Applied Bionics and Biomechanics 10 (2013) 11–18DOI 10.3233/ABB-2012-0072IOS Press
11
Suspension two-layered blood flow througha bell shaped stenosis in arteries
Amit Medhavi∗Department of Mechanical Engineering, Kamla Nehru Institute of Technology, Sultanpur, India
Abstract. The present study concerns with the effects of the hematocrit and the peripheral layer on blood flow characteristicsdue to the presence of a bell shaped stenosis in arteries. To account for the hematocrit and the peripheral layer, the flowing bloodhas been represented by a two-layered macroscopic two-phase (i.e., a suspension of red cells in plasma) model. The expressionsfor the flow characteristics, namely, the velocity profiles, the flow rate, the impedance, the wall shear stress in the stenotic regionand the shear stress at the stenosis throat have been derived. The quantitative effects of the hematocrit and the peripheral layeron these flow characteristics have been displayed graphically and discussed briefly.
The frequently occurring cardiovascular disease,stenosis or arteriosclerosis means narrowing of anybody passage, tube or orifice, is known to be respon-sible for many of the serious consequences (cerebralstrokes, myocardial infarction, angina pectoris, cardiacarrests). Although, the etiology of the initiation of dis-ease is not well understood, however, it is believed thatthe disease occurs due to the deposits of the cholesterol,fatty substances, cellular waste products, calcium andfibrin in the inner lining of an artery. It is also wellestablished that once the constriction has developed, itbrings about the significant changes in the flow field,particularly, the pressure distribution, the wall shearstress and the impedance (flow resistance). With theknowledge that the cardiovascular disease, stenosis isclosely associated with the flow conditions and otherhemodynamic factors, a large number of researchersincluding Young [41, 48], Young and Tsai [39], Caroet al. [7], Shukla et al. [29], Ahmed and Giddens [1],
∗Corresponding author: Amit Medhavi, Department of Mechan-ical Engineering, Kamla Nehru Institute of Technology, Sulanpur-228118, India. E-mail: [email protected].
Sarkar and Jayaraman [28], Pralhad and Schultz [27],Jung et al. [11], Liu et al. [13], Srivastava and Raas-togi [33, 34], Misra and Shit [21], Ponalagusamy [26],Layek et al. [12], Joshi et al. [10], Mekheimer and El-Kot [18], Tzirtzilakis [38], Mandal and coworkers [14,15], Politis et al. [23, 24], Singh et al. [30], Biswas andChakraborty [4, 5], Medhavi [17], Mishra and Siddiqui[20], Nadeem et al. [22], Mekheirmer et. al. [19], Pon-alagusamy and Selvi [25], Bandyopadhyay and Layek[2, 3], Srivastava et al. [36] and many others haveaddressed the stenotic development problems undervarious flow situations since the first investigation ofMann et al. [16].
Being a suspension of corpuscles, at low shearrates blood in general behaves like a non-Newtonianfluid in small diameter tubes. The experimental obser-vations of Cokelet [8] and theoretical investigationof Haynes [9] indicate that blood can no longer betreated as a single-phase homogeneous viscous fluidwhile flowing through narrow arteries (of diameter≤ 1000 �m). Skalak [31] concluded that an accu-rate description of the blood in small vessels requiresthe consideration of erythrocytes as discrete particles.In addition, Bugliarello and Sevilla [6], Cokelet [8]and Thurston [37] have shown experimentally that for
Fig. 1. The geometry of an arterial bell shaped stenosis.
blood flowing through small vessels, there is a cell-free plasma (Newtonian viscous fluid) layer and a coreregion of suspension of all the erythrocytes. Haynes [9]presented a two-fluid model for blood, consisting of acore region of suspension of all the erythrocytes as ahomogeneous Newtonian viscous fluid and a cell-freeplasma layer as a Newtonian fluid of constant viscos-ity (equal to the viscosity of water) and concluded thatthe significance of the peripheral layer increases withdecreasing blood vessel diameter. A brief discussionand survey on suspension modeling of blood flow hasbeen presented in Srivastava [32].
It is also known from the published literature thatstenoses may develop in series (multiple stenoses),may be of irregular shapes, overlapping, bell shaped,composite in nature, axially symmetric or non-symmetric, etc. The majority of the studies conductedhave used axially symmetric and non-symmetricstenoses. The present work is devoted to discuss theflow through a bell shaped stenosis assuming that theflowing blood is represented by a two-layered sus-pension model [32]. The theoretical model used toconduct the study enables one to observe simultane-ous effects of the hematocrit and the peripheral layeron flow characteristics of blood due to the presence ofa bell shaped stenosis in arteries. The artery length isconsidered large enough as compared to its radius sothat the entrance, end and special wall effects can beneglected.
