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Non-linear mathematical models for blood flow through tapered tubes
Applied Mathematics and Computation 188 (2007) 567–582 www.elsevier.com/locate/amc
Non-linear mathematical models for blood flow through tapered tubes D.S. Sankar a
a,*,1
, K. Hemalatha
b
Department of Mathematics, Crescent Engineering College, Vandalur, Chennai 600 048, India b Department of Mathematics, Anna University, Chennai 600 025, India
Abstract In this paper, the steady flow of blood through tapered tube has been analyzed assuming blood as (i) Casson fluid and (ii) Herschel–Bulkley fluid. The expressions for pressure drop, wall shear stress and resistance to flow have been obtained. The effects of tapering of the tube and the non-Newtonian nature of the fluid on pressure drop, wall shear stress and resistance to flow are discussed. For all fluids, the pressure drop increases with increasing angle of taper from 0.5° to 1° for a given value of yield stress h and tapered tube Reynolds number Rew. The resistance to flow as well as the wall shear stress increase with increasing yield stress for Herschel–Bulkley fluid and also for Casson’s fluid when the other parameters held constant. Both for Herschel–Bulkley fluid and Casson’s fluid, the wall shear stress as well as the resistance to flow increase with increasing axial distance for a given tapered tube Reynolds number Rew, angle of taper w and yield stress h. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Steady flow; Tapered tube; Herschel–Bulkley fluid; Casson fluid; Pressure drop; Resistance to flow
1. Introduction The analysis of blood flow through tapered tubes is very important in understanding the behaviour of the flow as the taper of the tube is an important factor in the pressure development. It has been pointed out that the blood vessels bifurcate at frequent intervals and although the individual segments of arteries may be treated as uniform between bifurcations, the diameter of the artery decreases quite fast at each bifurcation [1]. It has been observed that even for the small angles of taper (upto 2°), the effects of tapering of the blood vessels cannot be neglected [2]. As pointed out by How and Black [3], this study is also very useful for the design of prosthetic blood vessels as the use of grafts of tapered lumen has the advantage of surgical benefits, the blood vessels being wider upstream. The important hydrodynamical factor for tapered tube geometry is the pressure loss which leads to the diminished blood flow through the grafts. Several authors have reported theoretical and experimental study *
1
Corresponding author. E-mail address: [email protected] (D.S. Sankar). Address of the place where the work was carried out: Crescent Engineering College, Chennai 600 048, Tamil Nadu, India.
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.10.013
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Nomenclature K L1 L2 n p p r r Q Rew R R0 U0 u z z
coefficient of viscosity for non-Newtonian fluid axial distance of the cross section z ¼ 0 from cone apex axial distance of any cross section from cone apex power law index pressure dimensionless pressure radial distance dimensionless radial distance flow rate tapered tube Reynolds number radius of the tapered tube radius of the normal artery typical velocity axial velocity axial distance dimensionless axial distance
Greek letters D p pressure drop DP dimensionless pressure drop f ðsÞ flow curve for non-Newtonian fluid K resistance to flow K dimensionless resistance to flow h dimensionless yield stress s shear stress s dimensionless shear stress sB yield stress for Bingham fluid sC yield stress for Casson fluid sH yield stress for Herschel–Bulkley fluid sy yield stress sw wall shear stress sw dimensionless wall shear stress B l coefficient of viscosity for Bingham fluid C l coefficient of viscosity for Casson fluid H l coefficient of viscosity for Herschel–Bulkley fluid N l coefficient of viscosity for Newtonian fluid p l coefficient of viscosity for Power law fluid w angle of taper q density Subscript w wall shear (used for s)
of the pressure flow relationship for different fluids through tapered tubes such as Newtonian, Power law and Bingham [3–10]. However, it has been noticed in smaller blood vessels at low shear rates ð_c < 10=sÞ, the yield stress for blood is non-zero and blood behaves as a non-Newtonian fluid [11,12]. The non-Newtonian character of blood is typical in small arteries and veins where the presence of cells induce that specific behaviour [13]. It has been pointed out that in some diseased conditions e.g. patients with sever myocardial infarction,
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cerebrovascular diseases and hypertension, blood exhibits remarkable non-Newtonian properties [11,14]. Hence, it is appropriate to model blood as a non-Newtonian fluid when it flows through narrow arteries (of diameter 0.02–0.1 mm) at low shear rates ð_c < 10=sÞ [15–17]. Iida [18] reported that the velocity profiles in the arterioles having diameter less than 0.1mm are generally explained fairly by the Casson fluid model as well as by the Herschel–Bulkley fluid model. Thus, it is appropriate to model blood by (i) Casson fluid model and (ii) Herschel–Bulkley fluid model when it flows through narrow arteries at low shear rate. Hence, in this paper, we study the steady flow of blood through tapered arteries (with angle of tapering upto 2°) of diameter 0.02–0.1 mm at low shear rates ð_c < 10=sÞ modeling blood by (i) Casson fluid and (ii) Herschel– Bulkley fluid. The advantage of assuming blood as these two models is that the blood flow through larger arteries at high shear rates can also be studied from these two models as they can be reduced to Newtonian fluid model as a particular case of these models. Furthermore, modeling of blood by Bingham fluid and power law fluid can also be obtained from the present study as the particular cases of Herschel–Bulkley fluid model. In Section 2, the expressions for the pressure drop, wall shear stress distribution and resistance to flow have been obtained. Section 3 non-dimensionalizes these flow quantities. Variations of pressure drop, wall shear stress and resistance to flow with axial distance, yield stress, angle of tapering and tapered tube Reynolds number are discussed through graphs in Section 4. The results are summarized in the concluding Section 5. 2. Mathematical analysis Consider the steady state laminar flow of a non-Newtonian fluid in a uniformly tapering tube of circular cross section. The radius of the tapered tube is given by R ¼ RðzÞ ¼ R0 z tan w;
ð1Þ
where R0 is the radius at z ¼ 0, w is the angle of taper and the z axis is taken along the axis of the tapered tube. The geometry of the tapered tube is shown in Fig. 1. L1 is the axial distance of the cross section z ¼ 0 from the cone apex and L2 is the axial distance of any cross section at z from the cone apex. The general constitutive equation of a fluid with non-Newtonian behaviour for simple shear flow may be written as
d u ¼ f ðsÞ; dr
ð2Þ
where s is the shear stress, u is the axial velocity, du=dR is the shear rate and f ðsÞ is the flow curve characterizing the nature of the particular fluid. Using Eq. (2), Oka [8] and Oka and Murata [9] have obtained the following expression for the volumetric flow rate of a non-Newtonian fluid through a slightly tapering tube (w2 1) assuming the flow to be slow, steady and fully developed. These assumptions are true upto angle of taper 2°. Z pR3 ðzÞ sw ðzÞ 2 s f ðsÞ ds: Q¼ 3 ð3Þ sw ðzÞ 0
R0
R (z ) Ψ L2 L1 Fig. 1. Geometry of the tapered tube.
z
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Here sw ðzÞ is the wall shear stress which is given by sw ðzÞ ¼
R d p ; 2 dz
ð4Þ
where p is the pressure. 2.1. Pressure drop-flow rate expressions for Newtonian and power law fluids N is the coefficient of viscosity for For Newtonian fluid, the equation of the flow curve is f ðsÞ ¼ lsN , where l Newtonian fluid with dimension M L1 T1 and in this case Eq. (3) gives pR4 ðzÞ d p Q¼ ð5Þ 8 lN dz and the pressure drop is Z 8 lN Q z dz 8 lN Q 3 D p¼ ¼ ðL2 L3 1 Þ; 4 4 ð p R z Þ 3pw 0
ð6Þ
which is the result given by Walawender et al. [10] and How and Black [3]. Here RðzÞ ¼ R0 wz;
R0 ¼ wL1 ;
L2 ¼ L1 z:
ð7Þ
For a power law fluid f ðsÞ ¼
sn ; P l
ð8Þ n
P is the coefficient of viscosity for power law fluid with dimension ðML1 T 2 Þ T . In this case, How and where l Black [3] have obtained the pressure drop as 1=n P Qðn þ 3Þ 2n l 3=n 3=n D p ¼ nþ3 L2 L1 : ð9Þ p 3w n We present below the expression for pressure gradient and pressure drop when the non-Newtonian behaviour is characterized by (i) Casson’s fluid (ii) Herschel–Bulkley fluid. 2.2. Pressure drop-flow rate expressions for Casson’s fluid The flow curve for Casson’s fluid is ( pffiffiffi pffiffiffiffiffi 2 1 s sC for s P sC ; l C f ðsÞ ¼ 0 for s 6 sC ;
ð10Þ
C is the coefficient of viscosity for Casson’s fluid with dimension M L1 T1 and sC is the yield stress where l for Casson’s fluid. This equation represents a plug flow in regions where s 6 sC . Substituting Eq. (10) in Eq. (3), we get the volumetric flow rate " 1=2 4 # pR3 ðzÞsw ðzÞ 4 sC 16 sC 1 sC Q¼ 1þ : ð11Þ 4 lC 3 sw 7 sw 21 sw The above result is also given by Merrill et al. [7] and Oka [8]. Using the condition ðsC =sw Þ 1 in the non-plug 2 core region s P sC and omitting ðsC =sw Þ and higher powers of ðsC =sw Þ in Eq. (11), we get " 1=2 # pR3 ðzÞsw ðzÞ 4 sC 16 sC : ð12Þ Q¼ 1þ 4 lC 3 sw 7 sw
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From Eqs. (4) and (12), we get qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 16 pffiffiffiffiffi lC 1 sC sC 4 Q 147 R d p 7 pR3 : ¼ 2 dz 2 On simplifying Eq. (13), we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sC s2C d p 128 sC 8 Q lC 64 Q lCsC ¼ þ þ : dz 49 R R pR3 147 147 7R pR3
571
ð13Þ
ð14Þ
s2
sC C and 147 seem to be negligibly small, we cannot neglect them at this stage, Though the magnitude of the terms 147 since Eq. (14) is in the dimensional form. Hence, Eq. (14) is first non-dimensionalized and then the non-dimensional form of these terms are neglected because of their negligibly small magnitude. The resulting equation is then integrated between 0 and z to get the pressure drop. The details are given from Eq. (36) to Eq. (37). Due to the truncation of these terms in the non-dimensional form, the expressions for the pressure drop and hence the expressions for the wall shear stress and the resistance to flow are not derived for Casson fluid in the dimensional form but they are obtained directly in the non-dimensional form.
