Journal of Mathematical Control Science and Applications (JMCSA)Vol. 2, No. 2, December 2008, pp. 189-199

© International Science Press

* School of Applied Sciences and Mathematics, Universiti Teknologi Brunei, Jalan Tungku Link Gadong BE 1410, Brunei Darussalam

Two-fluid Non-linear Models for Blood Flow in Stenosed Arteries:A Comparative Study

D. S. Sankar*

Abstract: Pulsatile flow of blood through stenosed arteries is analyzed by assuming the blood as a two-fluidmodel with the suspension of all the erythrocytes in the core region as a non-Newtonian fluid and the plasma inthe peripheral layer as a Newtonian fluid. The non-Newtonian fluid in the core region of the artery is assumed asa (i) Herschel-Bulkley fluid and (ii) Casson fluid. Perturbation method is used to solve the system of non-linearpartial differential equations. The expressions for velocity, flow rate, plug core radius, wall shear stress andresistance to flow are obtained for two-fluid Casson model and the expressions for these flow quantities obtainedby Sankar and Lee (2006) for two-fluid Herschel-Bulkley model are used to get the data for comparison. It isnoted that the plug flow velocity and velocity distribution for the two-fluid Casson model are considerablyhigher than those of the two-fluid Herschel-Bulkley model for a given set of values of the parameters. Further, itis found that the pressure drop, plug core radius, wall shear stress and the resistance to flow are significantly verylow for the two-fluid Casson model than those of the two-fluid Herschel-Bulkley model. Thus, it is concludedthat the two-fluid Casson model is more useful than the two-fluid Herschel-Bulkley model to analyze the bloodflow through stenosed arteries.

Keywords: Two-fluid models; Pulsatile flow; Stenosed arteries; Non-Newtonian fluids; Resistance to flow

1. INTRODUCTION

There are many evidences that vascular fluid dynamics plays a major role in the development and progressionof arterial stenosis. Arteries are narrowed by the development of atherosclerotic plaques that protrude into thelumen, resulting arterial stenosis. When an obstruction developed in an artery, one of the most seriousconsequences is the increased resistance and the associated reduction of the blood flow to the particular vascularbed supplied by the artery. Thus, the presence of a stenosis leads to the serious circulatory disorder.

Several theoretical and experimental attempts were made to study the blood flow characteristics in thepresence of stenosis [1-8]. The assumption of Newtonian behavior of blood is acceptable for high shear rateflow through larger arteries [9]. But, blood, being a suspension of cells in plasma, exhibits non–Newtonianbehavior at low shear rate ( �� < 10/sec) in small diameter arteries [10]. In diseased state, the actual flow isdistinctly pulsatile [11, 12]. Many researchers studied the non–Newtonian behavior and pulsatile flow of bloodthrough stenosed arteries [1, 3, 9, 12].

Bugliarello and Sevilla [13] and Cokelet [14] have shown experimentally that for blood flowing throughnarrow blood vessels, there is a peripheral layer of plasma and a core region of suspension of all the erythrocytes.Thus, for a realistic description of the blood flow, it is appropriate to treat blood as a two-fluid model with thesuspension of all the erythrocytes in the core region as a non-Newtonian fluid and plasma in the peripheralregion as a Newtonian fluid.

Journal of Mathematical Control Science and ApplicationsVol. 5 No. 1 (January-June, 2019)

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190 Journal of Mathematical Control Science and Applications (JMCSA)

Sankar and Lee [15] have developed two-fluid model for pulsatile blood flow through arterial stenosistreating the fluid in the core region as Herschel-Bulkley fluid. Thus, in this paper, we extend this study to two-fluid Casson model and; compare these models and discuss the advantages of the two-fluid Casson model overthe two-fluid Herschel-Bulkley (H-B) model.

