On The Model of The Effect of External Magnetic Field on Blood Flow in Stenotic Artery

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3 (2), 2009, 161 - 164

©BEYKENT UNIVERSITY

On the Model of the Effect of External Magnetic

Field on Blood Flow in Stenotic Artery

Afgan ASLANOV

Department of Mathematics and Computing Beykent University, Istanbul, Turkey

afganaslanov@beykent.edu. tr Received: 03.08.2009, Accepted: 04.11.2009

Abstract

We consider the mathematical model of the effect of external magnetic field on blood flow in stenotic artery. We received an approximate analytical solution of the equation of velocity.

Keywords: Stenosis; Boundary value problems; Initial value problem; Blood

flow

Özet

Dış Manyetik Alanın Daralmış Damarlardaki Kan

Akışına Etkisinin Modellenmesi Hakkında

Bu çalışmada, daralmiş damarlarda kan akışına dış manyetik alanın etkisi araştırılmıştır. Burada hız denkleminin yaklaşık analitik çözümüne ulaşılmıştır.

Anahtar Kelimeler: Stenosis; Sınır-değer problemi; Başlangıç-değer

problemi; Kan akışı

Introduction

Stenosis are formed by substances depositing on vessel walls. The actual reason for formation of stenosis is not clearly known, but its effect over the flow characteristics has been studied by many researchers [1-3].

Many researchers [4-6] have studied blood flow in stenosed arteries by assuming that blood behaves like Newtonian fluid. Several researchers [7-9] studied the flow characteristics of blood in artery with mild stenosis by considering blood as Newtonian and non-Newtonian fluids. Chow and Soda [10] investigated the abnormal flow conditions caused by the presence of stenosis in arteries. Haldar and Ghosh [11] investigated the effect of an externally applied uniform magnetic field on the blood flow in single constricted blood vessel.

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Flow in Stenotic Artery

Shukla et al. [12] considered a two layer model in which the peripheral plasma layer and the core are both Newtonian in character. Shukla et al. [2] have again considered a two layer model in which fluids in both regions are non-Newtonian in character. Chandrasekhara and Rudraiah [13] observed that the overall effect of magnetic field is to decrease the resistance to flow due to irregular boundaries. Bali and Awasthi [14] have studied the effect of external magnetic field on blood flow in stenotic artery.

The aim of the present paper is to study the mathematical model related to the effect of external magnetic field on blood flow in stenotic artery and to find the analytical approximate solution of the equation of velocity.

Formulation Of The Problem

Consider a cylindrical blood vessel with the mild constriction in the presence of a uniformly applied magnetic field. We assume that the stenosis is symmetrical [14,6]. The radius of the vessel is given by

R(z) = Ro[1 {1 + c o s ^ ( z ' - d ' - d ' £ z' £ d' + L'

2 R Lo 2 (1)

= R

0

, elsewhere,

where R0 and R'(z) are the radius of the artery with and without stenosis

respectively, and ds is the maximum height of the stenosis such that

/ ( 2 R0) ^ 1, Lo is the length of the stenosis and d is the location of

stenosis.

For steady two-dimensional axisymmetric laminar flow of non-Newtonian fluid subjected to transverse applied magnetic field, the governing equation of motion and velocity are given by

dP' 1 d , , ,du .

T

, ^

t + — — M r —

7

) + J x B = 0, (2)

/ / / / / ? v 7

dz r dr dr

where

J' = a ( E ' + u x B'),

(3)

M=(r'/R )

(4)

(3)

M parameter depending upon the hematocrit value of the blood, E electric

field, B magnetic field, ^0 viscosity of plasma, a electrical conductivity.

The boundary conditions are:

du'

— = 0 at r = 0, u = 0 at r = R(z)

(5)

dr

To solve the above system of equations following non-dimensional variables are introduced:

z =

Z

-, d =

d

, L

o

= R =

R

, R

e

=

r U R

L L L R m

j-\/ f f f Tt n

P m u r R

o

P =

2

, m= , u = —, r =—,

e=—-r U

2

m U R

L

Then the equations (1)-(4) reduce to

r-

M

^-Ur + (1 - M)r "

(1+M)

— - H

2

u = R^e

dp

, (8)

dr dr dz

where (6)

R(z ) = [1 - A . {1 + c o s — (z - d -

L o

)}], d < z < d + L

0

,

2Ro

L,

l " 1' (7)

