![Ratio HT/HD as a function of tube diameter and discharge hematocrit. —, HD = 60%; --, HD = 45%; · · ·, HD = 20%. The curves are obtained from the modified function (Eq. (24)). Symbols @BULLET indicate the values obtained from Eq. (19) proposed by Pries et al. [16].](https://www.researchgate.net/profile/Aleksander-Popel/publication/11356004/figure/fig2/AS:349274404605956@1460284915413/Ratio-HT-HD-as-a-function-of-tube-diameter-and-discharge-hematocrit-HD-60--HD_Q320.jpg)
![Dimensionless thickness of the cell-free layer as a function of tube diameter and discharge hematocrit. Curves represent model results. Experimental points are taken from the studies of Bugliarello and Sevilla [3] and Reinke et al. [21]. Error bars correspond to standard deviation calculated from Bugliarello and Sevilla [3].](https://www.researchgate.net/profile/Aleksander-Popel/publication/11356004/figure/fig3/AS:349274412994572@1460284917204/Dimensionless-thickness-of-the-cell-free-layer-as-a-function-of-tube-diameter-and_Q320.jpg)
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Biorheology 38 (2001) 415–428 415
IOS Press
A two-phase model for flow of blood in
narrow tubes with increased effective
viscosity near the wall
Maithili Sharana,∗and Aleksander S. Popel b
aCentre for Atmospheric Sciences, Indian Institute of Technology, Hauz Khas, New Delhi, India
bDepartment of Biomedical Engineering, School of Medicine, Johns Hopkins University, Baltimore,
MD, USA
Received 2 August 2000
Accepted in revised form 20 August 2001
Abstract. A two-phase model for the flow of blood in narrow tubes is described. The model consists of a central core of
suspended erythrocytes and a cell-free layer surrounding the core. It is assumed that the viscosity in 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. A consistent system of nonlinear equations is solved numerically to estimate: (i) the effective dimensionless
viscosity in the cell-free layer (β), (ii) thickness of the cell-free layer (1 −λ) and (iii) core hematocrit (Hc). We have taken the
variation of apparent viscosity (µapp) and tube hematocrit with the tube diameter (D) and the discharge hematocrit (HD) from
in vitro experimental studies [16]. The thickness of the cell-free layer computed from the model is found to be in agreement
with the observations [3,21]. Sensitivity analysis has been carried out to study the behavior of the parameters 1 −λ,β,Hc,B
(bluntness of the velocity profile) and µapp with the variation of Dand HD.
Keywords: Mathematical model, two-phase blood flow, energy dissipation, relative viscosity, bluntness
1. Introduction
Quantitative understanding of the blood flow through arterioles and venules is necessary for assess-
ing the hemodynamic resistance and its regulation in the microcirculation as well as for analyzing mass
transport processes. When blood flows through tubes, the two-phase nature of blood as a suspension
becomes important as the diameter of the red blood cell (RBC) becomes comparable to the tube diam-
eter. The following are some of the effects observed in vitro and in vivo: (i) Fåhraeus–Lindqvist effect:
dependence of apparent viscosity on tube diameter; (ii) Fåhraeus effect: dependence of tube or vessel
hematocrit on tube diameter; (iii) Existence of a cell-free or cell-depleted layer near the wall; (iv) Blunt
velocity profile; (v) Phase separation effect: disproportionate distribution of red blood cells and plasma
at vessel bifurcations. Chien et al. [4] reviewed the studies on blood flow in narrow tubes. Pries et al.
[17] reviewed biophysical aspects of microvascular blood flow in vivo as well as in vitro. Numerous
models have been developed to interpret these effects (e.g., [9]). Here, we only discuss the models and
experimental data having direct relevance to our work.
*Address for correspondence: Professor Maithili Sharan, Centre for Atmospheric Sciences, Indian Institute of Technology,
Hauz Khas, New Delhi 110016, India. Fax: +91 11 686 2037; E-mail: mathilis@cas.iitd.ernet.in.
0006-355X/01/$8.00 2001 – IOS Press. All rights reserved
416 M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes
Nair et al. [12] used a two-phase model for the blood in modeling transport of oxygen in the arterioles.
They considered a cell-rich core surrounded by a cell-free plasma layer. In the cell-rich core, the radial
hematocrit distribution was expressed as a power law profile with maximum at the center of the tube.
