influence of mixed convection on blood flow of … · 2013-05-14 · nadeem and akbar [20] ... of...

INFLUENCE OF MIXED CONVECTION ON BLOOD FLOW OFJEFFREY FLUID THROUGH A TAPERED STENOSED ARTERY

by

Noreen Sher AKBAR a*, Tasawar HAYAT b, Sohail NADEEM b,

and Awatif A. HENDI c

a DBS&H, CEME, National University of Sciences and Technology, Islamabad, Pakistanb Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan

c Department of Physics, Faculty of Science, King Saud University, Riyadh, Saudi Arabia

Original scientific paperDOI: 10.2298/TSCI111121144A

Effect of heat and mass transfer on the blood flow through a tapered artery with ste-nosis is examined assuming blood as Jeffrey fluid. The governing equations havebeen modelled in cylindrical co-ordinates. Series solutions are constructed for thevelocity, temperature, concentration, resistance impedance, wall shear stress, andshearing stress at the stenosis throat. Attention has been mainly focused to the anal-ysis of embedded parameters in converging, diverging, and non-tapered situations.

Key words: Jeffrey fluid, blood flow, mixed convection

Introduction

The non-planer arterial configuration is associated closely with swirling blood flow. Ar-

teries (as a living tissues) require a supply of metabolites including oxygen and removal of waste

products. Blood is multi-component mixtures, consisting of plasma, red and white blood cells,

platelets, etc. [1]. The analysis of blood flow in arteries is very popular now a days because of

some cardiovascular diseases [2, 3]. The nature of blood itself sometimes has a key role in such

diseases. No doubt, it is concluded now that blood is a non-Newtonian fluid at low shear rate

(100/s). Humphrey and Delange [2] pointed out that a rheological properties of blood greatly de-

pends upon the change in flow configuration. He characterized blood as a mixture (i. e. solid and

fluid). Further Thurston [4] and Chien et al. [5] experimentally observed viscoelastic properties of

blood. Jones [6] found that the flow behaviour of blood changes from Newtonian to non-Newto-

nian character when there is a change in the diameter of vessel from large arteries to small

branches and capillaries. The flow varies with typical Reynold number from small arteries to large

arteries. Siddiqui et al. [7] examined the rheological effects on pulsatile flow of blood in a

stenosed artery. Mekheimer and El Kot [8] presented micropolar fluid model for axisymmetric

blood flow through an axially non-symmetric but radially symmetric mild stenosis tapered artery.

Unsteady flow analysis for blood as non-Newtonian fluid through tapered arteries with a stenosis

is studied numerically by Mandal [9] and Varshney et al. [10] considered the generalized power

law fluid model for blood flow in an artery having multiple stenosis. They carried out the numeri-

cal study under the action of transverse magnetic field. Power law fluid model for blood flow

through a tapered artery with a stenosis is recently developed by Nadeem et al. [11]. In another ar-

ticle, Nadeem and Akbar [12] revisited the flow analysis of ref. [11] for Jeffrey fluid. Mustafa et

al. [13] presented the numerical simulation of generalized Newtonian blood flow through a couple

Akbar, N. S., et al.: Influence of Mixed Convection on Blood Flow of Jeffrey Fluid ...THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 533-546 533

* Corresponding author; e-mail: [email protected]

of irregular arterial stenosis. The blood flow with an irregular stenosis in presence of magnetic

field has been looked at Abdullah et al. [14].

It is noted from existing literature that only few attempts investigate the blood flow in

presence of heat and mass transfer. Influence of pulsatile blood flow and heating scheme on the

temperature distribution with reference to hyperthermia treatment was studied by Khanafer et

al. [15]. Unsteady flow and mass transfer in models of stenotic arteries considering fluid-struc-

ture interaction was simulated by Valencia and Villanueva [16]. Back et al. [17] examined the

flow field and mass transfer analysis in arteries with longitudinal ridges. Heat and mass transfer

in a separated flow region for high Prandtl and Schmidt numbers have been studied by Ma et al.

