4.3. Effect of Radius
Cell migration is an essential part of the mechanism behind both the development of the CFL formation and the Fåhraeus effect, as it has been stated in the previous sections. These phenomena are mainly governed by excluded volume effects [
82,
83] and cell-cell interactions or collisions [
84,
85], which are enhanced by aggregation [
86,
87]. Erythrocytes appear in two structural forms of individual cells and aggregated cells, the distribution of which is strongly related to the stress field applied. The dynamic equilibrium shifts toward more individual cells when the applied shear-rate increases, which affects the overall configuration of the velocity field, inducing a distinguishable difference between narrow and wide tubes.
Figure 5 presents the profile of the axial velocity
along the radial position
of the tube, for radius ranging from 10–80 μm for an applied pressure gradient equal to
. Under these conditions, the dynamics of the system in narrow microtubes reveals that the velocity is more plug-like compared to wider ones, which is enhanced by the fact that in small tubes, aggregation is promoted [
86]. The discontinuity in the shear rate between the fluid in the core region and that in the peripheral layer indicates the presence of the CFL. As the radius increases, the applied shear rates increase too and lead to a progressive rouleaux breakdown into individual cells promoting a more parabolic profile, as illustrated in
Figure 5. In large tubes, the migration effects are less intense, and hence the CFL thickness is significantly smaller than that predicted for the narrower microtubes. When the applied shear stress is higher than the yield stress, the rouleaux network is broken, and the blood is free to flow like a liquid. This can also be evaluated by the structure parameter
which constitutes an indicator for the instantaneous state of blood within the tube.
Figure 6 demonstrates this thixotropic variable justifying the previous assertion. In general, blood is predicted to be in a fully structured state near the center of the tube, the extent of which is highly affected by the applied shear-rates. For very narrow tubes, such as those of
where the cross-section is comparable to RBC diameter, we observe two distinct zones with a sharp transition between them. The first zone is the RBC-rich region characterized by a fully structured state, and the second one is the plasma phase with
. It is evident that in this case, aggregation in the core is quite intense as blood does not demonstrate any change from its initial state and remains fully structured at steady-state conditions. The higher the imposed pseudo-shear-rate, the narrower is the plug flow region where blood depicts a fully structured form. At the center of the tube,
is always equal to unity since the shear-rate is zero there. On the contrary, as the distance from the center increases, the shear-rate attains higher values. Consequently, the breakdown term dominates, enforcing an abrupt decrease in blood aggregates. This behavior is apparent in microtubes with wider cross-sections where the aggregation of RBCs is relatively weak. In comparison with the same blood flowing in smaller microtubes,
attains lower values.
An indicator that demonstrates the deviation of the blunted velocity profile from a parabolic one is the parameter
. It characterizes the bluntness of the velocity profile in the core by correlating the average viscosities in the two phases given by
where
and
are the mean viscosity of the core region and the plasma phase, respectively. The mean shear viscosity of blood,
, is determined as
. Particularly, a value of
close to zero indicates that the velocity profile is nearly plug, while the pure parabolic profile is indicated by a
equal to unity, 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
increases as the diameter of the tube increases; in other words, the deviation from the parabolic profile decreases as the tube diameter increases. In the same Figure, we also illustrate the predictions of a two-phase model with a Newtonian representation for both blood and plasma. As it is expected, our predictions depict a considerable deviation from the pure Newtonian modeling, demonstrating a blunter profile for values of radius below
. The two curves converge above
of radius, predicting an almost equal bluntness parameter as the velocity profile reaches the parabolic form. Notably, our model predicts that the bluntness of the velocity profile is increased when
drops from
to
as well as the tube radius reduces from 1
to 1
. Further, for
to
in radius, the parameter
is increased from
to
. For further increase of the radius,
asymptotically reaches unity. Thus, the velocity profile becomes more parabolic when the tube diameter is increased.
