4.1. DoS and Spreading Degree Map
Figure 4 shows the calculated spreading degree maps for different channel heights and particle diameters. Local minima of the DoS, marked with green dots, indicate the positions at which the stream of particles experiences a local tightening; positions preceded and followed by regions with a lesser amount of focusing. In most of the cases, two local minima are found for the degree of spreading along the focused streak of particles at a given flow rate. The positions of the DoS’s local minima vary with the applied flow rate giving rise to stability valleys across the spreading degree map surface indicating the positions where the best local focusing conditions are achieved for a given flow rate. The general trend that can be inferred from the spreading degree maps is that the small curve region outputs smaller degree of spreading coefficients than those produced in the big curve, i.e., the DoS at the inlet (left hand side of the map) is always bigger than that at the outlet (right hand side), implying that the stream of particles defocus at the big turn when its action is considered globally.
The two vertical black dashed lines in the maps indicate the position at which the pseudo trajectory crosses the geometric boundary between the small and the big turns. The fact that these are not completely straight vertical lines is explained by the fact that the pseudo trajectory shape, as well as its relative position with respect to the walls of the channel, changes with the flow rate. These lines are useful to determine which regions of the spreading degree map correspond to the small and big turn regions. In some cases, the local minima (green dots) tend to first develop and remain inside the small curve region up to a given flow rate, beyond which the right local minimum enters the big curve of the serpentine. The left local minima are always contained within the small curve regardless of the particle size and the applied flow rate.
When comparing the focusing behavior of the same particle size for two different channel heights, we see how focusing for
and
particles is achieved faster in the
high channels (especially in the case of
particles) as the flow rate is increased (
Figure 4b-e). The spreading degree maps for the
particle shows how, for the
channel, particles are completely focused at around
whereas focusing occurs at
for the
channel. Still comparing the focusing behavior for a fixed particle size (
or
particles), we see how the DoS coefficient happens to reach smaller values (streak of particles more confined) in the
channel once focusing conditions are fully achieved, something that appears to be in concordance with its bigger particle confinement ratio value (
and
against
and
).
It should be noted that, for the
particles case, focusing takes place for particle Reynolds values below
in the
high channel (
, which is not surprising given that other authors reported the focusing of particles for
as well [
36,
39]. More remarkable is probably the fact that for the
channel
, yet full focusing is not quite achieved at this flow rate (
Figure 4e). This suggests that state diagrams, the majority of which are expressed in terms of the channel aspect ratio, represent a futile effort in trying to characterize inertial focusing conditions when these are used as a criterion to find a threshold to reach inertial focusing regime. Nonetheless, these results suggest that, while focusing happens at lower flow rates in the
channel, this comes at the expense of a less defined streak than the one obtained in the
channel. The flow field distribution in the lower aspect ratio channel (
,
) appears to have a greater influence in keeping particles (specially
particles) entrained in the Dean vortices at flow rates for which these same particles are focused in the
high channel. Likewise, as the flow rate is increased above the focusing threshold, bigger particle confinement ratios for a given particle size, guarantee a smaller DoS coefficient. Unsurprisingly,
particles achieve smaller DoS values (better focusing conditions) than
particles for a given channel height. This is mainly due to the strong dependence of the inertial lift force with the particle diameter (
).
The fact that
and
particles reach focusing conditions faster in the
channel for a given particle size seems to contradict the inertial force ratio argument,
, given its inverse relationship with the hydraulic diameter cubed,
[
36], where
is the biggest radius of curvature of the system,
is the particle’s diameter and
is a coefficient that depends on the channel Reynolds number, the normalized position of the particle in the section of the channel and the aspect ratio of the channel [
15,
35]. We are assuming here a small variation for the
coefficient between
and
with respect to the quotient
; variations of the normalized equilibrium positions and aspect ratios between each channel height are considered small whereas
is the same or varies very little between channel sizes at the applied flow rates. Additionally,
is assumed to vary significantly only beyond a given flow rate which, in the case of a straight channel, happens to lay somewhere around a corresponding channel Reynolds number of
[
14]. In our case the flow rate at this
is around
for both the
and the
high channels. If
magnitude is to be taken as a threshold for inertial focusing to occur, it would be expected that particles would focus faster (as the flow rate increases) in the
channel, again, assuming that the coefficient value
does not change significantly between each channel height and that it remains approximately constant below
. Another aspect in which these results reveal an inconsistency with the current theory is in the fact that focusing of
particles takes place at smaller values than the set threshold of the inertial force ratio,
[
36],
.
