multiphase flow of blood through arteries with a branch capillary

December 6, 2007 16:18 WSPC/170-JMMB

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Journal of Mechanics in Medicine and Biology Vol. 7, No. 4 (2007) 395–417 c World Scientific Publishing Company

J. Mech. Med. Biol. 2007.07:395-417. Downloaded from www.worldscientific.com by UNIVERSITY OF ALBERTA on 04/25/15. For personal use only.

MULTIPHASE FLOW OF BLOOD THROUGH ARTERIES WITH A BRANCH CAPILLARY: A THEORETICAL STUDY

J. C. MISRA∗ , S. D. ADHIKARY and G. C. SHIT Center for Theoretical Studies Indian Institute of Technology, Kharagpur Kharagpur-721302, India ∗[email protected]

Received 24 April 2007 Accepted 3 August 2007 In this paper, we present a theoretical analysis of the problem of hematocrit reduction (due to plasma skimming) in a capillary that emerges from an artery making an angle α with the parent artery. The analysis bears the potential to explore a variety of information regarding some phenomenological aspects of this important physiological problem. The flow is considered to consist of three distinct phases, viz., the peripheral plasma layer, the cell-depleted middle layer, and the core region which usually has a high concentration of erythrocytes. This study deals with both steady and pulsatile flow of blood, which is treated as a non-Newtonian fluid of Herschel–Bulkley type. A computational procedure is developed for a quantitative measure of the velocity profile, the volumetric flow rate, and the hematocrit of blood in a specific situation. The procedure also gives us an opportunity to examine the nature of variation of these important hemodynamic factors; this observation holds true irrespective of whether the flow of blood is steady or pulsatile. The study reveals that the velocity of blood in the parent artery reduces when the fluid index/yield stress increases. It is further revealed that the volumetric flow rate of blood in the capillary also decreases with an increase in the value of the fluid index/yield stress of blood. Keywords: Herschel–Bulkley model; multiphase flow; branch capillary; hematocrit.

1. Introduction The rheological properties of blood and its constituents play an important role in the study of the physiology of the cardiovascular system. Therefore, the rheological complexities of blood and its flow in the cardiovascular system have attracted serious attention from present-day researchers. From a physiological point of view, blood possesses some important characteristics because blood is a suspension of particles — erythrocytes, white blood cells, platelets, and other constituents — in ∗ Corresponding

author. 395



396

00239

J. C. Misra, S. D. Adhikary & G. C. Shit

an aqueous liquid called plasma. In large blood vessels, blood behaves more or less like a homogeneous Newtonian fluid; while in the case of narrow blood vessels, e.g. capillaries, it behaves in a non-Newtonian fluid manner. A further complication in describing the flow of blood lies in the fact that under certain conditions such as low shear rate, the platelets are activated, leading to certain chemical reactions which bring about significant changes to the flow behavior of blood. We thus intend to develop such a constitutive model that can depict its non-Newtonian behavior. In several investigations reported in the literature,1–5 the non-Newtonian model was considered to describe the flow behavior of blood in normal/pathological states. Misra et al.6,7 studied blood flow in the vicinity of a stenosis developed in an arterial segment by considering a non-Newtonian fluid model. Misra and Ghosh8,9 carried out an investigation on the pulsatile flow of a viscous fluid through a porous elastic vessel of variable cross-section. All of these studies bear strong potential for substantial applications in physiological fluid dynamics, particularly in the study of blood flow in normal/pathological states of blood vessels. Some of these are quite useful for a better understanding of the pathogenesis and proper treatment of various arterial diseases like myocardial infarction, stroke, etc. The literature on studies of the effects of physiological pulsatile flow and nonNewtonian behavior of blood through narrow vessels with bifurcation is scanty, although several studies related to the pulsatility of arterial flow have been reported.10,11 Aroesty and Gross12,13 theoretically analyzed the influence of pulsatile pressure on blood flow in small vessels by considering blood as a nonNewtonian fluid of Casson type. They studied the distribution of velocity and shear stress by considering Womersley parameter values less than 1.0. Iida and Murata14 analyzed the non-Newtonian effects of blood on pulsatile flow in arteries. They took into account the thickness of the plasma layer, taking it to be uniform, and assumed that the thickness does not change with time and that the flow is Newtonian. Hemodynamic characteristics of blood flow through arterial stenoses were numerically investigated by Moayeri and Zendehbudi.15 In this study, the pulsatile nature of flow was studied by using measured values of the flow rate and pressure for canine femoral artery, treating blood as a Newtonian fluid. The effect of change in values of the Womersley parameter in pulsatile blood flow was examined by Rohlf and Tanti,16 who treated blood as a Casson fluid. Tu and Deville17 presented a theoretical analysis of pulsatile blood flow in stenosed arteries, where the non-Newtonian behavior of blood was taken to be of Herschel–Bulkley type. The effects of arterial curvature, junctions, and bifurcations are of great importance in the physiology of cardiovascular systems. Several investigators18–21 studied the effect of hematocrit reduction in blood flow near bifurcations. Misra and Kar22 dealt with a mathematical analysis of branching in the microcirculatory system. Their study was, however, restricted to the particular case when blood enters from a feeding artery into a right-angled branch capillary. They considered a two-layer model, where the cell-free plasma layer near the wall of a micro-blood


