Effect of High Solid Concentration on Friction in a Transitional
and Turbulent Flow of Bioliquid Suspension
ARTUR S. BARTOSIK
Department of Production Engineering,
Kielce University of Technology,
Al. Tysiaclecia P.P. 7, 25-314, Kielce,
POLAND
Abstract: Some suspensions in nature have a complex structure and demonstrate a yield shear stress and a non-
linear relationship between the shear rate and the shear stress. Kaolin clay suspension is such an example in
engineering, whereas in nature it is blood. This study represents an innovative approach to simulate bioliquid
flow, similar to that of blood when the solid concentration is high. The objective of this study is to examine the
influence of high solid concentration of bioliquid, similar to blood, on energy losses and velocity profiles in
turbulent and transitional flow in a narrow tube. Using the analogy between the suspension of kaolin clay and
blood, the physical model and the mathematical model were formulated. The mathematical model comprises
continuity and time-averaged momentum equations, a two-equation turbulence model for low Reynolds
numbers, and a specially developed wall damping function, as such suspensions demonstrate the damping of
turbulence. Experimental data on blood rheology for solid concentrations equal to 43% and 70% by volume,
gathered from the literature, were used to establish a rheological model. The results of the simulations indicated
that an increase of solid concentration in bioliquid suspension from 43% to 70% causes an increase in wall
shear stress to approximately 10% and 6% for transitional and turbulent flow, respectively, and changes in
velocity profiles. Such simulations are important if an inserted stent or a chemical additive to the bioliquid
suspension is considered, as they can influence the shear stress. The results of the simulations are presented in
graphs, discussed, and conclusions are formulated.
Key-Words: Turbulent and transitional flow; suspension similar to blood; damping of turbulence, non-
Newtonian suspension; blood friction.
Received: October 8, 2022. Revised: March 8, 2023. Accepted: April 12, 2022. Published: May 18, 2023.
1 Introduction
Solid particles suspended in a liquid can have
different densities, shapes, degradation during
movement, and a tendency to settle. The mixture of
liquid and fine solid particles with a diameter of a
few microns is named ‘suspension’. It is common
for suspensions to exhibit a nonlinear relationship
between shear stress and shear rate, and this
depends mainly on the concentration of solid
particles in liquid and the properties of both phases,
[1].
An example of a suspension, which exists widely
in nature, is kaolin clay. Kaolin clay includes solid
particles such as kaolinite, quartz, mica, and water
as a carrier liquid, [2]. An example of a bioliquid
suspension is blood. The solid phase of human
blood constitutes cellular elements, usually more
than 30% by volume. There are three types of
cellular elements, for example, red blood cells
(erythrocytes), white blood cells (leukocytes), and
platelets. Erythrocytes play a dominant role, since
they contribute approximately 99% of all cellular
elements, [3], [4], [5]. Erythrocytes have a shape
similar to a disc with a diameter of approximately 7
m and a thickness of approximately 2 m and are
responsible for the transport of oxygen and carbon
dioxide, [6], [7], [8], [9]. Erythrocytes contribute
significantly to blood viscosity and strongly affect
blood rheology, [10]. The carrier liquid of cellular
elements in the blood is named plasma. Plasma is a
Newtonian liquid with a density and viscosity
similar to water because it contains about 90% of
water, [3].
When analysing predictions of bioliquid flow,
such as blood, in an aorta, one can mention the
research of [11], who considered a one-dimensional
numerical model, and, [12], who considered a two-
dimensional numerical model. They used the
Casson rheological model; however, their studies
focused only on laminar flow. Some researchers
have used turbulent models to predict wall shear
stress and energy losses in an aorta. For example,
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[13] used the large eddy simulation (LES) and the
model of the normalised group k-ε (RNG k-ε). The
authors assumed that the suspension is Newtonian
and made simulations only for a moderate
concentration of erythrocytes, [13].
Available studies on transitional and turbulent
blood flow are very rare because the dominant flow
in vessels is laminar. However, during intensive
physical exercises or the deceleration phase of the
blood flow, a transient or turbulent flow can appear,
[14], [15]. No studies were found that dealt with the
influence of a high solid concentration of
erythrocytes on frictional losses in transitional or
turbulent blood flows. Most studies consider
laminar blood flow or treat the blood as a
Newtonian liquid and consider the erythrocyte
concentration below 45% by volume, [16]. The
laminar flow is easier to model compared to a
transitional or turbulent one. Therefore, this study
presents modeling and simulations of a transitional
or turbulent bioliquid flow, similar to blood, with a
high solid concentration. The motivation to use such
a high solid concentration comes from the literature.
Available studies indicated that a high concentration
of erythrocyte count symptoms can include
shortness of breath, fatigue, headaches, or during
intense exercises, [17], [18], [19].
