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Dynamics and Mechanism of DNA Bending Proteins in Binding Site Recognition 1st Edition Yogambigai Velmurugu (Auth.)
Zhaohao Li · Xiaoyu Wang · Junwei Shen · Yuning Zhang
Cavity Dynamics and Splashing Mechanism in Droplets
SpringerBriefs in Energy
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Zhaohao Li · Xiaoyu Wang · Junwei Shen · Yuning Zhang
Cavity Dynamics and Splashing Mechanism in Droplets
Zhaohao Li
Key Laboratory of Power Station Energy Transfer Conversion and System Ministry of Education
North China Electric Power University
Beijing, China
Junwei Shen
Key Laboratory of Power Station Energy Transfer Conversion and System Ministry of Education
North China Electric Power University
Beijing, China
ISSN 2191-5520
SpringerBriefs in Energy
ISBN 978-3-031-54245-9
Xiaoyu Wang
Key Laboratory of Power Station Energy Transfer Conversion and System Ministry of Education
North China Electric Power University Beijing, China
Yuning Zhang
Key Laboratory of Power Station Energy Transfer Conversion and System Ministry of Education
North China Electric Power University Beijing, China
ISSN 2191-5539 (electronic)
ISBN 978-3-031-54246-6 (eBook) https://doi.org/10.1007/978-3-031-54246-6
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Preface
Cavitation activity within the droplets is of great significance in improving the performance of fuel atomization and other industrial applications. In this book, the characteristics of the cavitation bubble and the droplet dynamics are investigated based on the high-speed photography experiments, the bubble dynamics theory, and the numerical simulations. For the cavitation bubble dynamics, firstly, the movement trajectory of the cavitation bubble from the nucleation to the collapse is introduced. Secondly, the influence of droplets on the formation of cavitation bubble jets is analyzed. Thirdly, the paramount parameters dominating the collapse are theoretically given together with the propagation of shock waves within droplets. For the droplet splash dynamics, the droplet splash patterns are investigated with different eccentricities and radius ratio of bubbles and droplets. Based on the analysis of the flow field, the critical stability of the droplet surface is discussed, and the droplet breaking mechanism induced by cavitation bubble collapse is revealed. In addition, taking the vapor bubble and the diesel droplet as examples, the dynamic characteristics of several typical types of the droplet containing bubbles are analyzed. For vapor bubbles, the influences of vapor condensation on the propagation behaviors of gas flow and shock waves are discussed in detail together with the collapse mechanism of vapor bubbles. For diesel droplets, the influences of droplet viscosity, density, surface tension, and other physical properties on the growth, and the collapse stages of the cavitation bubble are analyzed.
Beijing, China
December 2023
Zhaohao Li
Xiaoyu Wang
Junwei Shen
Yuning Zhang
Acknowledgement This book was financially supported by the National Natural Science Foundation of China (Project No.: 51976056).
Chapter 1 Introduction
1.1 Research Background
Atomization technology can convert continuous liquid into discrete droplets. As shown in Fig. 1.1, the atomization technology has been widely utilized in many fields, e.g., fuel atomization [1], spay extinguishing [2], medical atomization [3], flue gas desulfurization and denitrification [4], wastewater jet flash evaporation [5], and atomization dust removal [6]. The introduction of atomization technology can reduce the particle size of droplets, promote the uniform distribution of small droplets, and increase the contact area between liquids and gases. Among them, cavitation atomization technology can effectively reduce the particle size of droplets by inducing the violent oscillation and the rapid collapse of cavitation bubbles within the droplets, which is worthy of attention.
