# Two-Way Coupling Fluid-Structure Interaction (FSI) Approach to Inertial Focusing Dynamics under Dean Flow Patterns in Asymmetric Serpentines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

_{0}= 1 μm/s) ≈ 1.8 × 10

^{−4}μL/min), u

_{0}and v

_{0}, and the variables obtained in the studied flow rate (Q(v

_{0}= 0.67 m/s) = 130 μL/min) u, v and w:

_{0}= 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.

## 3. Results and Discussion

#### 3.1. Stationary Solution for the Transverse Flow Field

#### 3.2. FSI Simulation

#### 3.3. Particle Rotational Velocity Components

_{1}).

_{1}component), while being under the influence of a secondary flow, is that the interface between the outward- and inward-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).

_{1}component, inverting its original spin direction.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Render that illustrates the position of the lateral mirror with respect to the serpentine; (

**b**) Zenithal (top) and lateral (bottom) views of the serpentine with an overlapped fluorescence streak image (false color) from inertially focused particles at a flow rate of 130 μL/min; (

**c**) The limited depth of field of the employed objective allows the streak of particles reflected in the mirror to focus on different focus planes (red planes).

**Figure 2.**(

**a**) Geometry’s dimensions of the serpentine; (

**b**) meshed simulation domain for the CFD simulation represented as a periodic unit of the serpentine. The channel is 73 μm in height; (

**c**) 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 (μm) represents the mesh element size.

**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 ($\overrightarrow{v}$) to a direction perpendicular to its corresponding negligible Dean flow streamline at that point. The perpendicular direction ($\widehat{n}$) is found using the variables u

_{0}and v

_{0}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 $\widehat{n}$.

**Figure 4.**(

**a**) Meshed simulation domain for the FSI simulation. The fluid enters the domain through one of the straight section ends (blue surface) and exits through the other side (red surface); (

**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 (green) at its initial position inside the channel.

**Figure 5.**Different stages of the movement of a Ø = 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.**(

**a**) Flow velocity magnitude for Q = 130 μL/min and h = 73 μm measured at z = 36.5 μm, at the middle height of the fluidic channel; (

**b**) Flow velocity magnitude for the same flow conditions measured at the y-z plane halfway through the small curve.

**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 8.**(

**a**) Isosurfaces of the transverse velocity flow modulus cut through y-z plane; (

**b**) Plane view of the cut section. The null transverse velocity paths are visible at the center of the Dean vortices.

**Figure 9.**(

**a**) Overlapping of the experimental image for a focused streak of Φ = 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 the particle (red line).

**Figure 10.**(

**a**) Angular velocity components of the particle as a function of the simulation time. ω

_{Total}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.**(

**a**) Velocity magnitude at the half height of the channel; (

**b**) 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, ω

_{1}, contained in the x-y plane pointing away from the small curve center.

**Figure 12.**Simulated transverse velocity magnitude and sense of the flow in the x-y plane. Reddish regions correspond to an outward-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, ω

_{2}; (

**c**) The flow around the particle in the center of the Dean vortex further induces a change in the particle’s rotation with a ω

_{3}component in the x direction; (

**d**) The rotating particle causes a disturbance (yellow arrows) at the interface between the outward and inward lateral flows with V

_{t}components, which in turn causes a modification in the angular velocity of the particle yielding to the ω

_{4}component.

**Figure 13.**(

**a**) z-component of the transverse flow field. Reddish regions correspond to flow moving upwards and vice versa for bluish regions; (

**b**)The vertical rotational component of the particle (ω

_{4−z}) induces a disturbance in this component of the lateral flow field (yellow arrows) that modifies the behavior of the rotating particle (ω

_{5}component).

**Figure 14.**Transverse velocity magnitude and sense of the flow in the x-y plane. (

**a**) General view of the channel with marked cross sections; (

**b**) The interface between the outward- and inward-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 (ω

_{2}, with positive z direction) is observed at the vicinity of the transverse inwards-outwards transition region.

**Figure 15.**(

**a**) Velocity magnitude at the half height of the channel; (

**b**) 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.

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**MDPI and ACS Style**

Pedrol, E.; Massons, J.; Díaz, F.; Aguiló, M. Two-Way Coupling Fluid-Structure Interaction (FSI) Approach to Inertial Focusing Dynamics under Dean Flow Patterns in Asymmetric Serpentines. *Fluids* **2018**, *3*, 62.
https://doi.org/10.3390/fluids3030062

**AMA Style**

Pedrol E, Massons J, Díaz F, Aguiló M. Two-Way Coupling Fluid-Structure Interaction (FSI) Approach to Inertial Focusing Dynamics under Dean Flow Patterns in Asymmetric Serpentines. *Fluids*. 2018; 3(3):62.
https://doi.org/10.3390/fluids3030062

**Chicago/Turabian Style**

Pedrol, Eric, Jaume Massons, Francesc Díaz, and Magdalena Aguiló. 2018. "Two-Way Coupling Fluid-Structure Interaction (FSI) Approach to Inertial Focusing Dynamics under Dean Flow Patterns in Asymmetric Serpentines" *Fluids* 3, no. 3: 62.
https://doi.org/10.3390/fluids3030062