INTRODUCTION
Many investigators (Chaturani and Kaloni, 1976; Chaturani and Upadhya,
1979; Shukla et al., 1980b; Majhi and Usha, 1984; Chaturani and
Biswas, 1983; Philip and Chandran, 1996) have theoretically studied the
flow of blood through uniform and stenosed tubes and analyzed the influence
of slip velocity or peripheral plasma layer thickness on the flow variables
such as velocity, wall shear stress and flow resistance. In these models,
the peripheral layer thickness and slip velocity are assumed a priori
based upon the experimental observations. Wang and Bassingthwaighte (2003)
have considered a two fluid model for blood flow through a slightly curved
tube by assuming a constant peripheral plasma layer. Using experimental
data, they have calculated curvefitted values of peripheral layer thickness
and the ratio of core viscosity to the peripheral plasma viscosity. It
would be of interest to obtain the analytic expression for them (slip
and peripheral layer thickness) in terms of the measurable flow variables
(flow rates, pressure gradient, etc.).
To understand the flow patterns in stenosed arteries, Young (1968), Macdonald
(1979) and Deshpande et al. (1979) etc., have analyzed the flow of blood
through an arterial stenosis. Lee and Fung (1970) have obtained the numerical
results for the streamlines and distribution of velocity, pressure, vorticity
and the shear stress for different Reynolds number in blood flow through locally
constricted tubes. In these models, the flow of blood is represented by onelayered
model. Bugliarello and sevilla (1970) and Bugliarello and Hayden (1963) have
experimentally observed that when blood flows through narrow tubes there exists
a cell free plasma layer near the wall. In view of their experiments, it is
preferable to represent the flow of blood through narrow tubes by a twolayered
model instead of onelayered model.
Blood Flow experiments (Bugliarello and Hayden, 1963; Bennet, 1967) indicate
the existence of slip at the tube wall. Nubar (1971), Brunn (1975) and
Hyman (1973) have also reported the existence of slip at the blood vessels
wall. Oldroyd (1956) has reviewed the several treatments of slip at the
walls of the capillary tubes.