Figure 8 and
Figure 9 demonstrate the prediction of the viscoelastic stresses for radius equal to
,
,
, and
with an imposed pressure gradient of
. The normal component of the stress tensor (
Figure 8) depicts the same pattern for all examined cases. As the shear rate gradually increases along the radial position,
progressively increases from zero to a maximum value, which is strongly dependent on the local shear rates. The pick of normal stress component is observed at the blood/plasma interface, followed by an abrupt decrease within the plasma phase. Within the CF Layer, the normal stress is finite but relatively insignificant compared to that in the core region for the imposed pressure gradient. A considerable contribution of normal stress in the plasma phase should occur under extremely high shear rates. However, the viscoelastic behavior must not be underestimated. The only observable differentiation in normal stress predictions for different
is the maximum value at the blood/plasma interface as well as the phase change location, i.e., the CFL thickness
. The maximum value for each case at the blood/plasma interface is
,
,
,
for
,
,
and
μm in radius, respectively. In
Figure 9, we present the spatial variation of the shear component of the viscoelastic stress tensor
for the same rheological conditions. The latter refers to the total shear stress applied to the system, as we assumed a negligible contribution from the solvent, highlighting a linear distribution along the radius of the microtube. Similarly to the normal stress prediction, the magnitude of
increases with
and
as a consequence of the appearance of higher values of shear rates. By comparing the maxima of normal and shear components, we observe that
is lower than
up to a tube radius equal to
. On the contrary, for cross-sections higher than
the normal stress contributes significantly to the total stress and overcomes the contribution of the shear component. Most of the blood constitutive modelling investigations do not present the normal stress prediction, and hence we are not able to make a comparison with other studies. Varchanis et al. [
56] in their work reported a significant contribution of normal stress in simple shear tests and compared their findings with those predicted by the ML-IKH model [
61], which was found to have similar behavior. The presence of normal stresses in suspensions is attributed to the intense interaction between the particles, whereas in plasma to protein stretching. Similar arguments are presented by Mall-Gleissle et al. [
88] for suspensions with viscoelastic matrix fluids.
The condition of whether blood is yielded or unyielded is defined through the von Mises criterion via the parameter
.
Figure 10a presents the spatial variation of
along the radial position
for microtubes of different radii. If the stress components present in the system are large enough to satisfy the von Mises yielding criterion, the quantity
acquires non-zero values indicating that the yield stress has been exceeded and blood is fluidized. On the contrary, a zero value of
indicates unyielded blood, like this depicted in the case with
. Near the center of this tube, the stresses are insignificant, resulting in an unyielded region, the size of which depends on the radius of the tube for the same imposed pressure gradient. Clearly,
in the CFL.
Figure 10b presents the relaxation time
variation along the radial position
. It is obvious that it follows similar dynamics with that depicted by the structure parameter due to our assumption that plastic viscosity depends on the instantaneous state of blood, and thus it is potentially a thixotropic property.
As we have already mentioned, the instantaneous state of the blood is defined through the parameter
. In the current formulation, we have assumed that our model includes a stress-controlled structural parameter in the sense that the thixotropic behavior of blood is controlled by the applied stresses via the von Mises criterion.
Figure 11a depicts the steady-state values of the mean structural parameter
in the core region of the tube, as a function of its radius. We observe a continuous deconstruction of the RBCs aggregates as the radius of the tube increases due to higher shear and extensional stresses (see
Figure 8 and
Figure 9). Across the whole range of the examined radii,
experiences a reduction from
for
to
for
, with a higher decrease up to
, while beyond this point, the average structural parameter continuously seems to approach an asymptote.
Figure 11b shows the variation of the fully structured fraction of the core region, where
is equal to unity. In particular, we use the quantity
, which stands for the percentage of the structured region out of the radius of the microtube. Also, it is used for quantifying the extension of the RBC aggregation. Blood aggregability exhibits an almost continuous reduction as blood flows in larger tubes. Indicatively, in a microtube of radius equal to
, we observe that fully structured material extends to
of
.
follows an abrupt reduction when
for which its corresponding value is
. As the radius increases further to
,
asymptotes to
approximately.
A comparison of our model results against plug flow predictions can be made by invoking the results of Gupta et al. [
89]. They reported the experimental observations of the velocity profiles in microtubes. In particular, they measured the region for which the velocity profile follows a plug pattern.
Figure 11c reports the experimental data of Gupta et al. [
89] along with our predictions for the normalized plug flow radius
as a function of the radius of the microtube. We observe an excellent agreement, from
up to
of radius with an overall deviation of about 4.5%. For tubes with a radius less than
, a non-monotonic behavior can be observed. This is mainly attributed to the variation of CFL with the radius of the tube, the width of which varies in a similar manner as
. At this point, it is necessary to underline the difference of
and
. The former is a feature of the velocity profile and highly correlated with the thickness of the CF Layer, while the latter represents the region where the RBCs are structured in aggregated forms.