4.2. Pathlines through Vortices’ Centerlines
As already pointed out in our previous study [
49], a joint analysis of fluid dynamics simulations and inertial focusing experimental results, reveals a strong tendency of the focused stream of particles to translate through Dean vortices’ centerlines. This behavior is maintained for a wide range of flow rates as the streak of particles moves towards the outer wall of the small turn following the vortices’ centerlines as the Dean number (flow rate) increased.
Figure 5 illustrates, from a zenithal view, how the fluorescence streaks of particles along the small curve approximately coincide with the vortices’ centerlines in the fluid dynamics (CFD) simulation—a computer algorithm that solves Navier–Stokes equations—as the flow rate varied and the central region of the vortices evolve in shape accordingly. The fluid-structure interaction (FSI) simulation, which incorporates the effects of the perturbed flow field through its interaction with the particle —coupled solving of Navier–Stokes and structural mechanics equations—shows how the
particle is contained at all times at the center of the upper vortex as it moves along the small curve (
Figure 6). These simulations were carried out in COMSOL (version 5.0, COMSOL Inc., Stockholm, Sweden).
The lateral displacement of a focused streak of particles in curved channels has already been reported in previous studies as either the flow rate is varied or the curvature of the channels changes along the pathlines as in a spiral [
21,
37,
50]. It is known that in our situation, when the focused streak is located in the central region of a curved channel, far from the lateral walls, said streak is in fact composed by two trains of particles mirrored in
, as opposed to the single streak that normally develops in the nearby region of the inner wall [
21,
37,
38,
50]. This situation implies that particles cannot occupy the half-height region of the channel, where Dean flows are stronger [
37,
51,
52,
53]. Instead, it is assumed that they will be located at regions with small Dean flows, specially as the flow rate is increased [
38]. In all studied cases in the literature, however, the idea that Dean drag/shear gradient lift force balance is a determinant factor is an extensively employed argument when trying to explain an unify all the physical processes that describe the focusing conditions of particles as the parameters of the problem change. This fact implies that, while working under inertial focusing conditions, particles being held in equilibrium in a stable trajectory inside curved channels are under the influence of secondary flows so that a net force (Dean drag) counteracts inertial lift forces.
The emergence of two stable trajectories, mirrored in
, in the
high serpentine [
49] supports the simulation results since each trajectory can be assigned to the upper and lower counter-rotating Dean vortices’ center. This fact suggests that particles in curved channels, focused under inertial focusing mechanisms in the exposed conditions, are partially shielded against the influence of Dean drag because they translate through regions with small
net local lateral flows, the centerlines of Dean vortices [
49,
53]. A direct implication of the prior statement is that the influence of Dean flow and its contribution to the modification of the focusing positions is minimal; the vortices’ centers play a decisive role in the particles’ pathline, whose development is not simply determined by the balance of two forces (shear gradient-induced lift and Dean drag) acting simultaneously on the particles.
The geometry of the channels, with its corresponding Dean vortices centerlines, will dictate the approximate trajectory of the resulting streaklines of particles as inertial focusing conditions are achieved. The natural question that arises is that if it is just due to coincidence that particles translate through centerlines of the vortices or if it may exist in some physical phenomena which imposes that such behavior must manifest if inertial focusing conditions are to be achieved.
