Two-Way Coupling FSI Approach to Inertial Focusing Dynamics Under Dean Flow Patterns in Asymmetric

The dynamics of a spherical particle in an asymmetric serpentine are studied by finite element method (FEM) simulations in a physically unconstrained system. The two-way coupled time dependent solutions illustrate the path of the particle along a curve where a secondary flow (Dean flow) has developed. The simulated conditions were adjusted to match those of an experiment for which particles were focused under inertial focusing conditions in a microfluidic device. The obtained rotational modes allowed to infer the influence of the local flow around the particle. We propose a new approach to find the decoupled secondary flow contribution employing a quasi-Stokes flow.


Introduction
Inertial migration effects have long been observed in macroscopic systems in studies regarding the flow of cells through capillaries and rheology experiments of spherical particles' suspensions [1][2][3][4].Saffman later provided the theoretical basis that attributed this apparent migration of particles and cells to the inner region of capillary tubes and blood vessels to non-linearities of the Navier-Stokes (N-S) equations [5].With the advent of microfluidic technologies, inertial migration has acquired significant attention thanks to its associated passive focusing capabilities in devices in which some form of suspended particulate is flowing in a driving fluid [6,7].Microfluidic devices composed of serpentines or spiral channels are particularly suitable for experiments in which efficient conditions for inertial focusing are sought because transverse flows, that spontaneously emerge in curved geometries, assist particles and cells in reaching stable positions faster [8][9][10][11][12][13].Experimental and theoretical developments in the physics of inertial migration have improved our understanding of this phenomena [14][15][16] and evolved towards the study of the underpinning mechanics for real-life conditions with finite Reynolds number (Re) and finite particle size approximations [17][18][19][20].
In parallel to the development of the theory of inertial migration of particles, numerical methods have emerged as a powerful tool to predict inertial forces and particle focusing positions.Aiming at reducing the computational complexity of the simulated dynamics of particles in confined flows, many authors have proposed the use of an iterative procedure according to which the variables of the problem are referred to a moving frame of reference fixed to the moving sphere [21][22][23][24][25].The velocity of the backwards-moving walls (the velocity of the moving particle in the original frame of reference) and the angular velocity of the particle are then iteratively updated until the particle moves force (in the streamwise direction) and torque free.This method allows to find the inertial force distribution over the particle in a section of the channel.The use of such physically constrained system comes at the expense of the need to know beforehand the velocity and spin direction of the analyzed particle, which makes this method unsuitable for complex channel geometries.Liu et al. proposed the use of an explicit formula for the lift force to predict the trajectories of particles in complex geometries using a Lagrangian tracking method [26].The coefficients of this formula, however, need to be determined through computational fluid dynamics (CFD) simulations of N-S equations solved for flows in straight channels.
