeable effect of permeability on the volumetric flow rate. Some of the found results show that the flow patterns in non-tapered region ( = 0), converging region ( < 0), and diverging region ( > 0), are effectively influenced by the presence of magnetic field and change in leaning of artery. Keywords: Inclined tapered artery, MHD equations, Stenotic region. AMS Subject Classification (1991): 760z05, 92c35
1. Introduction: Now-a-days many people are suffering from cardiovascular disease such as
76 Ajay Kumar et al
atherosclerosis (medically called stenosis) which is causes of death of people. A blockage by atherosclerosis, which is progressive vascular disease that causes collection of fatty substances, cholesterol, fibrin, calcium, and cellular waste. It is also known as plaque inside the wall of arteries. It leads to the narrowing of the internal space of the arteries and effects as carotid artery (major blood vessels in the neck that supply blood to the brain). When the plaque hardens and narrows down the arteries completely, then the blood and oxygen supply to the brain are reduced. This is one of the major contributing factors to strokes. This leads to block of brain function and the death of people. This is considerable evidence that vascular fluid dynamics plays an important role in the development and progression of arterial stenosis. The theory presented so far is enough for uniform visco-elastic tubes but, is inadequate for real arteries because they are not uniform. They suffer both continuous variation in cross sectional area and dispensability from repeated branching. If we take tapered tube the normal blood flow is disturbed. The main disadvantage is using a tapered geometry however, is the much greater energy losses which may leads to diminished blood flow through the tapered grafts. It is important therefore these losses are quantified and taken account in the design of tapered grafts. Various mathematical models have been investigated by several researchers to explore the behavior of blood under the magnetic field. A number of investigators like Darcy (1937) studied on the flow of fluids through porous media. Korchevskii and Marochnik (1965) discussed on magneto-hydrodynamic version of movement of blood. Young (1968) has presented the fluid mechanics of arterial stenosis. Sud et. al. (1974) gave an idea on effect of magnetic field on oscillating blood flow in arteries. Suri and Suri (1981) investigated on blood flow in branched arteries. Srivastava (1985) discussed on flow of couple stress fluid through stenotic blood vessels. Chaturani, and Pralhad R. N. (1985) have proposed blood flow in tapered tubes with biorheological applications. Belardinelli and Cavalcanti (1991) discussed on a new non-linear two-dimensional model of blood motion in tapered and elastic vessels. Ramamurty and Shanker (1994) studied on magneto-hydrodynamic effects on blood flow through a porous channel. Tzirtzilakis (2005) discussed on a mathematical model for blood flow in magnetic field. Mandal (2005) discussed an unsteady analysis of non-Newtonian blood flow through tapered arteries with a stenosis. If we apply a magnetic field to a moving electrically conducting liquid, it induces electric and magnetic fields. The interplay of these fields produces a body force known as Lorentz force which has a tendency to oppose the movement of the liquid. For the blood flow in arteries with arterial diseases like arteriosclerosis, influence of magnetic field may be utilized as a blood pump in carrying out cardiac operations, and in addition to this, the effect of vessels tapering together with the shape of stenosis on the flow character seem to be equally important and deserve special attention. Kumar and Kumar (2006) worked on numerical study of the axi-symmetric blood flow in a constricted rigid tube. Mekheimer and Kot (2008) have presented the micro-polar fluid model for blood flow through a tapered artery with a stenosis. Kumar and Kumar (2009) investigated on a mathematical model for Newtonian and non-
Mathematical modeling of blood flow 77