A parabolic velocity profile was taken for the plasma in the cell-free layer. Different velocities were
used for the plasma and RBCs in the cell-rich core. The parabolic profiles were slightly blunted and
differed by a constant called the slip. The thickness of the cell-free layer was chosen on the basis of
geometrical considerations alone in terms of RBC size and radius of the tube. However, the dependence
of the thickness of the cell-free layer on hematocrit was not taken into account.
Seshadri and Jaffrin [25] modeled the outer layer as cell-depleted, having a lower hematocrit than
in the core. The apparent viscosity and the mean tube hematocrit were taken from the measurements
obtained in glass tubes. The concentration of RBCs in the cell-depleted layer was assumed to be 50% of
that in the core. Gupta et al. [11] divided the outer layer into a cell-free plasma layer and cell-depleted
layer. In both these studies, the velocity profile in the core was assumed to follow a power law.
Pries et al. [16,18,19] derived empirical relationships of the relative apparent viscosity and mean tube
hematocrit as parametric functions of tube diameter and discharge hematocrit from in vitro [16,18] and in
vivo [19] data. The authors reported a significantly higher resistance in tubes with diameters below 40 µm
in vivo compared with predictions of the “in vitro viscosity law” for blood flow in smooth glass tubes [16].
Here the terms “in vitro viscosity law” and “in vivo viscosity law” refer to the empirical relationships of
relative viscosity with tube diameter and discharge hematocrit in vitro and in vivo, respectively.
Cokelet [5] commented on the “in vivo viscosity law” proposed by Pries et al. [19]. He pointed out that
the term “viscosity” for in vivo vessels may be misleading and the current values of the coefficients in the
“in vivo viscosity law” may not be correct since they include three effects: (i) those due to measurement
uncertainties in the in vivo data, (ii) limitations of the fluid dynamic relationships used in the mathemat-
ical models of microvascular network, and (iii) poor representation of the vessel as a cylindrical tube,
with idealized blood flow. Finally, it was surmised that the “in vivo viscosity law” proposed by Pries et
al. [19] is probably too simplified and needs more independent variables in it to realistically represent
the hemodynamics.
Pries and Secomb [20] responded to the issues raised by Cokelet [5] on the correctness of the results
and appropriateness of the terminology. They emphasized that (i) the network-based approach has been
valuable in demonstrating the substantial difference between blood flowresistance in vivo and in vitro for
vessels of corresponding diameters, (ii) demonstration of a thick glycocalycal layer on the endothelial
surface and exploration of its chemical and physical nature will have potentially far-reaching implications
for the understanding of microcirculatory physiology, and (iii) further work is required to improve the
“in vivo viscosity law” to be able to predict flow resistance in individual microvessels.
Damiano [6] has presented a semi-empirical model for the blood flow in glycocalyx-lined microvessels
greater than 20 µm in diameter. The model assumes a steady axisymmetric flow of a viscous fluid having
a smoothly varying cross-sectional viscosity distribution throughout the tube. The viscosity is assumed
to decrease monotonically from a maximum in the cell-rich core corresponding to a value for whole
blood obtained from rotational viscometry studies to the value near the tube wall corresponding to blood
plasma. Also, thickness of the cell-depleted region was assumed a constant and the variation in the core
hematocrit with the discharge hematocrit was not considered. Further, this model in glycocalyx-lined
arterioles does not fully explain the high apparent viscosity obtained from the “in vivo viscosity law”
[19].
Most mathematical descriptions of a steady blood flow in a cylindrical tube using a two-phase fluid
model yield three relations for apparent viscosity, tube hematocrit, and core hematocrit based on the
M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes 417
overall mass balance of the RBCs and blood in the tube. These three relations usually involve three
unknowns namely, the viscosity of the outer layer, core hematocrit and the thickness of cell-free plasma
layer. In most models, the viscosity in the outer layer is taken as the viscosity of the plasma with the result
that the system of three relations in two unknowns is overdetermined. The first two relations are sufficient
to determine the core hematocrit and the thickness of the cell-free layer yielding the core hematocrit to
be smaller than the discharge hematocrit in the tubes of diameters above 25 µm. On the other hand, the
first and third relations require that the core hematocrit be larger than the discharge hematocrit [9]. In
order to make the system of three relations consistent, it is necessary to introduce an additional unknown.