Given the complexity of simulating particle migration conditions, some authors have proposed the use of lattice Boltzmann methods (LBM) [27][28][29][30] due to its lower memory consumption with respect the more generally used FEM [31].Nonetheless, LBM is prone to cause instability issues in particle-fluid interaction simulations owing to its inherent inefficient representation of the solid boundary interface [32].Additionally, this method relies in analytical solutions [28] that, in some cases, are only valid for very small Reynolds numbers.In order to improve the treatment of particlefluid interactions at the boundary level, a coupling between LBM and the immersed boundary method (IBM) [33] was proposed (IB-LBM) [32].Employing the IB-LBM method, Jiang et at.studied the focusing conditions of spherical particles in a symmetric serpentine [34].The drawback of IB-LBM method is that the velocity field of the fluid and the particle are found by including a body force term into the lattice Boltzmann equation.This body force is in turn deduced from the deformation suffered by the boundary of the solid-fluid interface.Consequently, one must provide the simulation code with an appropriate stiffness factor, not too small in order to prevent flow perturbations caused by the deformation of the particle, but not too big so that no measurable distortion can be obtained preventing the simulation to converge [32].The requirement for a measurable deformation in the IB-LBM method may also be the cause that particles moving close to the center of symmetry of a given channel geometry, remain in those positions even when inertial focusing conditions are reached [34], a situation that is not reflected in the real experiment.
In this paper we propose a new 3d model scheme to simulate the influence of drag and inertial forces of the fluid over a particle with unrestricted degrees of freedom using a two-way coupling fluid-structure interaction (FSI) formulation.FSI models are normally employed in systems in which deformations of a solid structure are expected given their interaction with some fluid [35][36][37][38].Because of computational restrictions, this method is rarely employed in simulations in which one of its components translates a long distance, e.g., a particle moving through a microfluidic channel.The strength of this method, however, is that it provides a solution that fully describes the physics of the process since the particle and the flow are defined as unconstrained components during the simulation.We compared the trajectory of the particle along a curve of the serpentine in the simulation with experimental results obtained with a microfluidic device with the same geometric and flow parameters used in the numerical analysis.The simulation results from the free-rotating particle also allowed to find the angular velocity components of the particle providing a set of variables that are hardly attainable under postulated assumptions alone.Using a provided definition for the transverse flow, we present a phenomenological relationship about the behavior of the rotating particle and its interaction with the Dean and gradients of the flow.We believe that the outcome of this simulation and the provided descriptions, may represent a valuable insight in the behavior of inertially focused particles in serpentines that can be directly incorporated in the form of pre-set variables in less computation-intensive constrained systems.This would allow a substantial time improvement for simulations involving complex microfluidic systems in which some form of transverse flow is present.