= 1, elsewhere,

J = s(E + uU

0

B

0

), m = r , (9)

sB

2

Ri

H

2

= (Hartmann number). (10)

m

(4)

du

— = 0, at r = 0, u = 0 at r = R( z ).

dr

(11)

Solution

Now we consider the equation (8) with boundary conditions (11). For simplicity, let us suppose that the derivative dp/dz is a linear function of z. Thus we have

d

2

u 1 du

+ r

m

(au + bz + c) = 0,

dr

2

r dr

(12)

2

r> Cp ,

with boundary conditions (11), where

a = —H , R

e

£— = -bz — c.

dz

We first rewrite the equation (12) in the form

L(u) = -r

M

(au + bz + c),

where

L ( ) = ( ^ )

d r

( ^

d r (.)

)•

Applying

L"

1

(.)

= f ] ^ ( ) d r } R ( z )r V 0

dr to the first two terms of (12),

we find

L

- f

d u + - L (

r

1-

M

<di

)

dr

2

i~

M

^ > dr

L—

1 d

i~

M

dr

r1-M d

u

dr

-u(r, z) - u(R, z).

Then we have

u (r, z ) = u(R, z) — (r

M

(au + bz + c)).

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u (r, z ) = —L"

1

(r

M

(au + bz + c)).

Taking u0( r , z ) = 0, decomposition method admits the use of the recursive

relations [15-18]

Uo(r,z) = o, u

M

(r,z) = - L

1

( r

M

(A,)), i > o,

where

g (u, z) = au + bz + c

A

o = g

(o z )

=

bz

+

c, A

1 =

u

1

gu(o, z)

=

u

ı

a , A2 = u2gu (0 z ) + guu ^ z ) = U2a , A3 = U3a , . ..

Using the operator L-1 for the components of the approximate solution u = u0 + u1 + u2 + ... we obtain u1 =-L~\rM(Ao)) = - } -hûİV"(rM(bz + c))dr R( z )r V o

dr

r 2

_r

= - ( b z + c ) J 2 r- Md r = R( z ) b z + c ( r M + 2 - RM+2)

2(M + 2)

u = - i - V ( 4 ) ) = a

J-1M

İ J r '

r

b

r

~M

r

M

bzz

+

^

cc

^

R( z)' o

2(M + 2)

(r

M

+

2

- R

M

)dr

dr

a(bz

+

c)

J

r

M-1

if (r

M

+

3

-

rR

M

+

2

)dr

2(M+2) R

J

z) l

JV

,

dr =

J

r

3

r

l+

M

R

M

+

2

^

a(bz+c) r

2(M+2) r

J

z) V M + 4 2

dr

a(bz+c)

2(M + 2)

,4 - R4 ( rM+2 - Rm+2 )RM

4(M + 4) 2(M + 2)

2

(6)

u3 =- L~\r

M

(A,)) = - L~\r

M

(u

2

a))

J

fri - MrM

a

2

(bz + c)

J r

l

~

M

J 2(M + 2)

R ( z ) ' ^ 0 R 4 - R 4 ( RM+2 - RM +2) RM+2

4(M + 4) 2(M + 2)

dr

J J )

dr

a

2

(bz + c) r _ 1

2

(

M

+

2

)

Rfz )

^

f

r

i-

M

f

r

4

- R

4 ( RM+2 - RM +2) R M+2 v0 V

4(M + 4) 2(M + 2)

dr

J J

dr

a

2

(bz + c)

rf

1 r

rf

f

r

i-

M

f

2(M

+

2

)

R ( z )

r

v 0 v

r

5

- rR

4

(r

M

+

3

- rR

M

+

2

) R

M+2

4(M + 4) 2(M + 2)

dr

J J

dr

a

2

(bz+c) f r r

6 -

r

2

R

4

_ r

M

+

4

R

M

+

2

+ r

2

R

2M

+

4

^

' 2(M+2)

RJz)

r

l

~

M

V 24(M+4) 8(M+4) 2(M+2)(M+4) + 4(M+2)

dr

1+M r>4 ,.2M +3 j^M+2 ^+M R 2 M + 4 ^ r I r 5+M r i + M R r'l M +3R

a

2

(bz + c)

2(M + 2)

R

(

z)

V 24(M + 4) 8(M + 4) 2(M + 2)(M + 4) 4(M + 2)

+

dr

a

2

(bz + c)