Considering bioliquids, similar to blood
suspensions, and Kaolin clay suspensions, we can
find a similarity between them. For instance, both
suspensions have a similar size and shape of solid
particles and tend to form rouleaux at a low shear
rate, causing an increase in yield shear stress and
viscosity, [20], [21]. When the shear rate increases,
the erythrocyte rouleaux become progressively
disassociated. The term rouleaux (singular: rouleau)
indicates stacks or aggregation of erythrocyte cells
in blood, [22]. In the case of kaolin clay suspension,
such an aggregation is called a 'flocculation', [23],
[24].
Experiments with in vivo blood flow are very
difficult. The wall shear stress can be directly
estimated from phase contrast (PC) in magnetic
resonance (MR) measurements of blood velocity.
However, the exact position of the flow domain
(vessel) is not known because it moves when the
pump (heart) is working. Even a small error in
boundary identification significantly influences the
measurements. For this reason, we still face
difficulties in accessing reliable measurements of
blood friction. Such measurements are essential for
the validation of mathematical models. Several
researchers still work to develop an efficient method
for measuring wall shear stress in blood flow, such
as example, [25], [26], [27], [28]. For example, [28],
proposed the Clauser plot method to estimate the
shear stress of the wall in fully developed turbulent
steady flow using PC and MR. The authors stated
that although their method is valuable for correcting
MR-based wall shear stress extraction in a fully
developed steady turbulent flow, this method does
not apply to in vivo measurements, [28].
With reference to difficulties in measuring wall
shear stress in blood vessels, some researchers
propose specific suspensions containing solid
particles that closely approximate the flow
behaviour of erythrocytes, [29], [30]. As an
example, [31], studied several non-Newtonian
liquids to determine how closely they simulate the
flow behaviour of human blood. The authors
proposed their suspension with solid particles,
which approximates the flow behaviour of blood.
They added an appropriate number of disc-shaped
particles that stimulate red blood cells to the specific
suspension. Using a laser Doppler Anemometer, the
authors noted large differences in laminar velocity
profiles compared to Newtonian fluids, [30].
Despite the recent achievements in the research of
blood rheology, the effect of non-Newtonian blood
viscosity and yield shear stress on the
hydrodynamics of a blood flow, especially in
transient and turbulent flow, is still not well
understood.
This study deals with a bioliquid, whose physical
properties are similar to human blood. Taking into
account a proper rheological model together with
the apparent viscosity concept, continuity, and
RANS equations, and a turbulence model together
with an especially developed wall damping
function, simulations of transitional and turbulent
flow were performed. The objective of this study is
to examine the effect of high solid concentration
and, as a consequence, the yield shear stress on
friction in a transitional and turbulent flow of
bioliquid suspension, the physical properties of
which are similar to human blood. Such studies are
important if mechanical or chemical methods are
considered that could affect the flow rate in
bioliquid.
2 Physical Model
Taking into account the results of the available
studies in the literature, we can observe a similarity
between the shape of erythrocytes and the solid
particles in the clay kaolin suspension, [25]. Taking
into account the turbulent flow of the kaolin clay
suspension, one can find that the friction
predictions, using the RANS equations and the
apparent viscosity concept together with an
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adequate rheological model, are higher compared to
the measurements, [32]. The discrepancy is due to
the fact that the fine dispersive solid phase modifies
the viscous sublayer and the buffer layer, causing
damping of the turbulence at the wall. This
hypothesis was proposed by [33], [34]. Wilson and
Thomas reasoned that in turbulent flow with fine
solid particles, such as kaolin clay suspension, solid
particles are pushed away from a wall, causing
decreasing friction on a tube wall, [33], [34]. Taking
into account the hypothesis of [35], [36] proposed a
new damping function for the wall, including
dimensionless yield shear stress. This function,
together with the RANS equations and the two-
equation turbulence model, successfully predicts the
friction and velocity distribution in kaolin clay
suspensions.
For this study, it was assumed that there exists a
similarity between bioliquid suspension, similar to
blood, and kaolin clay. For an argument for such an
assumption, let us consider solid particles and
carrier liquids of kaolin clay and blood suspensions.
Comparing both suspensions, we can note that
Solid particles in kaolin suspension:
– solid particles are the only components
suspended in water; the shape of solid
particles shows similarity to the shape of
erythrocytes, [30]. Solid particles are
ridged.
Solid particles in blood:
– most solid particles in blood constitute
erythrocytes; 99% of all cellular elements
are erythrocytes, [4], [5]; therefore, it is
assumed that erythrocytes are the only solid
particles in blood; Erythrocytes are flexible.
Carrier liquid in kaolin:
– water is a carrier liquid for solid particles.
Carrier liquid in blood:
– plasma is a carrier liquid for erythrocytes;
however, plasma contains approximately
90% water, [3]; therefore, it was assumed
that water is a carrier liquid for erythrocytes.
It is seen above that the major difference
between both suspensions is the stiffness of the solid
particle since they are flexible in the case of blood
and ridged in the case of kaolin suspension.