In
Figure 11d, we observe the prediction of the steady blood mean relaxation time
as a function of the microtube radius
. It exhibits a continuous reduction as the radius of the tube increases. From
to
the relaxation time experiences a steep decrease from
to
, while for wider radii, the mean value approaches an asymptote. This behavior comes mainly from the fact that we have considered a plastic viscosity that depends on the instantaneous state of blood. Thus, as the radius increases, the microstructure of blood is more disintegrated.
One of the most important quantities in blood flow studies is the Wall Shear Stress
, which is the total shear stress exerted on the microtube’s wall. A proper calculation of WSS has an exceptional role, especially when blood flows in vivo because it is the stress applied on the internal Endothelial Cells (EC) surface. Vascular operations such as biochemical reactions are considerably affected by the WSS as it has been proved to be directly associated with Nitric Oxide (NO) production [
43] and calcium activation in Smooth Muscle Cells (SMC) [
90] by triggering the biochemical reactions that take place in vascular beds, leading to the regulation of vascular tone [
91]. Although in one-dimensional blood flows, the prediction of WSS is not complex, we offer a consistent model for a proper prediction of WSS in more complicated flows.
Figure 12 shows the distribution of ISS, WSS, INS, and WNS for different cross-sections of the arteriole for an imposed mean axial velocity equal to
. As it is expected, the wall shear stress demonstrates a non-linear dependence on arteriole radius. As we have neglected the solvent contribution in the plasma phase, the predicted
is related to the pure viscoelastic contribution of the proteinic phase. For a fixed mean velocity within the tube, the ISS and WSS demonstrate a gradual decrease until an asymptotic behavior is achieved. Our model predicts a significant contribution of normal stress for both interfacial and wall locations, which would be crucial for more complex blood flows such as arterial bifurcation or saccular aneurysms where the extensional phenomena are intense enough to promote the development of
. Both shear and normal stress components can be expressed as a function of the tube radius
in
through a non-linear relationship of the form
with
refer to
and
respectively. The corresponding coefficients are presented in
Table 5.
Figure 13 presents the maximum velocity
and interfacial velocity
as a function of the microtube radius. As blood flows under a constant pressure gradient in larger tubes, the axial velocity attains higher values, and hence the predicted maximum value demonstrates a continuous increase (
Figure 13a). However, interfacial velocity does not exhibit a monotonic behavior, as illustrated in
Figure 13b. This is attributed to the reduction of the aggregation effects, which enhance the transition of the velocity profiles from plug to parabolic ones. This behavior can also be justified by the predictions of velocity profiles for the two-phase blood flow in narrow tubes [
27,
46].
Migration effects in microcirculation are more clearly visible through the variation of the CFL thickness prediction and the evaluation of the discharged hematocrit.
Figure 14 demonstrates the steady-state values of
,
and
as a function of the tube radius
for
and
. Apparently, the tendency of the erythrocytes to migrate towards the center of the tube implies an interrelation between the hemodynamical properties such as the CFL thickness, the discharged hematocrit, and the apparent viscosity. As the tube radius decreases from
down to
the apparent viscosity (
) drops due to the CFL formation next to the wall, leading to a decrease in discharged hematocrit. As the location of the CFL interface is coincident with the region of the highest shear rate within the flow, the presence of such a layer can significantly reduce the apparent viscosity. Irrespectively of the pressure drop used in the current simulation, both quantities are affected considerably by the radius
.