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Multiphase Flow of Blood Through Arteries with a Branch Capillary

397

vessel reduces the hematocrit and the cells migrate to the core region, that depicts the non-Newtonian characteristics of blood. A two-layer mathematical model of blood flow through an artery with a sinusoidal constriction was presented by Haldar and Andersson.23 Their model consisted of a peripheral plasma layer free from red blood cells and a core region represented by a Casson fluid. Recently, Sharan and Pople24 investigated a two-phase model for blood flow of blood in narrow tubes. They assumed that the viscosity of the cell-free layer differs from that of plasma as a result of additional dissipation of energy near the wall caused by the red blood cell motion near the cell-free layer. Gupta et al.25 investigated a three-layered semiempirical model for the flow of blood and particulate suspension through narrow tubes without considering the branching effects. Misra and Ghosh26 developed a mathematical model for the study of Casson fluid flow in a narrow tube with a side branch by considering a three-layered model, taking into account the effect of hematocrit reduction. In the present investigation, multiphase flow of blood through a narrow artery is made for the microcirculatory system with an aim to study all situations where the arteriole may emerge from a feeding artery by making any arbitrary angle. The three phases are (1) the cell-free plasma layer, called the peripheral layer; (2) the cell-depleted middle layer; and (3) the core region, which usually has a high concentration of erythrocytes. While the cell-free plasma layer is considered as a power-law fluid, the cell-depleted middle layer is represented by the Herschel–Bulkley model (as per the observations mentioned in Whitmore27 ). The concentration of the core region in the neighborhood of the axis of the small artery is taken to be constant. The model developed here is particularly applicable to blood flow in microvessels that have a diameter of less than 200 µm and a shear rate below 10 s−1 . On the basis of our computational analysis, we present the variations of the velocity profile, volumetric flow rate, and hematocrit reduction in a branch capillary emerging out of a parent artery at an arbitrary angle. Both the cases of steady and pulsatile flows are dealt with. 2. Flow Analysis Let us denote the radii of the main artery and branch capillary as R∗ and R1∗ (< R∗ ), respectively. α is the angle between the the branch capillary and the main artery called the feeding artery. Blood is treated here as a Herschel–Bulkley fluid that flows under the action of a prescribed pressure gradient directed along the axis of the blood vessel. The analysis that follows is carried out under the consideration that the length of the blood vessel is quite large in comparison to its diameter. We use the cylindrical polar coordinates (r, φ, z), the pole being located at the vessel axis (cf. Fig. 1) which is taken to coincide with the axis of z. For a low Reynolds number, the governing equation of motion may be taken as ρ

∂p∗ 1 ∂(r∗ τ ∗ ) ∂u∗ = − − . ∂t∗ ∂z ∗ r∗ ∂r∗

(1)


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398

r

R

R0

z

Rp


ψ

R1

α

Fig. 1. Schematic diagram of an arterial segment with branch capillary.

The Herschel–Bulkley constitutive equation is −

∂u∗ 1 = (τ ∗ − τy )n , ∂r∗ µ

τ ∗ > τy

(2)

∂u∗ = 0, τ ∗ ≤ τy . (3) ∂r∗ The multiphase flow of blood (referred to in the preceding section) is characterized mathematically by the following system of equations: −

−

∂u∗ 1 = τ ∗n ∂r∗ µ

if R0∗ ≤ r∗ ≤ R∗

(4)

−

∂u∗ 1 ∗ n = (τ − τy ) ∗ ∂r µ1

if Rp∗ ≤ r∗ < R0∗

(5)

if r∗ < Rp∗ ,

(6)

τ ∗ = τy

where µ1 = µ(1 + k − kr∗ m ) ; R0∗ and Rp∗ are the quantities shown in Fig. 1. Let us introduce the following nondimensional variables: τ =

2τ ∗ p0 a

r=

r∗ a

u=

u∗ p0 a2 /2µp

t = t∗ w n−1 2 µp = µ ap0


00239


R∗ a R0∗ R0 = a Rp∗ Rp = a R1∗ and R1 = a ∂p∗ = −p0 p(t), ∂z ∗

399


R=

(7)

a being a characteristic radius; and p(t), a nondimensional pressure gradient along the axis of the tube that is taken to be a periodic function of time. Using the above nondimensional variables, Eq. (1) reduces to α∗ 2

1 ∂(rτ ) ∂u = 2p(t) − , ∂t r ∂r

0≤r≤R

(8)

2

w where α∗ 2 = = µap /ρ is the Womersley parameter. Equations (4)–(6) become

−

∂u = τ n, ∂r

R0 ≤ r ≤ R

(9)

−

1 ∂u = (τ − θ)n , Rp ≤ r < R0 ∂r (1 + k − kam rm )

(10)

−

∂u = 0, ∂r

(11)

0 ≤ r < Rp

2τ

where θ = p0 ya . Written mathematically, the boundary conditions for the present problem are u=0 τ

at r = R

is finite at r = 0.