The interface between the plasma and the core is not a smooth cylindrical surface as was assumed
in all previous models. The roughness of the surface due to the presence of the red blood cells leads
to an additional energy dissipation that is here modeled as an increase in the effective plasma viscosity
in the cell-free layer. This additional viscous drag will depend on the thickness of the cell-free layer
and the core hematocrit, among other factors. Thus, we hypothesize that the viscosity in the cell-free
layer is different from that of plasma due to the additional dissipation of energy caused by the pro-
trusion of red blood cells into the layer. In developing the model, we introduce the viscosity of the
cell-free layer as an unknown and use the available in vitro experimental data [16] to determine the
flow.
2. Mathematical model
Consider a two layer model (Fig. 1) for the blood flow within a cylindrical tube or vessel of radius
Rconsisting of a central core of radius rhand effective viscosity µc, which contains an erythrocyte
suspension of uniform hematocrit Hc, and a cell-free layer outside the core containing plasma with an
effective viscosity µo. The viscosity µo, in general, will be estimated to be higher than the true plasma
viscosity, µpl. The shear rates are assumed high enough so that the fluid in each region can be regarded as
Newtonian. Thus we consider flow of two immiscible Newtonian fluids, consisting of a cylindrical core
surrounded by a less viscous annular cell-free layer, with viscosity being discontinuous at the interface
between the two regions.
Fig. 1. Schematic diagram of the model.
418 M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes
2.1. Conservation equations and boundary conditions
The equations of motion for incompressible steady fully developed flow in a tube reduces to:
−∂p
∂z +µc
r
∂
∂rr∂uc
∂r =0, 0 r<r
h(1)
for the central core with red blood cells and
−∂p
∂z +µo
r
∂
∂rr∂uo
∂r =0, rh<r<R (2)
for the cell-free layer, where ucand uoare respectively the velocities in the core and the plasma layer in
the axial direction, pis the hydraulic pressure and rand zrepresent the radial and axial directions in the
tube.
The flow is subject to the following boundary conditions:
(a) Due to symmetry, the velocity gradient vanishes along the axis of the tube:
∂uc
∂r =0atr=0.(3a)
(b) No slip condition is assumed at the wall:
uo=0atr=R. (3b)
(c) The velocity and shear stress are continuous at the interface of plasma and the core:
(i) uc|r=rh=uo|r=rh,(3c)
(ii) µc∂uc
∂r
r=rh
=µo∂uo
∂r
r=rh
.(3d)
The solution to Eqs (1) and (2), subject to the boundary conditions (3a)–(3d), is given by:
uc(ξ)=PR
2
4µo1−λ2+µo
µcλ2−ξ2,0ξλ,
uo(ξ)=PR
2
4µo1−ξ2,λξ1,
(4)
where
ξ=r
R,λ=rh
Rand P=∆p
L
in which ∆prepresents the pressure drop along the length Lof the tube.
Velocity in the core can be expressed as:
uc(ξ)=umax1−Bξ2,(5)
M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes 419
where
umax =PR
2
4µo1−λ21−µo
µc,B=µo/µc
1−λ2(1 −µo/µc).(6)
The parameter Bis the bluntness of the velocity profile and represents the deviation from parabolic
flow. When µc→µo,B→1 and the velocity profile becomes parabolic throughout the entire cross
section of the tube.
The volumetric flow rate of the blood is given by:
Q=2πR2λ
0uc(ξ)ξdξ+2πR21
λ
uo(ξ)ξdξ. (7)
Overall mass balance of the cells in the tube is defined by:
QHD=2πR21
0ξu(ξ)h(ξ)dξ(8)
in which HDis the discharge hematocrit and
h(ξ)=Hc,0ξλ,
0, λ<ξ1.(9)
Evaluating the integrals in (7) and (8) with the velocity from Eq. (4), we get:
Q=πPR4
8µoµo
µcλ4+1−λ4, (10)
QHD=πPR4Hc
8µoµo
µcλ4+2λ21−λ2.(11)
Equation (10) can be rewritten as:
Q=πPR4
8µapp ,(12)
where µapp is the apparent total tube flow viscosity given by:
µapp =µoµo
µcλ4+1−λ4−1
.(13)
Notice that as λ→1, µapp →µc, which corresponds to the bulk viscosity of the blood in a large tube
at high shear rates.