Other typical atomization technologies include pressure atomization, gas flow atomization, electrostatic atomization, and so on. Their main characteristics are showninTable 1.1. The pressure atomization technology utilizes the way of pressurizing the liquid, so that it is sprayed from the nozzle with a high speed forming spray droplets. To obtain better atomization performance, the selected injection pressure is constantly increased. Wang [7] found that when the fuel injection pressure was increased from 200 to 250 MPa, fuel consumption could be saved by 3% with further 8% reduction of NOx emission. However, the high injection pressure increases the energy consumption of the system. To obtain a higher injection pressure, the nozzle diameter is greatly reduced. Therefore, the problem of blockage is easy to occur during the long-term operation. The gas flow atomizing nozzle contains two sets of pipelines, which spray liquid and gas at the same time respectively. It relies on the shear effect of gas on the liquid to divide the continuous medium to form spray droplets. The gas–liquid ratio is an important factor affecting the atomization performance of this technology. Yao et al. [8] found that when the gas–liquid mass ratio approaches two, the particle size of spray droplets reaches the minimum value. Hence, the working of the gas flow atomizing nozzle requires a great amount of gas
consumption. Furthermore, the installation of two sets of pipelines inside the nozzle increases the structure complexity and the cost of equipment.
Electrostatic atomization technology utilizes a high-voltage power supply to make the liquid surface obtain extra charges. The charged liquid is ejected from the nozzle under the action of electric field traction and is further broken into small droplets under the influence of electric disturbance. Due to the need to overcome the surface tension of the liquid, the electrostatic atomization technology requires a high voltage. There are some security risks in actual application, and corresponding prevention should be considered.
Cavitation atomization technology utilizes pressure waves or ultrasonic waves to induce cavitation bubbles inside the spray droplets. Under the action of highintensity shock waves released by cavitation bubble collapse, cavitation atomization
Fig. 1.1 Application of droplet atomization
Table 1.1 Comparison of three typical atomization technologies
technology can obtain spray droplets with small particle sizes with relatively low energy consumption. Unfortunately, due to the complex dynamic behaviors of cavitation bubbles inside the droplets, the mechanism of droplet breakup affected by the cavitation is still unclear.
Aiming at cavitation atomization technology, this book adopts high-speed photography experiments, bubble dynamics theory, and numerical simulation methods to reveal the interactions between dynamic behaviors of cavitation bubbles and droplets, the formation characteristics of shock waves and microjets. Furthermore, the mechanism of droplet action on the dynamic behaviors of the cavitation bubbles and the physical mechanism of droplet breakup induced by the cavitation bubble collapse are revealed.
1.2 Research Status
During the operation of cavitation atomization technology, cavitation bubbles and droplets interact with each other. Influenced by the collapse behaviors of cavitation bubbles, the droplets will splash and break into daughter droplets. Due to the influence of droplets, the growth and the collapse characteristics of cavitation bubbles are different from those near other boundaries. For example, Guo et al. [12] found that the dimensionless curvature between the gas–liquid interface and the bubble surface is a key parameter (e.g., affecting the local pressure peak).
In terms of bubble dynamics, the stability and the collapse characteristics of the cavitation bubble within a droplet have received extensive attentions. Zeng and Lee [13] employed a linear stability theory (without fluid viscosity) to discuss the stability of a cavitation bubble in a dimethyl ether droplet and its influencing parameters. On this basis, Zhang and Zhang [14] derived the dispersion equation of disturbance development during the growth of the cavitation bubble in a dimethyl ether droplet. They established a theoretical criterion for the cavitation bubble collapse, taking into account the influence of viscous force. Wang et al. [15] derived the ordinary differential equations describing the growth and the collapse of the cavitation bubble under the microgravity condition to analyze the motion and the curvature variation rules near the interface between cavitation bubbles and droplets. Obreschkow et al. [16] established a theoretical prediction model of cavitation bubble collapse time based on the Rayleigh–Plesset equation, ignoring surface tension, compressibility, viscosity, and mass transfer. Lv et al. [17] adopted the volume of fluid method to establish a theoretical model for the dynamics of the cavitation bubble inside a diesel droplet and analyzed the mechanism of forces on the growth and the collapse of cavitation bubbles.