(12) (13)

2.1. Steady flow For steady flow of blood, the governing Eq. (8) can be written as 1 ∂(rτ ) = 2ps , r ∂r

0≤r≤R

(14)

ps being the steady pressure gradient. Integrating Eq. (14) and using boundary condition (13), we get τ = ps r.

(15)


400

00239


Integrating Eqs. (9)–(11) and using boundary condition (12) and Eq. (15), we get u=

R0 ≤ r ≤ R

(16)

m+n+1 rn+1 θrn r nθrm+n nθ2 rn−1 kam − − + + 2 1+n ps 2ps 1 + k m + n + 1 ps (m + n) r2m+n+1 nθr2m+n n(n − 1)θ2 rm+n−1 k 2 a2m − + + 2p2s (m + n − 1) (1 + k)2 2m + n + 1 ps (2m + n) n(n − 1)θ2 r2m+n−1 (17) + , Rp ≤ r < R0 . 2p2s (2m + n − 1)

u = X1 −


pns (Rn+1 − rn+1 ), n+1

pns 1+k

By neglecting powers of θ higher than the second, Rpn+1 θRpn Rpm+n+1 nθRpm+n nθ2 Rpn−1 pns kam − − + + u = X1 − 2 1+k 1+n ps 2ps 1 + k m + n + 1 ps (m + n) n(n − 1)θ2 Rpm+n−1 nθRp2m+n Rp2m+n+1 k 2 a2m + − + 2p2s (m + n − 1) (1 + k)2 2m + n + 1 ps (2m + n) n(n − 1)θ2 Rp2m+n−1 + 2p2s (2m + n − 1)

,

0 ≤ r < Rp .

(18)

The volumetric flow rates of the three different layers under consideration are given by R ru dr Q1 = 2π R0

2πpns Rn+1 2 1 n+3 2 n+3 = (R − R0 ) − (R − R0 ) n+1 2 n+3 Q2 = 2π

R0

Rp

ru dr

X1 2 pn (R0 − Rp2 ) − s 2 1+k

1 (Rn+2 − Rpn+2 ) (n + 1)(n + 2) 0 nθ2 2 θ n+1 n+1 2 (R − Rp ) + 2 (R0 − Rp ) − ps (n + 1) 0 ps n kam 1 + (Rm+n+2 − Rpm+n+2 ) 1 + k (m + n + 1)(m + n + 2) 0

= 2π

−

nθ (Rm+n+1 − Rpm+n+1 ) ps (m + n)(m + n + 1) 0

(19)


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401

n(n − 1)θ2 m+n m+n (R − R ) p 2p2s (m + n)(m + n − 1) 0 k 2 a2m 1 + (R02m+n+2 − Rp2m+n+2 ) 2 (2m + n + 1)(2m + n + 2) (1 + k) +

nθ (R2m+n+1 − Rp2m+n+1 ) ps (2m + n)(2m + n + 1) 0 n(n − 1)θ2 + 2 (R02m+n − Rp2m+n ) 2ps (2m + n − 1)(2m + n)


−

Q3 = 2π

Rp 0

(20)

ru dr = πS0 (Rp )Rp2 .

(21)

Thus, the total volumetric flow rate is Q = Q1 + Q2 + Q3 . When blood flows from a parent artery to a branch capillary which emerges from the parent artery making an angle α to it, it is usually held that blood flow takes place predominantly from a cylindrical region into the branch capillary having its center on the inner surface of the artery. The geometry of the feeding artery is shown in Fig. 1. From the geometry of the stream tube, we have 2 R − R1 sin2 α + r2 , ψ = cos−1 2Rr where R1 is the radius of the branched artery. The volumetric flow rates of blood in the branch capillary for the three different layers are given by R ruψ dr Q1.cap = 2 R0

R2

=

pns n+1

=

2 pns (Rn+1 Z1 − Z2 )R R0 2 , n+1

R0 2

(Rn+1 − η

n+1 2

) cos−1

ηc + η √ 2R η

=

max(Rα ,Rp )

X1 Z1 −

pns 1+k

1 θ nθ2 Z2 − Z3 + 2 Z4 n+1 ps Ps

dη (22)

where ηc = R2 − R12 sin2 α, η = r2 ; R0 ruψ dr Q2.cap = 2


402

00239


kam 1+k

nθ n(n − 1)θ2 1 Z5 − Z6 + 2 m+n+1 ps (m + n) 2ps (m + n − 1) 1 nθ k 2 a2m Z8 − Z9 + 2 2m + n + 1 ps (2m + n) (1 + k) R20 n(n − 1)θ2 Z10 + 2 , 2ps (2m + n − 1) max(R2 ,R2 ) +

α

(23)

p


where Rα = R − R1 ; and Q3.cap =

Rp

Rα

=0

ruψ dr

if Rα < Rp if Rα > Rp .

