# Driving Forces of the Bubble-Driven Tubular Micromotor Based on the Full Life-Cycle of the Bubble

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Bubble Nucleation

_{LG}is the surface tension coefficient σ of the hydrogen peroxide solution required in the calculation. The equilibrium state is shown in Figure 1. According to Young’s equation of the contact angle

_{LG}and the solid and gas contact area A

_{SG}are respectively

_{c}can be obtained by the following formula

_{∞}is the standard atmospheric pressure, and P

_{G}is the internal pressure of the nucleation bubble.

_{c}= 4π${R}_{c}^{3}$/3. Assuming that the volume of fluid discharged along the left orifice is fV

_{c}, the volume of fluid discharged along the right orifice is (1 − f)V

_{c}. As can be seen from the figure, due to the incompressibility of the fluid, the volume V

_{c}of the nucleation bubble is not the volume of the fluid in the tube. The fluid discharged from the tube flows out from the two openings of the tube, respectively, wherein the ratio of the volume of the left outflow fluid to the total flow is f. The parameter f is obtained from FLUENT 18.0 (ANSYS).

_{1}, A

_{2}, and the function of bubble and oxygen production frequency q

_{H2O2}is the concentration of hydrogen peroxide and f is a function of time t.

#### 2.2. Bubble Growth

_{s}is the surface tension, F

_{qs}is the fluid resistance of the bubble, F

_{du}is the unstable force of the asymmetric growth of the bubble, F

_{b}is the buoyancy, F

_{sl}is the shearing force, F

_{h}is the force caused by the dynamic pressure, and F

_{cp}is the contact pressure. The subscript a, p denotes a semi-cone angle of a conical micromotor along the wall of the tube and perpendicular to the wall of the tube.

_{w}= 2Rsinθ. The volume of the bubble is expressed as $V=1/3\pi {R}^{3}(2+3\mathrm{cos}\theta -{\mathrm{cos}}^{3}\theta )$.

_{l}is the fluid density, μ is the fluid viscosity, R is the bubble radius, and V

_{y}is the fluid velocity at the bubble center point.

_{y}into the projection of surface tension on the coordinate axis, and considering the fluid resistance of small bubbles is 2/3 of the same size solid particles [31], the fluid resistance of the bubble is expressed as

_{b}is the density of the gas within the bubble, namely the oxygen density.

_{h}mainly acts on the circular surface where the bubble contacts the tube wall [32], namely

_{l}, velocity gradient $\kappa =\left|\frac{d{V}_{y}}{dy}\right|=\left|-\frac{64Q}{\pi {(2{R}_{\mathrm{max}}-L\mathrm{tan}\delta )}^{4}}y\right|$.

^{−10}, but the conversion efficiency of bubbles was not studied. Fomin et al. [20] then studied the bubble-driven tubular micromotor and found that the conversion force of the bubble is converted to a tubular motor driving force with a conversion coefficient of about 1/30. The driving force of the tubular micromotor in the stage of bubble growth is

#### 2.3. Bubble Slip

_{i}and the inner diameter H

_{i}of the tube wall at the nucleation point is ${H}_{i}=2{R}_{i}\mathrm{cos}\delta $, or ${H}_{i}=2{X}_{i}\mathrm{tan}\delta +2{R}_{\mathrm{min}}$.

_{i}− C/2 ≥0, X

_{i}+ C/2 ≤L.

#### 2.4. Bubble Ejection

_{1}and R

_{2}, respectively, and the volume of the two spheres is V

_{1}and V

_{2}. ${R}_{1}={R}_{\mathrm{max}}/\mathrm{cos}\delta $, ${H}_{1}={R}_{1}-{R}_{\mathrm{max}}\mathrm{tan}\delta ={R}_{\mathrm{max}}/\mathrm{cos}\delta -{R}_{\mathrm{max}}\mathrm{tan}\delta $

_{t}is the angle at which the bubble is ejected. The main change in bubble volume is reflected in the V

_{2}part,

_{2}portion,

_{w}is substantially close to zero. Consequently, the flow pressure, contact pressure, and shear lift force are zero

_{b}is the speed at which the bubble is ejected from the opening.

_{b}

#### 2.5. Micromotor Velocity

_{d}is the drag force experienced by the micromotor. The relationship between F

_{d}and Reynolds number, micromotor geometry, and the drag coefficient is obtained by the computational fluid dynamics software FLUENT 18.0 (ANSYS).

## 3. Results and Discussion

#### 3.1. Bubble Nucleation and Growth

_{jet2}= f (σ

^{2}). As can be seen from Figure 8b, when the viscosity changes from 0.1 mPa·s to 4.0 mPa·s, the driving force remains substantially unchanged. Thus, the fluid viscosity almost has no influence on the driving force in the stage. When the concentration of the hydrogen peroxide solution in Figure 9a changes, the speed does not change much, and the difference between the maximum and minimum values is only 0.62%. Basically, it can be considered that the change in the concentration of the peroxide solution almost has no effect on the driving force. Fitting the curve in Figure 9b to obtain the relationship between the driving force and the opening radius ${F}_{\mathrm{jet}2}=f({R}_{\mathrm{max}}^{3})$.The relationship with the micromotor length and semi-cone angle is linear, as demonstrated by Figure 10a,b. θ is the contact angle of the bubble with the micromotor wall, and φ is the inclination angle of the bubble under the action of the flow rate. The front and back contact angle α and β of the surface tension in Equation (18) are determined by both contact angels. Furthermore, Equation (19) used for calculation of the bubble radius R is related to the contact angle θ. Considering Equations (20)–(27) are all related to the bubble radius R, the formula of the driving force could be expressed as a function of θ and φ, namely f(θ) and g(φ).