[18]. Akbar and Nadeem [19] constructed the simulation of heat and chemical reactions for

Reiner-Rivlin fluid model of blood flow through a tapered artery subject to a stenosis. Recently

Nadeem and Akbar [20] studied the blood flow for Walter's B fluid model through a tapered ar-

tery with heat and mass transfer. Some important article related to the topic are included in refs.

[22-26].

The interest in the present article is to analyze the effects of mixed convection blood

flow of Jeffery through a tapered stenosid artery. Analysis has been carried out in the presence

of heat and mass transfer.

Problem development

We examine incompressible flow of Jeffrey fluid with constant viscosity m and density

r in a tube having length L. The cylindrical co-ordinate system (r, q, z) is chosen such that u and

w are the velocity components in the r and z directions, respectively. Here r = 0 is selected the

axis of the symmetry of the tubes. Mixed convection is considered in the presence of heat and

mass transfer by assigning the temperatureT0 and concentration C0 to the wall of the tube. Sym-

metry condition for both temperature and concentration is employed at the centre of the tube.

The consideration of stenosis is represented as [8]:

h(z) = d(z){1 – h1(bn–1(z – a) – (z – a)]n} for a � z � a + b (1)

otherwise h(z) = d(z),

with d(z) = d0 + xz (2)

In eqs. (1) and (2) d(z) is the radius of the tapered arterial segment in the stenotic re-

gion, d0 – the radius of the non-tapered artery in the non-stenotic region, x – the tapering param-

eter, b – the length of stenosis, n � 2

– a parameter determining the shape

of the constriction profile (n = 2,

gives symmetric stenosis), and a in-

dicates location of the stenosis (fig.

1).

The parameter h is given by:

hd

��

�*

( )

n

d b n

n

n

n

1

0 1

in which maximum height of steno-

sis located at z = a +b/nn/n–1)) [8], and

d* is the width of artery (minimum

diameter of artery at z).

Akbar, N. S., et al.: Influence of Mixed Convection on Blood Flow of Jeffrey Fluid ...534 THERMAL SCIENCE: Year 2013, Vol. 17, No. 2, pp. 533-546

Figure 1. Geometry of an axially non-symmetrical stenosisin the artery

The equations governing the flow are:

¶

¶

¶

¶

u

r

u

r

w

z� � � 0 (3)

r qqur

wz

up

r r rr

z rrr rz

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶�

�

��

� � � � � �

1( ) ( )S S

S(4)

r r aur

wz

wp

z r rr

zTrz zz

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶�

�

��

� � � � � �

1( ) ( ) (S S g � � �T C C0 0) ( )r ag (5)

rc ur

wz

Tu

r

w

r

u

z

wp rr rz rz zz

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶�

�

��

� � � � �S S S S

zk

T

r r

T

r

T

z� � �

�

��

��

¶

¶

¶

¶

¶

¶

2

2

2

2

1(6)

ur

wz

C DC

r r

C

r

C

z

DKT¶

¶

¶

¶

¶

¶

¶

¶

¶

¶�

�

��

� � � �

�

��

��

2

2

2

2

1

T

T

r r

T

r

T

zm

¶

¶

¶

¶

¶

¶

2

2

2

2

1� �

�

��

�� (7)

where p is the pressure, T – the temperature, C – the concentration of fluid, r – the density, k –

the thermal conductivity, cp – the specific heat at constant pressure, Tm – the temperature of the

medium, D – the coefficient of mass diffusivity, KT – the thermal-diffusion ratio, and g – the

acceleration due to the gravity.

