# Propulsion Mechanism of Flexible Microbead Swimmers in the Low Reynolds Number Regime

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{x}, was generated using a pair of coils that are powered by a DC power source (Guan Chun, New Taipei City, Taiwan). The magnetic beads that are magnetized by the static field H

_{x}tended to aggregate and randomly form various types of chain-like swimmers. A perpendicularly dynamic sinusoidal field, H

_{y}, with a maximum amplitude, H

_{p}, and an adjustable frequency, f, such that H

_{y}= H

_{p}sin(2πft), was generated using a pair of coils that are connected to an AC power supply (GWInstek APS-1102, GWInstek, New Taipei City, Taiwan) to constitute an overall oscillating field.

_{3}O

_{4}), which was embedded in polystyrene microspheres that were suspended in distilled water (Thermo Fisher Scientific Inc., Waltham, MA, USA: Dynabeads M-450 Epoxy and Dynabeads M-270 Epoxy). The mean diameters of the microparticles were d = 4.5 and 2.8 μm, and the respective magnetic susceptibilities were χ = 1.6 and 1.0. The microbeads had a saturation magnetization of M = 28,000–32,000 A/m and did not exhibit magnetic hysteresis or remanence, so these magnetic beads could be linearly magnetized or de-magnetized by applying or removing an external field. When the external applied field was removed, the microbeads dispersed again. The movement of the beads and the swimmers was recorded using an optical microscope that was connected to a high-speed camera with a maximum rate of capture of 200 frames per second.

## 3. Results and Discussion

#### 3.1. Non-Reciprocal Motion for a Microbead Swimmer

_{p}= 2000 A/m and H

_{d}= 1900 A/m that oscillates at a frequency of f = 10 Hz. All swimmers consisted of different numbers of magnetic microparticles of different sizes. Some previous investigations on the effect of the frequency have shown that a microbead chain or swimmer can be manipulated with a stable structure or efficient movement at a frequency of 7–10 Hz [35,36]. Therefore, all the swimmers were subjected to an oscillating frequency of f = 10 Hz in this study.

#### 3.2. Propulsion Mechanism for a Swimmer

#### 3.3. The Waveform That Generates Effective Propulsion for a Swimmer

#### 3.4. Theoretical Equation for the Net Thrust of the Microbead Swimmer

_{i}) when the swimmer moves along the axis of propulsion (x-axis). The orientation (θ

_{i}) of the ith particle to the axis of propulsion (x-axis) depends on the form of the wave and the particle’s position on the wave. As shown in Figure 7a, the velocity of the ith particle along the y-axis is V

_{i,y}and its surface is inclined to the x-axis by an angle of θ

_{i}. The transverse velocity of V

_{i},

_{v}consists of two components: A tangential velocity, V

_{i,y}sinθ

_{i}, and a normal velocity, V

_{i,y}cosθ

_{i}. The fluid imposes tangential (T

_{i,y}) and normal (N

_{i,y}) reactions on the surface of the ith particle during the transverse motion because of the resistance of the fluid. The forces N

_{i},y and T

_{i,y}have the components N

_{i},ysinθ

_{i}and T

_{i},ycosθ

_{i}, respectively, acting along the axis of propulsion. Thus, the resultant propulsive thrust F

_{i,y}along the axis of propulsion is given as:

_{N}and C

_{T}are the coefficients of resistance to the surface of the ith particle for a fluid of known viscosity, and d

_{i}is the diameter of the ith particle. As a result, the resultant forward thrust F

_{i,y}along the axis of propulsion is given as:

_{x}, which depends on the speed of the whole swimmer moving through the fluid. Figure 8b shows that the resultant drag is composed of N

_{i,x}sinθ

_{i}and T

_{i,x}conθ

_{i}, which are directly proportional to the velocity of the displacement and the viscosity of the fluid. As a result, the resultant drag F

_{i},

_{x}along the axis of propulsion is given as:

_{i}and T

_{i}along the axis of propulsion are expressed as N

_{i}sinθ

_{i}and T

_{i}cosθ

_{i}, respectively, and the net forward thrust of the ith particle (F

_{i}) is:

_{t}for a swimmer consisting of N particles at specific waveform is then given as:

#### 3.5. Comparison of the Propulsive Speed of Various Types of Microswimmers

_{p}= 2000 A/m and H

_{d}= 1900 A/m, and there was a constant frequency of f = 10 Hz. The swimmers swam toward the direction of their center of mass, which was located closer to the larger particles, so they moved to the right in Figure 8a,b and to the left in Figure 8c,d. The propulsive speed was measured by recording the moving trajectory of the swimmer. A comparison of the swimming effectiveness for the four swimmers is shown in Figure 9. It can be seen that the L2S2 swimmer moved farther than the rest of the other swimmers in 10 s. The maximum respective average velocities for the L1S3, L2S2, S1L1S3, and L3S1 swimmers were 1.38, 2.28, 1.55, and 0.9672 μm/s. These values of speed reveal that the swimmers can swim in an environment with a low Reynolds number of 1.77 × 10

^{–5}–3.74 × 10

^{–5}.