Fig. 1: 
Geometry of arterial stenosis with plasma layer and
axially variable slip velocity 
In view of theoretical and experimental
observations implying the existence of slip at the wall, it is improper
to ignore the slip in blood flow. It is also noted that in the literature,
there is no direct formula to calculate the slip velocity. It is, therefore,
worthwhile to find a formula for determining the slip velocity at the
wall.
Shukla et al. (1980a, b) have taken twolayered models and analysed
the influence of peripheral plasma viscosity on flow characteristics.
Chaturani and Kaloni (1976), Chaturani and Upadhya (1979) and Ponalagusamy
(1986) have considered the flow of blood represented by a twolayered
model and have obtained the apparent viscosity. In all these models, the
peripheral layer thickness is assumed a priori. It is, therefore, of interest
to obtain an analytic expression for the calculation of peripheral layer
thickness.
The focus of this investigation is to obtain analytical expressions for
slip velocity, peripheral layer thickness and core viscosity in terms
of measurable flow variables. (pressure gradient tube radius, flow rate
etc.).
Formulation of the problem: Consider an axially symmetric, steady,
laminar and fully developed flow of blood through an artery with mild
stenosis as shown in Fig. 1. Here the flow of blood
(incompressible fluid) is represented by a twolayered model (a core of
red blood cell suspension surrounded by a peripheral layer of plasma (Fig.
1)). It is assumed that the fluids of the peripheral layer and core
are Newtonian.
We shall take the cylindrical coordinate system ()
whose origin is located on the vessel (stenosed artery) axis. The consistency
function may
be written as
whereand
μ_{p} are the viscosities of the central core fluid and the
plasma respectively andare
the radii of the central core region and the artery in the stenotic region.
The nondimensional variables are
whereandare
velocity components in the axialand
radial directions,
the
pressure,
is the density, is
the radius of the normal artery, the
onefourth length of the stenoisisthe
average velocity in the normal artery region and is
the maximum height of the stenosis (Fig.1). The quantities
in the peripheral layer and in the central core are denoted by subscripts
p and c, respectively. ‘over a letter denotes the corresponding
dimensional quantity. As per discussion made by Young (1968), the appropriate
equations describing the flow in the case of a mild stenosis (δ_{s}/R<<1),subject
to the additional conditions (a) Re_{p }(δ_{s}/L_{0})<<1,
(b) 2R_{0}/L_{0} ~ o(1), are
for region 0≤r≤R_{1}(Z),
for region R_{1}(z)≤r≤R(z)
where
and
The boundary conditions are
where (= image) is the nondimensional slip velocity (axial)
and τ is the shear stress. It may be remarked that u_{s}
is a function of z. The geometry of the stenosis (nondimensional form)
is given by (Ponalagusamy, 1986),
Where n (≥2) is a parameter determining the shape of the stenosis,
R(z) is the radius of the artery in the stenotic region, L_{0}
is the length of the stenosis, d indicates its location and A is given
by
Here δ_{s} denotes the maximum heights of the stenosis
at
such that the ratio of the stenotic height to the radius of the normal
artery is much less than unity. It is of interest to note that an increase
in the value of n leads to the change of stenosis shape. When n = 2, the
geometry of stenosis becomes symmetri at
Solution: Using boundary conditions (7), the solutions of Eq.
(3) and (5) can be obtained as
where
The flow rate through the peripheral layer Q_{p} is defined as
which, on using Eq.(10), gives
Similarly, the flow rate through core region Q_{c} can be written
as
The total flow rate Q is,
From Eq. (14), we obtain
Integrating Eq. (15) and using the conditions p =
p_{0} at z = 0 and p = p_{1} at z = L (Fig.
1) and simplifying, we get
Where and
ë is the resistance to flow. The wall shear stress τ_{w}
can be defined as (in dimensionless form)
Which, on using Eq. (10) and (15),
gives
ANALYTIC EXPRESSIONS FOR SLIP VELOCITY, CORE VISCOSITY AND
PERIPHERAL LAYER THICKNESS
Btrunn (1975) has indicated that the introduction of a thin solvent layer
near the wall produces the same effect as that of the slip at the wall.
In the case of one layered model (R = R_{1}) with slip at the
wall, the flow rate Q_{1L}and wall shear stress (from
Eq. (14) and (18) can be obtained
as
where and
and
are the density and viscosity of the fluid when the flow is onelayered.
For the twolayered model without slip at the wall (u_{s }= 0),
the flow rate Q_{2L} and the wall shear stress (from
Eq. (14) and (18) can be obtained
as
Where is
the nondimensional peripheral layer thickness which is a function of
axial distance z. Since the two models (onelayered with slip and twolayered
without slip) represent the same phenomena, the flow rates and wall shear
stresses can be equated as
From Eq. (23) or (24), one can
obtain u_{s} as
From Eq. (21), the expression for core viscosity
_{ }can be obtained as
In Eq. (25) and (26), the peripheral
layer thickness is an unknown quantity which can be determined in the
following manner. For a twolayered model without slip at the wall (u_{s}
= 0), the expression for velocity in the core region is obtained from
Eq. (9) as
The centreline velocity U (at r = 0.0) from Eq. (27)
can be obtained as
Elimination of μ from Eq. (21) and
(28) gives
All the quantities on right hand side of Eq. (29)
are measurable experimentally, hence is known. Once we know the value
of δ,and
u_{s }can be calculated from Eq. (25) and (26).
RESULTS AND DISCUSSION
It has been noticed from Eq. (28) that the centerline
velocity U is always less than .
Since only those values of are
of interest which are real and less than or equal to unity, the following
condition can be established from Eq. (29).
And the Eq. (29) reduces to
Since the experimental values of pressure gradient, flow rate and centerline
velocity for flow through an arterial mild stenosis at different crosssections
for various values of stenotic height and shapes and red blood cells concentrations
are not available, the variation of slip velocity, peripheral layer thickness,
the core viscosity with the axial distance cannot be obtained. However,
to show the procedure and to see the accuracy of the method, we have used
the experimental data of flow through a uniform tube. First, we write
Eq. (25), (26) and (31) in the
dimensional form as
Where is
the pressure gradient and is
the peripheral plasma layer thickness in the normal artery region. For
blood with 6% and 40% red blood cell concentration, we have the following
data from Bugliarello and sevilla (1970) and Bugliarello and Hayden (1963).
Table 1: 
Comparison of peripheral layer thickness 

Table 2: 
Core fluid viscosity 

Table 3: 
Slip velocity 

Table 4: 
Wall shear stress 

Table 5: 
and
for resistanceat


For 40 μm Diameter
C = 40% and 
= 3.2 μm and 
25.5°C)
and 
For 66.6 μm Diameter
C = 6%, 
= 12.87 μm and 
25.5°C)
and