Model and Methods
In order to test the suitability of our simulation model, a microfluidic device with the same serpentine geometry was created employing conventional photolithography techniques.In addition to the conventional channels, the PDMS device also incorporated cavities to insert micromachined glass mirrors in the vicinity of the serpentine.The mirrors were oriented at 45° and its field of view covered the entire height of the fluidic channel.This allowed to visualize the two stable trajectories of inertially focused fluorescent polystyrene particles (Figure 1).For the simulation involving only the fluid, which was used to find the solution for the transverse flow, a 3d periodic unit of the serpentine (Figure 2) was used as the computational domain.Periodic flow conditions required to set a pressure difference between a source boundary, where the fluid enters the domain, and a destination group through which the fluid exits the domain.The pressure was set so that a flow rate of 130 µL/min was obtained during the simulation; the same used in the experiment with the fluidic device.We chose this geometry and flow rate for the simulation because the experimental results showed very good focusing conditions for the particles.The mesh also incorporated a finer mesh region inside the small turn in order to improve the simulation results in the region around which Dean vortex centers are expected to develop.Bounded flows with some degree of curvature may facilitate the emergence of a flow field distribution that is perpendicular to the streamwise flow called Dean flow.These flows originate from the difference in velocity at the inner region of the fluidic channel with respect the near-wall regions which in comparison, due to the parabolic velocity profile, tend to be negligible.When the flow rate is sufficiently low, the viscosity between flow elements in a moving mass of fluid suffices to force the streamlines of fluid to follow the curvature of the channel.
We propose a new approach by which the direction and magnitude of Dean flows can be numerically calculated at the pointwise level inside a simulation domain employing a FEM.The flow field is simulated at low and high flow rates and the differences between them are measured pointby-point in order to infer the magnitude and direction of the resulting transverse (Dean) flow.In our calculations, we chose the transverse components of the flow field at a given point to be the velocity field in the z direction (w), and the velocity field components in the x-y plane (u and v for x and y components respectively) projected along a direction perpendicular to the streamlines obtained for the simulation at low flow rates (Figure 3).This method can be applied to any channel geometry provided that the solutions for two different regimes (creeping flow and regular laminar flow) can be obtained.The analytical expression of the transverse flow field is then defined point-by-point and expressed in terms of the variables obtained in a simulation involving a low flow rate (Q(v0 =1 µm/s) ≈ 1.8 × 10 -4 µL/min), u0 and v0, and the variables obtained in the studied flow rate (Q(v0 =0.67 m/s) =130 µL/min ) u, v and w: Dean patterns are commonly represented as flow field projections in a section of a microfluidic channel [26,34,39,40].Such representation is indirectly based on the assumption that the streamlines of an hypothetical creeping flow are perpendicular to said plane (the tangent components to the plane are taken as the Dean flow components).Reference cross sectional planes (based on the geometry of the channel) to represent such flows can yield inaccurate results since, in general, it cannot be guaranteed that the streamlines of a flow, for which inertial effects can be neglected, are indeed perpendicular to the reference plane at any point.This is particularly true for the case of channels with complex geometries such as serpentines or mixers where new techniques have to be applied in order to visualize the complicated flow patterns that emerge [41,42].
Figure 4 depicts the 3d fluid domain employed during the FSI simulation process.Unlike the stationary case, we opted not to use a periodic domain in the simulation involving the interaction between the particle and the flow.The reason is that periodic flow conditions only allow the use of pressure differences between the source and destination boundaries of the domain.The problem of using a pressure difference as a driving source for the flow in a transient simulation is that the pressure difference between the source and the destination boundaries, the pressure needed to obtain the desired flow rate, fixes the time it takes the flow to achieve stationary conditions, the regime that is expected as the particle travels through the region of interest (the small curve).Unfortunately, in our geometry, the time it takes the flow to achieve such conditions is so long that the particle penetrates the region of interest while the flow hasn't yet achieved stationary conditions (stationary flow rate and velocity profiles).Instead, by fixing an inflow velocity field distribution as a boundary condition for the inlet modulated by a smooth time dependent function, we can artificially set the time it takes the fluid domain to achieve a fully developed flow, a time that will be short enough so that the particle enters the small turn long after this condition is achieved.With this new boundary condition, we designed a new domain that comprised a straight channel section, where the flow first develops, followed by a succession of big-small-big turns whose geometric parameters coincide with the microfluidic chip used to focus ⌀ =10 µm particles in the experimental section.Like in the CFD simulation, the flow rate was set at 130 µL/min.The