Extra stress tensor S in Jeffrey fluid is [12]:

S ��

�m

lg l g

1 1

2( � ��)* (8)

where m is the viscosity, l1 – the ratio of relaxation to retardation times, �g – the shear rate, and

l2* – the retardation time. Defining:

rr

dz

z

bw

w

uu

bu

up

d p

u bh

h

d� � � � � �

0 0 0

02

0 0

, , , , , ,d m

Re , ,~

, ,� � � �r

m m m mqq

buS

b

uS

d

uS

b

uSrr

rrrz

rzzz

zz0

0

0

0 0

S S S�

b

u

Sqq

m0

,

(9)

ll

qm a

n2

2 0 0

0

02

0

03

0

2� �

��

*

, , , Pr ,u

b

T T

T

u

c T

c

k

d C

p

pEc Cr

g,

Sr Sc Grg

� � ��

�r

m

m

rs

a

n

DK T

T C D

C C

C

d TT

m

0

0

0

0

03

0

2, , ,

where Re is the Raynolds number, Ec – the Eckert number, Pr – the Prandtl number, Sr – the

Soret number, Sc – the Schmidt number, Gr – the Grashof number, Cr – the local concentration

number, n – the kinematic viscosity, m – the dynamic viscosity, u0 – the velocity averaged over

the section of the tube of the width d0, and sis the concentration, and using eqs. (6) and (9) along

with the additional conditions [8]:

Re *d n

b

n

1

1�� 1 (10a)


d n

nO

n0

1

1

1�

~ ( ) (10b)

we have for mild stenosis (d*/d0 � 1) the following expressions:

¶

¶

¶

¶

u

r

u

r

w

z� � � 0 (11)

¶

¶

p

r� 0 (12)

¶

¶

¶

¶

¶

¶

¶

¶ ¶

p

z r r

r w

rw

w

r z�

��

�

��

�

�

�

�

��

��

��

�1

1 1

2

2

ll Gr Crq s� (13)

1 1

1 1

2

2

2

r rr

rB

w

rw

w

r z

w

rr

¶

¶

¶

¶

¶

¶

¶

¶ ¶

¶

¶

q

ll

�

��

� �

�

�

��

� �

�

��

�

��

��

��

��

��= 0 (14)

1 1 1

ScSr

r rr

r r rr

r

¶

¶

¶

¶

¶

¶

¶

¶

s q�

��

�

�

�

�

��

�

��

�

�

�

�

�� 0 (15)

where the Brickman number (Br) = EcPr. The boundary conditions are now given by:

¶

¶

¶

¶

¶

¶

w

r r rr� � � �0 0 0 0, ,

q sat (16a)

w r h z� � � �0 0 0, , ( )q s at (16b)

h z z z z zn( ) ( )[ ( ) ( ) ]� � � � � � � � �1 1 11 1 1 1 1x h s s s s (17)

and

hd

dd

s

xx

x f

1

1

0

1

0

1�

��

� � �

�n

n d

a

b

b

d

n

n, , ,

,

*

tg

(18)

in which f is called tapered angle. Further,

for converging tapering or (f < 0) non-ta-

pered artery (f = 0) and the diverging taper-

ing (f> 0) (fig. 2).

Solution of the problem

We have an interest to compute the series

solution by homotopy perturbation method

[21]. For that we write:

H q w q L w L w q L wp

z( , ) ( )[ ( ) ( )] ( ) ( )

(

� � � � � � ��

�

�

1 110 1

2

¶

¶l

l 1 1 11

2

1 1��

��

� � � � �

�

��l l q l s) ( ) ( )w

w

r z

¶

¶ ¶Gr Cr (19)


Figure 2. Geometry of the axially stenosed taperedartery

H q q L L q Lw

r( , ) ( )[ ( ) ( )] ( )q q q q

l� � � � �

�

�

��

�1

1

110

1

2

Br¶

¶�

�

��

�

��

��

��

��

��

�

�

��

��

l2

2

ww

r z

w

r

¶

¶ ¶

¶

¶(20)