In addition to the similarity between kaolin and
bioliquid, it should be noted that experiments
showed that erythrocytes migrate from the wall to
the centre of the blood vessel, leading to a cell-free
layer at the wall of the vessel, [37], [38], as a result
of which there can be a decrease in friction on the
wall of the aorta. This phenomenon is similar to that
in kaolin clay suspension, since Wilson and Thomas
reasoned that in the case of a fine-dispersive
mixture, the decrease of friction occurs on a tube
wall, [33], [34]. The same conclusion with respect
to blood flow was also found by [27]. The authors
used several turbulence models to predict fully or
partially developed turbulent flow in the aorta case
and observed consistently lower values of wall shear
stress from MR compared to CFD results, [27]. For
this reason, the successfully validated mathematical
model for kaolin clay suspension will be used to
predict transitional and turbulent bioliquid flow.
In [39], measurements indicated that the mean
diameters of the femoral artery in male and female
subjects are approximately 9.8 mm and 8.2 mm,
respectively. Therefore, for this study, a tube with a
rigid wall and an inner diameter equal to 8 mm was
arbitrarily chosen. The flow was assumed to be
steady, homogeneous, incompressible, axially
symmetric, and without circumferential swirls.
The physical model assumes that the physical
properties of the bioliquid suspension are similar to
those of human blood. Solid concentration is the
fundamental parameter that determines the physical
properties of the bioliquid. Taking into account the
measurements of [40], who conducted experiments
on human blood for moderate and high
concentrations of erythrocytes, it was decided that
two volumetric solid concentrations equal to 43%
and 70% will be considered. The density of blood is
approximately 1,060 kg/m3 and depends on the
concentration of erythrocytes and the temperature.
Human blood density was measured by several
researchers, such as, for example, [41], [42].
According to the [42], measurements, the blood
density at 37°C for two volumetric concentrations of
erythrocytes is stated in Table 1. The density was
assumed to be constant.
The rheological model for the bioliquid
suspension is a crucial point in a mathematical
model. The researchers applied a macroscopic
approach mainly and used the Couette viscometer to
determine the dependence of the shear rate on the
shear stress, [43], [44], [45], [46]. In [40], the
authors used radially symmetric coaxial cylinders
(Couette) and made measurements of shear stress
versus shear rate for erythrocyte concentrations
equal to 43% and 70% by volume. There are several
rheological models recommended for human blood,
like, for instance: Bingham, Ostwalda–de Waele,
Carreau, Casson, or Herschel-Bulkley. The
Bingham model is too simple because it includes
constant viscosity. Two- and three-parameter
models, such as Casson or Herschel-Bulkley,
describe the dependence of the shear rate on the
shear stress more accurately, [39], [40]. For this
study, the Casson model was arbitrarily chosen, as
some researchers recommend this model, as suitable
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and simpler compared to the Herschel-Bulkley
model, [40], [45], [46].
The Casson rheological model is described by
Equations (1) and (2) as follows, [47]:
for > o (1)
and for ≤ o (2)
Equation (2), however, meets a special situation if,
for some reason, the wall shear stress is below the
yield shear stress. This situation will not be
considered in this study.
Taking into account the concept of apparent
viscosity, [48], one can write.
(3)
Using Equations (1) and (3), the following
equation can be obtained for the apparent
viscosity.
(4)
Finally, the apparent viscosity is calculated as
follows.
(5)
Finally, Equation (5) was used in the
mathematical model. Equation (5) has to satisfy the
condition that WSS > YSS.
Based on the best match of the shear stress
predictions of the Casson model with measurements
performed by [40], for erythrocyte concentrations
equal to 43% and 70% by volume, the following
parameters of the Casson model were obtained,
respectively.
(6)
(7)
The rheological properties of the bioliquid are
collected in Table 1.
Table 1. Rheological and physical properties of
bioliquid for solid phase concentration equal to 43%
and 70% by volume and temperature 370 C.
Particle volumetric
concentration in
bioliquid
Casson model
Bioliquid
density
C
[%]
o,
[Pa]
∞,
[Pa s]
m,
[kg/m3]
43
0.0325
0.0039
1060.00
70
0.0850
0.0047
1066.20
Figure 1a and Figure 1b present the
measurements of [40], and the calculations using the
Casson model, described by Equations (6) and (7)
for two concentrations of erythrocytes (C=43% and
C=70%). It is seen that the calculated wall shear
stress and the apparent viscosity agree well with the
measurements. The Casson model in the form
described by Equations (6) and (7) was applied to
the mathematical model of the transient and
turbulent bioliquid flow.
Fig. 1a: Dependence of the shear rate on the
wall shear stress, [40], experimental data for
human blood with C=43% and C=70%, and
calculations using the Casson model described
by Equations (6) and (7).
Fig. 1b: Dependence of the shear rate on apparent
viscosity, [40], experimental data for human blood
with C=43% and C=70%, and calculations using the
Casson model described by Equations (6) and (7).