inlet velocity flow profile is approximated to that of a rectangular section channel by the expression: where step(t[1/s]) is a dimensionless smoothed step function included to improve the convergence at the beginning of the time dependent simulation.16  ⁄ is a normalizing factor at the inlet boundary condition so that  ⃗( ⃗, ) =  at the center of the inlet.Similarly, the expression in the numerator ensures that  ⃗( ⃗, ) decays from the center of the channel out down to 0 at the walls in accordance with the no slip wall boundary condition for the flow introduced in the simulation.Notice that in the selected reference frame  ⃗( ⃗, ) has only a y-component at the inlet (Figure 4a).A 0 Pa pressure condition is set at the outlet. is chosen so that, after stationary conditions are achieved, the desired flow rate is obtained in the channel (with our channel section, Q(v0 = 0.74 m/s) ≈ 130 µL/min).The flow rate is monitored by integrating the obtained flow fields across an arbitrary section of the channel.The length of the straight section ensures that the approximated velocity profile will have enough time to be fully developed at its end, before entering the first of the two big turns.Likewise, the big turn was completely included in the computational domain to ensure that the developed flow at the entry of the small turn was as similar as possible to the periodic case.
The mesh of the fluid domain was composed by two differentiated regions with two different mesh sizes (Figure 4b).The finer mesh region was restricted to the upper half of the channel.It was devoted to act as an "envelope" domain for a particle in the upper stable trajectory; it was placed in such a way that, during the simulations, the particle would remain inside this domain at all times.The finer mesh allows the solver to calculate the variables of the problem more precisely since it can resolve the gradients of pressure and velocity in the vicinity of the sphere more reliably.The domain also included a 10 µm in diameter spherical mesh (Figure 4c), representing a particle, whose material properties were adjusted to match those of polystyrene.The series of FSI simulations consisted of numerically solving pathline trajectories of these spherical particles with different initial positions.The initial and final positions in the resulting simulation were located well away from the entry and exit of the small turn of the serpentine so that the particle's behavior in this region was entirely captured.The mesh of the sphere itself was customized with lower element sizes than the predefined ones in order to obtain a reliable measurement of the calculated reaction forces.Since the flow and pressure profiles solutions are symmetric in z, the study of one of the two trajectories is enough to characterize the effects of the Dean flow when no interaction between particles is considered.In order to reduce the computational complexity and the number of degrees of freedom of the simulation, a coarser mesh was used in the rest of the fluid domain, a domain which only contained fluid throughout the simulation.It has to be stressed that due to the geometry of the problem and the complexity of the flow patterns arising in the channel, the 3D simulation could not make use of symmetries that would speed up the convergence of the solutions.Likewise, inertial terms in the N-S equations (which lead to non-linearities) could not be neglected in order for inertial migration effects to arise in the solutions.In our problem, the FSI formulation couples the flow field and pressure solutions from N-S equations for an incompressible flow to solid mechanics equations, namely, the reaction forces over the solid boundary of the particle.The use of the FSI formulation for the time-dependent study in COMSOL incorporates an arbitrary Lagrangian-Eulerian (ALE) method that takes into account the deformation of the mesh in the fluid domain caused by the movement of the particle being displaced under the influence of the surrounding flow.No constraint is applied on the particle, so it becomes a freely moving mesh.This circumstance imposes the use of an automatic remeshing feature at the mesh surrounding the particle, otherwise the fluid domain would be deformed indefinitely over time as the particle (the spherical mesh) moves along the channel (Figure 5).The transverse flow was decoupled from the main flow so the contribution of the Dean vortices can be inferred (Figure 7).In order to get a complete picture and to improve the understanding of the geometric complexity of the transverse flows, transverse velocity isosurfaces are also represented in moving inwards).No secondary Dean vortices were observed in the simulations of the small curve given their moderate aspect ratio [43].For heights near the middle of the channel, two well defined boundaries of negligible transverse flow between the small and big turns regions are visible.These boundaries are the consequence of a flow transition in these regions due to a rotation inversion of the Dean vortices between the small and big turns.For a region between the half height of the channel and its top wall (z=55.5 µm), a curved path with negligible transverse flow velocity is visible at the inner region of the small turn (Figure 7b).At z=17.5 µm an additional null velocity path can be found.These paths, which follow the curvature of the small turn, represent the position of the center of the upper and lower Dean vortices generated in the small turn (Figure 8).Some authors have highlighted what appears to be a tendency of focused particles to travel along vortices centerlines [34], which would explain the emergence of two stable trajectories of focused particles in some curved geometries, one for each upper and lower Dean vortices centerlines.