H q q L L q Lr r

rr

( , ) ( ) ( ) [ ( )] ( )s s s sq

� � � � ��

��

1

110 SrSc

¶

¶

¶

¶ �

�

�

�

��

��

��

(21)

and choose L = (1/r)(��r)[r(��r)] as the auxiliary linear operator. Further the initial guesses are

represented by:

w r zp

z

r hr z

r h10

02 2

10

2 2

4 4( , ) , ( , ) ,�

��

��

� �

��

��

�

¶

¶q s10

2 2

4( , )r z

r h� �

��

��

� (22)

Putting

w(r,q) = w0 + qw1 + q2w2 + ... (23)

q(r,q) = q0 + qq1 + q2q2 + ... (24)

s(r,q) = s0 + qs1 + q22

2s+ ... (25)

and then employing the procedure of ref. [19] for q = 1 we have the following results for veloc-

ity, temperature, and concentration fields:

w r zr h p

za r h a r h( , ) ( ) ( ) ( )�

��

��

� � � � � � �

2 2

1 22 2

33 3

41 l

¶

¶a r h

a r h a r h a r h a r

44 4

238 8

247 7

256 6

265

( )

( ) ( ) ( ) (

� �

� � � � � � � � �

� � � � � �

h

a r h a r h a r h

5

274 4

283 3

292 2

)

( ) ( ) ( ) (26)

q( , ) ( ) ( ) ( ) ( )r z a r h a r h a r h a r h� � � � � � � �54 4

66 6

5111 11

528 8 �

� � � � � � � � �a r h a r h a r h a r h a537 7

546 6

555 5

564 4

57( ) ( ) ( ) ( ) (r h3 3� ) (27)

s( , ) [ ( ) ( ) (r zr h

a r h a r h a� ��

��

� � � � � �

2 2

54 4

66 6

514

SrSc r h

a r h a r h a r h a r

11 11

528 8

537 7

546 6

555

� �

� � � � � � � �

)

( ) ( ) ( ) ( h

a r h a r h

5

564 4

573 3

)

( ) ( )]

�

� � � � (28)

Expressions (formulas) for a1 to a57 are given in the Appendix.

The volume flow rate Q is expressed as:

Q rw rh

� � d0

(29)

Through eqs. (26) and (29) one has:

d

d

p

z

Q

h

a

h� �

��

�

16

1

16

114

58

14( ) ( )l l

(30)

The pressure drop (Dp = p at z = 0 and Dp = –p at z = L) across the stenosis between the

section z = 0 and z = L can be obtained using the expression given below:

Dpp

zz

L

� ��

��

��

d

dd

0

(31)


Resistance impedance

The resistance impedance is given by:

~( ) ( ) ( )l � � � � � � ��

�

��

Dp

QF z z F z z F z z

a

ha

a b

a b

L

h0

1 1d d d (32)

in which

F zh

a

Q h( )

( ) ( )�

��

�

16

1

16

114

58

14l l

On simplification, eq. (32) yields:

~( )

( ) ( )( )l

l l� �

��

�

�

��

��

�L b

h

a

QF z z

a

a b16

1

16

114

58

1

d� (33)

Expression for the wall shear stress

The expression for dimensionless shear stress is:

~Srz

w

rw

w

r z�

��

�

��

�

�

�

�

��

1

1 1

2

2

ll

¶

¶

¶

¶ ¶(34)

The wall shear stress is of the form:

~Srz

r h

w

rw

w

r z�

��

�

��

�

�

�

�

��

��

��

1

1 1

2

2

ll

¶

¶

¶

¶ ¶(35)

The shearing stress at the stenosis throat i. e. the wall shear at the maximum height of

the stenosis located at z = (a/b) + [1/n(n/n–1)] can be expressed as:

~ ~t

ds rz

h�

� �S

1(36)

The final expressions for the dimensionless resistance to l, wall shear stress Srz and the

shearing stress at the throat ts are:

ll

l l l� � �

�

��

�

��

�

�

�

�

��

�

0 1

58

1

1

31

16

1

16

1

1b

L

a

Q LF

a

a

( )

b

z z��

��

( )d (37)

with

tt

tt

trz

rzs

sS� �

~

,~

0 0

(38)

l0 = 3L, t0 = 4Q (39)