2. MATHEMATICAL FORMULATION

Consider an axially symmetric, laminar, pulsatile and fully developed flow of blood (assumed to beincompressible) in the z direction through a rigid walled circular artery with an axially symmetric mild stenosis.The geometry of the arterial stenosis is shown in Fig. 1. We have used the cylindrical polar coordinates ( r, �, z).Blood is represented by a two-fluid model with the suspension of all the erythrocytes in the core region as aNon-Newtonian fluid and the plasma in the peripheral region as a Newtonian fluid. The non-Newtonian fluid inthe core region is represented by (i) Casson fluid model and (ii) Herschel-Bulkley fluid model. The geometry ofthe stenosis in the peripheral region (in dimensionless form) and core region are respectively given by

� � � � � � � �� �0

0 0 0 0

in the jormal artery region

2 1 cos 2 2 in P

RR z

R L z d L d z d L

��� � � �� � � � � � � � �� � �� (1)

� � � � � � � �� � in

0

10 0 0 0

in the normal artery region

2 1 cos 2 2C

RR z

R L z d L d z d L

� ��� � � �� � � � � � � � � �� � ��

(2)

Figure 1: Geometry of the Two-fluid Models with Arterial Stenosis

(a) Two-fluid Casson Model (b) Two-fluid Herschel-Bulkley Model

where R(z) and R1 are the radii of the stenosed artery with the peripheral region and core region respectively;R0 and �R0 are the radii of the normal artery and core region of the normal artery respectively; � is the ratio ofthe central core radius to the normal artery radius; L0 is the length of the stenosis; d indicates the location of thestenosis; �p and �C are the maximum projections of the stenosis in the peripheral region and core regionrespectively such that [ �p/ R0] << 1 and [ �C/ R0] << 1.

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Two-fluid Non-linear Models for Blood Flow in Stenosed Arteries: A Comparative Study 191

2.1 TWO-FLUID CASSON MODEL

2.1.1 Governing equations

It can be shown that the radial velocity is negligibly small and can be neglected for a low Reynolds numberflow. The basic momentum equations governing the flow are

� � � � � � � �� � � � in 11 0C C Cu t p z r r r r R z� � � � � � � � � � � � � (3)

� � � � � � � �� � � � � �in 11N N Nu t p z r r r R z r R z� � � � � � � � � � � � � (4)

where the shear stress � = � �� r z��= – �� r z�(since ��= �C or ��= �N); p is the pressure; uC and uN are the axialvelocity of the fluid in the core region and peripheral region, respectively; �C and �N are the shear stress of theCasson fluid and Newtonian fluid, respectively; �C and �N are the densities of the Casson fluid and Newtonianfluid respectively; t is the time. The relations between the shear stress and strain rate of the fluids in motion inthe core region (Casson fluid) and peripheral region (Newtonian fluid) are given by

� � � �1if andC C C y C y pu r R r R z� � �� � � � � � � � � � (5)

� � 0 if and 0C C y pu r r R� � � � � � � � (6)

� � � � � �1ifN N Nu r R z r R z� � �� � � � � (7)

where µC and µN are the viscosities of the Casson and Newtonian fluids respectively; �y is the yield stress; RP

is the plug core radius. The boundary conditions are

1

is finite and 0 at 0

0 at

and at

C C

N

C N C N

u r r

u r R

u u r R

� � � � �

� �

� � � � �

(8)

Since the pressure gradient is a function of z and t, we assume

� � � � � �p z q z f t� � � � (9)

where � � � �� �,0q z p z z� � � � . Since, any periodic function can be expanded in a Fourier sine series, it isreasonable to choose 1 sinA t� � as a good approximation for f( t ), where A and � are the amplitude andangular frequency of the flow respectively. We introduce the following non-dimensional variables

� � � � � � � �0 0 1 1 0 0 0 0 0 0, , , , , ,z z R R z R z R R z R z R r r R d d R L L R� � � � � �

� � � � 2 2 2 20 0 0 0, , , ,C C C C N N N N P Pq z q z q R R R R R� � � � � �� � � � � � �� � �

� � � �2 20 0 0 0 0 0, , 4 , 4 ,P P C C C C C N N NR R u u q R u u q R� � � � � � � � � �

� � � � � �0 0 0 0 0 02 , 2 , 2 ,C C N N yq R q R q R t t� � � � � � � � � � � (10)

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192 Journal of Mathematical Control Science and Applications (JMCSA)

where 0q is the negative of the pressure gradient in the normal artery, aC and aN are the pulsatile Reynoldsnumbers of the Casson fluid and Newtonian fluid respectively. Using the non-dimensional variables, Eqs. (1) -(5) are simplified to

� � � � � � � � � � � �14 2 if 0C C Cu t q z f t r r r r R z� � � � � � � � � � (11)