(24)

The total volumetric flow rate of blood in the branch capillary is then given by Qcap = Q1.cap + Q2.cap + Q3.cap . The hematocrit in the feeding artery is given by R H=

0

rcv (r)u(r) dr , R ru(r) dr 0

(25)

where cv (r) is the volume concentration of the red cells and u is the axial velocity of the blood. For a linear concentration change described by cv (r) = cv Θ(Rp − r) +

cv (R2 − r2 )Θ(R0 − r)Θ(r − Rp ), (R02 − Rp2 ) 0

the hematocrit in the feeding artery is obtained as Hf =

2 2 cv Rp3 R0 r r4 cv S0 (Rp ) + − X 1 2Q Q(R02 − Rp2 ) 2 4 2 n+3 1 R0 r rn+5 rn+4 pn θ R02 rn+2 − − − s − 1+k n+1 n+3 n+5 ps n+2 n+4 rn+3 1 R02 rm+n+3 nθ2 R02 rn+1 kam − + 2 + ps n+1 n+3 1+k m+n+1 m+n+3 R02 rm+n+2 rm+n+4 rm+n+5 nθ − − − m+n+5 ps (m + n) m + n + 2 m+n+4

(26)


00239


403

2 m+n+1 rm+n+3 n(n − 1)θ2 R0 r − 2p2s (m + n − 1) m + n + 1 m+n+3 1 R02 r2m+n+3 r2m+n+5 k 2 a2m − + (1 + k)2 2m + n + 1 2m + n + 3 2m + n + 5 2 2m+n+2 R0 r r2m+n+4 nθ − − ps (2m + n) 2m + n + 2 2m + n + 4


+

n(n − 1)θ2 + 2 2ps (2m + n − 1)

r2m+n+3 R02 r2m+n+1 − 2m + n + 1 2m + n + 3

R0 ;

(27)

Rp

while the expression for the hematocrit in the branch capillary is found in the form cv cv S0 (Rp )Zsh + Hcap = X1 (R2 Z1 − Z11 ) 2Qcap 2Qcap (R02 − Rp2 ) pn 1 θ nθ2 − s (R2 Z2 − Z12 ) − (R2 Z3 − Z13 ) + 2 (R2 Z4 − Z1 ) 1+k n+1 ps 2ps 1 kam nθ (R2 Z5 − Z14 ) − (R2 Z6 − Z15 ) + 1+k m+n+1 ps (m + n) k 2 a2m n(n − 1)θ2 (R2 Z7 − Z5 ) + + 2 2ps (m + n − 1) (1 + k) 1 nθ (R2 Z8 − Z16 ) − (R2 Z9 − Z17 ) × 2m + n + 1 ps (2m + n) +

R20 n(n − 1)θ2 2 (R Z − Z ) , 10 8 2p2s (2m + n − 1) max(R2 ,R2 ) p

(28)

α

where R

Zsh = Z1 |Rpα

if Rα < Rp

=0

if Rα > Rp .

(29)

The derived expressions for Zi (i = 1, 2, . . . , 17) are given in the Appendix. 2.2. Pulsatile flow To study the pulsatility of blood flow, we resort to a perturbation technique by writing u = u0 + u1 + · · ·

(30)

τ = τ0 + τ1 + · · ·

(31)


404

00239


R0 = R01 + R02 + · · ·

(32)

Rp = Rp1 + Rp2 + · · · ,

(33)

∗2

where = α is the perturbation parameter. The use of Eqs. (30) and (31) in Eq. (8) gives rise to


0 = 2p(t) −

1 ∂(rτ0 ) r ∂r

(34)

∂u0 1 ∂(rτ1 ) =− . ∂t r ∂r Integrating Eq. (34) and using boundary condition (13), we have τ0 = p(t)r.

(35)

(36)

Using Eqs. (34) and (35) in Eqs. (9)–(11), one obtains −

∂u0 = τ0 n ∂r

−

∂u1 = nτ1 τ0 n−1 , ∂r

if R0 ≤ r ≤ R

−

∂u0 1 n = (τ0 − θ) ∂r (1 + k − kam rm )

−

1 ∂u1 n−1 = nτ1 (τ0 − θ) , ∂r (1 + k − kam rm ) −

∂u0 =0 ∂r

−

∂u1 = 0, ∂r

if Rp ≤ r < R0

if 0 ≤ r < Rp .

(37)

(38)

(39)

The boundary conditions are u0 = 0, u1 = 0 at r = R u0 is continuous at r = R01 and r = Rp1 u1 is continuous at r = R02 and r = Rp2 .

(40)

From Eqs. (36)–(38) and using boundary condition (40), we obtain 2n−2 pn (t) n+1 1 p (t) n2 p(t) u= (R (R2n+2 − r2n+2 ) − rn+1 ) + n+1 n+1 (2n + 2)(n + 3) Rn+1 (Rn+1 − rn+1 ) , if R0 ≤ r ≤ R (41) − 2(n + 1) u = S0 (r) + (X3 + S1 (r)) ,

if Rp ≤ r < R0

u = S0 (Rp0 ) + (X3 + S1 (Rp1 )) ,

if 0 ≤ r < Rp .

The expressions for X1 , X2 , X3 , S0 , and S1 are shown in the Appendix.