Newtonian flow through tapered tubes. Again, Kumar and Kumar (2009) gave an idea on oscillatory MHD flow of blood through an artery with mild stenosis. Jain, M. et. al. (2010) have dealt a mathematical modeling of blood flow in an artery under MHD effect through porous medium. Bali and Awasthi (2011) gave an idea on mathematical model of blood flow in the small blood vessel in presence of magnetic field. Chakraborty et. al. (2011) studied on a suspension model blood flow through an inclined tube with an axially nonsymmetrical stenosis. Tripathi (2012) investigated a mathematical model for blood flow through an inclined artery under the influence of an inclined magnetic field. Again, Eldesoky (2012) has presented slip effects on the unsteady MHD pulsatile Blood flow through porous medium in an artery under the effect of body acceleration. Bhatnagar et. al. (2015) worked on a numerical analysis for the effect of slip velocity and stenosis shape on Non-Newtonian flow of blood. In this study, we are analyzing the characteristics of the blood flow through an inclined tapered porous artery with mild stenosis under the influence of an inclined magnetic field. This study can play a big role in the conclusion of axial velocity, shear stress and fluid acceleration in porous medium. This study is also useful for evaluating the role of porosity. The study is carried out by employing approximate analytical method. 2. Mathematical formulation: We consider one-dimensional study, laminar and fully developed flow of blood through inclined tapered artery, with the presence of mild stenosis. It is assume that the formation of stenosis which is symmetrical about the axis, but non symmetrical with respect to radial co-ordinates, and it depends upon the height and location of the constriction, formed at the innermost wall and the axial wall. It is assumed that the wall of the tapered tube is rigid. There is no loss of generally in considering a rigid artery as due to the formation of a stenosis, the elasticity of the arterial wall gets reduced. Further, the artery length is assumed to be large enough as compared to its radius, so that the entrance and special wall effects can be neglected. The geometry of an arterial non-symmetrical stenosis in a tapering wall can be expressed (Mekheiner and Kothari, 2008) as:
( )⎩⎨⎧ +≤≤−−−−
=−
otherwisezhbazaazazbzh
zRn
),()],)()((1[)( 21η
(1)
with zhzh ξ+= 0)( ; (2) where, ( )zR is the radius of the stenosed portion of arterial segment and )(zh is the radius of tapered arterial segment in the stenotic region, 0h is the radius of the non tapered artery in the non stenotic region, ξ is the tapering parameter, b is taking the length of the stenosis, and )2(≥n is being a parameter determining the shape constriction and referred to as a shape parameter.
78 Ajay Kumar et al
Here we are using the parameter η which is given by:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
−=
−
1
)1(1
0 nn
bh
n
n
δη (3)
where, δ denotes the maximum height of the stenosis to be found at:
)1/(1 −+= nnbaz (4)
Figure (1): Geometry an inclined stenosed artery with axially non-symmetrical stenosis
Mathematical modeling of blood flow 79
Figure (2): Geometry of construction
Here the body fluid is assumed to behave as a Newtonian fluid (Schlitchting and Gerstein 2004). The equation (as obtained from Navier-Stokes equation of motion for various fluids) describing the steady flow of Newtonian fluid is given by (Schlitchting and Gerstein 2004):
0=∂∂
rp
(5)
0=∂∂θp
(6)
3. Analytical Solution of the problem: As per the published literature and available physiological data, blood flow in the neighborhood of the vessel wall can be considered as Newtonian, if the shear rate of blood is high enough. However, the shear rate is very small towards the center of the artery (circular tube), the non-Newtonian behavior of blood is more evident (Mishra et. al 2007). The steady flow of blood through the cylindrical artery inclined at an angle α can write as follows:
0. =∇ V (7)
BXJVru
rrupF +−
⎭⎬⎫
⎩⎨⎧
∂∂+
∂∂+−∇=
κμμρ 1
2
2
(8)