Using these value, the peripheral layer thickness is computed (Table
1) for blood flow in 40 and 66.6 μm tube diameter from Eq. (46).
One can easily see from this table that the peripheral layer thickness,
obtained from the present analysis, has a good agreement with the experimental
observation (Bugliarello and sevilla, 1970; Bugliarello and Hayden, 1963),
the error is less than 1%.
With the help of the obtained values of the peripheral layer thickness,
the core viscosity and red blood cell concentration in the core have been
computed (Table 2). It is of interest to note that in
the present analysis, the core viscosity has been obtained by two methods.
The first by calculating from Eq. (33) and the second
by determining red blood cell concentration in the core and then using
concentration versus relative viscosity curve. A comparison of these two
values of the core viscosities shows a reasonably good agreement (difference
between them is 14 and 5% for 40 and 66.6 μm diameter tube respectively)
between them. Substituting the obtained values of peripheral layer thickness
and core viscosities in the Eq. (32), One can get two
values of the slip velocity (Table 3). Agreement between
these two values of slip velocity is reasonably good (difference up to
4%).
The wall shear stress
and flow resistance
have been computed by using the values of the peripheral layer thickness,
obtained in the present analysis and core viscosity, obtained by two methods
(mentioned above). Thus two values of
and are
quite close to each other (difference up to 3% Table 4
and 5).
CONCLUSIONS
A twolayered model of blood flow through a stenosed artery with axially
variable peripheral layer thickness and variable slip velocity at the
wall has been considered. The model consists of a core (red cell suspension)
surrounded by a peripheral plasma layer. Both the fluids (core and peripheral
layer) are assumed to be Newtonian having different viscosities. The analytic
expressions for peripheral layer thickness, core viscosity, slip velocity,
wall shear stress and resistance to flow have been obtained (Eq.
31, 26, 25, 18
and 16). It may be mentioned that we could not analyze
their (peripheral layer thickness, core viscosity, etc.,) variation with
the axial distance in the strenotic region because of the nonavailability
of the required experimental data (pressure gradient and centerline velocity
at difference crosssection of the stenosed arteries for various values
of stenotic heights and different shapes, flow rates and concentrations).
It is, therefore, of interest to conduct such experiments to provide the
required data which, in turn, will help in understanding the flow of blood
through a stenosed artery.
It is of interest to mention that measuring the thickness of peripheral
plasma layer experimentally is difficult because its thickness is not
constant even for the steady flow through uniform tubes, due to the random
motion of the suspended particle (red blood cell); whereas the reliable
values of pressure gradient, plasma viscosity and centerline velocity
can be measured for a given flow rate, tube size and concentration. Therefore,
it is preferable to use these reliable measurements for the computation
of the value of peripheral layer thickness (Eq. 34).
It is worth mentioning that there are two methods to determine the core
viscosity. The first method is by using Eq. (33) which,
in turn, uses the value of the obtained peripheral layer thickness (Eq.
34) and the experimental data for pressure grandient, plasma ciscocity
and the flow rate. The second method is by computing the red blood cell
concentration in the core and then using concentration vs viscosity curve.
It is important to note that the first method is more convenient than
the second. Further, the core viscosities obtained by tow methods differ
by 514%. The values of the apparent viscosity of blood, agreeability,
rigidity and deformability of red cells can be determined by the present
analysis more accurately than the other existing analyses (Das and Seshadri,
1970; Thomas, 1965) because in the present analysis, the core viscosity
is obtained by calculating the actual red cell concentration in the core
which is different from the concentration of whole blood.
The present analysis could also serve as the check for the experimentally
measured rheologic values of blood. It may be mentioned at this stage
that the variation of peripheral layer thickness, core viscosity and slip
velocity with the axial distance in the stenotic region has not been analyzed
due to the nonavailability of the experimental values of pressure gradient
and the centerline velocity at different crosssections of the stenosed
arteries for various values of stenotic heights, flow rates and concentrations.
It would be of interest to conduct such experiments to provide this vital
data which, in turn, could be useful in the understanding of the rheology
of blood. This rheologic information of blood in turn could be exploited
for the development of new diagnostic tools for many diseases such as
myocardial infarction, hypertension, renal, etc. (Dintenfass, 1977).