2.3. Pressure drop-flow rate expressions for Herschel–Bulkley fluid The flow curve for Herschel–Bulkley fluid is given by 1 n ðs sH Þ for s P sH ; f ðsÞ ¼ lH 0 for s 6 sH ;
ð15Þ
H is the coefficient of viscosity with dimension (M L1 T2)n T, n is the power index, sH is the yield where l stress for Herschel–Bulkley fluid. Again the above equation represents a plug flow in regions where s 6 sH Substituting Eq. (15) in Eq. (3) and simplifying we get the volumetric flow rate nþ1 " 2 # sH sH sH pR3 ðzÞsnw ðzÞ 2 2 1 Q¼ 1þ þ : ð16Þ H ðn þ 3Þ sw l ðn þ 2Þ sw ðn þ 1Þðn þ 2Þ sw Using the condition ðsH =sw Þ 1 in the non-plug core region s P sC and omitting ðsH =sw Þ2 and higher powers of ðsH =sw Þ, Eq. (16) reduces to
pR3 ðzÞsnw ðzÞ nðn þ 3Þ sH 1 Qffi : ð17Þ H ðn þ 3Þ l ðn þ 2Þ sw From Eqs. (4) and (17), the expression for pressure gradient is obtained as n 1=n H Qðn þ 3Þ d p 2l 1 2ðn þ 3Þ sH ¼ : nþ3 þ dz p ð n þ 2Þ R Rn
ð18Þ
This agrees with the result given by Chaturani and Ponnalagar Samy [14]. Integrating the above equation between 0 and z, we get. 1=n 2ð n þ 3Þ H Qðn þ 3Þ n 2n l L1 3=n 3=n L L log s D p¼ þ : ð19Þ H 2 1 3 wðn þ 2Þ L2 pwnþ3 Eq. (19) gives the pressure loss for the flow of Herschel–Bulkley fluid through a tapered tube with angle of tapering w. The following are obtained as particular cases from Eq. (19). H ¼ l N , we get the pressure loss for Newtonian fluid as (i) Newtonian fluid: When n = 1, sH ¼ 0 and l D p¼
8 lN Q 3 ; L2 L3 1 4 3pw
which is the same as Eq. (6).
ð20Þ
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H ¼ l P , we get the pressure loss as (ii) Power law fluid: When sH ¼ 0 and l 1=n 2n Q lP ðn þ 3Þ 3=n 3=n ; D p ¼ nþ3 L2 L1 p 3w n which is Eq. (9). H ¼ l B and sH ¼ sB , we get pressure loss for Bingham fluid as (iii) Bingham fluid: When n = 1, l 8sB 8 lB Q 3 L1 þ log D p¼ L2 L3 ; 1 4 3w L2 3pw
ð21Þ
ð22Þ
B is the coefficient of viscosity for Bingham fluid. where sB is the yield stress for Bingham fluid and l
2.4. Wall shear stress distribution Young [19] has pointed out that the variations of the resistance to flow and the wall shear stress with axial distance are physiologically important quantities. These variations in tapered tubes for Herschel–Bulkley fluid and for Casson’s fluid have not been studied earlier. Karino and Goldsmith [20] have pointed out that the wall shear stress plays an important role in determining aggregation sites of platelet. The wall shear stress is given by Eq. (4). For Herschel–Bulkley fluid, from Eqs. (4), (7) and (18) we get !1=n 3=n H Q ð n þ 3Þ 1 ð n þ 3Þ l sw ¼ sH : þ ð23Þ 3 3 ð n þ 2Þ 1 ðz=L1 Þ pw L1 The wall shear stress for Casson fluid is obtained in the non-dimensional form in Eq. (39). 2.5. Resistance to flow The resistance to flow is defined to be Dp K¼ : ð24Þ Q We obtain the resistance to flow for Casson fluid in the non-dimensional form in Eq. (43). From Eq. (19), we obtain the resistance to flow for Herschel–Bulkley fluid as # !1=n " 3=n H Qðn þ 3Þ 2n l L1 2ðn þ 3ÞsC L1 K¼ 1 þ : ð25Þ log 3 3 L2 L2 3wQ wðn þ 2ÞQ pw L1 3. Non-dimensionalization of flow quantities Define the typical velocity U 0 to be 2Q U0 ¼ : pw2 L21 The non-dimensional pressure loss DP, yield stress h and wall shear stress sw are defined by D p ; qU 20 sy h¼ ; qU 20 sw sw ¼ ; qU 20 DP ¼
ð26Þ
ð27Þ ð28Þ ð29Þ
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573
where sy represents the yield stress sC for Casson fluid or sH for Herschel–Bulkley fluid or sB for Bingham fluid. Defining the tapered tube Reynolds number (Rew) to be (see [3]) 1=n ð2R0 ÞU 02n1 ð30Þ Rew ¼ 8wq 2Kðn þ 3Þ and X ¼1
L2 z ¼ ; L1 L1
ð31Þ
N for Newtonian fluid, l P for power law fluid, l B for Bingham fluid, l C for Casson fluid where K represents l H for Herschel–Bulkley fluid. We get the following expressions for the non-dimensional pressure loss for and l different fluids. We also note that for Newtonian, Bingham and Casson fluids Rew is obtained by putting n = 1 in Eq. (30). (i) Herschel–Bulkley fluid " # 3=n n241=n 1 2ðn þ 3Þh 1 log DP ¼ 1 þ : 1X wðn þ 2Þ 1X 3Rew (ii) Power law fluid " # 3=n n241=n 1 DP ¼ 1 : 1X 3Rew (iii) Newtonian fluid " # 3 8 1 DP ¼ 1 : 3Rew 1X (iv) Bingham fluid " # 3 8 1 8h 1 log DP ¼ 1 þ : 3Rew 1X 3w 1X (v) Casson’s fluid: For Casson fluid, the non-dimensional form of pressure gradient is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # dp 128h 8 1 h 64 hw h2 ¼ þ : þ dz 49wð1 X Þ ð1 X Þ Rew ð1 X Þ3 147 7wð1 X Þ Rew ð1 X Þ 147
ð32Þ
ð33Þ
ð34Þ
ð35Þ
ð36Þ
The pressure drop can be obtained by integrating Eq. (36) between 0 and z with respect to z. The contributions h h2 of the terms 147 and 147 to the pressure gradient dp are negligibly small. Neglecting these terms in Eq. (36) and on dz integration, the pressure drop is obtained and is given by sffiffiffiffi" " # # 3 3=2 128h 1 8 1 128 h 1 log DP ¼ 1 þ pffiffiffiffiffiffiffiffi 1 : ð37Þ þ 49w 1X 3Rew 1X 21 Rew w 1 X Using Eqs. (26), (28) and (29) in Eqs. (23) and (14), we get wall shear stress for Herschel–Bulkley fluid as 3=n w231=n 1 ðn þ 3Þh sw ¼ ð38Þ þ 1X ðn þ 2Þ Rew and for Casson fluid as sffiffiffiffiffiffiffiffi 3 3=2 64h 4w 1 32 hw 1 sw ¼ þ þ ; 49 Rew 1 X 7 Rew 1 X
ð39Þ
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D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582 s2
sC C respectively. Note that to obtain Eq. (39), the non-dimensional form of the terms 147 and 147 in Eq. (14) are neglected because of their negligibly small magnitude. Let the non-dimensional resistance to flow be defined as
K¼
K ; KN
ð40Þ
where KN ¼
8qU 0 pw4 L21
ð41Þ
is the impedance for Newtonian fluid for Poiseuille flow multiplied by the factor w2/ReN, ReN being the Reynolds number for the Newtonian fluid. Using Eqs. (24), (28), (30), (31), (40) and (41) in Eqs. (25) and (37), we get the expression for resistance to flow for Herschel–Bulkley fluid as " # 3=n 221=n nw2 1 ðn þ 3Þhw 1 log K¼ 1 þ ð42Þ 1X 2ðn þ 2Þ 1X 3Rew and for Casson’s fluid as sffiffiffiffiffiffiffiffi" " # # 3 3=2 32wh 1 2w2 1 32w wh 1 log 1 þ 1 ; K¼ þ 49 1X 1X 21 Rew 1 X 3Rew
ð43Þ