In terms of droplet splash dynamics, scholars mainly focused on the droplet morphological characteristics and formation mechanisms. They have observed that the morphologies of droplets induced by cavitation bubbles include stable coronary structure [18], unstable coronary structure [19] and noncoronary structure [20]. To
better explain the dynamic characteristics, partition diagrams of the droplet deformation pattern is usually employed. Based on different methods of droplet generation and control, the observed droplet deformation patterns vary greatly. For example, the droplet morphologies summarized by Liang et al. [21] included annular, lateral jet and jet at the pole. The deformation characteristics described by Wang et al. [22] included spatter, ventilation, and steady state. Kobel et al. [23] classified the droplet dynamics induced by cavitation bubble collapse into microjets, spatter, and burst. Avila and Ohl [24] defined the droplet breakage modes as droplet atomization, unstable transparent water layer, stable transparent water layer, and droplet oscillation. To explain the experimental phenomenon, Zeng et al. [25] calculated the baroclinity of the droplet surface, analyzed the law of influence of vorticity, and revealed that the spattering phenomenon was caused by the Rayleigh–Taylor instability at the interface. Based on Zeng et al. [25], Wang et al. [22] further proposed theoretical criteria including the dimensionless parameter to define the differential characteristics of deformation modes.
1.3 Description of the Book
Figure 1.2 shows the chapter layout structure of the book. The main content of this book is introduced as follows: In this chapter, the research background and the application fields of cavitation atomization technology are introduced. The research of bubble dynamics and droplet splash dynamics are reviewed. In addition, the content of each chapter of the book is briefly introduced. Chapter 2 mainly introduces the primary research methods for investigating the dynamic behaviors of cavitation bubbles and droplets, including high-speed photography experiments, theoretical analysis of the bubble dynamics model, and numerical simulation based on OpenFOAM. In Chap. 3, the dynamic characteristics of cavitation nucleation, jet and oscillation and their main influencing factors are analyzed. In Chap. 4, several typical splashing modes of droplets are observed, and the theoretical basis of the critical stability is proposed based on the distribution characteristics of the flow field near the droplet surface. In Chap. 5, the dynamic behaviors between droplets containing air bubbles and vapor bubbles are compared, and the influences of vapor condensation on the gas flow and the shock wave propagation are discussed with the evolution mechanism of the vapor bubble revealed. In Chap. 6, the mechanism of various forces on the growth and the collapse of cavitation bubbles inside a diesel droplet are discussed. In Chap. 7, the primary concluding remarks of the present book are summarized.
Fig. 1.2 Structure of the present book
References
1. Shahnazari S, Astaraki MA, Sobati MA et al (2023) Atomization characteristics of different water/heavy fuel oil emulsions in a pressure-swirl injector. J Energy Inst 108:101204
2. Voytkov IS, Kuznetsov GV, Strizhak PA (2021) The critical atomization conditions of highpotential fire suppressant droplets in an air flow. Powder Technol 384:505–521
3. Broniarz-Press L, Ochowiak M, Matuszak M et al (2014) The effect of shear and extensional viscosity on atomization in medical inhaler. Int J Pharmaceut 468:199–206
4. Ma L, Duan XX, Wu JW et al (2022) Simultaneous desulfurization and denitrification of flue gas enabled by hydrojet cyclone. J Clean Prod 377:134205
5. Quan XJ, Wang FP, Zhao QH et al (2009) Air stripping of ammonia in a water-sparged aerocyclone reactor. J Hazard Mater 170:983–988