(42) (43)


00239


405

The volumetric flow rate is Q = Q1 + Q2 + Q3 R = 2π ru(r) dr + R0

R0

ru(r) dr +

Rp

Rp

0

ru(r) dr .

(44)

Using Eq. (26), the hematocrit in the feeding artery can be written as


Hf =

cv Q

0

Rp

ru dr +

cv Q(R02 − Rp2 )

R0

Rp

(R02 − r2 )ru dr.

(45)

The volumetric flow rate in the branch capillary is Qcap = Q1.cap + Q2.cap + Q3.cap R0 R =2 ruψ dr + R0

max{Rp ,Rα }

ruψ dr + Zpq

(46)

and the corresponding hematocrit in the branch capillary is given by Hcap

cv cv = Zpq + Qcap Qcap (R02 − Rp2 )

R0

max{Rα ,Rp }

(R02 − r2 )ruψ dr,

(47)

where Zpq =

Rp Rα

ruψ dr

=0

if Rα < Rp if Rα > Rp .

(48)

From Eq. (36) at r = Rp1 , τ0 = θ, we have Rp1 =

θ . p(t)

(49)

It is noted that the yield plane, which was initially located at r = Rp1 , will be displaced by a distance Rp2 . The new location of the yield plane can be described mathematically by τ 2 (Rp1 + Rp2 ) = θ2 .

(50)

Expanding Taylor’s series about Rp1 and using τ0 (Rp1 ) = θ, we have Rp2 = −

τ1 (Rp1 ) . p(t)

(51)

R01 and R02 can be determined by considering the continuity of u0 at r = Rp1 and that of u1 at r = Rp2 , and by employing the Newton–Raphson method.


406

00239


3. Results and Discussion


The motivation behind this study was to explain some aspects of multiphase flow of blood through the branch capillary and its parent artery. In the present investigation, blood was treated as a Herschel–Bulkley fluid model, which serves as a generalized form of Newtonian, Bingham plastic, and power-law models. By considering some particular values of the different rheological parameters, it is possible to derive the results for these models from the given analysis. For the present computational study, we took the following values for the different parameters involved in our study: R = 1.0 m = 0.05 k = 0.2 Rp = 0.1 R0 = 0.9 R1 = 0.15 ps = 1.0 A1 = 0.1 and radius of parent artery a = 0.0075 cm. In conformity to Merrill’s28 experimental finding that the yield stress of blood is 0.04 dyne/cm2 for 40% hematocrit, we considered the variation of the yield stress between 0.0 and 0.3 by taking different values of the fluid index n > 0. The analytical expressions derived in the previous section were computed numerically for different values of the rheological parameters listed above, and the results are presented graphically. Figures 2 and 3 illustrate the velocity distribution in the case of steady flow for different values of the fluid index n and for various values of the yield stress θ, respectively. One may observe from Fig. 2 that the axial velocity decreases with an increase in the value of the fluid index parameter. Figure 3 shows that as the yield stress decreases, the velocity at any location increases. In order to validate our theoretical procedure, the results obtained on the basis of the present study for n = 1 and θ = 0.0 (Newtonian case) were compared with those reported in Misra and Ghosh.26 The comparison in Fig. 4 shows that the results of the present study are in close conformity to the corresponding results presented earlier in Misra and Ghosh.26 Figure 5 gives the velocity distribution in the unsteady case (pulsatile flow of blood) at a particular instant of time for a given value of yield stress when the Womersley parameter is prescribed. Different values of n were examined. In the case of steady flow, the variation of the volumetric flow rate in the cell-depleted


00239


407

0.6

n = 0.75

0.5

n = 1.0 0.4

n = 1.5 n = 2.0

u 0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

Fig. 2. Distribution of axial velocity for different values of n at θ = 0.05 (steady flow).

0.4

θ = 0.0 0.35

θ = 0.05 θ = 0.1

0.3

0.25

θ = 0.2

0.2

u


0.3

0.15

0.1

0.05

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

r

Fig. 3. Distribution of axial velocity for different values of θ at n = 1.0 (steady flow).

1


408

00239

J. C. Misra, S. D. Adhikary & G. C. Shit 0.5

Present study 0.45

+++++++

Results of Misra & Ghosh26

0.4 0.35 0.3

u

0.2 0.15 0.1 0.05 0 0

0.2

0.4

0.6

0.8

1

r Fig. 4. Comparison of axial velocity for Newtonian model (n = 1.0, θ = 0.0) with Misra and Ghosh26 in the steady case.

0.6

n = 0.75

0.5

n = 1.0 n = 1.5

0.4

n = 2.0

0.3

u


0.25

0.2

0.1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

r

Fig. 5. Distribution of velocity in the parent artery with r for different values of n at θ = 0.05 and t = 0.05, = 0.04 (pulsatile flow).