_{p}= 1500 A/m and H

_{d}= 1900 A/m oscillating at a constant frequency of f = 10 Hz. T is the time when the swimmers began swimming at the highest speed and then cruised at a constant speed. Their moving trajectories are shown in Figure 12. The L2S3 swimmer traveled a farther distance than the L2S2 swimmer in 5 s. The average swimming velocity of the L2S3 swimmer was 2.24 μm/s, which is faster than the 1.9 μm/s for the L2S2 swimmer. These values of speed reveal that the swimmers can swim in an environment with a low Reynolds number of 3.12 × 10

^{–5}–4.38 × 10

^{–5}. The faster speed of the L2S3 swimmer is attributed to its longer flexible tail, which generates much more positive thrust within a period of oscillation. It is noted that the L2S2 swimmer was slower than the L2S2 case shown in Figure 8 because the dynamic field strength was lower at H

_{p}= 1500 A/m, so there was a smaller dipole moment at the head and the amplitude and flexibility were decreased. A microbead swimmer with a larger head that is subject to a stronger induced dipole moment and that has a moderately long flexible tail generates the greatest thrust.

#### 3.6. Crucial Factors for the Design of a Microbead Swimmer with Optimal Propulsive Efficiency

_{h}/L and A

_{t}/L to determine the waveform characteristics for a flexible swimmer, and studies the effect of these factors on the propulsion efficiency of a swimmer. Figure 10b illustrates the definitions of A

_{h}/L and A

_{t}/L, which are the respective ratios of the amplitude of the head and the amplitude of the tail of the swimmer (denoted as A

_{h}and A

_{t}, respectively) to the overall length of the swimmer (denoted as L). It has been shown that a larger head with a stronger dipole moment induces fast oscillation at the head and allows a wave to propagate from the head to the tail. A longer swimmer requires a larger (or longer) head to create a bending wave that gives greater propulsive efficiency. However, a larger (longer) head oscillates with a greater amplitude and produces much more negative thrust during the accelerating stage, as shown in Figure 5b. Effective locomotion requires a swimmer to have an oscillating head, but the amplitude is negatively correlated to swimming velocity.

_{h}/L and achieved its highest value between 0.015 and 0.02 for a value of A

_{h}/L = 0.26–0.28. This trend indicates that an increase in the oscillating amplitude initially enhances the propulsive efficiency of the swimmer. However, the dimensionless velocity begins to decrease when the value of A

_{h}/L is greater than 0.28, because a swimmer with a larger (longer) head generates greater negative thrust when the oscillating amplitude of the head is increased significantly. A similar trend was observed for the effect of the oscillating amplitude of the tail. Figure 13b shows that the value of the dimensionless velocity was greatest when the ratio of A

_{t}/L is 0.29–0.33. The dimensionless propulsion speed decreased thereafter as the value of A

_{t}/L increased. A higher value for A

_{t}/L than for A

_{h}/L produced the greatest propulsion, which implies that a one-dimensional magnetic microbead swimmer with a tail of higher beating amplitude dominates the effective locomotion in a low Reynolds number environment.

## 4. Conclusions

_{h}/L) of 0.26 to 0.28 produced the greatest swimming velocity. In contrast, a swimmer was faster when the ratio of the oscillating amplitude of the tail to the length of the swimmer (A

_{t}/L) was 0.29 to 0.33. This study determined the optimal configuration for a flexible microbead swimmer that produces the greatest swimming effectiveness, which can potentially be applied in a microfluidic system.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A schematic diagram of the experimental setup. A static directional magnetic field powered by a DC source was initially applied to create a linear microswimmer. An additional sinusoidal dynamic field from an AC power supply was then applied perpendicularly to generate an oscillating field, which created a waggling motion for the magnetic microswimmer.

**Figure 2.**The evolution of the distortion of the (

**a**) L1S3, (

**b**) L2S2, (

**c**) S1L1S3, (

**d**) L3S1 swimmers in an oscillating field (H

_{p}= 2000 A/m, H

_{d}= 1900 A/m, f = 10 Hz). The images were captured within an arbitrary period (denoted as P) of oscillation. All swimmers oscillated clockwise from the top position and anticlockwise when they reached the bottom, and then returned to the initial point.