6. Xiong W, Fu LD, Zhao G et al (2021) A new dust removal technology used in hot rolling mills: atomized spray with higher dedusting efficiency. J Clean Prod 311:127574
7. Wang Y (2018) Development of a droplet breakup model for ultra-high pressure fuel spays and numerical study of atomization mechanisms. Beijing Jiaotong University (in Chinese)
8. Yao KH, Jin Y, Zheng M et al (2020) Test study on spray performance of plain-jet airblast atomizer. Aeroengine 46:75–80 (in Chinese)
9. Shen LX, Fang GY, Wang SZ et al (2022) Numerical study of the secondary atomization characteristics and droplet distribution of pressure swirl atomizers. Fuel 324:124643
10. Hammad FA, Sun K, Che ZZ et al (2021) Internal two-phase flow and spray characteristics of outside-in-liquid twin-fluid atomizers. Appl Therm Eng 187:116555
11. Li SJ, Zhuo Z, He LJ et al (2019) Atomization characteristics of nano-Al/ethanol nanofluid fuel in electrostatic field. Fuel 236:811–819
12. Guo WL, Li HC, Wang JZ et al (2019) Research progress on interaction between a single cavitation and free surface, Chinese. J Theor Appl Mech 51:1682–1698 (in Chinese)
13. Zeng YB, Lee CF (2001) An atomization model for flash boiling sprays. Combust Sci Technol 169:45–67
14. Zhang P, Zhang YS (2009) Numerical modeling of dimethyl ether (DME) bubble growth and breakup. Chinese Sci Bull 54:2457–2463
15. Wang QG, Yao W, Quan XJ et al (2018) Validation of a dynamic model for vapor bubble growth and collapse under microgravity conditions. Int Commun Heat Mass 95:63–73
16. Obreschkow D, Kobel P, Dorsaz N et al (2006) Cavitation bubble dynamics inside liquid drops in microgravity. Phys Rev Lett 97:094502
17. Lv M, Ning Z, Sun CH (2016) Study on the growth and collapse of cavitation bubble within a droplet, Chinese. J Theor Appl Mech 48:857–866 (in Chinese)
18. Heijnen L, Su PAQ, Zhao X et al (2009) Cavitation within a droplet. Phys Fluids 21:091102
19. Thoroddsen ST, Takehara K, Etoh TG et al (2009) Spray and microjets produced by focusing a laser pulse into a hemispherical drop. Phys Fluids 21:112101
20. Martinez JPP, Juan JCRS, Rodriguez CB et al (2017) Controllable direction of liquid jets generated by thermocavitation within a droplet. Appl Optics 56:7167–7173
21. Liang Y, Jiang YZ, Wen CY et al (2020) Interaction of a planar shock wave and a water droplet embedded with a vapour cavity. J Fluid Mech 885:R6
22. Wang JZ, Li CH, Guo WL et al (2021) Rayleigh-Taylor instability of cylindrical water droplet induced by laser-produced cavitation bubble. J Fluid Mech 919:A42
23. Kobel P, Obreschkow D, de Bosset A et al (2009) Techniques for generating centimetric drop in microgravity and application to cavitation studies. Exp Fluids 47:39–48
24. Avila SRG, Ohl CD (2016) Fragmentation of acoustically levitating droplets by laser-induced cavitation bubbles. J Fluid Mech 805:551–576
25. Zeng QY, Avila SRG, Voorde ST et al (2018) Jetting of viscous droplets from cavitation-induced Rayleigh-Taylor instability. J Fluid Mech 846:916–943
Chapter 2 Research Methods
This chapter focuses on the research methods of dynamics of interactions between cavitation bubbles and droplets, including high-speed photographic experiments, theoretical analysis of bubble dynamics, and numerical simulation of OpenFOAM. From the experimental aspect, according to the droplet morphology produced in different modes, the droplets can be divided into hemispherical, ellipsoidal, and spherical droplets. The difference in the droplet morphology affects the analysis of dynamic behavior characteristics. From the theoretical view, based on the classical Rayleigh–Plesset equation, the dynamic equations of the cavitation bubble inside a droplet are developed considering the influence of surface tension and the viscous force. For the simulation, the governing equations are introduced together with the volume of fluid (VOF) and the large-eddy simulation (LES) methods.