respectively. 4. Results and discussion An extensive attempt has been made in this section to discuss the effects of tapering of the blood vessel and the non-Newtonian nature of the blood on the flow quantities pressure drop, wall shear stress distribution and resistance to flow. It is generally accepted that human blood is a non-Newtonian fluid when it flows through smaller diameter tubes at low shear rates. In smaller tubes, blood appears to possess both finite yield stress and shear dependent viscosity. The yield stress for of normal human blood is between 0.01 dyne/cm2 and 0.06 dyne/cm2 [21,22] and thus the value 0.04 is used for non-dimensional yield stress in this study. It is generally observed that the typical value of the power law index n for blood flow are taken to lie between 0.9 and 1.1 and we have taken the typical value of n to be 0.95 for n < 1 and 1.05 for n > 1 [15–17]. Since, the objective of the present study is to quantify the effects of tapering of the blood vessels for small angle of tapering on the various flow quantities, the range 0.5–1.25° is used for angle of tapering W in the present study. For easy comparison of the results given by How and Black [3] for power law fluid with the present results, the same values used by them for L1 and L2 are used here namely L1 (in cms) is taken as 53.18, 36.28, 26.11 and 20.61 corresponding to w = 0.5°, 0.75°, 1° and 1.25° of tapered angle and L2 (in cms) as 37.22, 20.29, 12.26 and 10.61 respectively. The variation of pressure drop DP against tapered tube Reynolds number (Rew) for Newtonian fluid, power law fluid with n = 0.95 and n = 1.05, Herschel–Bulkley fluid with n = 0.95 and n = 1.05, Bingham fluid and Casson’s fluid respectively are shown in Figs. 2–8 for different angles of taper w. It is generally observed that the pressure loss increases as the angle of taper w increases from 0.5° to 1° for all fluids. For tapered angle 1.25°, the pressure loss decreases and lies between those for w = 0.75° and 1° for Newtonian fluid and power law fluid whereas for Herschel–Bulkley fluid, Bingham fluid and Casson’s fluid after a certain value of Reynolds number, the pressure loss is less than that for w = 0.75° as seen in Figs. 2–8. It is to be noted that the last four figures (Figs. 5–8) are for fluids with non-zero yield stress. The maximum pressure loss occurs when w = 1° for all fluids. We observe that the pressure drop for all the fluids at the angle of taper w = 1.25° is lower than that of w = 1°, because the distance between the pressure ports when w = 1.25° is shorter than that for w = 1°. Figs. 2–8 depict the simultaneous effects of the angle of tapering of the blood vessel and the tapered tube Reynolds number on pressure drop for different fluids.
D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582
575
25
ψ = 1o
15
ψ = 1. 25 o
10
ψ = 0. 75 o
ΔP
20
ψ = 0. 5 o
5
0 1
2
3
4
5
6
7
8
9
10
Re ψ
Fig. 2. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Newtonian fluid.
30
ΔP
25 20
ψ = 1o
15
ψ = 1.25 o ψ = 0.75 o
10
ψ = 0. 5 o 5 0 1
2
3
4
5
6
7
8
9
10
Re ψ
Fig. 3. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Power law fluid with n = 0.95.
25 20
ψ = 1o 15
ΔP
ψ = 1 .2 5 o ψ = 0 .7 5 o ψ = 0 .5 o
10
5 0 1
2
3
4
5
6
7
8
9
10
R eψ Fig. 4. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Power law fluid with n = 1.05.
The variation of pressure drop with tapered tube Reynolds number for different fluids for the angle of taper 1° is shown in Fig. 9. It is observed that there is not much of a difference between the values for the pressure
576
D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582 35 30
ΔP
25
ψ = 1o
20
ψ = 1.25 o
15
ψ = 0.75 o
10
ψ = 0 .5 o
5 0 1
2
3
4
5
6
7
8
9
10
Reψ
Fig. 5. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Herschel–Bulkley fluid with n = 0.95. and h = 0.04.
30 25
ψ = 1º
20 ΔP
ψ = 1.25º 15
ψ = 0.75º
ψ = 0.5º
10 5 0 1
3
5
7
9
Re ψ
Fig. 6. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Herschel–Bulkley fluid with n = 1.05 and h = 0.04.
30 25 ψ = 1o 20 ΔP
ψ = 1.25o 15 ψ = 0.75o
10
ψ = 0.5o
5 0 1
3
5
7
9
Re ψ
Fig. 7. Variation of pressure loss with tapered tube Reynolds number for different angles of tapering for Bingham fluid with h = 0.04.
loss for n = 0.95 and for n = 1.05 in the case of power law fluid. The same behaviour is observed in the case of Herschel–Bulkley fluid too. The pressure loss for power law fluid with n = 0.95 is slightly higher and for
D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582
577
50 45 40
ψ=1
35
ψ = 1.25
30 ΔP
o
ψ = 0.75
25
o
o
ψ = 0.5
20
o
15 10 5 0 1
3
5
7
9
Reψ
Fig. 8. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Casson fluid with h = 0.04.
n = 1.05, it is slightly lower than that for Newtonian fluid as seen from Fig. 9. That is, the pressure loss decreases with increasing ‘n’. In general for power law fluids as ‘n’ increases the pressure loss decreases. Fig. 9 shows the simultaneous effects of tapered tube Reynolds number and the non-Newtonian nature of the blood on pressure drop. It is observed from Figs. 2–9 that for all other fluids with yield stress, the pressure losses are much higher compared to those of Newtonian and power law fluids. The highest pressure drop is observed for Casson’s fluid for all angles of taper and for all values of tapered tube Reynolds number. For Herschel–Bulkley fluid with n = 0.95 and n = 1.05 and Bingham fluid, the pressure losses are almost the same for a given angle of taper w and a given tapered tube Reynolds number Rew. It is generally observed that for all fluids, the pressure losses are much higher for values of Rew upto 2 and for each fluid the pressure loss is almost the same constant value as the tapered tube Reynolds number Rew increases from 6. The variation of wall shear stress with axial distance for tapered tube Reynolds number 5 and yield stress 0.04 are shown in Figs. 10–12 for different values of w for Herschel–Bulkley fluid with n = 0.95 and 1.05 and for Casson’s fluid respectively. It is clear that for each value of n, the wall shear stress increases with increasing
50 45
Casson fluid with θ = 0.04
40 Herschel - Bulkley fluid with n = 0.95 and θ = 0.04
35
Bingham fluid with θ = 0.04
ΔP
30
Herschel - Bulkley fluid with n = 1.05 θ = 0.04
25 Power law fluid with n = 0.95
20 Newtonian fluid
15
Power law fluid with n = 1.05
10 5 0 1
3
5
7
9
Reψ Fig. 9. Variation of pressure loss with tapered tube Reynolds number for different fluids with angle of taper w = 1° and h = 0.04.
578
D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582 0.9 0.8 0.7
τw
0.6 o
ψ = 1.25
0.5
o
ψ=1
0.4
o
ψ = 0.75
0.3
o
ψ = 0.5
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0. 7
X Fig. 10. Variation of wall shear stress with axial distance for different angles of taper for Herschel–Bulkley fluid with n = 0.95, Rew = 5 and h = 0.04.
0.7 0.6
τw
0.5 ψ = 1.25o
0.4
o
ψ=1
0.3
ψ = 0.75o
0.2
o
ψ = 0.5
0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X Fig. 11. Variation of wall shear stress with axial distance for different angles of taper for Herschel–Bulkley fluid with n = 1.05, Rew = 5 and h = 0.04.
1.2 1 0.8
τw
o
ψ = 1.25
0.6
o
ψ=1
ψ = 0.75o
0.4
ψ = 0.5o
0.2 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X Fig. 12. Variation of wall shear stress with axial distance for different angles of taper for Casson fluid with h = 0.04.