The tube hematocrit HTis defined by:
HT=21
0h(ξ)ξdξ, (14)
420 M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes
which yields:
HT=λ2Hc.(15)
Eliminating Qfrom Eqs (10) and (11), we get:
Hc
HD=(µo/µc)λ4+1−λ4
(µo/µc)λ4+1−λ4−(1 −λ2)2.(16)
From (16), the ratio Hc/HD1 if and only if λ1. This shows that the core hematocrit must be
larger than the discharge hematocrit to ensure that the core diameter is less than the tube diameter.
The maximum velocity umax can be expressed in terms of apparent viscosity and average velocity of
the blood (U) as:
umax
U=2µapp
µo1−λ2+µo
µcλ2.(17)
The shear stress at the wall is obtained as:
τw=PR
2=4µappU
R, (18)
i.e., the expression for the shear stress at the wall in this two-phase flow model is the same as that in
homogeneous flow if the bulk viscosity is replaced with the apparent viscosity. Equations (13), (15)
and (16) allow us to express µapp,HTand Hcin terms of λ,HD,µoand µc. In general, the tube diameter,
D, and discharge hematocrit, HD, may be regarded as independent variables, while µapp and HTare
available from experimental data in terms of Dand HD. Furthermore, we show below how an empirical
expression for µccan be derived from µapp. Thus, Eqs (13), (15) and (16) can be used to estimate the
dimensionless core radius, λ, the core hematocrit, Hc, and the effective viscosity, µo, of the outer region.
2.2. Model closure using empirical data
In the model, we would like to include the dependence of apparent viscosity of blood on tube diameter
and hematocrit (Fåhraeus–Lindqvist effect) and the reduction of tube hematocrit relative tothe discharge
or feed hematocrit (Fåhraeus effect). Quantitative description of these effects is required in Eqs (13), (15)
and (16) to estimate λand Hc. Pries et al. [18] have developed a parametric description of the reduction
of the tube hematocrit relative to discharge hematocrit on the basis of broad literature data on human red
blood cells suspension flows through glass tubes with different diameters. Pries et al. [16] have used their
earlier form of HTwith HDin vitro:
HT
HD=HD+(1 −HD)1+1.7e−0.35D−0.6e−0.01D, (19)
where Dis the diameter of the tube in microns. The coefficient in the first exponent was 0.415 in [18]
instead of 0.35.
M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes 421
Pries et al. [16] have derived an empirical equation for the variation of relative apparent blood viscosity
(µrel) with HDand D:
µrel =µapp
µpl =1+(µrel,0.45 −1) (1 −HD)C−1
(1 −0.45)C−1, (20)
where
µrel,0.45 =220e−1.3D+3.2−2.44e−0.06D0.645 , (21)
C=0.8+e−0.0753D−1+1
1+10−11D12 +1
1+10−11D12 .(22)
Equation (20) was developed on the basis of comprehensive analysis of data available in the literature
and their own experimental data obtained in a capillary viscometer. The combined database contains
measurements at high shear rates in tubes with diameters ranging from 3.3 to 1978 µm at hematocrits up
to 90%. The data were corrected for differences in suspending medium viscosity and temperature [16].
We estimate the viscosity in the core as the bulk viscosity from the relations (20)–(22) by letting
D→∞:
µc(Hc)
µpl =1+2.2(1 −Hc)−0.8−1
(1 −0.45)−0.8−1.(23)
In the formulation of a two-phase model for the blood flow, most of the investigators have assumed
the viscosity in the cell-depleted layer to be the same as the plasma viscosity used in the empirical
relation (20). This assumption yields an overdetermined system of three Eqs (13), (15) and (16) with two
unknowns Hcand λ.
If one assumes µo=µpl, then Eqs (13) and (15) are sufficient to determine Hcand λ.However,by
solving Eqs (13) and (15), one finds HcHDfor D22 µmandHc>H
Dfor D<22 µm. This
result is not consistent with the one based on Eq. (16). In some of the studies [27], Eqs (13) and (16)
were used to estimate Hcand λ. Although Eq. (16) ensures that the core hematocrit is larger than the
discharge hematocrit, the system of three Eqs (13), (15) and (16), with two unknowns is overdetermined.