Numerical results and discussion

Our interest is to analyze the effects of the ratio of relaxation to retardation time l1, re-

tardation time l2, the stenosis shape n, and maximum height of the stenosis d for converging ta-

pering, diverging, and non-tapered arteries in Jeffrey fluid. This object has been achieved

through the plots (3) to (11). The variation of axial velocity for l1, n, l2, and d in converging, di-

verging, and non-tapered arteries are displayed in figs. 3(a) to 3(d). We observed that due to in-

crease in l1, n, and d, the velocity profile decreases. The velocity increases due to increase in

l2. It is also seen that for the case of converging tapering the velocity gives larger values as com-


pared to the case of diverging tapering and non-tapered arteries. Figures 4(a) to 4(c) show how

the converging, diverging and non-tapered arteries influence on the wall shear stress trz. Inter-

estingly with an increase in l1 and d the shear stress increases and decreases with an increase in

l2. It is also seen that the stress yield diverging tapering with tapered angle f > 0, converging ta-

pering with tapered angle f < 0, and non-tapered artery with tapered angle f= 0. In figs. 5(a) and

5(b) it has been noticed that the impedance resistance increases for converging, diverging, and

non-tapered arteries when we increase l1. Such resistance decreases upon increasing l2. We

also observed that resistive impedance in a diverging tapering appear to be smaller than in con-

verging tapering because the flow rate is higher in the former case when compared with the later.

Impedance resistance attains its maximum values in the symmetric stenosis case (n = 2). Figures

6(a) and 6(b) illustrate the variation of shearing stress at the stenosis throat ts with d. The shear-

ing stress at the stenosis throat decreases with an increase in Q and l1. Figures 7(a) and 7(b) de-

pict the variation of temperature profile for different values of Brinkman number and ratio of re-

laxation to retardation time l1. It is observed that with an increase in Br, the temperature profile

decreases, while increases with an increase in ratio of relaxation to retardation time l1.

Temperature profile has the large values for converging tapering when compared with the di-

verging and non-tapered arteries. Figures 8(a) and 8(b) are prepared to see the variation of con-

centration profile for Brinkman number, ratio of relaxation to retardation time, l1 and Soret


Figure 3. Variation of velocity profile for (a) l1 = 0.3, n = 2, and l2 = 0.5, (b) l1 = 0.3, n = 2, and d = 0.5,(c) l1 = 0.3, l2 = 0.5, and d = 0.5; the other parameters are Br = 0.3, s1 = 0.0, z = 0.5, Q = 0.3, Gr = 0.5,Cr = 0.3, Sc = 0.5, and Sr = 0.5

number. It is found that with an increase in Brinkman number, and Soret number, the concentra-

tion profile increases. However it decreases because of an increase in ratio of relaxation to retar-

dation time l1. It is also observed that concentration profile has an opposite behavior as com-

pared to the temperature profile. Figures 9 to (11) show the streamlines for different values of n,


Figure 5. Variation of resistance for (a) l1 = 0.3, (b) l2 = 0.4; the other parameters are Q = 0.3, L = 1,s1 = 0.0, Br = 0.5, z = 0.5, Gr = 0.5, Cr = 0.3, Sc = 0.5, and Sr = 0.5

Figure 4. Variation of wall shear stress for(a) l1 = 0.3, and l2 = 0.5, (b) l1 = 0.3 andd = 0.5, (c) l1 = 0.3, and d = 0.5; the otherparameters are Br = 0.3, s1 = 0.0,z = 0.5, n = 2, Q = 0.3, Gr = 0.5, Cr = 0.3, Sc = 0.5,and Sr = 0.5


Figure 6. Variation of shear stress at the stenosis throat for (a) Q = 0.5, (b) l1 = 0.3; the other parametersare Br = 0.3, s = 0.0, z = 0.5, n = 2, d = 0.3, Gr = 0.5, Cr = 0.3, Sc = 0.5, Sr = 0.5, and l2 = 0.3