4.3. The Origin of the Dean-Induced Inertial Lift Force
Given that in this approach the Dean drag contribution to inertial focusing is being relativized, the emergence of a force, whose existence has never been postulated, to balance conventional Dean drag and inertial forces is necessary in order to take account for the modified focusing positions in the serpentine with respect to the rectangular section straight channel case. We speculate about the possibility that regular lift forces, that stem from the parabolic velocity profile in the streamwise direction, may have a Dean-induced counterpart in curved channels whose action would explain the apparent coincidence that stable inertial focusing equilibrium trajectories approximately match Dean vortices’ centerlines. The origin of this force, that would act over the particles in superposition with conventional lift forces, could be attributed to the velocity profile of the secondary flows and its finite shear gradient.
Due to the characteristic velocity profile of the Dean flow in a section of the channel, with an inflexion point between the inwards/outwards (upwards/downwards) moving lateral flow, and, by inferring the net sign of the product between the shear rate and its gradient along a radial direction [
17,
41,
54], it is expected that this Dean-induced lift force is always perpendicularly oriented to the main flow and directed towards the center of the vortex (
Figure 7). This way, the vortical structure of the Dean pattern, perturbs the trajectory of the particles not only because of the imparted drag but also because of the inertial effects it induces to the particles. This is a similar effect to the one that takes place in an expansion/contraction channel, in which particles translate through streamlines under the influence of a vortex (and, hence, the shear gradient of its flow profile) that develops in an expanded chamber in the vicinity of the main channel [
55,
56,
57]. Like in the case of wall effect and shear-gradient lift forces, this Dean-induced inertial lift is expected to be stronger in the regions of the serpentine where velocity gradients are stronger.
CFD simulations reveal that under some circumstances, particularly in the small curve at low flow rates, the position of the centerlines of the vortices (represented as isosurfaces in
Figure 5) approximately coincide with the maximum streamwise velocity magnitude streamlines—the vortex centerlines and maximum velocity points overlap—. As the flow rate is increased, the paths of the centerlines and the streamlines start to behave differently. The tendency of the focused particles is, however, to follow the centerlines of the vortices as the flow rate is varied up to a certain limit. Although the shear gradient-induced lift can be seen as the dominant contribution to inertial focusing at low flow rates, i.e., a particle’s path follows the region with the strongest (vertical) shear gradient (similarly to what happens in a low aspect ratio straight channel), it lacks an additional mechanism (which can be provided by the proposed Dean-induced lift force) to explain the seeming affinity of focused particles to remain at the center of the vortices (as shown in the FSI simulation with
particles). In the general case, when the streaklines do not necessarily coincide with the region of maximum vertical shear rate, the action of a Dean-related force must be considered.
In order to quantify the strength of the proposed Dean-induced lift force, named as
, it can be noted that a theoretically deduced formula for the lift forces for small Reynolds number in a two dimensional parabolic shear flow of the form
is
where
is the density of the fluid,
is the maximum velocity of the fluid,
is the particle diameter,
is the hydraulic diameter,
and
are the shear rate and the shear gradient (second derivative) of the velocity profile and
and
are parameters that modulate the strength of the wall effect,
, and shear gradient-induced lift,
, respectively, according to the position of a particle in the section of a channel [
54].
,
and
depend on a parameter
, the normalized position of the particle’s center with respect to the channel, so that its position directly influences the value of
. According to Equation 3 convention, the velocity profile is expressed with respect to a frame of reference whose origin is located at the particle’s center. In addition, by convention, a positive force points towards the direction of increasing
and vice versa.