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axial distance and also with increasing angle of taper. In each case upto X = 0.2, there is not much variation for different angles of taper, but the difference become pronounced as X increases further. Figs. 10–12 show the clear effects of tapering of the blood vessel on wall shear stress distribution for different fluids. For different fluids, the variation of wall shear stress with tapered tube Reynolds number Rew for X = 0.3 and w = 0.75° is shown in Fig. 13. It is observed that for all fluids, the wall shear stress decreases with increasing tapered tube Reynolds number Rew and this decrease is higher for value of Rew upto 2 and for each fluid the wall shear stress is almost the same constant value as Rew increases further from 6. For the given values of axial distance X, angle of tapering w and tapered tube Reynolds number Rew, the wall shear stress is maximum for Casson’s fluid and for Herschel–Bulkley fluid with n = 0.95 as well as n = 1.05, there is no significant difference in the values of wall shear stress when the yield stress, axial distance and tapered tube Reynolds number are held constant. The same behaviour is also noticed for Newtonian fluid and power law fluid with n = 0.95 as well as for n = 1.05. Fig. 13 shows the simultaneous effects of non-Newtonian nature and tapered tube Reynolds number of the fluid on wall shear stress distribution. The variation of resistance to flow with axial distance for Herschel–Bulkley fluid with n = 0.95 and Casson’s fluid are shown in Figs. 14 and 15. It is observed that in each case as the angle of taper increases the resistance to flow increases and also as X increases (i.e. the pipe diameter decreases) the resistance increases. This is an obvious result, since an increase in the angle of taper increases the pressure drop and since
0.45 0.4 Casso n fluid θ = 0.04
0.35
Herschel - B ulkley fluid with n = 0.95 and θ = 0.04
τw
0.3
B ingham fluid with θ = 0.04
0.25
Herschel - B ulkley fluid with n = 1.05 and θ = 0.04
0.2
P o wer law fluid with n = 0.95
0.15
Newto nian fluid P o wer law fluid with n = 1.05
0.1 0.05 0 1
2
3
4
5
6
7
8
9
10
Re ψ
Fig. 13. Variation of wall shear stress with tapered tube Reynolds number for different fluids with X = 0.3 and w = 0.75°.
1.8 1.6 1.4
ψ = 1.25o
1
3
10 Λ
1.2
ψ = 1o
0.8 0.6
ψ = 0.5o
0.4
ψ= 0.75o
0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
X Fig. 14. Variation of resistance to flow with axial distance for different values of angle of taper for Herschel–Bulkley fluid with h = 0.04 and Rew = 5.
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2.5
103 Λ
2
1.5 ψ = 1.25o ψ = 1o
1 ψ = 0.75o ψ = 0.5o
0.5
0 0
0.1
0.2
0.3
0.4
0.5
0.6
X
Fig. 15. Variation of resistance to flow with axial distance for different angles of taper for Casson fluid with h = 0.04 and Rew = 5.
0.7 0.6 Casson fluid with θ = 0.04
0.5 Bingham fluid with θ = 0.04
3
10 Λ
Herschel - Bulkley fluid with n = 1.05 and θ = 0.04
0.4
Herschel - Bulkley fluid with n = 0.95 and θ = 0.04 Power law fluid with n = 1.05
0.3 0.2
Newtonian fluid
0.1
Power law fluid with n = 0.95
0 1
2
3
4
5
6
7
8
9
10
Reψ
Fig. 16. Variation of resistance to flow with tapered tube Reynolds number for different fluids with X = 0.3, w = 0.75°.
the flow is fully developed for the same flow rate, the flow resistance increases as the pipe diameter decreases. Figs. 14 and 15 depict the effects of tapering of the blood vessel on resistance to flow for Casson fluid and Herschel–Bulkley fluid. Fig. 16 depicts the variation of resistance to flow for different fluids with X = 0.3 and w = 0.75°. It is to be noted that for each fluid and given values of axial distance X, tapered tube Reynolds number Rew and angle of taper w, the resistance to flow decreases with increase in tapered tube Reynolds number and this decrease is higher upto Rew value 2 and is almost the constant value once Rew exceeds the value 6. When the parameters X, Rew and w are held constant, the resistance to flow is maximum for Casson fluid. For Bingham fluid and Herschel–Bulkley fluid with n = 0.95 as well as for n = 1.05, there is no significant difference in the values of resistance to flow. The same behaviour is observed for Newtonian fluid and power law fluid with n = 0.95 as well as n = 1.05. Fig. 16 shows the simultaneous effects of non-Newtonian nature of the blood and the tapered tube Reynolds number on resistance to flow. 5. Conclusions Non-Newtonian fluid models with non-zero yield stress namely Casson’s and Herschel–Bulkley fluids are used to study the flow aspects of blood flow through tapered tubes with small angles of taper. Estimates of
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pressure drop that provides important information in the design of tapered grafts are given. Apart from these aspects, expressions for wall shear stress and resistance to flow are also analyzed. For all fluids, the non-dimensional pressure drop increases with increasing angle of taper from 0.5° to 1° for a given yield stress h and tapered tube Reynolds number Rew. Also the non-dimensional pressure losses are much higher for fluids with non-zero yield stress compared to those of Newtonian and power law fluids for all angles of taper and for all tapered tube Reynolds number. The highest pressure drop is observed for Casson’s fluid. It is observed that the pressure loss increases with yield stress in each case. For each fluid, the pressure losses are much higher only upto tapered tube Reynolds number value 2 and remains almost the same value once the Reynolds number exceeds 6. Both for Herschel–Bulkley fluid and Casson’s fluid the wall shear stress as well as the resistance increase with increasing axial distance for a given tapered tube Reynolds number Rew, angle of taper w and yield stress h and also with increasing angle of taper and the values are much higher in the case of Casson’s fluid. For Herschel–Bulkley fluid, both resistance and wall shear stress decrease as ‘n’ increases for fixed values of other parameters. The resistance as well as the wall shear stress increase with increasing yield stress for Herschel– Bulkley fluid and also for Casson’s fluid when the other parameters held constant. Since, the blood flow through narrow arteries is often pulsatile, the extension of this study to pulsatile flow could be more useful and will have wider applications to cardiovascular studies. Acknowledgement The authors thank Prof. P. Chaturani, Department of Mathematics, Indian Institute of Technology, Bombay, for his valuable suggestions and timely help in the preparation of the paper. References [1] A.P. Dwivedi, T.S. Pal, L. Rakesh, Micropolar fluid model for blood flow through a small tapered tube, Indian J. Technol. 20 (1982) 295–299. [2] P. Chaturani, R. Pralhad, Blood flow in tapered tubes with biorheological applications, Biorheology 22 (1985) 303–314. [3] T.V. How, R.A. Black, Pressure losses in non-Newtonian flow through rigid wall tapered tubes, Biorheology 24 (1987) 337–351. [4] S. Chakravarthy, P.K. Mandal, Two dimensional blood flow through tapered arteries under stenotic conditions, Int. J. Non-Linear Mech. 35 (2000) 779–793. [5] T.V. How, R.A. Black, D. Annis, Comparison of pressure losses in steady non-Newtonian flow through experimental tapered and cylindrical arterial prostheses, J. Biomed. Eng. 10 (1988) 225–230. [6] P.K. Mandal, An unsteady analysis of non-Newtonian blood flow through tapered arteries with stenosis, Int. J. Non-Linear Mech. 40 (2005) 151–164. [7] E.W. Merrill, A.M. Benis, E.R. Gilliland, T.K. Sherwood, E.W. Salzman, Pressure flow relations of human blood in hallow fibers at low shear rates, J. Appl. Physiol. 20 (1965) 954–967. [8] S. Oka, Pressure development in a non-Newtonian flow through a tapered tube, Biorheology 10 (1973) 207–212. [9] S. Oka, T. Murata, Theory of the steady slow motion of non- Newtonian fluids through a tapered tube, Jpn. J. Appl. Phys. 8 (1969) 5–8. [10] Jr. Walawender, C. Tien, L.C. Cerny, Experimental studies on the blood flow throughout tapered tubes, Int. J. Eng. Sci. 10 (1972) 1123–1135. [11] S. Chien, Hemorheology in clinical medicine, Rec. Adv. Cardiovascular Dis. 2 (Suppl.) (1981) 21–26. [12] J.B. Shukla, R. Parihar, S.P. Gupta, Biorheological aspects of blood flow through artery with mild stenosis: effects of peripheral layer, Biorheology 17 (1980) 403–410. [13] C. Tu, M. Deville, Pulsatile flow of non-Newtonian fluids through arterial stenosis, J. Biomech. 29 (1996) 899–908. [14] P . Chaturani, R. Ponnalagar Samy, A study of non-Newtonian aspects of blood flow through stenosed arteries and its applications in arterial diseases, Biorheology 22 (1985) 521–531. [15] D.S. Sankar, K. Hemalatha, Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – a mathematical model. App. Math. Modeling, in press. [16] D.S. Sankar, K. Hemalatha, A non-Newtonian fluid model for blood flow through a catheterized artery - steady flow, App. Math. Modeling, in press. [17] D.S. Sankar, Hemalatha, Pulsatile flow of Herschel–Bulkley fluid through stenosed arteries – a mathematical model, Int. J. Non-Linear Mech., in press. [18] N. Iida, Influence of plasma layer on steady blood flow in microvessels, Jpn. J. Appl. Phys. 17 (1978) 203–214. [19] D.F. Young, Fluid mechanics of arterial stenosis, J. Biomech. Eng. (Trans. ASME) 101 (1979) 157–175.
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[20] T. Karino, H.L. Goldsmith, Flow behaviour of blood cells and rigid spheres in annular vortex, Philos. Trans. Roy. Soc., London B 279 (1977) 413–445. [21] J.N. Kapur, Mathematical Models in Biology and Medicine, East West Press Pvt. Ltd., New Delhi, India, 1992. [22] E.W. Merrill, Rheology of human blood and space speculations on its role in vascular homeostasis, in: P.N. Sawyer (Ed.), Biomechanical Mechanisms in Vascular Homeostasis and Intravascular Thrombus, Appleton Century Crafts, New York, 1965.