The heart of an average person, under normal
conditions, beats about 72 times per minute. During
every heartbeat, the ventricles pump about 70 ml of
blood. This means that the heart pumps about 5
litres of blood per minute. The increase in oxygen
consumption must be met primarily by an increase
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 20 40 60 80 100 120 140
WSS, Pa
dU/dy, 1/s
EXP Wells and Merrill C=43%
Casson Model
EXP Wells and Merrill C=70%
Casson Model
0.000
0.010
0.020
0.030
0.040
0.050
0.060
0.070
0 20 40 60 80 100 120 140
µ
app, Pa s
dU/dy, 1/s
EXP Wells and Merrill C=43%
Casson Model
EXP Wells and Merrill C=70%
Casson Model
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in blood flow, which can even increase five times
during exercise, [49]. For such an extreme flow rate,
it is reasonable to consider transitional and even
turbulent flow, [50]. However, it was arbitrarily
assumed that the maximum Reynolds number
cannot exceed 5,000.
3 Mathematical Model
The mathematical model constitutes continuity and
randomly averaged Navier-Stokes equations
(RANS). As the flow is axially symmetrical, the
cylindrical coordinates (x, r, ) were applied. In this
case, the ox coordinate lies on the symmetry axis of
the aorta, while the or coordinate is the radial
distance from the symmetry axis. According to the
assumption that the bioliquid flow is axially
symmetric and without circumferential swirls, the
velocity components V and W are equal to zero.
Taking into account the assumption stated in the
physical model, the continuity equation can be
described as follows.
(8)
Equation (8) indicates that the flow is fully
developed, which means that the velocity
distribution
in the ox direction does not change
because the density is constant. Therefore, including
Equation (9) in the time-averaged Navier-Stokes
equations, we can write the final form in cylindrical
coordinates as follows.
(9)
The dashed lines in the continuity and momentum
equations represent the time-averaged quantities.
The apparent viscosity in Equation (9) is calculated
using Equation (5), while the component of the
turbulent stress tensor is designated taking into
account the Boussinesque hypothesis, [51], which is
an indirect method and describes the turbulence
using turbulent viscosity (t), which is a scalar
quantity. The component of the turbulent stress
tensor in Equation (9) was expressed as follows.
(10)
In [52], [53], the authors successfully examined
several k- models for low Reynolds numbers for
homogeneous and pseudo-homogeneous
suspensions. Studies have shown that the Launder
and Sharma k- turbulence model for low Reynolds
numbers is one of the first and most widely used
models and has been shown to agree well with
experimental and DNS data for a wide range of
turbulent flow problems, performing better than
other k–ε models, [54], [55], [56], [57]. In addition
to that, the model has great potential to predict non-
Newtonian flows, [58]. Therefore, in [59], a
turbulence model was chosen to calculate the
component of the turbulent stress tensor. Launder
and Sharma performed dimensional analyses with
the assumption that turbulent viscosity (t) depends
on the kinetic energy of turbulence (k) and its
dissipation rate (), and the density of the fluid ().
In [59], the authors proposed the following
expression for turbulent viscosity.
(11)
The function f in Equation (11) is called the
turbulence damping function or the wall damping
function and causes a reduction in turbulent
viscosity if a distance from the wall of a tube
approaches zero. This function was developed
empirically by matching predictions with
measurements, [59]. The function is as follows.
(12)
The turbulent Reynolds number (Ret) in
Equation (12) was developed from a dimensionless
analysis, [59], as follows.
(13)
The equations for the kinetic energy of
turbulence and its dissipation rate were derived from
the Navier-Stokes equations using a time-averaged
procedure, [59], [60]. Taking into account the
assumptions made in the physical model, the final
form of both equations is as follows.
(14)
(15)
It must be emphasised that taking into account
the mathematical model in the form presented above
to simulate suspension flow such as kaolin clay, the
predicted friction is higher than the measurements.
As noted above, the Wilson and Thomas hypothesis
indicates that for such suspensions the fine solid
particles modify the viscous sublayer and the buffer
layer, causing the particles to pull away from the
wall of the tube toward the symmetry axis, [33],
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[34]. Several researchers reached a similar
conclusion on blood flow, [27], [37], [38]. In [27],
authors used several turbulence models to predict
fully or partially developed turbulent flow in an
aorta case and observed consistently lower values of
wall shear stress from MR compared to CFD results.
For this reason, Equation (12), which describes the
wall damping function, was replaced by Equation
(16), proposed by [35], [36].
= 0.09 EXP
(16)
The wall damping function (16) depends on the
yield and wall shear stresses and, together with the
above mathematical model, was validated for
several suspensions in a wide range of solid
concentrations, yield stresses, tube diameters, and
Reynolds number, and provided fairly good
predictions of friction and velocity profiles, [35],
[36], [52], [53]. This function is important at a close
distance from a tube wall and its importance is
negligible in the core region. If the ratio of the yield
stress to the wall shear stress is zero, the damping
function (16) approaches the standard function
defined by Equation (12).