00239


409

65

60

n = 2.0

n = 1.5


Q3

Q1 + Q2

55

n = 1.0

50 n = 0.75

45

40

35 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

R0 − Rp

Fig. 6. Variation of the ratio of the volumetric flow rate with thickness of the cell-depleted middle layer for different values of n at θ = 0.05 (steady flow).

middle layer is shown in Fig. 6. It reveals that the said flow rate increases with an increase in the value of the fluid index parameter. Figure 7 gives the flow rate variation in the cell-depleted middle layer when the fluid index equals unity for different nonzero values of the yield stress parameter. These results in fact correspond to the Bingham plastic model. The time variation of the volumetric flow rate for the pulsatile flow of blood in the feeding artery is shown in Fig. 8 for different n. The flow rate variations in the branch capillary for various values of n are presented in Figs. 9 and 10. It may be observed that when the branch capillary emerges from the parent artery making a right angle to it, when the flow of blood is steady for all values of n, the volumetric flow rate decreases as the yield stress increases. It is also found that with the increase in n, the flow decreases when the flow of blood is steady. In the case of pulsatile flow, the flow rate variation is, however, very different from that of steady flow. Figure 11 gives the variation of hematocrit in the feeding artery for steady flow of blood with yield stress for different values of n. Hematocrit variation in the branch capillary with time is shown in Fig. 12 for the four different values of n. It is to be noted that at any instant of time, hematocrit in the branch capillary decreases with an increase in the value of fluid index parameter.


00239


410

54 θ = 0.2

52

θ = 0.15

50

θ = 0.1 θ = 0.05

48

Q3

Q1 + Q2

44

42

40

38

36 0

0.1

0.2

0.3 R0 − Rp

0.4

0.5

0.6

0.7

0.8

Fig. 7. Variation of the ratio of the volumetric flow rate with thickness of the cell-depleted middle layer for different values of θ at n = 1.0 (steady flow).

0.9 n = 0.75

n = 1.0

n = 1.5

n = 2.0

0.85

0.8

0.75

0.7

Q


46

0.65

0.6

0.55

0.5

0.45 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

t/100 Fig. 8. Variation of volumetric flow rate in parent artery with time for different values of n at θ = 0.05, = 0.04 (pulsatile flow).


00239


411

0.0007

0.0006

0.0005 n = 0.75 n = 1.0

Q cap

n = 1.5

0.0003 n = 2.0

0.0002

0.0001

0 0

0.05

0.1

0.15

0.2

0.25

0.3

θ

Fig. 9. Variation of volumetric flow rate of branch capillary with θ for different values of n at α = π2 (steady flow). 0.0048

0.0046

0.0044

0.0042

0.004

Qcap


0.0004

0.0038 n = 0.75

0.0036 n = 1.0

0.0034

n = 1.5 n = 2.0

0.0032 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t Fig. 10. Variation of volumetric flow rate of branch capillary with time for different values of n when α = π6 , = 0.04 (pulsatile flow).


412

00239

J. C. Misra, S. D. Adhikary & G. C. Shit 0.012

0.011

0.01

Hf

0.008

0.007

n = 0.75 n = 1.0

0.006

n = 1.5 n = 2.0 0.005 0.05

0.1

0.15

0.2

0.25

0.3

θ Fig. 11. Variation of hematocrit in parent artery with θ for different values of n (steady state). 0.00484

n = 0.75

0.00482

n = 1.0

0.0048

0.00478

Hcap


0.009

n = 1.5

0.00476

0.00474 n = 2.0

0.00472

0.0047 0

1

2

3

4

5

6

7

8

9

10

t Fig. 12. Variation of hematocrit in branch capillary with time for different values of n at θ = 0.05, = 0.04, α = π6 (pulsatile flow).


00239


413

Acknowledgment The authors are thankful to the Council of Scientific and Industrial Research, New Delhi, for their financial support of this investigation.