Non-linear mathematical models for blood flow through tapered tubes D.S. Sankar a
a,*,1
, K. Hemalatha
b
Department of Mathematics, Crescent Engineering College, Vandalur, Chennai 600 048, India b Department of Mathematics, Anna University, Chennai 600 025, India
Abstract In this paper, the steady flow of blood through tapered tube has been analyzed assuming blood as (i) Casson fluid and (ii) Herschel–Bulkley fluid. The expressions for pressure drop, wall shear stress and resistance to flow have been obtained. The effects of tapering of the tube and the non-Newtonian nature of the fluid on pressure drop, wall shear stress and resistance to flow are discussed. For all fluids, the pressure drop increases with increasing angle of taper from 0.5° to 1° for a given value of yield stress h and tapered tube Reynolds number Rew. The resistance to flow as well as the wall shear stress increase with increasing yield stress for Herschel–Bulkley fluid and also for Casson’s fluid when the other parameters held constant. Both for Herschel–Bulkley fluid and Casson’s fluid, the wall shear stress as well as the resistance to flow increase with increasing axial distance for a given tapered tube Reynolds number Rew, angle of taper w and yield stress h. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Steady flow; Tapered tube; Herschel–Bulkley fluid; Casson fluid; Pressure drop; Resistance to flow
1. Introduction The analysis of blood flow through tapered tubes is very important in understanding the behaviour of the flow as the taper of the tube is an important factor in the pressure development. It has been pointed out that the blood vessels bifurcate at frequent intervals and although the individual segments of arteries may be treated as uniform between bifurcations, the diameter of the artery decreases quite fast at each bifurcation [1]. It has been observed that even for the small angles of taper (upto 2°), the effects of tapering of the blood vessels cannot be neglected [2]. As pointed out by How and Black [3], this study is also very useful for the design of prosthetic blood vessels as the use of grafts of tapered lumen has the advantage of surgical benefits, the blood vessels being wider upstream. The important hydrodynamical factor for tapered tube geometry is the pressure loss which leads to the diminished blood flow through the grafts. Several authors have reported theoretical and experimental study *
1
Corresponding author. E-mail address: [email protected] (D.S. Sankar). Address of the place where the work was carried out: Crescent Engineering College, Chennai 600 048, Tamil Nadu, India.
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.10.013
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Nomenclature K L1 L2 n p p r r Q Rew R R0 U0 u z z
coefficient of viscosity for non-Newtonian fluid axial distance of the cross section z ¼ 0 from cone apex axial distance of any cross section from cone apex power law index pressure dimensionless pressure radial distance dimensionless radial distance flow rate tapered tube Reynolds number radius of the tapered tube radius of the normal artery typical velocity axial velocity axial distance dimensionless axial distance
Greek letters D p pressure drop DP dimensionless pressure drop f ðsÞ flow curve for non-Newtonian fluid K resistance to flow K dimensionless resistance to flow h dimensionless yield stress s shear stress s dimensionless shear stress sB yield stress for Bingham fluid sC yield stress for Casson fluid sH yield stress for Herschel–Bulkley fluid sy yield stress sw wall shear stress sw dimensionless wall shear stress B l coefficient of viscosity for Bingham fluid C l coefficient of viscosity for Casson fluid H l coefficient of viscosity for Herschel–Bulkley fluid N l coefficient of viscosity for Newtonian fluid p l coefficient of viscosity for Power law fluid w angle of taper q density Subscript w wall shear (used for s)
of the pressure flow relationship for different fluids through tapered tubes such as Newtonian, Power law and Bingham [3–10]. However, it has been noticed in smaller blood vessels at low shear rates ð_c < 10=sÞ, the yield stress for blood is non-zero and blood behaves as a non-Newtonian fluid [11,12]. The non-Newtonian character of blood is typical in small arteries and veins where the presence of cells induce that specific behaviour [13]. It has been pointed out that in some diseased conditions e.g. patients with sever myocardial infarction,
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cerebrovascular diseases and hypertension, blood exhibits remarkable non-Newtonian properties [11,14]. Hence, it is appropriate to model blood as a non-Newtonian fluid when it flows through narrow arteries (of diameter 0.02–0.1 mm) at low shear rates ð_c < 10=sÞ [15–17]. Iida [18] reported that the velocity profiles in the arterioles having diameter less than 0.1mm are generally explained fairly by the Casson fluid model as well as by the Herschel–Bulkley fluid model. Thus, it is appropriate to model blood by (i) Casson fluid model and (ii) Herschel–Bulkley fluid model when it flows through narrow arteries at low shear rate. Hence, in this paper, we study the steady flow of blood through tapered arteries (with angle of tapering upto 2°) of diameter 0.02–0.1 mm at low shear rates ð_c < 10=sÞ modeling blood by (i) Casson fluid and (ii) Herschel– Bulkley fluid. The advantage of assuming blood as these two models is that the blood flow through larger arteries at high shear rates can also be studied from these two models as they can be reduced to Newtonian fluid model as a particular case of these models. Furthermore, modeling of blood by Bingham fluid and power law fluid can also be obtained from the present study as the particular cases of Herschel–Bulkley fluid model. In Section 2, the expressions for the pressure drop, wall shear stress distribution and resistance to flow have been obtained. Section 3 non-dimensionalizes these flow quantities. Variations of pressure drop, wall shear stress and resistance to flow with axial distance, yield stress, angle of tapering and tapered tube Reynolds number are discussed through graphs in Section 4. The results are summarized in the concluding Section 5. 2. Mathematical analysis Consider the steady state laminar flow of a non-Newtonian fluid in a uniformly tapering tube of circular cross section. The radius of the tapered tube is given by R ¼ RðzÞ ¼ R0 z tan w;
ð1Þ
where R0 is the radius at z ¼ 0, w is the angle of taper and the z axis is taken along the axis of the tapered tube. The geometry of the tapered tube is shown in Fig. 1. L1 is the axial distance of the cross section z ¼ 0 from the cone apex and L2 is the axial distance of any cross section at z from the cone apex. The general constitutive equation of a fluid with non-Newtonian behaviour for simple shear flow may be written as
d u ¼ f ðsÞ; dr
ð2Þ
where s is the shear stress, u is the axial velocity, du=dR is the shear rate and f ðsÞ is the flow curve characterizing the nature of the particular fluid. Using Eq. (2), Oka [8] and Oka and Murata [9] have obtained the following expression for the volumetric flow rate of a non-Newtonian fluid through a slightly tapering tube (w2 1) assuming the flow to be slow, steady and fully developed. These assumptions are true upto angle of taper 2°. Z pR3 ðzÞ sw ðzÞ 2 s f ðsÞ ds: Q¼ 3 ð3Þ sw ðzÞ 0
R0
R (z ) Ψ L2 L1 Fig. 1. Geometry of the tapered tube.
z
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Here sw ðzÞ is the wall shear stress which is given by sw ðzÞ ¼
R d p ; 2 dz
ð4Þ
where p is the pressure. 2.1. Pressure drop-flow rate expressions for Newtonian and power law fluids N is the coefficient of viscosity for For Newtonian fluid, the equation of the flow curve is f ðsÞ ¼ lsN , where l Newtonian fluid with dimension M L1 T1 and in this case Eq. (3) gives pR4 ðzÞ d p Q¼ ð5Þ 8 lN dz and the pressure drop is Z 8 lN Q z dz 8 lN Q 3 D p¼ ¼ ðL2 L3 1 Þ; 4 4 ð p R z Þ 3pw 0
ð6Þ
which is the result given by Walawender et al. [10] and How and Black [3]. Here RðzÞ ¼ R0 wz;
R0 ¼ wL1 ;
L2 ¼ L1 z:
ð7Þ
For a power law fluid f ðsÞ ¼
sn ; P l
ð8Þ n
P is the coefficient of viscosity for power law fluid with dimension ðML1 T 2 Þ T . In this case, How and where l Black [3] have obtained the pressure drop as 1=n P Qðn þ 3Þ 2n l 3=n 3=n D p ¼ nþ3 L2 L1 : ð9Þ p 3w n We present below the expression for pressure gradient and pressure drop when the non-Newtonian behaviour is characterized by (i) Casson’s fluid (ii) Herschel–Bulkley fluid. 2.2. Pressure drop-flow rate expressions for Casson’s fluid The flow curve for Casson’s fluid is ( pffiffiffi pffiffiffiffiffi 2 1 s sC for s P sC ; l C f ðsÞ ¼ 0 for s 6 sC ;
ð10Þ
C is the coefficient of viscosity for Casson’s fluid with dimension M L1 T1 and sC is the yield stress where l for Casson’s fluid. This equation represents a plug flow in regions where s 6 sC . Substituting Eq. (10) in Eq. (3), we get the volumetric flow rate " 1=2 4 # pR3 ðzÞsw ðzÞ 4 sC 16 sC 1 sC Q¼ 1þ : ð11Þ 4 lC 3 sw 7 sw 21 sw The above result is also given by Merrill et al. [7] and Oka [8]. Using the condition ðsC =sw Þ 1 in the non-plug 2 core region s P sC and omitting ðsC =sw Þ and higher powers of ðsC =sw Þ in Eq. (11), we get " 1=2 # pR3 ðzÞsw ðzÞ 4 sC 16 sC : ð12Þ Q¼ 1þ 4 lC 3 sw 7 sw
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From Eqs. (4) and (12), we get qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=2 16 pffiffiffiffiffi lC 1 sC sC 4 Q 147 R d p 7 pR3 : ¼ 2 dz 2 On simplifying Eq. (13), we get sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sC s2C d p 128 sC 8 Q lC 64 Q lCsC ¼ þ þ : dz 49 R R pR3 147 147 7R pR3
571
ð13Þ
ð14Þ
s2
sC C and 147 seem to be negligibly small, we cannot neglect them at this stage, Though the magnitude of the terms 147 since Eq. (14) is in the dimensional form. Hence, Eq. (14) is first non-dimensionalized and then the non-dimensional form of these terms are neglected because of their negligibly small magnitude. The resulting equation is then integrated between 0 and z to get the pressure drop. The details are given from Eq. (36) to Eq. (37). Due to the truncation of these terms in the non-dimensional form, the expressions for the pressure drop and hence the expressions for the wall shear stress and the resistance to flow are not derived for Casson fluid in the dimensional form but they are obtained directly in the non-dimensional form.