Figure 14a demonstrates a non-monotonic behavior of the computed cell-free layer width across the wide range of the examined cross-sections. CFL is determined by a balance between lateral migration and mass diffusion caused by cell–cell interactions. To this end, as the tube diameter decreases below a specific value, red cell migration becomes restricted due to strong interaction between RBCs in the core. This is the reason why we observe the non-monotonic behavior for the microtubes of radius below
. Although our continuum model agrees very well with the empirical observations of Pries et al. [
92], it may cease to be valid once the tube radius becomes comparable to the diameter of the individual RBCs. To validate our prediction, we invoke the results of a related study in which a different constitutive model for the description of blood rheology has been used. In that work, Moyers-Gonzalez and Owens [
46] conducted blood flow simulations regarding the CFL thickness under various hemodynamical conditions. Unfortunately, we are not able to completely compare our model results with those reported in their work because they simulated blood flows with different values of the discharged hematocrit and not with its core counterpart. As we impose a constant value of the core hematocrit in the current work, the only comparable CFL thickness is that predicted for a microtube with a radius of about
. At this radius, the discharged hematocrit predicted by our model is almost equal to 0.2 which is one of the examined values of the discharged hematocrits in [
46]. Our model predicts
while Moyers-Gonzalez and Owens reported a value equal to
, i.e., a discrepancy of only 1.04%. As the radius increases from
to
the CFL thickness follows a monotonic decrease as it is expected. When a radius lies in this range, migration phenomena become inappreciable leading to a more parabolic velocity profile and hence the apparent viscosity resembles that given by the Poiseuille law. These phenomena seem to be quite weak above a radius of
where an asymptote is approached. For a more detailed description of the interplay between the apparent viscosity and the CFL formation, the reader is referred to
Appendix A. Determining the CFL thickness is of significant importance not only for a two phase blood flow simulation but also for other processes. To this end we provide a mathematical expression for the evaluation of
in
as a function of the microtube radius
given by
where
with
are adjustable parameters, which are presented in
Table 6. The radius
is in
.
Figure 14b describes the effective migration through the display of the
as a function of the radius of the microtube, corresponding to a fixed value of
. It is obvious that a constant
does not imply a constnant
for small radii. As it is expected, the increased CFL thickness in a narrower microtube indicates intensification of the migration of the erythrocytes, yielding a considerably lower value of the discharged hematocrit corresponding to a core hematocrit equal to
. Indicatively, from
up to
of radius, we observe a remarkable increase of
from an extremely low value of about
to
respectively. Consequently, it is safe to claim that above a
radius, the migration effects are relatively insignificant when blood flows in vitro.
4.4. Effect of Pressure Gradient
Another significant model parameter is the flow rate or the applied pressure-gradient because it impacts the velocity profile [
93] and, consequently, the instantaneous state of blood microstructure. In the following Figures, we demonstrate the effect of the imposed pressure gradient
on the velocity profile, the viscoelastic stress distribution, and the steady structure parameter
. The simulation refers to a microtube with a radius of
while the pressure gradient ranges from
to
with a constant core hematocrit equal to
.
Figure 15 demonstrates the steady-state profile of the axial velocity
along the radial position
, when blood flows under the aforementioned rheological conditions. As it is expected, the imposed pressure gradient affects the bluntness of the velocity profile, indicating a plug-like flow for and a more parabolic-like behavior for
. However, the variation is limited to profile skewness and the magnitude of the maximum velocity. Particularly, as the bluntness of the profile increases, the axial velocity field acquires higher values. However, the CFL thickness remains nearly constant, as it is implied by the experimental observations of Pries et al. [
73], who argued that migration effects are not significantly affected by the applied shear rates.
To further elucidate the impact of the pressure gradient on the rheological behavior of blood, we present the steady-state values of the structure parameter distribution along the radial position of the tube
(
Figure 16). For all imposed pressure gradients, the blood is initially at rest with
. Beginning from the same state, we distinguish four different responses of the blood regarding its final microstructural configuration. When
and
, blood does not exhibit any change from its initial state, meaning that the stress has not exceeded the yield-stress value to disintegrate the rouleaux. Thus, in these cases, we have two distinct areas, a core region with
, and a plasma phase with
. Further increase of pressure gradient causes a partial deconstruction of rouleaux. Particularly, for a pressure gradient equal to
,
demonstrates a monotonic decrease from
to
. It is obvious that the total stress near the center of the tube does not surpass the blood yield stress, and thus, blood remains in an unstructured state. Interestingly, with an imposition of a pressure gradient of one order of magnitude higher than the previous one, the thixotropic parameter
experiences a steep decrease caused by the higher stress applied. In the vicinity of the blood/plasma interface, where the shear rates are high, the structure parameter approaches an asymptote at a low value. The size of the region where
has an almost constant value is associated with the imposed pressure-gradient. The higher the J, the wider the region where
is maintained at a constant low value. This is more obvious in the case with
, where the microstructure of blood demonstrates a significant collapse. Here, blood has a constant value of about
from
to
. In any case, blood never becomes fully unstructured in the core region, i.e.,
never reaches
, irrespective of the intensity of the imposed pressure gradients.