**Figure 3.**Waveform evolutions for the (

**a**) L1S3, (

**b**) L2S2, (

**c**) S1L1S3, and (

**d**) L3S1 swimmers shown in Figure 2. The dots mark the theoretical position of the center of mass of the swimmer, while the numbers represent the sequential waveforms of the swimmer within a full oscillating period. The interval of each waveform is 5P/30, as shown in the sequential images in Figure 2.

**Figure 4.**Diagram of the net thrust generation mechanism for a microbead swimmer in a low Reynolds number environment: (

**a**) The forces that are imposed on an oscillating swimmer and (

**b**) a schematic diagram of the net thrust resulting from the projection of the normal (N) and tangential (T) components along the axis of the thrust. (

**c**) Schematic diagram of the four stages within a period of oscillation for a swimmer.

**Figure 5.**Schematic diagram of sequential images showing the direction of the thrust for the head and tail of a swimmer within the first half-period of oscillation. (

**a**) Initial position. (

**b**) The head and tail generate negative and positive thrust, respectively. (

**c**) The head and tail simultaneously generate positive thrust. (

**d**) The head and tail generate positive and negative thrust, respectively.

**Figure 6.**Sequential images of the direction of thrust for the head and tail of a swimmer within a period of oscillation for the (

**a**) L1S3, (

**b**) L2S2, (

**c**) S1L1S3, and (

**d**) L3S1 swimmers in Figure 2. The numbers at the top of the figure correspond to the number of the curve in Figure 3. The green and red arrows represent the positive and negative thrust, respectively, of the head and tail of the swimmer.

**Figure 7.**Schematic diagram of the forces imposed on the ith particle (

**a**) when moving vertically along the axis of propulsion (x-axis) at the velocity of V

_{i,y}, or (

**b**) when moving along the x-axis at the velocity of V

_{i},

_{x}.

**Figure 8.**Sequential images of the (

**a**) L1S3, (

**b**) L2S2, (

**c**) S1L1S3, (

**d**) L3S1 swimmers in an oscillating field (H

_{p}= 2000 A/m, H

_{d}= 1900, A/m, f = 10 Hz). T represents the time when the swimmer began swimming at the highest speed and then cruised at a constant speed, and S represents second. A static single particle was used as a reference point to observe the locomotion of the swimmer, as well as for evidence of the uniformity of the applied field.

**Figure 9.**Trajectories of the swimmers in Figure 8 moving more 10 s.

**Figure 10.**(

**a**) Schematic diagram of the configuration of the different types of swimmers. (

**b**) Definition of the oscillating amplitude of the head (A

_{h}) and the tail (A

_{t}) for a swimmer. A

_{h}is the distance measured from the tip of the head particle to the center of the mass of the swimmer; A

_{t}is the distance between the tip of the tail particle and the center of the mass of the swimmer; L represents the overall length of the swimmer.

**Figure 11.**Sequential images of the (

**a**) L2S2 and (

**b**) L2S3 swimmers in an oscillating field with a strength of H

_{p}= 1500 A/m and H

_{d}= 1900 A/m and a frequency of f = 10 Hz. T represents the time when the swimmer started swimming at the greatest velocity and then cruised at a nearly constant speed, while S refers to second.

**Figure 12.**Trajectories for the swimmers shown in Figure 11 in a period of 5 s.

**Figure 13.**Distribution of the dimensionless swimming velocity in terms of the ratio of maximum amplitude of oscillation for (

**a**) the head (A

_{h}) and (

**b**) the tail (A

_{t}) to the length of the swimmers.

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**MDPI and ACS Style**

Li, Y.-H.; Chen, S.-C.
Propulsion Mechanism of Flexible Microbead Swimmers in the Low Reynolds Number Regime. *Micromachines* **2020**, *11*, 1107.
https://doi.org/10.3390/mi11121107

**AMA Style**

Li Y-H, Chen S-C.
Propulsion Mechanism of Flexible Microbead Swimmers in the Low Reynolds Number Regime. *Micromachines*. 2020; 11(12):1107.
https://doi.org/10.3390/mi11121107

**Chicago/Turabian Style**

Li, Yan-Hom, and Shao-Chun Chen.
2020. "Propulsion Mechanism of Flexible Microbead Swimmers in the Low Reynolds Number Regime" *Micromachines* 11, no. 12: 1107.
https://doi.org/10.3390/mi11121107