Finally, the mathematical model constitutes
partial differential Equations (9), (14), (15) and
complementary relations (8)-(11), (13), and (16).
Taking into account the bioliquid properties, stated
in Table 1, simulations of transitional and turbulent
flow were performed. The constants presented in
Equations (14) and (15) are the same as those
proposed by Launder and Sharma and are the
following: C1=1.44; C2=1.92; k=1.0; =1.3, [59].
4 Numerical Computations
The mathematical model is dedicated to the
transitional and turbulent flow of suspension,
similar to blood, which exhibits non-Newtonian
characteristics of the yield shear stresses. The set of
equations has four dependent variables, namely, the
velocity component U(r), the static pressure p(x),
the kinetic energy of the turbulence k(r), and its
dissipation rate (r). However, we have only three
partial differential equations, namely (9), (14), and
(15). For this reason, computations are performed
for a priori known ∂p/∂x. In such an approach, the
dependent variables are U(r), k(r), and (r). For
known ∂p/∂x the equation set is closed, that is, the
number of dependent variables is equal to the
number of partial differential equations. For the
assumed value of ∂p/∂x, computations of U(r), k(r),
and (r) are performed based on which the Reynolds
number is calculated. If the Reynolds number is
beyond the assumed range the new value of ∂p/∂x is
used until reaching the estimated range of Re.
The following boundary conditions were applied
to the mathematical model.
– for the wall of the aorta, r=R:
U=0, k=0 and =0; (17)
– for the aorta symmetry axis, r=0:
U/r=0, k/r=0, /r=0; (18)
which means that for r=R there is no sleep for the
dependent variables, while on the symmetry axis
(r=0) axially symmetric conditions were applied to
all the dependent variables.
The set of differential equations was computed
using the final volume method, with an iteration
procedure, [61], and its computer code. The
calculations were carried out for 80 nodal points not
uniformly distributed in the radius of the tube. Most
of the nodal points were located in the vicinity of a
tube wall to ensure the convergence process. The
number of nodal points was set experimentally to
ensure nodally independent computations. The
iteration cycles were repeated until a convergence
criterion, defined by Equation (19), was achieved.
(19)
where is the general dependent variable that
=U, k, , and ‘j’ is the nodal point, and ‘n’ is the
iteration cycle, while is the nodal point ‘j’
after the (n-1) iteration cycle.
5 Results of the Simulations
Numerical simulations were performed using the
mathematical model described by Equations (5),
(8)-(11), (13)-(16) for two sets of bioliquid
concentrations equal to 43% and 70% by volume
and rigid horizontal tube with inner diameter D=8
mm. The rheological and physical properties of the
bioliquid suspension are presented in Table 1.
Because the bioliquid yield shear stress is relatively
low, it was assumed that the Reynolds number is the
same as that for Newtonian liquids and is described
as follows.
(20)
Assuming that there is a transitional flow for
2,300<Re<4,000, and a turbulent one for Re>4,000,
the numerical simulations were performed for
Reynolds numbers in the range of 2,615 to 5,000.
The maximum Reynolds number equal to Re =
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5,000 was arbitrarily chosen as sufficiently high for
humans under intensive exercises.
Taking into account Equation (5) one can
calculate the dependence of WSS on the apparent
viscosity of the bioliquid suspension for two solid
concentrations equal to 43% and 70%; see Figure
2a. It is obvious that the apparent viscosity
decreases as the WSS increases. However, the
decrease rate depends on the solid concentration and
is higher for C=70% than for C=43%. The relative
difference in apparent viscosities is higher by ab.
25% for C=70% compared to C=43%.
Fig. 2a: Dependence of wall shear stress on apparent
viscosity – Eq. (5); Simulations of apparent
viscosity for bioliquid suspensions for C=43% and
C=70%.
Analysing Figure 2a one can see that for
bioliquid suspension with C=43%, the WSS is in the
range from 12 to 27.8 [Pa], while for C=70% it is 16
to 42. This is because in both cases the Reynolds
number is in the range from Re=2,615 to Re=5,000.
To illustrate this, Figure 2b shows the same results
as Figure 2a, but refers to the Reynolds number. In
this case, predictions were made using a
mathematical model for transitional and turbulent
flow.
Fig. 2b: Dependence of Reynolds' number on
apparent viscosity – numerical predictions;
Simulations of apparent viscosity for bioliquid
suspensions for C=43% and C=70%.
Analysing Figures 2a and 2b, one can see that for
bioliquid with C=43% the first point has coordinates
(Re=2,885; WSS=12 [Pa]) and the last one
(Re=4,961, WSS=27.8 [Pa]) while for C=70% it is
(Re=2,615, WSS=16 [Pa]) and (Re=4,964, WSS=42
[Pa]), respectively.
