# Numerical and Experimental Analyses of Three-Dimensional Unsteady Flow around a Micro-Pillar Subjected to Rotational Vibration

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{s}~ (2ν/ω)

^{1/2}, where ν and ω are the kinematic viscosity and angular frequency, respectively [13]. However, when an obstacle is present in the flow field, the interaction between the oscillating bulk fluid and the obstacle creates vorticity and causes net-momentum transfer. This is similar to the Reynolds shear stress that arises from the correlation of the fluctuating velocity components in turbulent flows, and it appears as an additional forcing term in the averaged Navier‒Stokes equations. As a result, the steady time-averaged velocity is generated, although the applied periodic forcing does not have a mean component. Because it requires no net displacement or a pressure gradient to drive the flow, the SS is expected to simplify and miniaturize microfluidic systems without introducing external pumps or tubing.

## 2. Numerical Procedures

#### 2.1. Computational Domain and Governing Equations

^{*}. No-slip conditions were applied at the bottom and top walls, i.e., y

^{*}= 0 and 2δ

^{*}, as well as the surface of the pillar. Periodic boundary conditions were employed in the x and z directions. This condition corresponds to the case where the geometry shown in Figure 1a repeats in these two directions. The present configuration is chosen because it is relatively simple (the two parallel walls and the periodic placement of cylinders between the walls), while the truncation of the cylinder at the middle of the channel causes complex three-dimensionality of the resulting flow.

^{*}and T

^{*}, the relative displacement of the substrate

**x**

^{*}

_{R}is given as

**x**

^{*}

_{R}as

^{*}and the oscillation period of T

^{*}, the generalized dimensionless forms of Equations (1) and (2) on a rotating frame are expressed as

_{i}due to the acceleration/deceleration of the moving coordinate, which is given by

#### 2.2. Volume Penalization Method

#### 2.3. Numerical Methods and Conditions

^{*}= 800 µm, which is equal to the spacing of pillars in the experiment. This dimension was set to be large enough so that induced flow profiles of neighboring pillars do not interact. The height of the domain was 2δ

^{*}= 200 µm. The numbers of modes employed in the current simulation were (N

_{1}, N

_{2}, N

_{3}) = (64 × 33 × 64) in x, y, z directions, respectively. 3/2 rule was used for removing aliasing errors, so that the non-linear terms were evaluated in 1.5 times finer physical grid points in each direction. Throughout this work, the vibration amplitude was A = 4 µm and frequency was f = 1000 Hz. Accordingly, dimensionless numbers were Re = 10 and St = 12.4/π, respectively. The numerical time step was set to be Δt = 1.0 × 10

^{−4}, which indicates that t

^{−1}= 10

^{4}time steps are required to compute the velocity field for one oscillation period (t = 1). The computation was started from a stationary flow at t = 0, and the rotational vibration was applied for t = 60 to achieve a fully developed velocity field. After the transient period, the flow field became completely periodic in one oscillation cycle. All statistics shown below were obtained by integrating the velocity data over one oscillation period after the flow field had reached the statistically steady state.

#### 2.4. Derivation of Steady Streaming Flow (Time-Averaged Velocity Field)

## 3. Experimental Procedure

#### 3.1. Fabrication of Micro-Pillar Array

#### 3.2. Experimental Setup and Conditions

#### 3.3. PIV System

#### 3.4. PIV Analysis

#### 3.5. Horizontal Visualization

## 4. Numerical Results

#### 4.1. Instantaneous Velocity Field

_{max}= 2πAf ~ 25.2 mm/s. Although not shown in the figure, the comparable speed was also observed at the vicinity of no-slip boundaries (e.g., the upper and lower walls).

#### 4.2. Time-Averaged Flow Field

_{hor}| of the average velocity. It is clear that a net flow is induced around the pillar in both cases. However, the radial peak position of the velocity is closer to the pillar in the former case (r ~ 120 μm in the Eulerian method, but r ~ 130 μm in the Lagrangian method). Moreover, the peak value is about two times greater in Eulerian method compared to the Lagrangian method. The result clearly indicates that the averaged flow field depends on the averaging methods. Because the mass transport and the paths of suspended molecules/particles are governed by the Lagrangian trajectories, the Lagrangian averaging is necessary to predict the above-mentioned transport phenomena in micro devices. In the following sections, the Lagrangian velocity obtained from the present simulation will be compared with the averaged translational velocity of fluorescence particles in the experiment in order to validate the present numerical code.

## 5. Comparison with experimental results

#### 5.1. Results of PIV Measurement

#### 5.2. Comparison of Radial Velocity Profile

_{hor}| in Figure 9. The profiles were averaged in the azimuthal direction of the pillar. The magnitude of the peak velocity obtained by Eulerian method is twice as large as the other, and the peak position of the velocity is closer to the pillar, as qualitatively seen in Figure 7. On the other hand, the velocity distributions obtained by the Lagrangian method and PIV measurement showed a similar trend in terms of the peak position (r = 130 µm; 30 µm away from the pillar wall) and the decay of the profile. There is a difference between the two profiles close to the pillar; this difference could be partly caused by the difficulty in resolving the high-shear region in both simulation and PIV. The size of the single grid in the simulation corresponds to be 8.3 μm, and the size of the window for image correlation in PIV corresponds to be 6.4 μm. Particle images tended to blur in the region close to the wall due to the 3D flow described later. Nonetheless, a good agreement in the decaying profile after the peak supports the validity of the present simulation.