� � � �11 2 if andC C C pu r R r R z� � � � � � � � � � � � (12)

0 if and 0C C pu r r R� � � � � � � � (13)

� � � � � � � �� �� �

� � � �14 2

if1 2

N N N

N N

u t q z f t r r rR z r R z

u r

�� � � � � � � � � � ���� � � � � �

(14)

where

f(t) = 1 + A sin t (15)

The boundary conditions (in the dimensionless form) are

�C is finite and �uC /�r = 0 at r = 0

�C = �N and uC = uN at r = R1 (16)

uN = 0 at r = R

The geometry of the stenosis in the peripheral region and core region (in the dimensionless form) are given by

� � � � � � � �� �� �0 0 0

1 in the normal artery region

1 2 1 cos 2 2 inP

R zL z d L d z d L

��� � � �� � � � � � � � �� � ��(17)

� � � � � � � �� �� �10 0 0

in the normal artery region

2 1 cos 2 2 inC

R zL z d L d z d L

���� � � �� � � � � � � � � �� � �� (18)

The non-dimensional volume flow rate Q is given by

� �� �0

4 , ,R z

Q u r z t rdr� � (19)

where 40 0 0( 8 ),Q Q R q Q� � � is the volume flow rate.

2.1.2 Method of Solution

When we non-dimensionalize the constitutive Eqs. (1) and (2), �C and �N occur naturally and these aretime dependent and hence, it is more appropriate to expand the Eqs. (11)-(14) about �C and �N. Let usexpand the plug core velocity up, the velocity in the core region uC in the perturbation series of �C as below(where �C << 1)

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Two-fluid Non-linear Models for Blood Flow in Stenosed Arteries: A Comparative Study 193

up (z, t) = uop (z, t) + �C u1p (z, t) + … (20)

uC (r, z, t) = u0C (r, z, t) + �C u1C (r, z, t) + … (21)

Similarly, one can expand uN, �P, �C, �N and RP in powers of �C and �N, where �N << 1. Using the perturbationseries in Eqs. (11) and (12) and then equating the constant terms and �C terms, the differential equations of thecore region become

� � � � � � � �� � � �

0 0 1

0 0 0 1 1 0

2 , (2 )

2 2 , 2 1

C C C

C C C C C C

r r q z f t r u t r r r

u r u r

� � � � � � � � � � �

�� � � � � �� � � �� � � � � � �(22)

Similarly, using the perturbation series expansions in Eq. (14) and then equating the constant terms and �N

terms, the differential equations of the peripheral region become

� � � � � � � �0 0 1

0 0 1 1

2 , (2 )

2 , 2N N N

N N N N

r r q z f t r u t r r r

u r u r

� � � � � � � � � � ��� � � � �� � � �

(23)

Substituting the perturbation series expansions in Eq. (16) and then equating the constant terms and �C and �N

terms, we get

0 1 0 1

0 0 1 1 0 0 1 1 1

0 1

and are finite and 0, 0 at 0

, , , at

0 at

p p P P

C N C N C N C N

N N

u r u r r

u u u u r R

u u r R

� � � � � � � � �

� � � � � � � � �

� � �

(24)

Solving the system of Eqs. (22) and (23) using Eq. (24) for the unknowns u0C, u1C, �0C, �1C, u0N, u1N, �0N, �1N, onecan obtain

�0p = �R0p, �0C = , �r, �0N = �r (25)

u0N = ��R2 (1– �2) (26)

u0C = �R2 {(1 –��2) + �2 [(1 – �12) – (8/3) �1

1/2 (1 – �13/2) + 2�1 (1–��1)]} (27)

u0p = �R2 {(1 –��2) + �2 [(1 – �2) – (8/3) �11/2 (1 – �3/2) + 2�1 (1 –��)]} (28)

�1p = �BR3 {(1/4)�(1 –��2) + �3 �1[(1/4) –(1/3) �11/2 + (1/12) �1

2]} (29)

�1C = – � BR3 {(1/4)� (1– �2)

– (1/8)�3 [2�1 – �13 – �1

� �1–1 – (8/21)�1

1/2 (7�1 – 4�15/2 – 3�1

7/2 �1–1)]} (30)