Equation 3 was derived under the assumption that
and
. When these limitations are removed (by means of numerical calculations for instance), the magnitude of the lift force changes dramatically although the shape of the theoretically deduced formula is preserved [
17]. This will serve us to prove that the magnitude of the proposed lift force, that emerges as a result of the Dean flow profile, is indeed strong enough to be considered during inertial focusing effects when compared to the conventional lift force magnitude caused by the streamwise parabolic velocity profile. We used the same conditions employed during the FSI simulation, for whom inertial focusing is known to occur in the real experiment, i.e., a channel with
section and a flow rate of
. From these,
and
parameters are obtained from the flow profiles and the magnitude of
with respect to
can be estimated.
Figure 8 reveals how the velocity profiles from the opposing inwards and outwards Dean flows can be approximated around the maxima by a parabolic profile, which guarantees a proper adoption of
and
from Equation 3 to obtain an estimate of the Dean-induced lift force magnitude at least at regions away from the vortices centers (where the profile has an inflexion point).
Fitting results for the simulation of the profile from
Figure 8 are shown in
Figure 9. The values of the
coefficients for the streamwise and transverse (Dean flow) profiles, which were evaluated across the channel height, can generally be expressed as
, where
is the normalized particle height in the channel. The values for the
coefficients, which are constant, were
,
and
. The term
makes reference to transverse flows moving to/away from the center of curvature of the small curve. It should be noted that the outwards Dean flow was defined at
, the region bounded by null transverse velocities (
Figure 9b).
By taking the product
, for the wall-effect lift, and
, for the shear-induced lift, the direction (the sign of the resulting force) of the contributing inertial forces can be inferred from
Figure 9. Since
, the sign of
completely determines the direction of
. The direction of
, on the other hand, is determined by the product
(
). For example,
and
(
Figure 9a). Since
,
(pointing towards increasing
) and
(pointing towards decreasing
). The same procedure can be followed to deduce the direction of
and
caused by the streamwise velocity profile and the
inwards-moving Dean flow profile.
The explicit lift formula from Equation 3 and the provided
and
coefficients can be used to obtain the strength of the Dean-induced lift and the conventional lift forces caused by the curvature of the streamwise velocity profile. This study, however, needs to be performed by segments in the fluidic channel. For instance, in the central region, we have that the contributing
and
factors for the Dean velocity profiles are
and
. By making the proper substitutions in Equation 6.3 we found that:
where the values for
and
were obtained from the simulation and the factor
, which is defined in the central region of the channel (
), can be found in the literature [
54]. The Dean-induced lift force in the upper region of the fluidic channel can be defined as
where
in the upper (lower) region of the channel is
and
. The coefficient
for the upper region (
) is
(yellow straight line in the upper region of
Figure 9a).
Equations 4 and 5 together with
Figure 10 reveal how the proposed Dean-induced lift has a comparable order of magnitude than that of the shear-induced and wall-effect lift forces therefore representing an important contribution to inertial migration effects. According to Equation 3, it may appear that no Dean-induced effects can be as strong as those produced by conventional lift due to their dependence with the maximum velocity of the flow squared (
). As demonstrated, a thorough study of the involved velocity gradients and second derivatives, should be considered as well. As mentioned earlier, the magnitude of the lift forces is expected to be much smaller than the one predicted theoretically when finite particle sizes and
are considered. This effect, however, will be noticeable for both, the Dean-induced and the conventional lift, so a direct comparison between the magnitudes of both forces is still reasonable.
4.4. Focusing Mechanism and Early Effects of Dean-Induced Inertial Lift Force
In order to improve our understanding of inertial focusing mechanisms in an asymmetric serpentine by means of Dean drag and inertial forces, a descriptive approach, supported by simulations and experimental observations, about the initial stages of focusing at the beginning of the serpentine is provided in this section.
When focusing conditions are achieved (minimum flow rate and correct particle size/channel geometry ratio), particles entering the serpentine at random positions are subjected to both, the influence of shear gradient-induced lift and Dean drag. Besides the effects produced by these two forces, an additional effect caused by this hypothetical Dean-induced lift force would be the gradual inward displacement of particles to the center of either the upper or lower vortices. Like in the case of the conventional shear gradient lift, this force is entirely gradient-induced in nature.