2.3. Pressure drop-flow rate expressions for Herschel–Bulkley fluid The flow curve for Herschel–Bulkley fluid is given by 1 n ðs sH Þ for s P sH ; f ðsÞ ¼ lH 0 for s 6 sH ;
ð15Þ
H is the coefficient of viscosity with dimension (M L1 T2)n T, n is the power index, sH is the yield where l stress for Herschel–Bulkley fluid. Again the above equation represents a plug flow in regions where s 6 sH Substituting Eq. (15) in Eq. (3) and simplifying we get the volumetric flow rate nþ1 " 2 # sH sH sH pR3 ðzÞsnw ðzÞ 2 2 1 Q¼ 1þ þ : ð16Þ H ðn þ 3Þ sw l ðn þ 2Þ sw ðn þ 1Þðn þ 2Þ sw Using the condition ðsH =sw Þ 1 in the non-plug core region s P sC and omitting ðsH =sw Þ2 and higher powers of ðsH =sw Þ, Eq. (16) reduces to
pR3 ðzÞsnw ðzÞ nðn þ 3Þ sH 1 Qffi : ð17Þ H ðn þ 3Þ l ðn þ 2Þ sw From Eqs. (4) and (17), the expression for pressure gradient is obtained as n 1=n H Qðn þ 3Þ d p 2l 1 2ðn þ 3Þ sH ¼ : nþ3 þ dz p ð n þ 2Þ R Rn
ð18Þ
This agrees with the result given by Chaturani and Ponnalagar Samy [14]. Integrating the above equation between 0 and z, we get. 1=n 2ð n þ 3Þ H Qðn þ 3Þ n 2n l L1 3=n 3=n L L log s D p¼ þ : ð19Þ H 2 1 3 wðn þ 2Þ L2 pwnþ3 Eq. (19) gives the pressure loss for the flow of Herschel–Bulkley fluid through a tapered tube with angle of tapering w. The following are obtained as particular cases from Eq. (19). H ¼ l N , we get the pressure loss for Newtonian fluid as (i) Newtonian fluid: When n = 1, sH ¼ 0 and l D p¼
8 lN Q 3 ; L2 L3 1 4 3pw
which is the same as Eq. (6).
ð20Þ
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H ¼ l P , we get the pressure loss as (ii) Power law fluid: When sH ¼ 0 and l 1=n 2n Q lP ðn þ 3Þ 3=n 3=n ; D p ¼ nþ3 L2 L1 p 3w n which is Eq. (9). H ¼ l B and sH ¼ sB , we get pressure loss for Bingham fluid as (iii) Bingham fluid: When n = 1, l 8sB 8 lB Q 3 L1 þ log D p¼ L2 L3 ; 1 4 3w L2 3pw
ð21Þ
ð22Þ
B is the coefficient of viscosity for Bingham fluid. where sB is the yield stress for Bingham fluid and l
2.4. Wall shear stress distribution Young [19] has pointed out that the variations of the resistance to flow and the wall shear stress with axial distance are physiologically important quantities. These variations in tapered tubes for Herschel–Bulkley fluid and for Casson’s fluid have not been studied earlier. Karino and Goldsmith [20] have pointed out that the wall shear stress plays an important role in determining aggregation sites of platelet. The wall shear stress is given by Eq. (4). For Herschel–Bulkley fluid, from Eqs. (4), (7) and (18) we get !1=n 3=n H Q ð n þ 3Þ 1 ð n þ 3Þ l sw ¼ sH : þ ð23Þ 3 3 ð n þ 2Þ 1 ðz=L1 Þ pw L1 The wall shear stress for Casson fluid is obtained in the non-dimensional form in Eq. (39). 2.5. Resistance to flow The resistance to flow is defined to be Dp K¼ : ð24Þ Q We obtain the resistance to flow for Casson fluid in the non-dimensional form in Eq. (43). From Eq. (19), we obtain the resistance to flow for Herschel–Bulkley fluid as # !1=n " 3=n H Qðn þ 3Þ 2n l L1 2ðn þ 3ÞsC L1 K¼ 1 þ : ð25Þ log 3 3 L2 L2 3wQ wðn þ 2ÞQ pw L1 3. Non-dimensionalization of flow quantities Define the typical velocity U 0 to be 2Q U0 ¼ : pw2 L21 The non-dimensional pressure loss DP, yield stress h and wall shear stress sw are defined by D p ; qU 20 sy h¼ ; qU 20 sw sw ¼ ; qU 20 DP ¼
ð26Þ
ð27Þ ð28Þ ð29Þ
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573
where sy represents the yield stress sC for Casson fluid or sH for Herschel–Bulkley fluid or sB for Bingham fluid. Defining the tapered tube Reynolds number (Rew) to be (see [3]) 1=n ð2R0 ÞU 02n1 ð30Þ Rew ¼ 8wq 2Kðn þ 3Þ and X ¼1
L2 z ¼ ; L1 L1
ð31Þ
N for Newtonian fluid, l P for power law fluid, l B for Bingham fluid, l C for Casson fluid where K represents l H for Herschel–Bulkley fluid. We get the following expressions for the non-dimensional pressure loss for and l different fluids. We also note that for Newtonian, Bingham and Casson fluids Rew is obtained by putting n = 1 in Eq. (30). (i) Herschel–Bulkley fluid " # 3=n n241=n 1 2ðn þ 3Þh 1 log DP ¼ 1 þ : 1X wðn þ 2Þ 1X 3Rew (ii) Power law fluid " # 3=n n241=n 1 DP ¼ 1 : 1X 3Rew (iii) Newtonian fluid " # 3 8 1 DP ¼ 1 : 3Rew 1X (iv) Bingham fluid " # 3 8 1 8h 1 log DP ¼ 1 þ : 3Rew 1X 3w 1X (v) Casson’s fluid: For Casson fluid, the non-dimensional form of pressure gradient is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # dp 128h 8 1 h 64 hw h2 ¼ þ : þ dz 49wð1 X Þ ð1 X Þ Rew ð1 X Þ3 147 7wð1 X Þ Rew ð1 X Þ 147
ð32Þ
ð33Þ
ð34Þ
ð35Þ
ð36Þ
The pressure drop can be obtained by integrating Eq. (36) between 0 and z with respect to z. The contributions h h2 of the terms 147 and 147 to the pressure gradient dp are negligibly small. Neglecting these terms in Eq. (36) and on dz integration, the pressure drop is obtained and is given by sffiffiffiffi" " # # 3 3=2 128h 1 8 1 128 h 1 log DP ¼ 1 þ pffiffiffiffiffiffiffiffi 1 : ð37Þ þ 49w 1X 3Rew 1X 21 Rew w 1 X Using Eqs. (26), (28) and (29) in Eqs. (23) and (14), we get wall shear stress for Herschel–Bulkley fluid as 3=n w231=n 1 ðn þ 3Þh sw ¼ ð38Þ þ 1X ðn þ 2Þ Rew and for Casson fluid as sffiffiffiffiffiffiffiffi 3 3=2 64h 4w 1 32 hw 1 sw ¼ þ þ ; 49 Rew 1 X 7 Rew 1 X
ð39Þ
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sC C respectively. Note that to obtain Eq. (39), the non-dimensional form of the terms 147 and 147 in Eq. (14) are neglected because of their negligibly small magnitude. Let the non-dimensional resistance to flow be defined as
K¼
K ; KN
ð40Þ
where KN ¼
8qU 0 pw4 L21
ð41Þ
is the impedance for Newtonian fluid for Poiseuille flow multiplied by the factor w2/ReN, ReN being the Reynolds number for the Newtonian fluid. Using Eqs. (24), (28), (30), (31), (40) and (41) in Eqs. (25) and (37), we get the expression for resistance to flow for Herschel–Bulkley fluid as " # 3=n 221=n nw2 1 ðn þ 3Þhw 1 log K¼ 1 þ ð42Þ 1X 2ðn þ 2Þ 1X 3Rew and for Casson’s fluid as sffiffiffiffiffiffiffiffi" " # # 3 3=2 32wh 1 2w2 1 32w wh 1 log 1 þ 1 ; K¼ þ 49 1X 1X 21 Rew 1 X 3Rew
ð43Þ
respectively. 4. Results and discussion An extensive attempt has been made in this section to discuss the effects of tapering of the blood vessel and the non-Newtonian nature of the blood on the flow quantities pressure drop, wall shear stress distribution and resistance to flow. It is generally accepted that human blood is a non-Newtonian fluid when it flows through smaller diameter tubes at low shear rates. In smaller tubes, blood appears to possess both finite yield stress and shear dependent viscosity. The yield stress for of normal human blood is between 0.01 dyne/cm2 and 0.06 dyne/cm2 [21,22] and thus the value 0.04 is used for non-dimensional yield stress in this study. It is generally observed that the typical value of the power law index n for blood flow are taken to lie between 0.9 and 1.1 and we have taken the typical value of n to be 0.95 for n < 1 and 1.05 for n > 1 [15–17]. Since, the objective of the present study is to quantify the effects of tapering of the blood vessels for small angle of tapering on the various flow quantities, the range 0.5–1.25° is used for angle of tapering W in the present study. For easy comparison of the results given by How and Black [3] for power law fluid with the present results, the same values used by them for L1 and L2 are used here namely L1 (in cms) is taken as 53.18, 36.28, 26.11 and 20.61 corresponding to w = 0.5°, 0.75°, 1° and 1.25° of tapered angle and L2 (in cms) as 37.22, 20.29, 12.26 and 10.61 respectively. The variation of pressure drop DP against tapered tube Reynolds number (Rew) for Newtonian fluid, power law fluid with n = 0.95 and n = 1.05, Herschel–Bulkley fluid with n = 0.95 and n = 1.05, Bingham fluid and Casson’s fluid respectively are shown in Figs. 2–8 for different angles of taper w. It is generally observed that the pressure loss increases as the angle of taper w increases from 0.5° to 1° for all fluids. For tapered angle 1.25°, the pressure loss decreases and lies between those for w = 0.75° and 1° for Newtonian fluid and power law fluid whereas for Herschel–Bulkley fluid, Bingham fluid and Casson’s fluid after a certain value of Reynolds number, the pressure loss is less than that for w = 0.75° as seen in Figs. 2–8. It is to be noted that the last four figures (Figs. 5–8) are for fluids with non-zero yield stress. The maximum pressure loss occurs when w = 1° for all fluids. We observe that the pressure drop for all the fluids at the angle of taper w = 1.25° is lower than that of w = 1°, because the distance between the pressure ports when w = 1.25° is shorter than that for w = 1°. Figs. 2–8 depict the simultaneous effects of the angle of tapering of the blood vessel and the tapered tube Reynolds number on pressure drop for different fluids.