Figure 7. Variation of temperature profile for (a) l1 = 0.5, (b) Br = 0.3; the other parameters are Q = 0.3,s = 0.0, z = 0.5, n = 2, d = 0.3, Gr = 0.5, Cr = 0.3, Sc = 0.5, Sr = 0.5, and l2 = 0.3

Figure 8. Variation of concentration profile for (a) Sr = 0.5, (b) l1 = 0.3; the other parameters are Q = 0.3,s = 0.0, z = 0.5, n = 2, d = 0.3, Gr = 0.5, Cr = 0.3, Sc = 0.5, and l2 = 0.3

l1, and l2. Streamlines for different values of the stenosis shape n is shown in fig. 9. Here it is

noticed that the size of the trapping bolus increases when we increase the stenosis shape. Figures

10 and 11 are plotted to see the streamlines for different values of ratio of relaxation to retarda-

tion time l1 and retardation time l2. Here the size of the trapping bolus increases with an in-


Figure 9. Streamlines for different values on n (a) n = 2, (b) n = 4; the other paramters are Q = 0.3, d = 0.1,l1 = 0.2, Gr = 0.4, Sc = 0.5, Sr = 0.5, Cr = 0.6, and l2 = 0.3

Figure 10. Streamlines for different values of l1 (a) l1 = 0.1, (b) l1 = 0.3; the other paramters are Q = 0.3,d = 0.1, n = 2, Gr = 0.4, Sc = 0.5, Sr = 0.5, Cr = 0.6 and l2 = 0.3

Figure 11. Streamlines for different values of l2 (a) l2 = 0.1 (f) l2 = 0.3; the other parameters are Q = 0.3,d = 0.1, l1 = 0.2, Gr = 0.4, Sc = 0.5, Sr = 0.5, Cr = 0.6, and n = 2

crease in ratio of relaxation to retardation time while it decreases when retardation time in-

creases.

Conclusions

This study examines the mixed convection effects on blood flow of Jeffrey through a

tapered stenosed artery. The main points of the performed analysis are as follows.

� The influence of relaxation to retardation times l1, shape of the stenosis n, and height of the

stenosis d on the velocity profile is qualitatively similar.

� The velocity profile increases when retardation time l2 increases.

� The velocity in converging tapering gives larger values when compared with the cases of

diverging and non-tapered arteries.

� The resistive impedance in a diverging tapering is smaller than converging tapering.

� Shear stress increases by increasing l1 and d. However the shear stress decreases with an

increase in l2.

� It is observed that with an increase in Brinkman number the temperature profile decreases.

The temperature profile increases with an increase in ratio of relaxation to retardation times

l1.

� Concentration profile has an opposite behavior in comparison to the temperature profile.

Appendix

Here we have:

aP

z

P

za

P

z1

0 01

22

01��

��

��

��

�

��

�

��

�

d

d

d

d

d

d( ) ,

(l

1

4

16

1

1 0

14

��

�

l

l

),

( )

d

d

P

z

Q

h

aa h h h

aa

32 1

2 2 2

42 1

36 12 12 100 64 64� � � � � � �

��

l lGr Br Gr Br,

aP

za

P

z

h5

1

0

2

12

2 10

2

1

1

64 128� �

�

�

��

�

��

Br d

dBr

d

dl

ll

( )

aP

za

P

z6

2

1

0 17

0 1

1

1

288

1� �

�

�

��

�

��

�

��

�

��Br d

d

d

d

l

l

l l,

2 498

4 7, aa a

�

aa a h a a h

93 7

1

21 5 6

4

136

116

1

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where primes denotes the derivatives with respect to z.

Acknowledgment

T. Hayat as a visiting Professor recognizes the support of Global research Network for

Computational Mathematics and King Saud University.

References

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Paper submitted: November 22, 2011Paper revised: November 24, 2011Paper accepted: December 12, 2011


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