Stationary solution for the transverse flow field
(a) (b) (c)

FSI simulation
One of the purposes of the FSI simulation was to find the numerical solution that would best describe the movement of a ⌀ =10 µm particle along the trajectory found in the experimental section at Q=130 µL/min.An overlap between the averaged fluorescence streak image from the experiment and an animation provided by the time dependent solution of the moving sphere, was used as a comparison method between the experiment and the simulation (Figure 9).Since no height information of the particles can be extracted from a zenithal view of the fluorescence streak, lateral images of the small curve were taken using the inserted mirrors in the microfluidic device.This allowed to obtain the approximate height of the trajectory and use it as a guess for the particles' initial position in the simulation.Since the height of the stream of particles could not be measured with enough precision, slightly different initial positions for the spherical mesh were tested during the FSI simulations in order to find the best fit with the experimental data.

Particle rotational velocity components
Data from the FSI simulation was analyzed to obtain the angular velocity components of the particle as it moves along its trajectory in the upper half region of the small curve (Figure 10a) under the influence of the upper Dean vortex.Given the observed changes in the direction and strength of the angular velocity vector in this region of the channel, we can distinguish between different particle behavior regimes (Figure 10b, c, d).The obtained rotation rates are in the kilohertz range, the same that has been observed in straight channels with conventional flow rates [21,44].The particle, initially at rest at the beginning of the simulation, gradually starts to spin while being displaced by the drag in the streamwise direction producing an angular velocity vector contained in the x-y plane (section 1 in Figure 10b and Figure 11).Taking into account its proximity to the upper wall, the spin direction of the particle is the one that might be expected for an inertially focused particle under the influence of a vertical parabolic shear gradient [22,45].Viscous forces exerted by the fluid located between the particle and the inner region of the channel (which moves faster than the particle) and by the fluid between the particle and the upper wall (which moves slower than the particle) both create a torque that add up to the same direction creating a net spin over the particle (ω1).Right after entering the small curve, the angular velocity vector momentarily deviates downwards indicating the emergence of a clockwise spin rotation around the z axis (if we look the particle from above).This component reaches its maximum in the transition boundary between inwards and outwards lateral moving flow (with respect the center of rotation of the small curve) (Figure 12a, b) at the plane where the particle is moving.Once the particle leaves this region, the velocity vector returns to the x-y plane.In the next region of the trajectory, as the particle approximates halfway of the small turn, the influence of the upper Dean vortex starts to show up as the particle is forced to rotate in the direction of the Dean vortex [34].As for the total angular velocity vector of the sphere, it gradually moves counterclockwise (section 2 and 3 in Figure 10b) under the influence of this new component.Since the particle translates along the centerline of the Dean vortices, the flow revolving around the particle (a flow that is perpendicular to the streamwise flow) forces it to acquire an angular velocity component tangent to its trajectory pointing upstream (Figure 12c) along the regions where Dean flows are stronger.This is reflected by the fact that the resulting angular velocity vector is no longer perpendicular to the trajectory (section 3 in Figure 10b).A side effect of the rotation of the particle due to the shear gradient (causing a strong ω1 component), while being under the influence of a secondary flow, is that the interface between the outwards and inwards-moving flow suffers a distortion caused by the no-slip boundary condition of the rotating particle; the surface of the rotating sphere drags the surrounding fluid altering this interface.This distortion causes the torque to suffer a change on its balance that causes the velocity vector to move downwards (Figure 12d) increasing its negative z-component (clockwise rotation of the particle as seen from above).Simultaneously, as the particle moves along the curve and this clockwise rotation in the z direction increases in magnitude, it further induces a distortion of the Dean flow in the z direction around the particle modifying the torque balance again (Figure 13).This additional distortion modifies the angular velocity vector by forcing it to move inwards to the center of curvature (section 3 in Figure 10b).Apparently, this induced torque is strong enough to counteract the effect of the shear-induced ω1 component, inverting its original spin direction.As the particle approaches the end of the small curve, the strength of the Dean flow gradually decreases and the distortions of the flow are greatly attenuated.As it can be seen in Figure 14b, the interface between inwards and outwards-moving flow remains relatively unaltered.However, similarly to what it is observed at the inlet of the small curve, the gradients that the particle encounters as it exits the small curve, induce a strong counterclockwise rotation around the z-axis (Figure 14c).At the outlet of the small curve, the particle also encounters regions with big velocity magnitudes, hence, stronger gradients in the vertical direction.The cumulative effect of the reduction in Dean intensity and the pronounced streamwise flow gradient increase in the vertical direction, originates a restitution of the original particle spin pointing outwards (Figure 15, section 4 in Figure 10b).Like in the case of the particle entering the small curve and, since the particle is moving far from the lateral walls, the shear-induced rotation effect is expected to be stronger in the vertical direction; because of the parabolic velocity profile, the lagging-leading flow differences at each side of the particle are expected to be much greater in the vertical direction.As the particle enters the big curve, the angular velocity vector gradually moves to the x-y plane again.The shear in the vertical direction induces an angular velocity vector contained in the x-y plane.