_{jet2}and σ as well as R

_{max}, the final expression of the driving force in the stage is

#### 3.2. Driving Force of the Full Life-Cycle of the Bubbles

_{l}= 10

^{3}kg/m

^{3}, the viscosity μ = 0.9 mPa⋅s, the surface tension coefficient σ = 30 mN/m. The geometry of the tubular micromotor has: length L = 100 μm, larger opening radius R

_{max}= 10 μm and semi-cone angle δ = 3.2°. The driving forces at each stage are shown in Table 1. A phase diagram of the bubble in the full life cycle of bubbles can be found in Figure 11.

#### 3.3. The Influence of Driving Force on Speed in Each Stage

_{l}= 1130 kg/m

^{3}, viscosity μ = 0.9 mPa⋅s, surface tension coefficient σ = 30 mN/m; micromotor length L = 100 μm, larger opening radius R

_{max}= 10 μm, semi-cone angle δ = 3.2°; solution density ρ

_{g}= 1.33, contact angle α = 45°, β = 36°, angle of inclination φ = 4.5°. The viscosity μ = 0.9 mPa⋅s that we used falls into the general viscosity of the fluid ranges from 0.1–4 mPa·s. The generation rate of bubbles in the chemical reaction is n = 9.8 × 10

^{−4}m/s. For example, when the random nucleation point is in the middle of the tube, the stage of bubble ejection has a shorter duration. The velocity of the tubular micromotor in the various stages of the bubble is as shown in Figure 13.

_{max}= 10 μm, semi-cone angle δ = 3.2°. When moving in 5% peroxide solution, the average speed of motion is 450.93 μm/s which is in accordance with 431 μm/s reported by Li et al [5].

_{l}= 1130 kg/m

^{3}, viscosity μ = 0.9 mPa⋅s, surface tension coefficient σ = 30 mN/m, micromotor length L = 100 μm, larger opening radius R

_{max}= 10 μm, semi-cone angle δ = 3.2°, solution density ρ

_{g}= 1.33, contact angle α = 45°, β = 36°, angle of inclination φ = 4.5°.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 4.**The contact of the bubble with the inner wall of the micromotor. (

**a**) The volume of several parts as the bubble is in full contact with the inner wall of the micromotor. (

**b**) The geometry of the bubble and the forces it experiences.

**Figure 8.**Relationship between driving force and (

**a**) surface tension coefficient (

**b**) fluid viscosity.

**Figure 9.**Relationship between driving force and (

**a**) concentration of hydrogen peroxide C

_{H2O2}and (

**b**) opening radius.

**Figure 10.**Relationship between driving force and (

**a**) micromotor length (

**b**) tangent semi-cone angle.

**Figure 11.**A phase diagram of bubbles in the stage of (

**a**) bubble nucleation (

**b**) bubble growth (

**c**) bubble slip (

**d**) bubble ejection, where red represents the gas and blue represents the fluid.

**Figure 13.**The velocity of the micromotor in the full life-cycles of bubbles, namely bubble nucleation, growth, slip, and ejection. Inset: ejection stage.

**Figure 15.**The velocity of the tubular micromotor in the stage of bubble nucleation, slip, and ejection.

**Table 1.**The driving forces, average speed, and duration of the bubble-driven micromotor at different stages.

Stages | Equation | Driving Force | Average Speed (μm/s) | Duration (ms) |
---|---|---|---|---|

Nucleation | (14) | 0.03958 | 6.808 × 10^{−4} | 0–4.2 |

Growth | (56) | 2995 | 1011 | 4.2–5.25 |

Slip | (33) | 18.83 | 191.4 | 5.25–8.97 |

Ejection | (49) | 2360 | 15.65 | 8.97–8.98 |

Surface Area | Upstream Ring | Backflow Ring | Outer Wall | Inner Wall |
---|---|---|---|---|

Drag force (nN) | 0.028 | 0.043 | 0.002 | 0.191 |

Drag force (nN) | 0.264 |

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**MDPI and ACS Style**

Lin, Y.; Geng, X.; Chi, Q.; Wang, C.; Wang, Z.
Driving Forces of the Bubble-Driven Tubular Micromotor Based on the Full Life-Cycle of the Bubble. *Micromachines* **2019**, *10*, 415.
https://doi.org/10.3390/mi10060415

**AMA Style**

Lin Y, Geng X, Chi Q, Wang C, Wang Z.
Driving Forces of the Bubble-Driven Tubular Micromotor Based on the Full Life-Cycle of the Bubble. *Micromachines*. 2019; 10(6):415.
https://doi.org/10.3390/mi10060415

**Chicago/Turabian Style**

Lin, Yongshui, Xinge Geng, Qingjia Chi, Chunli Wang, and Zhen Wang.
2019. "Driving Forces of the Bubble-Driven Tubular Micromotor Based on the Full Life-Cycle of the Bubble" *Micromachines* 10, no. 6: 415.
https://doi.org/10.3390/mi10060415