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575
25
ψ = 1o
15
ψ = 1. 25 o
10
ψ = 0. 75 o
ΔP
20
ψ = 0. 5 o
5
0 1
2
3
4
5
6
7
8
9
10
Re ψ
Fig. 2. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Newtonian fluid.
30
ΔP
25 20
ψ = 1o
15
ψ = 1.25 o ψ = 0.75 o
10
ψ = 0. 5 o 5 0 1
2
3
4
5
6
7
8
9
10
Re ψ
Fig. 3. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Power law fluid with n = 0.95.
25 20
ψ = 1o 15
ΔP
ψ = 1 .2 5 o ψ = 0 .7 5 o ψ = 0 .5 o
10
5 0 1
2
3
4
5
6
7
8
9
10
R eψ Fig. 4. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Power law fluid with n = 1.05.
The variation of pressure drop with tapered tube Reynolds number for different fluids for the angle of taper 1° is shown in Fig. 9. It is observed that there is not much of a difference between the values for the pressure
576
D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582 35 30
ΔP
25
ψ = 1o
20
ψ = 1.25 o
15
ψ = 0.75 o
10
ψ = 0 .5 o
5 0 1
2
3
4
5
6
7
8
9
10
Reψ
Fig. 5. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Herschel–Bulkley fluid with n = 0.95. and h = 0.04.
30 25
ψ = 1º
20 ΔP
ψ = 1.25º 15
ψ = 0.75º
ψ = 0.5º
10 5 0 1
3
5
7
9
Re ψ
Fig. 6. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Herschel–Bulkley fluid with n = 1.05 and h = 0.04.
30 25 ψ = 1o 20 ΔP
ψ = 1.25o 15 ψ = 0.75o
10
ψ = 0.5o
5 0 1
3
5
7
9
Re ψ
Fig. 7. Variation of pressure loss with tapered tube Reynolds number for different angles of tapering for Bingham fluid with h = 0.04.
loss for n = 0.95 and for n = 1.05 in the case of power law fluid. The same behaviour is observed in the case of Herschel–Bulkley fluid too. The pressure loss for power law fluid with n = 0.95 is slightly higher and for
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577
50 45 40
ψ=1
35
ψ = 1.25
30 ΔP
o
ψ = 0.75
25
o
o
ψ = 0.5
20
o
15 10 5 0 1
3
5
7
9
Reψ
Fig. 8. Variation of pressure loss with tapered tube Reynolds number for different angles of taper for Casson fluid with h = 0.04.
n = 1.05, it is slightly lower than that for Newtonian fluid as seen from Fig. 9. That is, the pressure loss decreases with increasing ‘n’. In general for power law fluids as ‘n’ increases the pressure loss decreases. Fig. 9 shows the simultaneous effects of tapered tube Reynolds number and the non-Newtonian nature of the blood on pressure drop. It is observed from Figs. 2–9 that for all other fluids with yield stress, the pressure losses are much higher compared to those of Newtonian and power law fluids. The highest pressure drop is observed for Casson’s fluid for all angles of taper and for all values of tapered tube Reynolds number. For Herschel–Bulkley fluid with n = 0.95 and n = 1.05 and Bingham fluid, the pressure losses are almost the same for a given angle of taper w and a given tapered tube Reynolds number Rew. It is generally observed that for all fluids, the pressure losses are much higher for values of Rew upto 2 and for each fluid the pressure loss is almost the same constant value as the tapered tube Reynolds number Rew increases from 6. The variation of wall shear stress with axial distance for tapered tube Reynolds number 5 and yield stress 0.04 are shown in Figs. 10–12 for different values of w for Herschel–Bulkley fluid with n = 0.95 and 1.05 and for Casson’s fluid respectively. It is clear that for each value of n, the wall shear stress increases with increasing
50 45
Casson fluid with θ = 0.04
40 Herschel - Bulkley fluid with n = 0.95 and θ = 0.04
35
Bingham fluid with θ = 0.04
ΔP
30
Herschel - Bulkley fluid with n = 1.05 θ = 0.04
25 Power law fluid with n = 0.95
20 Newtonian fluid
15
Power law fluid with n = 1.05
10 5 0 1
3
5
7
9
Reψ Fig. 9. Variation of pressure loss with tapered tube Reynolds number for different fluids with angle of taper w = 1° and h = 0.04.
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D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582 0.9 0.8 0.7
τw
0.6 o
ψ = 1.25
0.5
o
ψ=1
0.4
o
ψ = 0.75
0.3
o
ψ = 0.5
0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0. 7
X Fig. 10. Variation of wall shear stress with axial distance for different angles of taper for Herschel–Bulkley fluid with n = 0.95, Rew = 5 and h = 0.04.
0.7 0.6
τw
0.5 ψ = 1.25o
0.4
o
ψ=1
0.3
ψ = 0.75o
0.2
o
ψ = 0.5
0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X Fig. 11. Variation of wall shear stress with axial distance for different angles of taper for Herschel–Bulkley fluid with n = 1.05, Rew = 5 and h = 0.04.
1.2 1 0.8
τw
o
ψ = 1.25
0.6
o
ψ=1
ψ = 0.75o
0.4
ψ = 0.5o
0.2 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X Fig. 12. Variation of wall shear stress with axial distance for different angles of taper for Casson fluid with h = 0.04.
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axial distance and also with increasing angle of taper. In each case upto X = 0.2, there is not much variation for different angles of taper, but the difference become pronounced as X increases further. Figs. 10–12 show the clear effects of tapering of the blood vessel on wall shear stress distribution for different fluids. For different fluids, the variation of wall shear stress with tapered tube Reynolds number Rew for X = 0.3 and w = 0.75° is shown in Fig. 13. It is observed that for all fluids, the wall shear stress decreases with increasing tapered tube Reynolds number Rew and this decrease is higher for value of Rew upto 2 and for each fluid the wall shear stress is almost the same constant value as Rew increases further from 6. For the given values of axial distance X, angle of tapering w and tapered tube Reynolds number Rew, the wall shear stress is maximum for Casson’s fluid and for Herschel–Bulkley fluid with n = 0.95 as well as n = 1.05, there is no significant difference in the values of wall shear stress when the yield stress, axial distance and tapered tube Reynolds number are held constant. The same behaviour is also noticed for Newtonian fluid and power law fluid with n = 0.95 as well as for n = 1.05. Fig. 13 shows the simultaneous effects of non-Newtonian nature and tapered tube Reynolds number of the fluid on wall shear stress distribution. The variation of resistance to flow with axial distance for Herschel–Bulkley fluid with n = 0.95 and Casson’s fluid are shown in Figs. 14 and 15. It is observed that in each case as the angle of taper increases the resistance to flow increases and also as X increases (i.e. the pipe diameter decreases) the resistance increases. This is an obvious result, since an increase in the angle of taper increases the pressure drop and since
0.45 0.4 Casso n fluid θ = 0.04
0.35
Herschel - B ulkley fluid with n = 0.95 and θ = 0.04
τw
0.3
B ingham fluid with θ = 0.04
0.25
Herschel - B ulkley fluid with n = 1.05 and θ = 0.04
0.2
P o wer law fluid with n = 0.95
0.15
Newto nian fluid P o wer law fluid with n = 1.05
0.1 0.05 0 1
2
3
4
5
6
7
8
9
10
Re ψ
Fig. 13. Variation of wall shear stress with tapered tube Reynolds number for different fluids with X = 0.3 and w = 0.75°.