Conclusions
We presented CFD and FSI simulation results for the flow in an asymmetric serpentine and an unrestricted free spherical particle.A custom-made microfluidic device was fabricated in order to compare the numerical results with the experiment conducted at the same conditions.The micromirrors inserted in the microfluidic device were useful to verify the existence of two stable trajectories, mirrored in z, for the focused particles and to make an estimation of the height of the streak to use it as the initial conditions in the FSI time-dependent simulation.
We also introduced a new methodology, entirely based in the results of numerical simulations, to obtain the transverse flow magnitude in a curved geometry.The fact that this method is based on the measurable differences between a creeping flow and the flow containing transverse lateral flows, enables to find the direction and magnitude of the transverse flow at any point in the fluidic domain without any geometric bias such as the ones that may be introduced by section planes when Dean patterns are represented.
FSI results suggest that particles focused under inertial focusing conditions tend to be entrained in the centerlines of the Dean vortices, something already observed by Jiang et al. [34].with the IB-LBM.This fact seems to be relativizing the importance of Dean drag contribution to inertial focusing conditions in curved channels; particles appear to have a preference to be focused at regions with small local lateral flow intensities.The comparison between the trajectory found in the simulation and the experimental one shows good agreement between them validating the join application of FEM+FSI+ALE+remeshing features in COMSOL.
The obtention of the rotational components of the translating sphere revealed the complexity of the particle's dynamics as it moves along its trajectory.This behavior clearly differs from the one observed in straight channels due to the presence of Dean flows in the small curve.Provided that particles seem to be translating along Dean vortices centerlines under focusing conditions, this fact can already be used as a simplification of the simulation by means of constrained variables because centerlines can be obtained with conventional CFD simulations alone.The elucidation of the theory behind the apparent movement of particles through vortices' centerlines, will require further studies (its response to flow rate changes, particle's confinement ratio, ...).We believe that our analysis on angular velocity components can be incorporated to simpler simulation models to infer, at least as a first approximation, the evolution of the angular velocity of the particle on its path through a curved geometry suppressing the need to perform computationally intensive simulations with coupled solvers.The use of a complex structure (the asymmetric serpentine) has probably improved our understanding of the particle's behavior in a way that it probably couldn't be achieved if we had simulated focusing conditions in a simpler geometry (a spiral or a curved channel for instance), for which Dean inversions and abrupt gradient changes hardly occur.

Figure 1 .
Figure 1.(a) Render that illustrates the position of the lateral mirror with respect the serpentine; (b) Zenithal (top) and lateral (bottom) views of the serpentine with an overlapped fluorescence streak image (false color) from inertially focused particles; (c) The limited depth of field of the employed objective allows to focus the streak of particles reflected in the mirror at different focus planes (red planes).

Figure 2 .
Figure 2. (a) Meshed simulation domain for the CFD simulation represented as a periodic unit of the serpentine.The channel is 73 µm in height; (b) Inner region of the simulation domain.Part of the mesh has been removed to expose the insides of the mesh emphasizing the effects of the finer mesh regions in the small curve.Color scale (in microns) represents the mesh element size.

Figure 3 .
Figure 3. (a) Obtained streamlines for a flow with negligible Dean flow (black lines) and for Q=130 µL/min (red lines) for h=73 µm measured at z=36.5 µm.Dean flows at this height displace the streamlines outwards in the central region of the curves; (b) Decomposition, at any given point, of a flow velocity vector (  ⃗ ) to a direction perpendicular to its corresponding negligible Dean flow streamline at that point.The perpendicular direction ( ) is found using the variables u0 and v0 from the solution at low flow rates; (c) Once the flow rate is increased, Dean flow appears and the velocity vector, now with v and u components, can be projected along vector  .

Figure 4 .
Figure 4. (a) Meshed simulation domain for the FSI simulation.The fluid enters the domain through one of the straight section ends; (b) Detail of the different sized meshes.The upper one corresponds to the region of the domain where the particle will translate through the curve; (c) Zoomed section of the mesh showing the particle domain at its initial position inside the channel.