#### 5.3. Comparison of Normal Velocity Profile

_{hor}| in r-y plane obtained in the numerical simulation (Lagrangian averaging) and the PIV measurement are shown in Figure 10a,b, respectively. They showed a similar trend in terms of the peak position of r = 130 μm at 40 < y < 80 μm, which slightly moves closer to the pillar near the top (y ~ 100 μm). This is attributed to the strong three-dimensionality of the flow around the tip of the pillar, as will be shown later. The magnitude of the velocity decays rapidly as the position gets closer to the bottom wall.

_{s}= (2ν/ω)

^{1/2}[13]. From its analytical solution, the effect of the wall motion decreases down to ~5% (1/e

^{3}) at y = 3δ

_{s}. In the present condition, 3δ

_{s}corresponds to 53.4 µm by substituting ν = 1.0 × 10

^{−6}m

^{2}/s (25 °C) and ω = 2πf = 6.28 × 10

^{3}rad/s. The height is indicated by dotted lines in Figure 10 and Figure 11. Since the net flow discussed in the present system is induced by the relative velocity between the fluid and the substrate, the maximum average flow field occurs outside the Stokes layer. In contrast, the fluid within the Stokes layer thickness 3δ

_{s}is dragged by the motion of the wall due to the viscous effect so that the relative velocity diminishes on the bottom wall.

#### 5.4. Three-Dimensionality of the Flow

_{hor}|, the presence of the vertical velocity component has a strong impact on the averaged velocity field as discussed in Figure 10.

## 6. Conclusions

## Supplementary Materials

_{hor}| calculated with different grid resolutions in the numerical simulation. The distribution calculated with a low-resolution grid (48 × 25 × 48) differ significantly, but those calculated with medium (96 × 49 × 96; used in the main results) and high-resolutions (144 × 73 × 144) were similar., Figure S2: Comparison of the radial profile of |V

_{hor}| calculated with different window size in PIV analysis. One pixel in images corresponds to 0.8 μm. Supporting Movie 1: Animation of the velocity field (u

_{1}, u

_{3}) at y = 50 μm obtained from the numerical simulation., Supporting Movie 2: Motion of a tracer particle depicted in Figure 13a in the main text., Supporting Movie 3: Motion of a tracer particle depicted in Figure 13b in the main text.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) Schematic of coordinate system and computational domain. (

**b**) Circular vibration of the micro-pillar.

**Figure 3.**(

**a**) Assembly of the micro-pillar plate. (

**b**) Experimental setup for application of circular vibration.

**Figure 5.**2D vector plot of the instantaneous velocity field at phases of (

**a**) 0°, (

**b**) 90°, (

**c**) 180°, (

**d**) 270° within the one rotational cycle. White arrows indicate the instantaneous moving directions of the pillar.

**Figure 7.**2D vector plots of the averaged velocity flow field calculated by the (

**a**) Eulerian and (

**b**) Lagrangian approaches.

**Figure 8.**(

**a**) Raw image of fluorescent tracer beads overlaid for sixty cycles. (

**b**) 2D vector plot of the averaged velocity flow field obtained from PIV measurement.

**Figure 9.**Distribution of the mean horizontal velocity magnitude of particles in radial direction. (▲) Eulerian averaging and (●) Lanrangian averaging of the simulation result. (◆) PIV measurement. PIV was repeated three times with different setup, and the average and standard deviation are shown.

**Figure 10.**Contour plots of mean horizontal velocity magnitude |V

_{hor}| in r-y plane of (

**a**) numerical result (Lagrangian averaging) and (

**b**) PIV measurement. The dashed line indicates the Stokes layer thickness calculated based on the present experimental condition.

**Figure 13.**Comparison of particle trajectories obtained in the experiment and the numerical simulation. (

**a**) Ascending motion near the apex of the pillar. (

**b**) Descending and ascending motion.

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**MDPI and ACS Style**

Kaneko, K.; Osawa, T.; Kametani, Y.; Hayakawa, T.; Hasegawa, Y.; Suzuki, H.
Numerical and Experimental Analyses of Three-Dimensional Unsteady Flow around a Micro-Pillar Subjected to Rotational Vibration. *Micromachines* **2018**, *9*, 668.
https://doi.org/10.3390/mi9120668

**AMA Style**

Kaneko K, Osawa T, Kametani Y, Hayakawa T, Hasegawa Y, Suzuki H.
Numerical and Experimental Analyses of Three-Dimensional Unsteady Flow around a Micro-Pillar Subjected to Rotational Vibration. *Micromachines*. 2018; 9(12):668.
https://doi.org/10.3390/mi9120668

**Chicago/Turabian Style**

Kaneko, Kanji, Takayuki Osawa, Yukinori Kametani, Takeshi Hayakawa, Yosuke Hasegawa, and Hiroaki Suzuki.
2018. "Numerical and Experimental Analyses of Three-Dimensional Unsteady Flow around a Micro-Pillar Subjected to Rotational Vibration" *Micromachines* 9, no. 12: 668.
https://doi.org/10.3390/mi9120668