�1N = – � BR2 R1 {([1/4)�1 – (1/8)�2 �1–1 – (1/8)�2�1

3]

+�1–1 �2 [(1/8) – (1/7) �1

1/2 + (1/56)�14]} (31)

u1N = – � BR3 R1 {[1/4) �–1 (1–�2) – (1/4)�3 log �–1 – (1/16) �–1 (1–�4)]

– �3 log � [(1/4) – (2/7) �11/2 + (1/28)�1

4]} (32)

u1C = – � BR3 R1 {[3/16)�–1 – (1/4)� + (1/16)�3 + (1/4)�3 log �]

– �3 log � [(1/4( – (2/7)�11/2 + (1/28)�1

4]

+ � (1–�2)[(1/4)(1–�12) – (1/3)�1

1/2 (1 –������)] (33)

+ �3 [(1/4) (1 – �12) – (1/3)�1

1/2 (1–�13/2)–(1/16)(1– �1

4)

+ (53/294)�11/2 (1 – �1

7/2) – (1/3) (1 – �12) + (4/9) �1 (1 –��1

3/2)

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194 Journal of Mathematical Control Science and Applications (JMCSA)

–(8/63)�1 (1 –��13) – (1/28)�1

4 log �1 + (1/14)�19/2 (1 – �1

–1/2)]}

u1P = – � BR3 R1 {((3/16)�–1 –(1/4)� + (1/16)�3 + (1/4)�3 log �)

– �3 log � ((1/4) – (2/7)�11/2 + (1/28) �1

4)

+ � (1 – �2)[1/4)(1– �12) – (1/3) �1

1/2 (1 – �13/2)] (34)

+ �3 [(1/4)(1 – �12) – (1/3)�1

1/2 (1 – �13/2) – (1/16) (1– �1

4)

–(53/294)�11/2 (1 – �1

7/2) – (1/3)(1 – �12) + (4/9) �1 (1 – �1

3/2)

+ (8/63) 2�1 (1– �13) – (1/28) �1

4 log �1 + (1/14) �19/2 (1– �1–

1/2)]} (34)

where � = q(z)f(t), k2 = r�r0pz � = R0p = � / [q(z)f(t)], B = [1/f(t)](df(t)/dt), � = r/R, �1 = r/R1, � = R1/R, � = k2/R, �1

= k2/R1, and � = R0p/R1. The wall shear stress �w can be obtained as below.

�w = (t�N + �N�1N)r = R = �0w + �N �1w

= � {R – 1/8) BR3 �N (1 – �4) – (1/8) BR13 �N � [1 – (8/7) �1

1/2 + (1/7) �14]} (35)

Using Eqs. (26)-(28) and (32)-(34) in Eq. (19), the volume flow rate is obtained as

Q = � R4 {1 – �2) (1 + 3�2) + �4 [1– (16/7)�11/2 + (4/3)�1 – (1/21)�1

4]}

– �C � BR3 R13 {[3/8)�–1 – (1/2)� + (1/8)�3 + (1/2) �3 log �]

–��3 log � [(1/2) – (4/7)�11/2 + (1/14)�1

4]

+ �(1 – �2) [(1/4) – (2/7)�11/2 + (1/28) �1

4] (36)

+ �3 [(1/6) – (30/77)�11/2 + (8/35)�1 – (1/3) �1

3/2 + (1/14) �14

+ (5/21) �19/2 – (41/770)��1

6 – (1/14) �16 – log �1 + (1/14) �1

4 (1–�12) log k]}

– �N � BR5 R1 {[(1/6)�–1 – (3/8)� + (5/24)�5 – (1/2)�3 (1–�2) log R1]

+ �4 (1–�2) (1 + 2 log R1) [(1/4) – (2/7) �11/2 + (1/28) �1

4]}

The shear stress �C = �C + �H �1C at r = Rp is given by

��0C + �C �1C�r = Rp = � (37)

Using the Taylor’s series of �0C and �1C about R0p and using �0C�r=R0p = �, we get

R1p = – �1C�r=R0P/� (38)

Using Eqs. (25), (30) and (38) in the two term approximated perturbation series of RP, the expression for RP canbe obtained as

Rp = k2 – (1/4) B�C R3 [�2 (1–�2) + �3 (�1 – 4�13/2/3 + �1

3/3)] (39)

The resistance to flow is given by

� = [�p f(t)]/Q (40)

where �p is the pressure drop. When R1 = R, the present model reduces to the single fluid Casson model and insuch case, the expressions obtained in the present model for velocity uC, shear stress �C, wall shear stress �w,flow rate Q and plug core radius Rp are in good agreement with those of Chaturani and Ponnalagar Samy [12].