2(M + 2)

r

-

R

S

+M (

r2+M -

R

2

+

M

) R

4

(

r 2 M+ 4 -R2 M+4

)

RRA+2

(

r M+ 2 - R 2 +m

)

R 2m+ 4 ^

24(M+4)(M+6) 8(M+4)(M+2) 4(M+4)(M+2)

2

4(M+2)

2

Then the solution u = u0 + u1 + u2 +... of the problem (8), (11) has the form

r

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U = — R

e

£ dp

2M + 4 dz

( r M+2 — RM+2 ) —

H

2

R

e

£ dp

2M+4 dz

+2 \ t t M + 2 r4 — R4 ( RM+2 — RM+2 ) RM

4(M + 4)

2(M + 2)

H

4

R

e

£ dp

2 (M + 2 ) dz

r 6

+M

— R

(

z

)6+M ( r

M

+

2

— R( z )

M

+

2

) R

4

24(M + 4)(M + 6) 8(M + 4)(M + 2)

( r

2M

+

4

— R(z )

2M

+

4

) R

M

+

2

( r

M

+

2

— R(z)

M

+

2

) R(z)

2M+4 "N

4(M + 4)(M + 2)

2

+

4(M + 2)

2

+...

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Conclusion

Looking at the importance of the different factors in the understanding of blood flow through stenotic artery, it may be said that the present solution could be useful for investigating blood flow. More interesting models can be studied by considering the effect of magnetic field.

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REFERENCES

[1] Haldar H., Bull. Math. Effects of the shape of stenosis on the resistance to blood flow through an artery, Biol. 47, 545-547, 1985.

[2] Shukla J.B., Parihar R.S. and Rao B.R.P., Biorheological aspects of blood flow through artery with mild stenosis: effects of peripheral layer, Biorheology 17, 403-410, 1980.

[3] Shukla J.B., Parihar R.S. and Rao B.R.P., Effect of stenosis on non-Newtonian flow of blood in an artery, Bull. Math. Biol. 42, 283-294, 1980.

[4] Forrester J.H. and Young D.F., Flow through a converging and diverging tube and the implication in occlusive vascular disease, J. Biomech. 3, 297-316, 1970.

[5] MacDonald D.A., On steady flow through modeled vascular stenosis, J. Biomech. 12, 13-20, 1979.

[6] Young D.F., Fluid mechanics of arterial stenoses, J. Biomech. 101, 157-175, 1979 [7] Devanathan R. and Parvathomma S., in: Proc. of 1st Int. Conf. on Physiological Fluid Dynamics. 69-74, 1983.

[8] Tandon P.N. and Misra J.K., Microstructural and peripheral layer viscosity effects on flow of blood through artery with mild stenosis, in: Proc. of Int. Conf. on Phys. Chem. Hydrodynamics, NY Academy of Science, vol. 39, 404, 1982.

[9] Tandon P.N., Siddiqui S.U. and Pal T.S., Pulsatile blood flow through an axi-symmetric time dependent developing stenotic tube, Int J. Tech. 24, 190-194, 1986. [10] Chow J.C. and Soda K., Laminar flow in tubes with constriction, Phys. Fluids 15 [10], 1700, 1972

[11] Haldar K. and Ghosh S.N., Effect of a magnetic field on blood flow through an indented tube in the presence of erythrocytes, Indian J. Pure Appl. Math. 25(3), 345-352, 1994.

[12] Shukla J.B., Parihar R.S. and Rao B.R.P., Effect of peripheral layer viscosity on blood flow through the artery with mild stenosis, Bull.Math. Biol. 42, 797-805, 1980. [13] Chandrasekara B.C. and Rudraiah N., MHD flow through a channel of varying gap, Indian J. Pure Appl. Math. 11 (8), 1105-1123, 1980.

[14] Bali B. and Awasthi U., Effect of a magnetic field on the resistance to blood flow through stenotic artery, Appl. Math. Comp. 188, 1635-1641, 2007.

[15] Adomian G., Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, Boston, MA, 1994.

[16] Wazwaz A.M., A new algorithm for solving differential equations of Lane-Emden type, Appl. Math. Comput. 118, 287-310, 2001.

[17] Wazwaz A.M., A new method for solving singular value problem in the second order ordinary differential equations, Appl. Math. Comput. 128, 45-57, 2002. [18] Wazwaz A.M., Adomian decomposition method for a reliable treatment of the