From the balance of forces acting on the
suspension that flows in a horizontal tube with a
constant inner diameter D and length L, we can get
the following.
(21)
The term p/L in Equation (21) represents the term
∂p/∂x in the momentum Equation (9) and it is
named the pressure gradient or the frictional head
loss and shows the energy losses in a flowing
suspension. Equation (21) shows that for constant
tube diameter, the frictional head loss (p/L) and
the wall shear stress (w) are qualitatively the same.
Blood flow plays an important role in oxygen
transport and is highly dependent on friction. Higher
friction causes a lower flow rate for the same
pressure generated by the pump (heart). Therefore,
decreasing frictional head loss in flowing blood is a
great challenge of fluid mechanics and
biomechanics. In reference to the reduction of blood
friction, we recognise two main approaches. The
first one is the mechanical approach, which mainly
deals with increasing vessel diameter, i.e. by
inserting a stent with a higher inner diameter
compared to a natural vessel. The second approach
uses chemical additives in order to decree blood
viscosity and as a consequence increase the flow
rate.
To see the effect of the concentration of solid
particles in bioliquid suspension on wall shear stress
(WSS), proper simulations were performed for
C=43% and C=70%. Figure 3 shows the results of
the predicted WSS versus the flow rate for bioliquid
suspension similar to blood with C=43% and
C=70% and for water at a constant temperature
equal to 20 °C. It is seen that the data for water are
much lower than those for bioliquid.
0.0040
0.0042
0.0044
0.0046
0.0048
0.0050
0.0052
0.0054
0.0056
12 17 22 27 32 37 42
µ
app, Pa s
WSS, Pa
Eq.(5) C=43%
Eq.(5) C=70%
0.0040
0.0042
0.0044
0.0046
0.0048
0.0050
0.0052
0.0054
0.0056
2 500 3 000 3 500 4 000 4 500 5 000
µ
app, Pa s
Reynolds Number
Numer.Pred. C=70%
Numer.Pred. C=43%
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DOI: 10.37394/232013.2023.18.2
Artur S. Bartosik
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Fig. 3: Simulations of the dependence of flow rate
on WSS for bioliquid suspension similar to blood
for C=43% and C=70%, and for water.
The numerical simulations presented in Figure 3
demonstrate that suspension with C=70% (o=0.085
[Pa]) exhibits higher wall shear stress compared to
C=43% (o=0.0325 [Pa]) which is not surprising.
Average relative differences are about 10% and 6%
in the transitional (2,300<Re<4,000) and turbulent-
flow regimes (Re>4,000), respectively. Assuming
that there is a fully turbulent suspension flow for
Re>4,000, it was calculated that for suspension with
C=70% the WSS=30 [Pa] for Re=4,000, while for
C=43% the WSS=26 [Pa].
The friction factor simulations for two different
solid concentrations (two different yield stresses) in
the suspension flow rate range of 5 to 9 [l/min] are
presented in Figure 4a. The results show that for a
constant flow rate, the friction factor is significantly
higher for C=70% than for C=43%. The average
relative difference is about 7% and depends on the
flow rate. Simulations of the friction factor in the
Reynolds number range of 2,615 to 5,000 are
presented in Figure 4b. In Figure 4b, it is seen that
the friction factor is the same for both solid
concentrations. This is consistent with an earlier
assumption that because the YSS is relatively low,
the Reynolds number can be the same as that for a
Newtonian liquid.
Fig. 4a: Dependence of bioliquid flow rate on
friction factor; Simulations of friction factor in
bioliquid suspension for C=43% and C=70%.
Fig. 4b: Dependence of Reynolds' number on
friction factor; Simulations of friction factor in
bioliquid suspension for C=43% and C=70%.
6 Discussion
Experiments with in vivo blood flow are extremely
difficult. Difficulties arise with the decrease in the
dimension of the flow domain even when new
technologies are applied, [62]. The exact position of
the flow domain (vessel) is not known beforehand
because it is moving when the pump (heart) is
working. Even a small error in boundary
identification significantly influences the
measurements. For this reason, we still face
difficulties in accessing reliable measurements of
blood friction. Such measurements are essential for
the validation of mathematical models.
It was mentioned in the physical model that some
researchers observed a similarity between the shape
of erythrocytes and the clay kaolin particles, [30].