2.2 Two-Fluid Herschel-Bulkley Model

The basic momentum equations governing the flow and the constitutive equations in the non-dimensional formare

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Two-fluid Non-linear Models for Blood Flow in Stenosed Arteries: A Comparative Study 195

�H (�uH/�t) = 4q (z) f (t) – (2/r)(�(r�H)/�r) if 0 � r � R1 (z) (41)

�H (�uN/�t) = 4q (z) f(t) – 2/r)(�(r�N)/�r) if R1(z) � r � R(z) (42)

� �� � � �11 2 if andnH H H pu r R r R z� � � � � � � � � � � � (43)

�uH/�r = 0 if �H � � and 0 � r � Rp (44)

�N = – (1/2) (�uN /�r) if R1 (z) � r � R(z) (45)

The boundary conditions (in dimensionless form) of this model are similar to the boundary conditions ofthe two-fluid Casson model given in Eq. (8). Eqs. (41) – (45) are also solved using perturbation method with thehelp of the appropriate boundary conditions as in the case of the two-fluid Casson model. The details of thederivation of the expressions for shear stress, velocity, flow rate, plug core radius, wall shear stress and resistanceto flow are given in Sankar and Lee [15].

3. RESULTS AND DISCUSSION

The objective of the present analysis is to compare and bring out advantages of the two-fluid Casson model overthe two-fluid Herschel-Bulkley model. It is observed that the typical value of the power law index n for bloodflow models is taken as 0.95 [3]. The value 0.1 used for the non-dimensional yield stress � in this study. Thoughthe range of the amplitude A is from 0 to 1, we have used the value 0.5. The value 0.5 is used for the pulsatileReynolds number �H, �C and pulsatile Reynolds number ratio � of both the two-fluid models [11]. The value ofthe ratio � of central core radius ��R0 to the normal artery radius R0 in the unobstructed artery is generally takenas 0.95 [15]. Following Shukla et al. [16], relations R1 = �R and �c = ��p are used to estimate R1 and �c. Themaximum thickness of the stenosis in the peripheral region �P is taken as 0.1 [11]. The steasy flow rate QS valueis taken as 1.0 [12]. It is observed that in the expression of the flow rate for the two-fluid Casson model, f(t),R and � are known and Q and q(z) are the unknowns to be determined. A careful analysis of the flow rateexpression reveals the fact that q(z) is the pressure gradient of the steady flow. Thus, if steady flow is assumed,then the expression for the flow rate can be solved for q(z) [3,12]. For steady flow, the expression for flow rateof the two-fluid Casson model reduces to

� � � � � � � � � � � � � �74 34 2 2 4 4 4 3

1 1 1 1 14 3 16 7 4 3 1 21 0SR R R R y R y R y R y Q y� �� � � � � � � � � � �� �� � (46)

The similar equation for the two-fluid Herschel-Bulkley model is

(R2 – R12)�4(�/�)2 + (R2 – R1

2)� y3 + [4/(n + 2)(n + 3)]

{(n + 2)(R1y)n+3 – n(n + 3)� (R1y)n+2 + (n2 + 2n – 2) �n+2} – QS y

3 = 0 (47)

The variation of pressure drop in a time cycle for the two-fluid Herschel-Bulkley (H-B) and Casson modelswith � = �P = 0.1, A = 0.5 and � = 0.95 is shown in Fig. 2. It is observed that for both the two-fluid models thepressure drop increases as time t (in degrees) increases from 0° to 90° and then it decreases as t increases from90° to 270° and again the pressure drop increases as t increases further from 270° to 360°. The pressure drop ismaximum at 90° and minimum at 270°. It is found that at any time, the pressure drop is considerably very lowfor the two-fluid Casson model than that of the two-fluid H-B model while all the other parameters held constant.Fig. 3 depicts the variation of the plug core radius with axial distance for the two-fluid H-B and Casson modelswith � = �P = 0.1, A = 0.5 and � = 0.95. It is noticed that the plug core radius decreases as the axial variable

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196 Journal of Mathematical Control Science and Applications (JMCSA)

increases from 4 to 5 and it increases symmetrically when the axial variable increases from 5 to 6. It is noted thatfor a given set of values of the parameters, the plug core radius values are significantly very low for the two-fluid Casson model than that of the two-fluid H-B model.