2.1 High-Speed Photography Experiment
2.1.1 Hemispherical Droplet
Figure 2.1 shows the experimental platform for the dynamics of the hemispherical droplet containing cavitation bubbles, and the experimental method refers to Ref. [1]. A microinjection pump is utilized to transport the liquid from the bottom of the vertical transparent pipe to the top. Due to the surface tension of the liquid, a millimeter-magnitude hemispherical droplet floating on the pipe outlet is formed. The droplet size is changed by adjusting the injection time of the microinjection pump. After the Nd: YAG pulsed laser is focused, the cavitation bubble is generated inside a droplet. The size of the cavitation bubble is changed by adjusting the energy attenuator to control the input laser energy. The dynamic behaviors of the droplet and the cavitation bubble are recorded by a high-speed camera, which is controlled synchronously with the laser generator by a digital delay generator. Through the
reconstruction of high-speed photographic images, the whole dynamic process of the cavitation bubble and the splash trajectory of the hemispherical droplet are obtained in detail. The relevant instrument parameters are shown in Table 2.1.
Figure 2.2 shows the definition of radius ratio parameters of the hemispherical droplet. Among them, point O not only represents the laser focus position, but also represents the cavitation bubble birth position. The white area represents the hemispherical droplet, and the blue area represents the cavitation bubble. Affected by the gravity, the droplet is not an ideal spherical structure, but an approximate hemispherical structure. Based on this, the cavitation radius ratio is defined in Eq. (2.1).
Device name
Model
High-speed camera X113
Laser generator
Penny-100A-SC
Digital delay generator DG535
High-power focusing lens
LMH-10X-532
Light emitting diode Bolang
Microinjection pump Lande
Acrylic material supporting pipe –
Parameter
60,000 FPS
Maximum energy: 100 mJ
Delay resolution: 5 ps
Effective focal length: 20 mm
Power: 300 W
Flowrate: 0.1 ml/h
Outside diameter: 5.06 mm
Inside diameter: 3.05 mm
Fig. 2.1 Experimental platform for dynamic characteristics of a hemispherical droplet containing cavitation bubble
Table 2.1 Main instrument performance parameters of the hemispherical droplet experimental platform
Fig. 2.2 Schematic diagram of radius ratio parameter definition
where “λ” represents the ratio of the equivalent radius of the cavitation bubble to the equivalent radius of the droplet. “Rbx ” and “Rby ” indicate the dimensions of the long and the short axes of the cavitation bubble. “Rdx ” and “Rdy ” indicate the dimensions of the long and the short axes of the droplet.
Figure 2.3 shows the definition of the eccentricity of the bubble in the hemispherical droplet. Let us take the definition of eccentricity in the Y -axis direction as an example. In the Y -axis direction, the distance between the droplet center Od and the cavitation bubble birth position Ob is d bY , and the distance between the droplet surface and the droplet center is d dY . Based on this, the eccentricity in the Y -axis direction is defined by Eq. (2.2). Similarly, the eccentricities in the X -axis direction and the 45° axis direction could be also obtained, as shown in Eqs. (2.3) and (2.4) respectively.
Fig. 2.3 Eccentricity definition diagram
2.1.2 Ellipsoidal Droplet
Figure 2.4 shows the experimental platform for the dynamics of ellipsoidal droplets containing cavitation bubbles, and the experimental method refers to Ref. [2]. Furthermore, acoustic transducers and reflectors are utilized to create sound fields. With the sound field added, an ellipsoidal suspended droplet is formed, and the droplet size could be changed by adjusting the frequency of the transducer. The generation and the control methods of the cavitation bubble are similar to those in the hemispherical droplet experiment platform, and the Nd: YAG pulsed laser is also utilized for the bubble generation. During the experiments, the laser emission energy is up to 5 mJ. The frame rate of the high-speed camera for recording dynamic behaviors is 1 Mfps.