An additional unknown can be introduced to form a consistent system of equations. We have first
attempted to introduce a parameter, s, representing the slip between the plasma and red blood cells in the
core following the approach of Nair et al. [12]. In solving Eqs (13), (15) and (16), we predicted that the
red blood cells are lagging behind the plasma in the core in the tubes with D25 µm whereas the red
blood cells move faster than the plasma in the core in the tubes with D<25 µm. These predictions are
not supported by experimental observations. We also considered another approach where the outer layer
is cell-depleted with hematocrit Hoand viscosity µo. The viscosity µoas a function of Hois calculated
from Eq. (23) by replacing Hcby Ho. In this case, there is a well-defined system of three equations with
three unknowns Hc,Hoand λ. However, the model breaks down when Hoapproaches either Hcor HT.
Thus, we propose a different solution to the problem. Due to the geometrical roughness of the interface
between the core and the cell-free plasma layer and occasional presence of red blood cells near the wall,
the energy dissipation in the plasma layer results in an effective viscosity that may be larger than the
plasma viscosity. This normalized effective viscosity, β=µo/µpl may depend on the thickness of the
422 M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes
cell-free layer and the core hematocrit. We introduce βas an unknown in the model and solve Eqs (13),
(15) and (16) with (20)–(23) and a slightly modified version of (19) for three unknowns Hc,λand β.
Since the equations are nonlinear, they are solved numerically using an iterative technique.
3. Results and discussion
Using Eq. (19) for tube hematocrit as a function of discharge hematocrit and tube diameter, we found
that the value of βbecomes less than one for large values of tube diameter. For example, β<1for
D>250 µmandHD=20%. Equation (19) for HTwith HDwas obtained by Pries et al. [16] by
extrapolating the data for tubes of smaller diameter. To avoid the problem for large tube diameter, we
have refitted a function of HT/HDin terms of HDand Dby lowering the value at one point (HD=20%,
D=595 µm) by <1.5% using nonlinear regression technique (Sigma Plot):
HT
HD=HD+(1 −HD)1+0.3871e−0.1779D−0.603e−0.0111D−0.0187e−9.06.10−11D.(24)
Figure 2 shows that there is a close agreement between the curves computed from Eq. (19) and the
modified function (24) for HD=20%, 45% and 60%.
Bugliarello and Sevilla [3] measured the cell velocity distribution in fine glass tubes under steady flow
conditions. We have taken data for the dimensionless thickness of cell-free layer for HD=10%, 20% and
40% from their Figs 4–6. From these figures, we have also calculated the mean and standard deviation
for the thickness in each case and they are shown with the error bars in Fig. 3. Also, shown in the figure
is thickness based on the measurements from Reinke et al. [21] for HD=45%. The dimensionless
thickness of the cell-free layer computed from the model is in agreement with the available observations.
Fig. 2. Ratio HT/HDas a function of tube diameter and discharge hematocrit. —, HD=60%; - -, HD=45%; ···,HD=20%.
The curves are obtained from the modified function (Eq. (24)). Symbols ◦indicate the values obtained from Eq. (19) proposed
by Pries et al. [16].
M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes 423
Fig. 3. Dimensionless thickness of the cell-free layer as a function of tube diameter and discharge hematocrit. Curves represent
model results. Experimental points are taken from the studies of Bugliarello and Sevilla [3] and Reinke et al. [21]. Error bars
correspond to standard deviation calculated from Bugliarello and Sevilla [3].
Fig. 4. Variation of plasma layer thickness in microns with diameter (D) in microns and discharge hematocrit (HD).
—, HD=60%; - -, HD=45%; ···,HD=20%.
The dimensionless thickness of the cell-free layer decreases as the diameter of the tube or discharge
hematocrit increases.
These results can be recast to show dimensional thickness. For HD=45%, dimensional thickness
of the cell-free layer increases from 2.8 to 4.3 µm when the tube diameter increases from 20 to about
70 µm (Fig. 4). A further increase in tube diameter decreases thickness of the cell-free layer. This trend
of dimensional thickness of cell-free layer with diameter is consistent with that obtained from the earlier
models [11]. As expected, thickness of the cell-free layer decreases as hematocrit increases. The peak
value of thickness reduces from 6.8 to 3 µm as the hematocrit is increased from 20 to 60%. The in-
creasing trend in dimensional thickness of the cell-free layer with tube diameter for HD=45%, 30%,
424 M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes
Fig. 5. Variation of relative viscosity of the cell-free layer (β) with diameter (D) in microns and discharge hematocrit (HD).