2. Formulation of the problem
Consider the axisymmetric flow of blood in anartery of circular cross-section of radius R with anaxisymmetric bell shaped stenosis. Assuming that the
flowing blood is represented by a two-layered suspen-sion model consisting of a central layer of suspensionof all the erythrocytes (i.e., a suspension of red cells inplasma) of radius R1 and a peripheral layer of plasma (aNewtonian viscous fluid) of thickness (R − R1). Thestenosis geometry [21] and the shape of the centrallayer, assumed to be manifested in the arterial segment,are respectively described in Figs. 1 and 2, as
(R(z), R1(z))
R0
= 1− (δ, δ1)
R0exp
(−m2ε2z2
R20
), −L0 ≤ z ≤ L0,
= (1, α), otherwise, (1)
where R0 is the radius of the arterial segment in thenon-stenotic region, R(z) is the radius of the stenosedportion located at the axial distance z from the left endof the segment, � is the depth of stenosis at the throatand m is a parametric constant, � is the relative lengthof the constriction defined as the ratio of the radius tothe half length of the stenosis, i.e., � = R0/L0.
The equations describing a two-layered suspensionblood flow [32] in the case of a mild stenosis (�/R0<< 1), are given as
(1 − C)dp
dz= (1 − C)
�s(C)
r
∂
∂ r
(r
∂
∂ r
)uf
+ CS (up − uf), 0 ≤ r ≤ R1, (2)
Cdp
dz= CS (uf − up), 0 ≤ r ≤ R1, (3)
dp
dz= �0
r
∂
∂r
(r∂
∂r
)u0, R1 ≤ r ≤ R, (4)
where r is the radial coordinate measured normal tothe artery axis and p denotes the pressure, (uf, up)are the axial velocity of (fluid, particle) phases in thecore region (0 ≤ r ≤ R1), (�0, u0) are (viscosity, axialvelocity) of fluid (plasma) in the peripheral region(R1 ≤ r ≤ R), �S(C) ∼= �S is the suspension viscos-ity (apparent or effective viscosity) in the core region,C denotes the constant [35] volume fraction density ofthe particles (called hematocrit), S is the drag coeffi-cient of interaction exerted by one phase on the other,and the subscripts f and p denote the quantities associ-ated with the plasma (fluid) and erythrocyte (particle)phases, respectively. The limitations and the usefulness
A. Medhavi / Suspension two-layered blood flow 13
of the present theoretical model are discussed brieflyin Srivastava [32]. The expression for the viscosity ofsuspension, �s and the drag coefficient of interaction,S for the present study are selected [35] as
�s∼= �s (C) =
�o
1 − qC,
q = 0.07 exp
[2.49C +
(1107
T
)exp (−1.69C)
],
(5)
S = 4.5(�o /a2o)
4 + 3[8C − 3C2] 1/2 + 3C
(2 − 3C)2 , (6)
where T is measured in absolute scale of the tempera-ture (K), �o is the constant plasma viscosity and ao isthe radius of an erythrocyte.
The boundary conditions are the standard no slipconditions of velocities and the shear stresses at thetube wall and the interface, and are stated as
u0 = 0 at r = R, (7)
u0 = uf and �p = �f at r = R1, (8)
∂uf
∂r= ∂up
∂r= 0 at r = 0, (9)
where �p = �o∂u0/∂r and �f = (1 − C)�s∂uf/∂r are theshear stresses of the peripheral and central layers,respectively.
3. Analysis
The expressions for velocities, u0, uf and up obtainedas the solutions of Equations (2)–(4), subject to theboundary conditions (7)–(9), are given as
u0 = − R20
4�0
dp
dz
{(R/R0)2 − (r/R0)2
}, R1 ≤ r ≤ R,
(10)
uf = − R2o
4 (1 − C) �0
dp
dz
[�{
(R1/R0)2 − (r/R0)2}
+ (1 − C){
(R/R0)2 − (r/R0)2}]
,
0 ≤ r ≤ R1, (11)
up =− R2o
4(1 − C) �0
dp
dz
{�[(R1/R0)2 − (r/R0)2
]
+ (1 − C)[(R/R0)2 − (r/R0)2
]+ 4(1 − C) �0
SR20
},
0 ≤ r ≤ R1, (12)
where � = �0/ �s.The flow flux, Q is now calculated as
Q = 2 �
{∫ R
R1
ru0 dr
+R1∫
0
r[(1 − C) uf + C up
]dr
⎫⎬⎭
= − �R4o
8(1 − C)�0
dp
dz
{(1 − C)
[(R/R0)4 − (R1/R0)4
]
+ � (R1/R0)4 + � (R1/R0)2}
, (13)
with � = 8C(1 − C)�0 /SR20, a non-dimensional sus-
pension parameter.Using the fact that the total flux is equal to the sum
of the fluxes across the two regions (peripheral andcore), one determines the relations [32]: R1 = α R and�1=��. In view of these relations, the pressure drop,�p (= p at z = − L, − p at z = L) across the steno-sis between the sections z = −L and z = L, using theexpression for (−dp/dz) obtained from Equation (13),is derived as
�p =L∫
- L
(− dp
dz
)dz = 8 (1 − C) �0 Q
� R4o
�, (14)
where � =−L0∫−L
[ (z)]R/Ro=1dz +Lo∫
−L0
(z) dz
+L∫
Lo
[ (z)
]R/R0=1 dz,
(z) = 1
(R/R0)4 + ��2 (R/R0)2 ,
= (1 − C)(1 − �4) + ��4.