Figure 5 .
Figure 5. Different stages of the movement of a ⌀ =10 µm spherical domain (green sphere) through the fine (rainbow) mesh.The last picture shows the remeshing of the fine mesh.Bluish facets in the fine mesh represent low mesh quality regions.

Figure 6 Figure 6 .
Figure 6 illustrates the simulation results for the velocity magnitudes of the flow at a given flow rate and channel height (Q=130 µL/min, h=73 µm).The velocity magnitude distribution clearly reveals the asymmetric character of the flow; the maximum values of the velocity tend to be displaced in the streamwise direction.

Figure 8 .
As expected, secondary flows direction and intensities vary along the height of the small curve having its maximum intensity in the middle height region of the channel (corresponding to flow moving radially outwards) and opposite direction near the top and bottom boundaries (flow Preprints (www.preprints.org)| NOT PEER-REVIEWED | Posted: 6 August 2018 doi:10.20944/preprints201808.0114.v1

Figure 7 .Figure 8 .
Figure 7. Modulus of the transverse flow field calculated at different heights.The transverse flow field strength is an order of magnitude weaker in the big curve.(a) At z=36.5 µm (middle height) the magnitude for an outwards-moving flow is maximum; (b) At z=55.5 µm a null velocity region (the center of the upper Dean vortex) is visible in the central region of the small curve; (c) The magnitude of the velocity in the small curve increases again as we move upwards.This time the flow is moving inwards, in the direction of the center of curvature.

Figure 9 .
Figure 9. (a) Overlapping of the experimental image for a focused streak of ⌀ =10 µm particles and positions of the particle calculated with the FSI simulation along the small (white circles).The overlapped image is represented in false color to improve the visibility of the streak; (b) Composed perspective of the 3D trajectory of particle (red line).

Figure 10 .
Figure 10.(a) Angular velocity components of the particle as a function of the simulation time.ωTotal is the total angular velocity magnitude; (b) Angular velocity vector along the trajectory of the particle as seen from a zenithal view; (c, d) Angular velocity vector seen from different perspectives in COMSOL's frame of reference.

Figure 11 .
Figure 11.Left: Velocity magnitude at the half height of the channel.Right: Vertical section of the fluidic channel at the inlet region of the small curve showing the velocity magnitude distribution.The shear in the vertical direction induces an angular velocity vector, ω1, contained in the x-y plane pointing away from the small curve center.

Figure 12 .
Figure 12.Simulated transverse velocity magnitude and sense of the flow.Reddish regions correspond to outwards-moving flow (moving away from the center of curvature of the small curve) and vice versa for the bluish regions.(a) General view of the channel with marked cross sections; (b) Transverse flow field gradients in the x-y plane induce particle's rotation, ω2; (c) The flow around the particle in the center of the Dean vortex further induces a change in the particle's rotation with a ω3 component in the x direction; (d) The rotating particle causes a disturbance (yellow arrows) at the interface between outwards and inwards lateral flows with Vt components which in turn causes a modification in the angular velocity of the particle yielding to ω4 component.

Figure 13 .
Figure 13.Left: z-component of the lateral flow field.Reddish regions correspond to flow moving upwards and vice versa for bluish regions.Right: The vertical rotational component of the particle (ω4-z) induces a disturbance in this component of the lateral flow field (yellow arrows) that modifies the behavior of the rotating particle (ω5 component).

Figure 14 .
Figure 14.Transverse velocity magnitude and sense of the flow.(a) General view of the channel with marked cross sections; (b) The interface between the outwards and inwards-moving flow is no longer distorted so no modification of the angular velocity vector is produced in this region; (c) A strong induced angular velocity (ω2, with positive z direction) is observed at the vicinity of the transverse inwards-outwards transition region.

Figure 15 .
Figure 15.Right: Velocity magnitude at the half height of the channel.Left: Vertical section of the fluidic channel at the outlet region of the small curve showing the velocity magnitude distribution.The shear in the vertical direction induces an angular velocity vector contained in the x-y plane.