Boundary conditions for the problem stated above may be listed as: ( ) ( )
0
0,0
0
==
==∂∂
==
ratfiniteu
ratru
wallarteryzRratu
(9)
The above equation (8) transformed in the term as:
80 Ajay Kumar et al
uBuru
rru
zpg 2
02
2 1sin σκ
μμαρ −−⎭⎬⎫
⎩⎨⎧
∂∂+
∂∂+
∂∂−= (10)
where, g is the acceleration due the gravity, μ is the viscosity of blood, κ is the permeability of porous medium, ρ is the fluid density, α is the inclination of an artery, 0B is an applied magnetic field with an inclination θ . The non-dimensional variables are:
μσκκ
μρ
μ
δ2
02
0
0
20
20
00
0
20
0000
,,,,
,,,,,
BRM
gRu
FR
RuR
ubRp
p
uuu
uvbv
RRR
Rzz
Rrr
re =====
=====
(11)
where, u and v are velocity components in the axial z and radial r directions, pthe pressure, ρ is the density, 0R is the radius of the normal arteryδ is the maximum height of the stenosis, eR is the Reynolds number, rF is the Froude number, M is the Hartmann number. Substituting (11) in (10), we can get a dimensionless form for (8) as follows:
( ) uMru
rru
zp
FR
r
e ⎟⎠⎞
⎜⎝⎛ +−
⎭⎬⎫
⎩⎨⎧
∂∂+
∂∂+
∂∂−=⎟⎟
⎠
⎞⎜⎜⎝
⎛θ
κμα 22
2
2
cos11sin (12)
As the flow is steady and axisymmetric, let the solution for ( )tru , and p be set in the forms:
( ) ( )rutru =, and Pzp =
∂∂− (13)
where, P is a constant. Substitute equation (13) in (12), we can have a second order ordinary differential equation as follows:
PFR
udrdu
rdrud
r
e −=−+ αβ sin1 22
2
(14)
where ⎟⎠⎞
⎜⎝⎛
+=
θκκβ 22
2
cosM The solution of second order differential equation (14) can be written as follows:
( ) ( )( ) ⎟⎟
⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛−= 1sin
0
02
hJrJP
FRru
r
e
ββαβ (15)
where, 0J is the modified Bessel’s function of the zero order The volumetric flow rate Q of fluid is the stenotic region is given by:
( )∫=h
drrruRQ0
02π (16)
Substitute ( )ru from (15) into (16) an them integrating with respect to r , we obtain:
Mathematical modeling of blood flow 81
( )( )⎟⎟⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
hJhJhhP
FRQ
r
e
ββ
βα
β 1
12
2 2sin1
(17)
where, 1J is the modified Bessel’s function of the order one The wall shear stress is defined by:
( )hr
r drdur
=⎥⎦⎤
⎢⎣⎡−−= μτ (18)
which, on using (15) gives: ( )( )⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−=
hJhJP
FR
r
er β
βαβμτ
1
1sin (19)
4. Result and Discussion: Most of the theoretical result such as permeability, inclination angle of artery ( )α , the inclination of magnetic field ( )θ , wall shear stress, shear stress at the stenosis throat, axial velocity, and volumetric flow rate are obtained in the numerical analysis with computational illustrations. Out of these results, only the numerical solution of wall shear stress and relative local pressure gradient are shown graphically for different values of Reynolds number ( )eR , Froude number ( )rF and Hartmann number ( )M for better understanding of the problem. All graphs are plotted by using mathematical software MATLAB, for the value
200,1,0,43,2,1.0,1.0 toheightstenosisbabandMFR re ======= δσ
.002.0,0,002.0tan,11,6,2 −=== φξandnparametershape Figure (3) depicts that the variation of wall stressτ , with shape parameter n and permeability. It is evident that the wall stress decreases with increases of permeability parameter.
Figure (3): Effect of permeability on the shearing stress for different values of shape parameter ( )n and permeability ( )κ .
82 Ajay Kumar et al
The wall shear stress attains maximum value in case of radically symmetric stenosis ( )2=n and starts diminishing as stenosis losing its symmetric i. e. stenosis shape parameter n becoming larger. By the inspection of the figure 4, it can be noticed that the converging region ( )0<ξ , the stress will be more as compared to the diverging region ( )0>ξ and non tapered region ( )0=ξ that is wall shear stress increases with the increase in tapering angle( )ξ .
Figure (4): The result of artery inclination ( )α on the shearing stress for different values of tapering angle ( )ξ
Figure (5): Variation of shearing stress at the stenosis throat ( )δ with different inclination ( )α for different values of permeability ( )κ .