The first and third integrals in the expression for� obtained above are straight forward whereas theevaluation of the second integral in closed form is a
14 A. Medhavi / Suspension two-layered blood flow
formidable task and thus will be evaluated numeri-cally. Using now the definitions from Srivastava andRastogi [34], the expression for the impedance (flowresistance), � the wall shear stress, �w and the shearstress at the stenosis throat, �s are obtained in theirnon-dimensional form as
� = (1 − C)
{1 − L0/L
+ ��2
+ 1
L
L0∫−L0
dz
(R/R0)4 + ��2(R/R0)2
⎫⎪⎬⎪⎭, (15)
�w = (1 − C)
(R/R0)3 + ��2(R/R0), (16)
�s = (1 − C)
(1− � /R0)3 + ��2(1− � /R0), (17)
where
� = �/ �0, (�w, �s) = (�w, �s) / �0,
� = �p/Q, �w = ( − R/2)dp/dz,
�s = [ − (R/2)(dp/dz)]R/R0=(1−�/R0),
�0 = 16�0 L/� R40, �0= 4�0Q/� R3
0,
�0 and �0 are the impedance and shear stress in anormal (no stenosis) artery for a Newtonian fluid (i.e.,C = 0), and (�, �w,�s) are (impedance, wall shear stress,shear stress at the stenosis throat) in their dimensionalform.
When the core mixture behaves like a Newtonianfluid of constant viscosity, �1 (different from �o), theresults obtained above reduce to the case of a two-fluidmodel of Newtonian fluid as
�t = �
⎧⎪⎨⎪⎩1 − L0/L + (1/L)
L0∫−L0
dz
(R/R0)4
⎫⎪⎬⎪⎭, (18)
�wt = �
(R/R0)3 , (19)
�st = �
(1 − �/R0)3 , (20)
with � = 1/[1 − (1 − �′) �4], �′= �o/ �1. The sec-ond subscript t denotes the quantities associated withthe two-fluid model of Newtonian fluids.
In the absence of the peripheral layer (i.e., � = 1),the expressions for the flow characteristics obtained inEquations (15)–(18), derive the corresponding resultsfor the case of a single-layered macroscopic two-phaseblood flow as
�m = (1 − C)
{1 − L0/L
� + �
+ 1
L
L0∫−L0
dz
� (R/R0)4+ � (R/R0)2
⎫⎪⎬⎪⎭, (21)
�wm = (1 − C)
�(R/R0)3+ � (R/R0), (22)
�sm = (1 − C)
� (1− � /R0)3+ � (1 − �/R0), (23)
The second subscript m stands for the quantitiesassociated with the flow of a single-layered macro-scopic two-phase blood flow. Further, it is interestingto note that in the absence of the particle phase inthe core region, the two-phase fluid in the core regionreduces to the same fluid as in the peripheral regionand consequently the role of the interface automati-cally disappears and one obtains the expressions for theblood flow characteristics for a single-layered Newto-nian fluid as
�N= 1 − L0/L + 1
L
L0∫−L0
dz
(R/R0)4 , (24)
�wN = 1
(R/R0)3 , (25)
�sN = 1
(1− �/R0)3 , (26)
where the second subscript N stands for single-layeredNewtonian fluid.
4. Numerical results and discussion
In order to discuss the results of the study quan-titatively, computer codes are developed to evaluateanalytical results obtained in Equations (2.19)–(2.21)at the temperature of 37◦C in an artery of radius0.01 cm for various parameter values [34, 35, 40]
A. Medhavi / Suspension two-layered blood flow 15
Fig. 2. The shape of the central layer.