1.8 1.6 1.4
ψ = 1.25o
1
3
10 Λ
1.2
ψ = 1o
0.8 0.6
ψ = 0.5o
0.4
ψ= 0.75o
0.2 0 0
0.1
0.2
0.3
0.4
0.5
0.6
X Fig. 14. Variation of resistance to flow with axial distance for different values of angle of taper for Herschel–Bulkley fluid with h = 0.04 and Rew = 5.
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D.S. Sankar, K. Hemalatha / Applied Mathematics and Computation 188 (2007) 567–582 3
2.5
103 Λ
2
1.5 ψ = 1.25o ψ = 1o
1 ψ = 0.75o ψ = 0.5o
0.5
0 0
0.1
0.2
0.3
0.4
0.5
0.6
X
Fig. 15. Variation of resistance to flow with axial distance for different angles of taper for Casson fluid with h = 0.04 and Rew = 5.
0.7 0.6 Casson fluid with θ = 0.04
0.5 Bingham fluid with θ = 0.04
3
10 Λ
Herschel - Bulkley fluid with n = 1.05 and θ = 0.04
0.4
Herschel - Bulkley fluid with n = 0.95 and θ = 0.04 Power law fluid with n = 1.05
0.3 0.2
Newtonian fluid
0.1
Power law fluid with n = 0.95
0 1
2
3
4
5
6
7
8
9
10
Reψ
Fig. 16. Variation of resistance to flow with tapered tube Reynolds number for different fluids with X = 0.3, w = 0.75°.
the flow is fully developed for the same flow rate, the flow resistance increases as the pipe diameter decreases. Figs. 14 and 15 depict the effects of tapering of the blood vessel on resistance to flow for Casson fluid and Herschel–Bulkley fluid. Fig. 16 depicts the variation of resistance to flow for different fluids with X = 0.3 and w = 0.75°. It is to be noted that for each fluid and given values of axial distance X, tapered tube Reynolds number Rew and angle of taper w, the resistance to flow decreases with increase in tapered tube Reynolds number and this decrease is higher upto Rew value 2 and is almost the constant value once Rew exceeds the value 6. When the parameters X, Rew and w are held constant, the resistance to flow is maximum for Casson fluid. For Bingham fluid and Herschel–Bulkley fluid with n = 0.95 as well as for n = 1.05, there is no significant difference in the values of resistance to flow. The same behaviour is observed for Newtonian fluid and power law fluid with n = 0.95 as well as n = 1.05. Fig. 16 shows the simultaneous effects of non-Newtonian nature of the blood and the tapered tube Reynolds number on resistance to flow. 5. Conclusions Non-Newtonian fluid models with non-zero yield stress namely Casson’s and Herschel–Bulkley fluids are used to study the flow aspects of blood flow through tapered tubes with small angles of taper. Estimates of
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pressure drop that provides important information in the design of tapered grafts are given. Apart from these aspects, expressions for wall shear stress and resistance to flow are also analyzed. For all fluids, the non-dimensional pressure drop increases with increasing angle of taper from 0.5° to 1° for a given yield stress h and tapered tube Reynolds number Rew. Also the non-dimensional pressure losses are much higher for fluids with non-zero yield stress compared to those of Newtonian and power law fluids for all angles of taper and for all tapered tube Reynolds number. The highest pressure drop is observed for Casson’s fluid. It is observed that the pressure loss increases with yield stress in each case. For each fluid, the pressure losses are much higher only upto tapered tube Reynolds number value 2 and remains almost the same value once the Reynolds number exceeds 6. Both for Herschel–Bulkley fluid and Casson’s fluid the wall shear stress as well as the resistance increase with increasing axial distance for a given tapered tube Reynolds number Rew, angle of taper w and yield stress h and also with increasing angle of taper and the values are much higher in the case of Casson’s fluid. For Herschel–Bulkley fluid, both resistance and wall shear stress decrease as ‘n’ increases for fixed values of other parameters. The resistance as well as the wall shear stress increase with increasing yield stress for Herschel– Bulkley fluid and also for Casson’s fluid when the other parameters held constant. Since, the blood flow through narrow arteries is often pulsatile, the extension of this study to pulsatile flow could be more useful and will have wider applications to cardiovascular studies. Acknowledgement The authors thank Prof. P. Chaturani, Department of Mathematics, Indian Institute of Technology, Bombay, for his valuable suggestions and timely help in the preparation of the paper. References [1] A.P. Dwivedi, T.S. Pal, L. Rakesh, Micropolar fluid model for blood flow through a small tapered tube, Indian J. Technol. 20 (1982) 295–299. [2] P. Chaturani, R. Pralhad, Blood flow in tapered tubes with biorheological applications, Biorheology 22 (1985) 303–314. [3] T.V. How, R.A. Black, Pressure losses in non-Newtonian flow through rigid wall tapered tubes, Biorheology 24 (1987) 337–351. [4] S. Chakravarthy, P.K. Mandal, Two dimensional blood flow through tapered arteries under stenotic conditions, Int. J. Non-Linear Mech. 35 (2000) 779–793. [5] T.V. How, R.A. Black, D. Annis, Comparison of pressure losses in steady non-Newtonian flow through experimental tapered and cylindrical arterial prostheses, J. Biomed. Eng. 10 (1988) 225–230. [6] P.K. Mandal, An unsteady analysis of non-Newtonian blood flow through tapered arteries with stenosis, Int. J. Non-Linear Mech. 40 (2005) 151–164. [7] E.W. Merrill, A.M. Benis, E.R. Gilliland, T.K. Sherwood, E.W. Salzman, Pressure flow relations of human blood in hallow fibers at low shear rates, J. Appl. Physiol. 20 (1965) 954–967. [8] S. Oka, Pressure development in a non-Newtonian flow through a tapered tube, Biorheology 10 (1973) 207–212. [9] S. Oka, T. Murata, Theory of the steady slow motion of non- Newtonian fluids through a tapered tube, Jpn. J. Appl. Phys. 8 (1969) 5–8. [10] Jr. Walawender, C. Tien, L.C. Cerny, Experimental studies on the blood flow throughout tapered tubes, Int. J. Eng. Sci. 10 (1972) 1123–1135. [11] S. Chien, Hemorheology in clinical medicine, Rec. Adv. Cardiovascular Dis. 2 (Suppl.) (1981) 21–26. [12] J.B. Shukla, R. Parihar, S.P. Gupta, Biorheological aspects of blood flow through artery with mild stenosis: effects of peripheral layer, Biorheology 17 (1980) 403–410. [13] C. Tu, M. Deville, Pulsatile flow of non-Newtonian fluids through arterial stenosis, J. Biomech. 29 (1996) 899–908. [14] P . Chaturani, R. Ponnalagar Samy, A study of non-Newtonian aspects of blood flow through stenosed arteries and its applications in arterial diseases, Biorheology 22 (1985) 521–531. [15] D.S. Sankar, K. Hemalatha, Pulsatile flow of Herschel–Bulkley fluid through catheterized arteries – a mathematical model. App. Math. Modeling, in press. [16] D.S. Sankar, K. Hemalatha, A non-Newtonian fluid model for blood flow through a catheterized artery - steady flow, App. Math. Modeling, in press. [17] D.S. Sankar, Hemalatha, Pulsatile flow of Herschel–Bulkley fluid through stenosed arteries – a mathematical model, Int. J. Non-Linear Mech., in press. [18] N. Iida, Influence of plasma layer on steady blood flow in microvessels, Jpn. J. Appl. Phys. 17 (1978) 203–214. [19] D.F. Young, Fluid mechanics of arterial stenosis, J. Biomech. Eng. (Trans. ASME) 101 (1979) 157–175.
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[20] T. Karino, H.L. Goldsmith, Flow behaviour of blood cells and rigid spheres in annular vortex, Philos. Trans. Roy. Soc., London B 279 (1977) 413–445. [21] J.N. Kapur, Mathematical Models in Biology and Medicine, East West Press Pvt. Ltd., New Delhi, India, 1992. [22] E.W. Merrill, Rheology of human blood and space speculations on its role in vascular homeostasis, in: P.N. Sawyer (Ed.), Biomechanical Mechanisms in Vascular Homeostasis and Intravascular Thrombus, Appleton Century Crafts, New York, 1965.