Figure 17 shows the normal viscoelastic stress distributions for different pressure gradients. An increase in
from
to
causes an increase in stress magnitude in both core and plasma regions. Regarding the RBC-rich central region,
demonstrates a continuous non-linear increase as the distance from the center is increased too. In the plasma phase, we can observe that the developed stress is quite insignificant, but when blood flows at higher velocities, as those depicted by the case with
, our model predicts an observable contribution of normal stress in the plasma phase. In this case, the rheological behavior is quite reasonable as we do not expect a considerable viscoelastic contribution from plasma. However, further increase in
yields a considerable normal stress distribution along the plasma layer, which is comparable to that predicted for the RBCs. This observation is in excellent agreement with the findings in the work of Varchanis et al. [
60]. They predicted a pronounced normal stress, caused by the extension of plasma proteins, especially in high shear rates. This extra elastic contribution to the rheological response of whole blood may have a significant impact on the red blood cell deformation and interaction when flowing in microtubes. The effect of the pressure gradient on the viscoelastic shear stress distribution is presented in
Figure 18. As expected,
implies a linear dependence on the radial position
along the tube as it is the total stress, the magnitude of which is strongly associated with the imposed pressure gradient.
Figure 19a presents the
parameter for different pressure gradients. It reveals that the pressure gradient has a significant impact on the state of blood regarding its yielded or unyielded regions. Since plasma does not demonstrate plasticity,
has a dual role indicating both the unyielded blood and the absence of plasticity in the annulus region. For the lowest imposed
, the predicted behavior implies that
throughout the tube, i.e., unyielded blood. Increasing
to
, results in partial fluidization with an unyielded region up to
of the tube and a yielded region from this point up to the phase change location
. On the contrary, the remaining two cases depict the total fluidization of blood. Although, the transition from the unyielded
to completely yielded blood
is quite steep, under intermediate conditions we would observe situations with partial fluidization. For the cases with
to
, the region with zero
is the plasma layer which is totally yielded demonstrating zero plasticity.
The developed stresses are quite sensitive to pressure gradient imposition, as it is depicted in
Figure 20a,b, which demonstrates the viscoelastic stresses at the blood/plasma interface and on the microtube wall. We can observe the monotonic variation of stress when the pressure gradient is ranging between
to
for a microtube with a radius equal to
and a constant core hematocrit equal to
. From
Figure 20a, we can see that ISS is always lower than the WSS. In
Figure 20b, INS attains higher values than WNS does, as it is expected. The viscoelastic contribution of blood is more significant than that of pure plasma, and therefore, we cannot observe any excess of WNS.
Figure 21 demonstrates the prediction for the interfacial velocity
and the maximum attainable velocity at the center of the tube
under the aforementioned rheological conditions. For low-pressure gradients up to
, the velocity profiles are almost plug and thus the
and the
acquire the same values. As the imposed pressure gradient increases, we observe that the two curves begin to deviate from each other up to
where the velocity profile tends to be parabolic, the predictions for the
and the
are
and
respectively.
In
Figure 22a,b, we illustrate the mean microstructural configuration
and the plug flow radius
, respectively. The flow conditions correspond to a pressure gradient range of 10–
, for microtube with a radius of
with a constant core hematocrit equal to
. Regarding
Figure 22a, we can observe a quite expectable dependence of the mean structural variable on the pressure gradient
. For extremely low-pressure gradients, and hence low flow rates, the blood structure presents no change from the initial fully structured state up to a critical value of about
. From this point onward, the microstructure of blood starts to disintegrate, and
experiences an abrupt and continuous reduction from
to
. From
Figure 22b, we can observe that the prediction of the normalized plug-flow radius
as a function of the imposed pressure gradient is quantitatively similar to that presented for the mean value of
(
Figure 22a). Particularly, we observe two distinct responses of blood regarding the normalized region for which the blood velocity presents a plug profile. Initially, for low-pressure gradients, the velocity profile is plug throughout the core region, and hence the
is constant and about
. As the pressure gradient increases from
to
the velocity profile gradually obtains a more parabolic pattern, and thus the plug flow region is reduced dramatically to
which corresponds to the extreme pressure gradient value of our simulations.