Fig. 3. � vs δ/Ro for different, C.
selected as: L0(cm) = 1; L(cm) = 1, 2, 5; C = 0, 0.2,0.4, 0.6; �/R0 = 0, 0.05, 0.10, 0.15, 0.20. Some ofthe critical results obtained are displayed graphicallyin Figs. 3–8. In view of the fact that the peripherallayer thickness strongly depends on the core suspen-sion viscosity (i.e., on erythrocyte concentration; [6,32]), we choose 2a0 (diameter of a red cell) = 8 �m,the peripheral layer thickness, � (�m) ∼= �(C) = 6.18,4.67, 3.60, 3.12, 2.58, 2.18 corresponding to the hema-tocrit, C = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, respectively [9].The value of the parameter, � is then calculated fromthe relation: � = 1 − �/R0.
The resistance to flow, � increases with the hema-tocrit, C as well as with the stenosis height, �/R0(Fig. 3). The impedance, � decreases with the
Fig. 4. � vs δ/Ro for different C and L.
increasing length of the tube which in terns implies thatthe impedance, � increases with the stenosis length,2L0 (Fig. 4). The blood flow characteristic, � increasessteeply with the hematocrit, C for any given set of otherparameters (Fig. 5). The flow characteristic, � assumeslower magnitude in two-layered analysis than its cor-responding value in single-layered model (Figs. 2–5).
At any axial distance the wall shear stress in thestenotic region, τw increases with the hematocrit, Cand stenosis height, �/R0 (Fig. 6). The blood flowcharacteristic, increases rapidly in the up stream of thestenosis throat and attains its peak magnitude at thethroat located at z/L0 = 0, it then decreases rapidly inthe down stream of the throat and attains its approachedvalue (i.e., at z/L0 = −1) at the end point of the
16 A. Medhavi / Suspension two-layered blood flow
Fig. 5. � vs C for different δ/Ro.
Fig. 6. �w vs z/Lo in stenotic region for different δ/Ro.
constriction profile located at z/L0 = 1 (Fig. 6). It isto note here that for small stenosis height, �/R0 (≤0.1, 19% stenosis by area reduction), the magnitude ofthe shear stress, τw in two-fluid analysis follow closelythe magnitude of the shear stress, τw in one-fluid anal-ysis but considerable difference between the two isclearly observed increasing stenosis size. In addition,one notices that the peak point of the shear stress intwo-layered analysis occurs slightly right to the peakpoint of the shear stress in one-layered analysis. Theshear stress at the stenosis in throat, �s also increaseswith the hematocrit, C and the stenosis height, �/R0(Fig. 7). An inspection of Figs. 2–4, 6 and 7 reveals
Fig. 7. �s vs δ/Ro for different C.
Fig. 8. � vs δ/Ro for different stenosis geometry.
that the shear stress at the stenosis throat, �s possessesthe characteristics similar to that of the flow resistance,� with respect to any parameter.
To emphasize further on the significance of thepresent work, a comparison of the results (impedance)obtained in the case of the present bell shaped steno-sis with those obtained in an axisymmetric stenosis[34] and axially non-symmetric stenosis [33] has beenpresented in Fig. 8 for the same values of the vari-ous parameters. For any stenosis height, �/R0 the flowresistance, � assumes considerably higher magnitudein the present bell shaped stenosis as compared to othergeometries (symmetric or non-symmetric).
A. Medhavi / Suspension two-layered blood flow 17
The condition that �/R0 << 1 limits the usefulnessof the present study to very early stages of the vesselconstriction, which allows the use of fully developedflow equations and closed form solutions; the use ofparameter �/R0 is restricted to the value up to 0.15(i.e., 28% stenosis by area reduction) as beyond thisvalue a separation in the flow may occur [40].
5. Conclusions
To observe the effects of hematocrit on blood flowcharacteristics due to the presence of a mild stenosis, amacroscopic two-phase model of blood has been usedto discuss the flow through a bell shaped stenosis. Theblood flow characteristics (the flow resistance, the wallshear stress in the stenotic region and the shear stressat the stenosis throat) increase with the hematocrit aswell as with the stenosis size (length and height). Theshear stress at the stenosis throat possesses the charac-teristics similar to that of the impedance with respectto any parameter. The two-phase fluid (particle-fluidsuspension) seems to be more sensitive to the steno-sis than a single-phase fluid. The flow characteristicsassume considerably higher magnitude in the presentbell shaped stenosis than its corresponding valve inaxisymmetric and non-symmetric stenoses. The flowcharacteristics assume lower magnitude in two-fluidanalysis than its corresponding magnitude in one-fluidstudy which concludes that the peripheral layer helpsin functioning of diseased arteries.
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