Erythrocytes and kaolin particles have similar sizes
and shapes and tend to form rouleaux at a low shear
rate, causing an increase in yield shear stress and
viscosity, [20], [21]. Another similarity is associated
with the tendency of solid particles to migrate from
the wall to the centre of the blood vessel, leading to
10.00
14.00
18.00
22.00
26.00
30.00
34.00
38.00
42.00
4.00 5.00 6.00 7.00 8.00 9.00 10.00
WSS, Pa
QV, l/min
Numer.Pred. C=43%
Numer.Pred. C=70%
Numer.Pred. Water
0.033
0.035
0.037
0.039
0.041
0.043
4.00 5.00 6.00 7.00 8.00 9.00 10.00
l
QV, l/min
Numer.Pred. C=43%
Numer.Pred. C=70%
0.033
0.035
0.037
0.039
0.041
0.043
2500 3000 3500 4000 4500 5000
l
Reynolds Number
Numer.Pred. C=43%
Numer.Pred. C=70%
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Artur S. Bartosik
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Volume 18, 2023
a cell-free layer on the wall of the vessel, [27], [37],
[38], as a result, a decrease in friction can appear on
the wall of the tube, [63]. This phenomenon was
also observed by [33], [34], in the case of kaolin
clay suspension. For this reason, the mathematical
model, which was well examined for the suspension
of kaolin, has been used in this study to predict the
bioliquid flow.
In the literature, no predictions for transitional or
turbulent blood flow with a high concentration of
erythrocytes, like 70%, were found, including the
concept of damping of turbulence. This study
presents the results of simulations of suspensions
similar to those of blood, assuming an analogy
between kaolin clay and blood suspensions. Of
course, there are also some differences since
erythrocytes are deformable, whereas solid particles
of kaolin clay are not. The aorta and other vessels
are flexible, whereas industrial transport of kaolin
clay exists in rigid pipelines. Despite these
differences, it is valuable to examine the influence
of the high solid concentration and as a consequence
the yield shear stress, on energy losses in a
transitional, and turbulent flow of the suspension
similar to blood in a narrow tube.
The crucial point in this study is the range of
flow rates and the range of Reynolds numbers. It is
known that in some circumstances, such as physical
activity, the flow of human blood in an aorta could
be transitional or turbulent. In the analysis of
turbulent flow, it was assumed that the Reynolds
number does not exceed 5,000. The reason for such
an assumption comes from the research of [49],
[50]. The authors noted that increased oxygen
consumption must be met primarily by increased
blood flow, which can increase even five times
during exercise. This means that we could expect
that in some circumstances the blood flow rate may
exceed 25 [l/min], as the typical human blood flow
rate is approximately 5 [l/min], [49]. In this study,
the flow rate was arbitrarily chosen in the range
between 5 and 9 [l/min], which seems reasonable.
The studies of blood flow available in the
literature deal with the concentration of erythrocytes
below 45%, [12], [43], [64]. However, if the
concentration of erythrocytes increases, which is
due to physical activity or disease, the shear stress
of the product also increases. When analysing
Equation (5), it is seen that an increase in the yield
shear stress results in an increase in apparent
viscosity. To analyse this, Equation (5) will be used
to calculate the apparent viscosity for an arbitrarily
chosen range of YSS, that is, from 0=0 to 0=0.10
[Pa], and for solid concentration C=70%, and WSS
equal to w=5, w=15 and w=60 [Pa]. The range of
WSS from 5 to 60 [Pa] was intentionally chosen
according to laminar, transitional, and turbulent
flow, respectively. The results of the calculations
are presented in Figure 5a. When analysing Figure
5a, it can be concluded that the apparent viscosity
increases as YSS increases. Therefore, for the same
density of suspension, tube diameter, and YSS, the
apparent viscosity in laminar flow is higher than in
transitional or turbulent flow. Additionally, a lower
value of WSS results in a higher influence of YSS
on apparent viscosity; see Figure 5a. Finally, it can
be concluded that the importance of the yield shear
stress decreases with increasing Reynolds number.
This is even more pronounced if we take into
account the fact that, in the case of transitional and
turbulent flow, the importance of apparent viscosity
has a secondary meaning, since turbulent viscosity
tends to dominate.
Fig. 5a: Dependence of YSS on apparent viscosity
for the WSS chosen; Importance of YSS in bioliquid
suspension for constant D=0.008m, and C=70%.
Fig. 5b: Dimensionless velocity profiles of bioliquid
suspensions for the chosen YSS and constant flow
rate Qv=7.5 [l/min]; Importance of YSS in bioliquid
suspension for constant D=0.008 [m], and C=70%.
Figure 5b presents the influence of the chosen
YSS on dimensionless velocity profiles in the
turbulent flow of the bioliquid suspension for a
0.0045
0.0047
0.0049
0.0051
0.0053
0.0055
0.0057
0.0059
0.0061
0.0063
0.0065
0.00 0.02 0.04 0.06 0.08 0.10
µ
app, Pa s
YSS, Pa
Numer.Pred for WSS=5 Pa
Numer.Pred. for WSS=15 Pa
Numer.Pred. for WSS=60 Pa
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0.00 0.20 0.40 0.60 0.80 1.00
U/Umax
y/R
Numer.Pred. for YSS=0.00 Pa
Numer.Pred. for YSS=0.20 Pa
Numer.Pred. for YSS=1.00 Pa
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DOI: 10.37394/232013.2023.18.2
Artur S. Bartosik
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Volume 18, 2023
constant flow rate equal to 7.5 [l/min] and for a
constant tube diameter D=8 mm, solid concentration
C = 70% and viscosity ∞=0.0047 [Pa s]. It is seen
that with an increase in YSS, the velocity profiles
change significantly. If the YSS increases the
velocity profiles have a clear tendency to change
shape from flat to parabolical one - see Figure 5b. It
is worth highlighting that the influence of damping
of turbulence, expressed by Equation (16), on the
frictional loss and velocity profile is not substantial
because the YSS compared to the WSS is relatively
small and its importance decreases with increasing
Reynolds number.