—, HD=60%; - -, HD=45%; ···,HD=20%.
Fig. 6. Ratio of viscous dissipation β=ε/εo, with irregular (rectangular or sinusoidal) surface to that with the flat surface
versus dimensionless thickness of the cell-free layer, rr/a =0.2. —, rectangular; - -, sinusoidal.
16% computed from the model is found to be consistent qualitatively with the measurements [26] in
microvessels of diameters less than 40 µm.
Figure 5 reveals that in a tube of fixed diameter, the effective viscosity in the outer layer relative to that
of plasma, β, increases as the hematocrit increases. This is understandable because reduction in thickness
of cell-free layer and an increase in dissipation of energy due to increase in red blood cell concentration
will cause an increase in the viscosity of the outer layer. The computed values of βare between 1 and 2.
The value β=1 means that the effective viscosity of the cell-free layer is the same as that of plasma,
thus there is no additional dissipation of energy in the cell-free layer due to the core. The parameter β
first increases with D, attains a maximum value (1.46 at D∼
=70 µmforHD=45%) beyond which it
M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes 425
decreases as Dincreases. This trend is similar to that of thickness of the cell-free layer in microns with
D(Fig. 4).
To assess the magnitude for additional energy dissipation in a fluid layer confined between a smooth
and a rough surface (representing the tube wall and the plasma–RBC core, respectively), we consider
steady Stokes flow of a viscous fluid in a two-dimensional channel. The upper surface is a flat plate and
the lower surface has rectangular or sinusoidal protrusions (Fig. 9). The upper surface is moving with a
constant velocity. We calculate the viscous energy dissipation, ε, in the fluid layer and compare it to the
energy dissipation εoin a flat layer of the same average width. The ratio ε/εois used to compare with
the magnitude of β=µo/µpl. The details of the computations are described in the Appendix.
The ratio β=ε/εois greater than one (Fig. 6) and its range is in qualitative agreement with the
range of βobtained in the present model. However, the model predicts that the relative viscosity near the
wall decreases monotonically as the thickness of the wall-layer increases, whereas Figs 4 and 5 indicate
that there are two possible values of βfor a given thickness of the cell-free layer. Thus, the simple
calculations described in the Appendix should only be considered as qualitative; a quantitative analysis
awaits a detailed computational study of the dynamics of deformable red cells suspended in a Newtonian
viscous fluid.
The parameter Bin Eq. (6) is the bluntness of the velocity profile in the core and 1 −Brepresents
the deviation from parabolic flow. This parameter depends on the thickness of the cell-free layer and
viscosities of the core and the outer layer. B=0 corresponds to plug flow. The velocity profile becomes
parabolic when B→1, i.e., viscosities in the core and the plasma layer are the same and equal to
the bulk viscosity. Figure 7 reveals that the bluntness parameter Bincreases as the diameter of the
tube increases. In other words, the deviation from the parabolic profile decreases as the tube diameter
increases. For HD=45%, the deviation from the parabolic profile is increased from 0.04 to 0.32 when
the tube diameter is reduced from 295 to 55 µm. Further, for D=100 µm, the bluntness parameter is
increased from 0.81 to 0.90 when the hematocrit is reduced from 60 to 20%. Thus, the velocity profile
becomes more parabolic when the tube diameter is increased or hematocrit is decreased. A number of
experimental studies in vitro [3] and in vivo [8,10,13,22] have reported that velocity profiles were blunted
and became more parabolic with increased tube diameter or reduced hematocrit. Pittman and Ellsworth
Fig. 7. Variation of bluntness parameter (B) with diameter (D) in microns and discharge hematocrit (HD). —, HD=60%;
--,HD=45%; ···,HD=20%.
426 M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes
Fig. 8. Variation of the ratio of core hematocrit to discharge hematocrit with diameter (D) in microns and discharge hematocrit
(HD). —, HD=60%; - -, HD=45%; ···,HD=0%.