3.1 Plug Flow Velocity

The variation of the plug flow velocity in a time cycle for the two-fluid Casson and H-B models with � = �P =0.1, A = 0.5, � = �H = �C = 0.5, �N = 0.25, � = 0.95 and z = 5 is depicted in Fig. 4. It is seen that the plug flowvelocity decreases as time t (in degrees) increases from 0° to 90° and then it increases as t increases from 90° to270° and then again it decreases from 270° to 360°. The plug flow velocity is minimum at 90° and maximum at270°. It is noted that the plug flow velocity is considerably higher for the two-fluid Casson model than that ofthe two-fluid H-B model.

Figure 2: Variation of Pressure Drop in a Time Cycle for the Two-fluid Casson and H-B Models

Figure 3: Variation of Plug Core Radius with Axial Distance for the Two-fluid Casson and Herschel-Bulkley Models

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Two-fluid Non-linear Models for Blood Flow in Stenosed Arteries: A Comparative Study 197

3.2 Wall Shear Stress

Fig. 5 shows the variation of the wall shear stress in a time cycle for the two-fluid Casson and H-B models with� = �P = 0.1, A = 0.5, � = �H = �C = 0.5, �N = 0.25, � = 0.95 and z = 5. The behaviour of the wall shear stress isjust reversed for the two-fluid models that we observed in Fig. 4 for the plug flow velocity.

Figure 4: Variation of Plug Flow Velocity in a Time Cycle for the Two-fluid Casson and TwoFluid Herschel-Bulkley Models

Figure 5: Variation of Wall Shear Stress in a Cycle for the Two-fluid Casson and Two Fluid Herschel-Bulkley Models

3.3 Velocity Distribution

The velocity distributions for the two-fluid H-B and Casson models with � = �P = 0.1, A = 0.5, � = �H =�C = 0.5, �N = 0.25, � = 0.95 and t = 45° are sketched in Fig. 6. One can notice the plug flow around the tubeaxis for both the fluid models. It is further recorded that for a given set values of the parameters, asignificantly high magituge velocity profile is found for the two-fluid Casson model model than the two-fluidH-B model.

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198 Journal of Mathematical Control Science and Applications (JMCSA)

4. CONCLUSION

The pulsatile blood flow through stenosed arteries is analyzed assuming blood as a (i) two-fluid Casson modeland (ii) two-fluid Herschel-Bulkley model. It is observed that the velocity distribution for the two-fluid Cassonmodel is considerably higher than that of the two-fluid Herschel-Bulkley fluid model for a given set of values ofthe parameters. Further, it is noticed that the pressure drop, plug core radius, wall sheat stress and the resistancto flow are significantly very low for the two-fluid Casson model than those of the two-fluid Herschel-Bulkley

3.4 Resistance to Flow

The variation of resistance to flow with peripheral layer stenosis height for the two-fluid Casson and H-Bmodels with � = �P = 0.1, A = 0.5, a = aH = aC = 0.5, �N = 0.25, � = 0.95 and t = 45° is shown in Fig. 7. It is seenthat the resistance to flow increases non-linearly with the increase of the peripheral stenosis height. It is ofinterest to note that for any value of the stenosis height, the resistanc to flow is considerably very low for thetwo-fluid Casson model than that of the H-B model.

Figure 7: Variation of Resistance to Flow with Peripheral Layer Stenosis Height for the Two- fluidCasson and Two-fluid Herschel-Bulkley Models

Figure 6: Velocity Distribution for the Two-fluid Casson and Two-fluid Herschel-Bulkley Model

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Two-fluid Non-linear Models for Blood Flow in Stenosed Arteries: A Comparative Study 199

model. It is of interest to note that the difference between the estimates of the two-fluid Casson model and the two-fluid Herschel-Bulkley model is sustantial. Thus, it is concluded that the two-fluid Casson model will have more applicability in analysing the blood flow through stenosed arteries.

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