7 Conclusions
Human blood flow is extremely complex, and there
is no universal model for its predictions. Based on
simulation made for bioliquid suspensions similar to
blood in transitional or turbulent flow with moderate
and high solid concentrations, using the apparent
viscosity concept, Casson rheological model, and
turbulence model for low Reynolds numbers, and
the wall damping function, the following
conclusions can be formulated.
1. The Casson model is suitable to predict bioliquid
rheology for moderate (C=43%) and high solid
concentrations (C=70%); see Figure 1a and
Figure 1b.
2. If the solid phase concentration increases, the
yield shear stress, the apparent viscosity, and the
wall shear stress also increase; see Figure 1a,
Figure 1b, Figure 2a, Figure 2b, and Figure 3.
3. The increase in solid concentration from C=43%
(o=0.0325 [Pa]) to C=70% (o=0.085 [Pa]),
causes an increase in wall shear stress of
approximately 10% and 6% for transitional and
turbulent flow, respectively, while compared to
water, the differences are substantial; see Figure
3.
4. The influence of the yield shear stress (or solid
concentration) on energy losses cannot be
neglected if a transitional or turbulent flow of the
bioliquid suspension is considered; see Figure 3
and Figure 4a. However, the importance of the
yield shear stress decreases with increasing
Reynolds number; see Figure 5a.
5. For the same flow rate, the friction factor of the
bioliquid suspension with C=70% is higher than
for C=43% and the average relative difference is
approximately 8%.
6. Changes in the yield shear stress result in
changes in the shape of the velocity distribution.
If the YSS increases, the velocity profiles have a
clear tendency to change shape from flat to
parabolic; see Figure 5b.
7. The influence of turbulence damping, expressed
by Equation (16), on frictional loss and the
velocity profile is not substantial because the
YSS compared to the WSS is relatively small
and its importance decreases with increasing
Reynolds number.
The paper contributes to the discussion between
scientists as to whether the rheological properties of
blood are important or not in the prediction of the
shear stress of the blood wall or the velocity
profiles.
When the results of bioliquid flow simulations
are taken into account, it is permissible to conclude
that the frictional losses increase with increasing
solid concentration or YSS. The inclusion of a
proper rheological model to predict laminar
bioliquid flow is essential, while for transitional and
turbulent flow it is less important, as the increase in
wall shear stress for the parameters studied was
approximately 10% and 6%, respectively.
Future studies are needed to better understand
the flowing nature of bioliquid, which is similar to
that of blood; especially, attention is needed on the
process of particle migration from the tube wall to
the symmetry axis and its effect on WSS. However,
this requires reliable measurements on a vessel wall.
Other future studies should focus on a better
understanding of the bioliquid flow near the vessel
wall during the acceleration and deceleration
phases. Also, the influence of temperature on the
rheological properties of bioliquid and flow
structure is very desired.
Nomenclature
C concentration of solid particles by volume,
%
Ci constants in the Launder and Sharma
turbulence model, i=1, 2
f turbulence damping function at the tube
wall
k kinetic energy of turbulence, m2/s2
L tube length, m
p static pressure, Pa
Qv flow rate, l/min
r radial distance from symmetry axis / axial
coordinate, m
R inner tube radius, m
Re Reynolds number
u’, v’ fluctuating components of suspension
velocity in ox and or directions, m/s
U suspension velocity component in x
direction, m/s
V suspension velocity component in r
direction, m/s
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Volume 18, 2023
W suspension velocity component in
direction, m/s
WSS wall shear stress, Pa
x axial coordinate, m
y distance from the tube wall, m
YSS yield shear stress, Pa
Greek symbols
share rate; shear deformation, strain rate, s−1
rate of dissipation of kinetic energy of
turbulence, m2/s3
angle around the symmetry axis of the tube
/ tangential coordinate, deg
friction factor
∞ suspension viscosity, Pa s
suspension density, kg/m3
k effective Prandtl-Schmidt number for k
effective Prandtl-Schmidt number for
0 shear stress / yield shear stress, Pa
general dependent variable, =U, k,
Subscripts:
app apparent viscosity
b bulk (cross section average value)
t turbulent
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WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.2
Artur S. Bartosik
E-ISSN: 2224-347X
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23.
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DOI: 10.37394/232013.2023.18.2
Artur S. Bartosik
E-ISSN: 2224-347X
23
Volume 18, 2023