References 1. Rodkiewicz CM, Sinha P, Kennedy JS, On the application of a constitutive equation for whole human blood, J Biomech Eng 112:198–206, 1990. 2. Chaturani P, Pralhad RN, Blood flow in tapered tubes with rheological applications, Biorheology 22:303–314, 1985. 3. Luo XY, Kuang ZB, A study on the constitutive equation of blood, J Biomech 25:929– 934, 1992. 4. Grigioni M, Daniele C, D’Avenio G, Pontrelli G, The role of wall shear stress in unsteady vascular dynamics, Proc Inst Mech Eng [H] 7:204–212, 2002. 5. Nakamura N, Sawada T, Numerical study on the flow of a non-Newtonian fluid through an asymmetric stenosis, J Biomech Eng 110:1129–1141, 1988. 6. Misra JC, Patra MK, Misra SC, A non-Newtonian fluid model for blood flow through arteries under stenotic conditions, J Biomech 26:137–143, 1993. 7. Misra JC, Shit GC, Blood flow through arteries in a pathological state: A theoretical study, Int J Eng Sci 44:662–671, 2006. 8. Misra JC, Ghosh SK, Pulsatile flow of a viscous fluid through a porous elastic vessel of variable cross-section: A mathematical model for hemodynamic flows, Comput Math Appl 46:447–457, 2003. 9. Misra JC, Ghosh SK, Pulsative flow of a couple stress fluid through narrow porous tube of elliptical cross-section: A model for blood flow in a stenosed arteriole, Eng Simul 15:849–864, 1998. 10. Creff R, Andre P, Batina J, Dynamic and connective results for a developing laminar unsteady flow, J Num Methods Fluid 5:139–153, 1985. 11. Krijger JKB, Pulsating entry flow in a plane channel, J Appl Math Phys (ZAMP) 42:139–153, 1991. 12. Aroesty J, Gross JF, The mathematics of pulsatile flow in small vessels: I. Casson theory, Microvasc Res 4:1–12, 1972. 13. Aroesty J, Gross JF, Pulsatile flow in small blood vessels: I. Casson theory, Biorheology 9:33–43, 1972. 14. Iida N, Murata T, Theoretical analysis of pulsatile blood flow in small vessels, Biorheology 35:377–384, 1980. 15. Moayeri MS, Zendehbudi GR, Effect of elastic property of the wall on flow characteristics through arterial stenosis, J Biomech 36:525–535, 2003. 16. Rohlf K, Tanti G, The role of Womersley number in pulsatile blood flow: A theoretical study of the Casson model, J Biomech 34:141–148, 2001. 17. Tu C, Deville M, Pulsatile flow of non-Newtonian fluid through arterial stenosis, J Biomech 29:899–908, 1996. 18. Das RN, Seshadri V, A semi-empirical model for flow of blood and other particulate suspensions through narrow tubes, Bull Math Biol 37:459–469, 1975. 19. Perktold K, Resch M, Florian H, Pulsatile non-Newtonian flow characteristics in a three-dimensional human carotid bifurcation model, J Biomech Eng 113:464–475, 1991. 20. Perkkio J, Keskinen R, Hematocrit reduction in bifurcations due to plasma skimming, Bull Math Biol 45(1):41–50, 1983.



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21. Lau Z, Yang WJ, A computer simulation of non-Newtonian blood flow at an aortic bifurcation, J Biomech 26:37–49, 1993. 22. Misra JC, Kar BK, A mathematical analysis of blood flow from a feeding artery into a branch capillary, Math Comput Model 15(6):9–18, 1991. 23. Haldar K, Andersson HI, Two-layered model of blood flow through stenosed arteries, Acta Mech 117:221–228, 1996. 24. Sharan M, Pople AS, A two-phase model for flow of blood in narrow tubes with increased effective viscosity near the wall, Biorheology 38:415–428, 2001. 25. Gupta BB, Nigam KM, Jaffin MY, A three-layer semi-empirical model for flow of blood and other particulate suspensions through narrow tubes, J Biomech Eng 104:129–135, 1982. 26. Misra JC, Ghosh SK, Flow of a Casson fluid in a narrow tube with a side branch, Int J Eng Sci 38:2045–2077, 2000. 27. Whitmore RL, Rheology of the Circulation, Pergamon Press, New York, 1968. 28. Merrill EW, Rheology of blood, Physiol Rev 49(4):863–888, 1969.

Appendix pns pn X1 = (Rn+1 − R0n+1 ) − s n+1 1+k

nθ2 R0n−1 R0n+1 θR0n + − 1+n ps 2p2s

+

m+n+1 kam R0 nθR0m+n n(n − 1)θ2 R0m+n−1 − + 1 + k m + n + 1 ps (m + n) 2p2s (m + n − 1)

+

2m+n+1 R0 nθR02m+n n(n − 1)θ2 R02m+n−1 k 2 a2m − + . (1 + k)2 2m + n + 1 ps (2m + n) 2p2s (2m + n − 1)

X2 =

np pn−1 n+1 p pn+1 (R − R0n+1 ) − n+1 k+1 +

nR0n+1 (n − 1)θR0n − n+1 p

n(n − 1)θR0m+n kam nR0m+n+1 n(n − 2)θ2 R0n−1 − + 2 2p 1+k m+n+1 p(m + n)

nR02m+n+1 n(n − 1)(n − 2)θ2 R0m+n−1 k 2 a2m + + 2p2 (m + n − 1) (1 + k)2 2m + n + 1 n(n − 1)(n − 2)θ2 R02m+n−1 n(n − 1)θR02m+n + − p(2m + n) 2p2 (2m + n − 1) Z1 =

cos−1

Z2 =

η

n+1 2

η + ηc √ 2R η

cos−1

dη.

η + ηc √ 2R η

dη.

.


00239


Z3 = Z4 =

η

Z5 =

η


Z6 =

η

Z7 =

η

Z8 =

η

Z9 =

η

Z10 =

η

Z11 =

η

Z13 =

η

Z14 =

η

Z15 =

η

Z16 =

η

Z17 =

n−1 2

m+n+1 2

m+n 2

η

η + ηc √ 2R η

cos−1

cos

m+n−1 2

−1

2m+n 2

cos−1

n+2 2

−1

cos

cos

−1

cos

2m+n+3 2

cos−1

2m+n+2 2

cos−1

dη. dη. dη.

dη.

η + ηc √ 2R η

dη.

dη.

η + ηc √ 2R η

dη.

η + ηc √ 2R η

η + ηc √ 2R η

η + ηc √ 2R η

−1

dη.