Fig. 9. Simulation domain for computing dissipation. Lower surface is a series of rectangular (—) or sinusoidal (- -) blocks.
[14] used the dual-slit method to determine velocity profiles in arterioles and venules (30–140 µm) of the
hamster retractor muscle. They reported the value of Bbetween 0.18 and 0.97 with the hematocrit in the
range 35–40% [8]. From the present model, we find the value of Bbetween 0.49 and 0.9. The velocity
profiles obtained by Ellsworth and Pittman [8] were significantly more blunt as compared to a parabolic
profile, possibly a result of the averaging effect of the method of determining velocity [2,14].
Figure 8 indicates that the core hematocrit is larger than the discharge hematocrit for all D.Thisis
consistent with the relation obtained from (16). For a fixed HD, core hematocrit decreases as the tube
diameter increases and approaches the discharge hematocrit.
4. Conclusions
A two-phase model for the flow of blood in the narrow tubes is described. The model consists of a
central core of suspended erythrocytes and a cell-free layer surrounding the core. It is assumed that the
viscosity in the cell-free layer may differ from that of plasma as a result of dissipation of energy near the
wall from the core due to the roughness of the surface between the core and the cell-free plasma layer. A
consistent system of nonlinear equations is solved numerically to estimate β,λand Hc. As a first step,
we have taken the variation of apparent viscosity and tube hematocrit with the tube diameter and the
discharge hematocrit in vitro from Pries et al. [16].
The thickness of the cell-free layer computed from the model is in good agreement with experimental
observations [3,21]. Sensitivity analysis has been carried out to study the trend of the parameters 1 −λ,
β,B,Hcand µapp with the variation of Dand HD. The proposed model is physically consistent and
works for the tubes of diameter in the range 20 D300 µm.
M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes 427
For smaller tubes the dimension of the red blood cell becomes comparable to the tube diameter, thus
the continuum approach breaks down. Discrete models have been developed for multiple rigid particles
[24] in a circular tube, and for red blood cells – like particles in shear flow [7,15]. Models for discrete
red blood cells in capillary size tubes where the cells move in a single file are well developed [23]. At
present, theoretical description of blood flow in tubes of 10–20 µm diameter remains unresolved.
Acknowledgments
The authors thank Dr. Arjun Vadapalli for computing the dissipation in the two-dimensional channel
using the finite element package PDE-Flex. Supported in part by NIH grants HL 52684 and HL 18242.
The authors would like to thank the reviewers for their valuable comments.
Appendix
In the present study, we have considered the blood as a two-phase fluid consisting of a central core
of erythrocytes and a cell-free or cell-depleted layer outside the core. The rough surface between the
core and the cell-free layer and occasional presence of red blood cell near the wall lead to an additional
energy dissipation which could be represented as an increase in the effective viscosity in the outer layer.
Toassess the magnitude for additional energy dissipation in a fluid layer confined between a smooth and
a rough surfaces, we consider steady Stokes flow of a viscous fluid in a two-dimensional channel. The
upper surface is a flat plate and the lower surface has rectangular or sinusoidal protrusions (Fig. 9). The
upper surface is moving with a constant velocity.
The equations for Stokes flow in a two-dimensional channel are:
∂u
∂x +∂v
∂y =0,
−∂p
∂x +µpl∂2u
∂x2+∂2u
∂y2=0,
−∂p
∂y +µpl∂2v
∂x2+∂2v
∂y2=0,
where uand vare the components of the velocity along xand ydirections, pis the hydraulic pressure
and µpl is the viscosity of the fluid.
Viscous dissipation is defined as:
ε=µpl 2∂u
∂x2
+∂v
∂y2+∂v
∂x +∂u
∂y2dxdy.
These equations are solved using the finite element package PDE-Flex (PDE Solutions Inc., Fremont,
CA). The tolerance is taken as 10−7. The no-slip condition is imposed at the lower surface and the peri-
odic boundary conditions are taken at both ends of the tube. We calculate the viscous energy dissipation,
ε, in the fluid layer and compare it to the energy dissipation εoin a flat layer of the same average width.
428 M. Sharan and A.S. Popel / A two-phase model for flow of blood in narrow tubes
Figure 6 shows that the ratio ε/εodecreases as the thickness of the cell-free layer increases. The ratio
is greater than one implying the additional dissipation of energy due to the surface roughness.
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