The migration of particles moving in the central region of a given curve (between the upper and lower vortices centers) is strengthen by the addition of this Dean-induced lift force with the conventional shear gradient lift since they both push particles away from the central region. For particles located between the walls and a vortex center, however, the action of this force is different since the shear due to the Dean in those regions forces particles to move away from the walls. Although its effect in this situation is analogous, Dean-induced lift is not expected to be short ranged like the wall-effect lift since the former is entirely related to the geometry of the Dean flow profile in the near-wall region.
A possible explanation of how particles are gradually focused to the final focused trajectory is that, as unfocused particles pass through the succession of big and small turns, they are alternatively displaced axially (counterclockwise-clockwise) under the influence of the drag from the different Dean flow patterns that the particles encounter along the serpentine [
53]. The effect of this vortex center-directed Dean-induced inertial force is, however, additive at each successive curve since it periodically pushes particles to the center of the vortices no matter of its rotation direction as they encounter the Dean vortices at each big-small curves. Particles will remain in a stationary trajectory once they reach a focusing position for which a balance between the different involved forces is achieved, a position that happens to be near the Dean vortices centerlines. Due to its Dean flow direct relationship, Dean-induced inertial lift effect will be minimal in those regions where Dean flow gradients are small (mainly at the vorticity transition regions) and maximal in those regions where they are stronger (halfway through the small curves).
The mechanisms by which vortex center-directed migration effects are attributed to this Dean-induced inertial force may be used to explain why inertial focusing can also occur in symmetric serpentines [
1,
47,
48,
53] since no
net Dean drag effect—the product of the joint effect of the flow pattern in the small and the big curves—is required to reach inertial focusing conditions more efficiently [
36]. Inertial focusing in symmetric serpentines generally produce two (as seen from above) focused streaks. The selective focusing to a single streak in an asymmetric serpentine is essentially taking place in the small curve as it is being favored by the stronger gradients in the small curve for 1) the streamwise flow (due to the smaller section of the small curve) and 2) the smaller curvature radius (which increases the Dean flow intensities and, hence, the strength of the Dean-induced lift force in the small curve). The use of a symmetric serpentine allows particles to focus at two coexisting stable trajectories, the corresponding to curves to the right and to the left in the serpentine and, for the appropriate flow rate, it allows the focusing of particles into a single trajectory. This is because an increase in the flow rate will eventually led the trajectories (which are initially closer to the inner walls of the left and right curves) to move outwards form the center of curvature until the two stable trajectories merge in a single one [
47,
53].
The existence of an additional Dean-induced inertial lift force may represent an additional argument to explain why focusing of particles occurs faster in curved geometries than in straight channels where no lateral flows are present and the migration rate of particles is exclusively attributed to the shear gradient-induced lift and the wall-effect inertial lift. Although Dean drag may be seen as an ancillary effect that assists particles to sweep the section of the channel to reach their equilibrium position faster [
36], the inertial forces that Dean flow induces to the particles may be seen as an active component that speeds up this process or even makes inertial focusing possible when Dean flows are present because they speed up the translation of particles towards the vortices centerlines.
Early effects of inertial migration and Dean drag can already be noticed in the lateral view of the serpentine close to the inlet. Surprisingly, particles tend to be quickly entrained in the upper and lower vortices, much before the particles are completely focused. This fast separation of particles into two separate populations, suggests that the action of the shear gradient-induced lift in the vertical direction (whose gradient is stronger due to the aspect ratios of the channels) causes particles to be displaced vertically being entrained in the upper and lower vortices emptying the central region of the channel in the vertical direction.