Mathematical modeling of blood flow 83
Figure 6: Variation of axial velocity ( )u with ( )z and height of stenosis ( )δ for different values of permeability ( )κ It is quite interesting to observe from the figure (5) that as the variation of shearing stress at the stenosis throat for different values of inclination of artery. It has been noticed that with the increase of inclination ( )α , the shearing stress ( )τ increases. It is also observed that the variation of permeability affects the stress inversely. Figure (6) illustrates that variation of axial velocity u with z and the height of stenosis δ for different value permeability ( )κ . It is clear that the axial velocity possesses reverse behavior on either side of the centre line of the artery. It shows that an increase in axial velocity with the increase of permeability ( )κ ,
Figure (7): Variation of axial velocity u with magnetic field for different values of tapering angle ξ .
84 Ajay Kumar et al
Figure (8): Variation of axial velocity u with z for different values of artery angleξ
In figure (7), with the increase of magnetic field, the axial velocity shows a reverse behavior. It is observed that with the increase of magnetic field, the curve representing the axial flow velocity do shift towards the origin for a converging region, while they shift away from the origin for a non tapered and diverging tapered artery. It is quite interesting to observe from the figure (8) that as the argumentation in the axial velocity shows the remarkable changes with the inclination of artery. It has been observed that with the increase of inclination artery, the curve representing the axial flow velocities does shift towards the origin.
Figure (9): Axial velocity with the inclination of magnetic field ( )θ , for3/,2,2,1.0,1.0 πακ ===== andMFR re
Mathematical modeling of blood flow 85
Figure (10): Variation of volumetric flow rate with the tapering angle ( )ξ , for different values of ( )M and for fixed values of
3/,2,1.0,1.0 πακ ==== andFR re Figure (9) reveals that the variation of axial velocity with the inclination of magnetic field ( )θ . It has been observed that the volumetric flow rate will increase and will be more for diverging region as compared to converging region. It is seen that the increase of inclination of magnetic field, the curve does shift towards the origin. Figure (10) shows an increasing behavior of volumetric flow rate from converging to diverging region. The volumetric flow rate for an inclined artery will decrease with the Hartmann number for the fixed values of Reynolds number
( )( ) ( ) .3/,2
,1.0,1.0)(number Reynoldsπακ ==
==ninclinatioarteryandtypermeabili
FnumberFroudeR re
The volumetric flow rate for an inclined artery will be greater in diverging region ( > 0) as compared to converging region ( < 0).
86 Ajay Kumar et al
Figure (11) : Variation of volumetric flow rate with the angle of inclination α of artery for converging region ( < 0), diverging region ( > 0) and non-tapered region ( = 0)
Figure (12) : Variation of volumetric flow rate with the angle of magnetic field θ for converging region ( < 0), diverging region ( > 0) and non-tapered region ( = 0) Figure (11) illustrates that the variation of volumetric flow rate with the angle of inclination ( )α of artery for converging region ( < 0), diverging region ( > 0) and non-tapered region ( = 0), It is seen that the increases of inclination ( )α , the volumetric flow rate decreases. Figure (12) depicts that the variation of volumetric flow rate with the angle of magnetic field ( )θ will increases for all converging region ( < 0), diverging region ( > 0) and non-tapered region ( = 0).
Mathematical modeling of blood flow 87
Conclusion: In the present investigation, we have developed a mathematical model of blood flow in an inclined tapered artery, with the presence of mild stenosis under the MHD effect through porous medium. Analytical expressions of flow variables are obtained and variations of shear stresses at stenotic wall resistance to flow are shown graphically. This investigation can play a vital role in the determination of axial velocity, shear stress, and permeability in particular situations. Since, this study has been carried out for a situation when the human body is subjected to an external magnetic field. The study is also useful for evaluating the roll of porosity. It is observed that magnetic field reduces the flow characteristics amazingly. Also the height of stenosis significantly affects the shearing stress, wall shear stress and axial velocity. This investigation may be helpful for the practitioners to treat the hypertension patient through magnetic therapy and to understand the flow of blood under stenotic conditions. REFERENCES: 1. Bali, R. and Awasthi, U. (2011): “Mathematical model of blood flow in the
small bloodvessel in presence of magnetic field, ” Applied Mathematics, vol. 2, pp. 264–269.
2. Belardinelli, E. and Cavalcanti, S. (1991): “A new non-linear two-dimensional model of blood motion in tapered and elastic vessels”, Computers in Biology and Medicine, vol. 21, pp. 1-13.
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