η + ηc √ 2R η

η + ηc √ 2R η

n+3 2

m+n+2 2

cos−1

η + ηc √ 2R η

cos

2m+n−1 2

dη.

η + ηc √ 2R η

−1

cos−1

m+n+3 2

cos

2m+n+1 2

η + ηc √ 2R η

cos−1 −1

η cos−1

Z12 =

n

η 2 cos−1

dη.

η + ηc √ 2R η η + ηc √ 2R η

dη. dη.

η + ηc √ 2R η η + ηc √ 2R η

dη. dη. dη.

415


416

00239


n+1 r θrn pns nθ2 rn−1 − + 1+k 1+n ps 2p2s m+n+1 r nθrm+n n(n − 1)θ2 rm+n−1 kam − + + 1 + k m + n + 1 ps (m + n) 2p2s (m + n − 1) 2m+n+1 r nθr2m+n n(n − 1)θ2 r2m+n−1 k 2 a2m − + + . (1 + k)2 2m + n + 1 ps (2m + n) 2p2s (2m + n − 1)


S0 (r) = X1 −

n+1 r (n − 1)θrn (n − 2)θ2 rn−1 X2 − + − 2 n+1 np 2p2

S1 (r) = −

npn−1 1+k

+

kam 1+k

+

k 2 a2m (1 + k)2

+

(n − 1)(n − 2)θ2 r2m+n−1 2p2 (2m + n − 1)

×

rm+n+1 (n − 1)θrm+n (n − 1)(n − 2)θ2 rm+n−1 − + m+n+1 p(m + n) 2p2 (m + n − 1)

r2m+n+1 (n − 1)θr2m+n − 2m + n + 1 p(2m + n) +

pn−1 p 1+k

−

n (n + 1)(n + 3)

r2n+2 (n − 1)θr2n+1 (n − 1)(n − 2)θ2 r2n − + 2n + 2 p(2n + 1) 4p2 n

kam + 1+k

(n − 1) + (n + 2)

θrm+2n+1 (n − 1)θ2 rm+2n − p(m + 2n + 1) p2 (m + 2n)

ka2m + (1 + k)2

r2m+2n+2 (n − 1)θr2m+2n+1 − 2m + 2n + 2 p(2m + 2n + 1)

(n − 1)(n − 2)θ2 r2m+2n + 2p2 (2m + 2n)

rm+2n+2 (n − 1)θrm+2n+1 (n − 1)(n − 2)θ2 rm+2n − + m + 2n + 2 p(m + 2n + 1) 2p2 (m + 2n)

ka2m + (1 + k)2

kam + 1+k

θr2n+1 (n − 1)θ2 r2n − p(2n + 1) p2 n

(n − 1)θ2 r2m+2n θr2m+2n+1 − p(2m + 2n + 1) p2 (2m + 2n)

kam θ2 rm+2n k 2 a2m θ2 r2m+2n n(n − 2) θ2 r2n + + − 2(n + 1) p2 2n (1 + k)p2 (m + 2n) (1 + k)2 p2 (2m + 2n) kam n − (1 + k)(m + n + 1)(m + n + 3)

rm+2n+2 (n − 1)θrm+2n+1 − m + 2n + 2 p(m + 2n + 1)


00239


+

(n − 1)(n − 2)θ2 rm+2n 2p2 (m + 2n)

+

kam 1+k

(n − 1)θr2m+2n+1 r2m+2n+2 − 2m + 2n + 2 p(2m + 2n + 1)

kam n(n − 1) (n − 1)(n − 2)θ2 r2m+2n + 2p2 (2m + 2n) (1 + k)(m + n)(m + n + 2) θrm+2n+1 (n − 1)θ2 rm+2n θr2m+2n+2 kam − × + p(m + 2n + 1) p2 (m + 2n) 1 + k p(2m + 2n + 2) (n − 1)θ2 r2m+2n kam n(n − 2) − − p2 (2m + 2n) (1 + k)2(m + n − 1)(m + n + 1) 2 2m+2n+2 θ r (n − 1)θ2 r2m+2n − × p(2m + 2n + 2) p2 (2m + 2n) 2m+2n+2 r (n − 1)θr2m+2n+1 k 2 a2m n − − 2 (1 + k) (2m + n + 1)(2m + n + 3) 2m + 2n + 2 p(2m + 2n + 1) (n − 1)(n − 2)θ2 r2m+2n k 2 a2m n(n − 1) + + 2 2 2p (2m + 2n) (1 + k) (2m + n)(2m + n + 2) θr2m+2n+1 (n − 1)θ2 r2m+2n − × p(2m + 2n + 1) p2 (2m + 2n) θ2 r2m+2n k 2 a2m n(n − 1)(n − 2) − . (1 + k)2 (2m + n − 1)(2m + n + 1) p2 (2m + 2n) +


417

X3 =

2n−2 1 p (t) n2 p(t) 2n+2 (R2n+2 − R02 ) n+1 (2n + 2)(n + 3) Rn+1 n+1 − (Rn+1 − R02 ) − S1 (R02 ). 2(n + 1)