When observing the microfluidic channels azimuthally, the first sign of focusing is a progressive particle clearing at the inner region of the small curves and the corresponding outer region of the big curves (
Figure 11), creating two differentiated regions, one in which there are no particles flowing within and another one in which particles seem to be entrained in a convective motion around the vortices’ centers due to the Dean drag. As particles enter the serpentine from the inlet, their position gradually evolves as they move along the channel until the inner region of the small curve and the outer region of the big curve are no longer accessible by any particle. It would appear that inertial lift forces, working in conjunction with Dean drag, prevent particles from leaving this convective region once they enter inside it after the particles have traveled a short distance in the serpentine. For instance, for the
in height channel and the
particles, the convective region is almost fully developed after the fourth big-small curve doublet when a flow rate of
is applied (
Figure 11).
By inspecting
Figure 11, it appears that the particle-free region formation is particularly fast in the small curve as the streaklines at the inner region of the curve are pushed to their final position. This effect is hard to explain by taking into account Dean drag alone. Dean drag may explain the apparent outward-moving tendency of the streaks but only if particles are located close to the central region of the channel in the first and successive small curves, where Dean flows are directed outwards. This behavior suggests that, indeed, particles are being confined, due to the effect of the described Dean-induced lift, to a region where the influence of the Dean flow drives particles away from the inner wall in the small curve.
CFD simulations reveal how the lateral flow field in the big curve exhibits a marked asymmetric pattern as the width of the outwards moving streamlines in the central region increases from the center of curvature out, while the upper and lower lateral flow fields behavior is the opposite (
Figure 12). Given this lack of symmetry, stable upper and lower trajectories in the big curve (white dots in
Figure 12) appear to be closer to each other than in the small curve. Since the flow velocity is an order of magnitude higher in the small curve, it can be assumed that the stable points for the small curve will be dominant over the stable points of the big curve given the proportionality of lift forces to the maximum velocity squared, the shear and its gradient (Equation (3)). Since the equilibrium trajectory in the small curve is closer to the upper (lower) wall, as particles are gradually driven to the equilibrium points of the small curve, they will be eventually exposed, to a larger extent, to the inner flow of the big curve as they move from the small to the big curve speeding up the process of particle clearing in the outer region of the big curve. In addition, the simulations also reveal how upwards/downwards moving transverse flow is stronger in the inner region of the channel’s section in the small curve (
Figure 12). This fact implies that particles will experience a stronger Dean-induced lift with a horizontal component in the inner region of the small curve, a circumstance that probably contributes to clear this region of particles faster.
In what appears to be a second focusing process, the rest of the fluorescence streaklines in the convective region (red streaklines in
Figure 11) are gradually displaced to the final equilibrium trajectory as particles move through the fluidic serpentine until a single stable streak is observed at the end (blue streakline in
Figure 11). This is, however, a relatively slow process when compared to the convective region formation since, under the studied circumstances, it takes almost the entire serpentine length (24 doublets) to obtain a completely focused streak of particles. The slow translation to the final equilibrium trajectory may be explained by the fact that, at this stage, the dynamics of unfocused particles is mainly governed by the influence of the Dean drag which can either drive particles closer or away from equilibrium positions. As long as particles do not approximate the equilibrium position, they will be entrained inside the Dean flows in the convective region.
In describing the particles’ completely focusing state, we can argue that the slight particle focusing variations observed in the first section of this paper are a direct consequence of the interaction between all the involved forces, including the hypothesized Dean-induced lift force. As particles travel along a given path, the level of confinement will depend on the particularities of the flow at each position in a very complex fashion (
Figure 13). The variation of the Dean flow strength along said path will dictate the variation in the Dean drag and the Dean-induced lift force accordingly. In the same way, changes in the streamwise velocity profile, given the continuous change in channel’s section and curvature, will cause variations in the conventional lift forces. It is worth noticing that, under this new perspective, Dean drag is no longer an active component once the particles are completely focused —as suggested by the force ratio threshold argument (
)— but merely a circumstantial effect during the focusing process in a curved channel; Dean-induced lift ultimately drives particles away from regions where transverse flows are stronger to regions with null net lateral flow and negligible Dean drag.