Figure 2.5 shows the definition of geometric parameters of the ellipsoidal droplet. The radius ratio and the eccentricity of the X -axis and the Y -axis are defined in Eqs. (2.5–2.7). In addition, to explore the energy of forming a cavitation bubble, the energy parameter E is defined in Eq. (2.8).
Fig. 2.4 Experimental platform for dynamics characteristics of an ellipsoidal droplet containing a cavitation bubble
Fig. 2.5 Definition of geometric parameters of an ellipsoidal droplet
where “Φ ” represents the radius ratio of an ellipsoidal droplet. “ε X ” and “ε Y ” represent the eccentricities of the X -axis and the Y -axis, respectively. “δ X ” and “δ Y ” represent the distance between the center of cavitation bubble and the center of droplet on the X -axis and the Y -axis, respectively. “κ ” represents the percentage of laser energy utilized for cavitation bubble nucleation. “E l ” represents the energy of the laser emitted. “V d ” indicates the volume of the cavitation bubble. “pa ” indicates environmental pressure.
2.1.3 Spherical Droplet
Figure 2.6 shows the experimental platform for the dynamics of spherical droplets containing cavitation bubbles. Compared with the experimental platform of hemispherical and ellipsoidal droplets, the feature of this experimental platform is that the droplet relies on the microgravity environment to maintain the spherical profile.
Fig. 2.6 Experimental platform for dynamic characteristics of a spherical droplet containing cavitation bubble. Reprinted with the permission from Ref. [4] Copyright (2009) (Springer Nature)
The experiments were carried out during a manned parabolic flight by the European Space Agency [3, 4]. Specifically, the gravity in this environment is about 0.02–0.05 g, where “g” represents the acceleration of gravity. The droplet size can be controlled by adjusting the power of the micropump that injects a droplet into the experimental platform. The formation of the cavitation bubble inside the droplet depends on the discharge of the electric spark between two thin electrodes. Figure 2.7 shows the basic principle and the spherical droplet containing the cavitation bubble. The high-speed camera recording the dynamic behaviors was set to be 12,000 fps.
The geometric contour of the spherical droplet is more regular, which provides great convenience for the definition of relevant parameters. The radius ratio and the eccentricity are defined in Eqs. (2.9) and (2.10).
where “α ” represents the radius ratio of the spherical droplet. “ε ” represents eccentricity. “Rb,max ” represents the maximum radius of the cavitation bubble. “Rd,min ” represents the initial radius of the droplet. “d ” represents the distance between the center of the droplet and the center of the cavitation bubble.
Fig. 2.7 Schematic diagram of a spheroidal droplet containing cavitation bubble in a microgravity environment: a cavitation bubble generation method; b high-speed photography. Reprinted with the permission from Ref. [4] Copyright (2009) (Springer Nature)
2.2 Bubble Dynamics Theory
2.2.1 Bubble Dynamics Model
The establishment of the bubble dynamics model is based on Ref. [1]. Figure 2.8 shows the physical model of a spherical droplet containing a cavitation bubble. Based on the classical Rayleigh–Plesset equation considering the influence of surface tension and the viscous force, the dynamic model suitable for the bubble in a spherical droplet is expanded from the point of view of force balance. The kinetic equation of cavitation bubble oscillation is shown in Eq. (2.11).
where “p0 ” represents the ambient pressure. “Rb ” represents the instantaneous radius of the cavitation bubble. “Rd ” represents the instantaneous radius of the droplet. “Rb0 ” represents the instantaneous radius of the cavitation bubble at equilibrium. “Rd 0 ” represents the instantaneous radius of the droplet at equilibrium. “μL ” and
Fig. 2.8 Physical model of a spherical droplet containing cavitation bubble
“ μG ” represent the viscosity coefficients of the liquid and the gas, respectively. “ρ L ” represents the liquid density. “σ ” represents the surface tension at the gas–liquid interface.
Assuming that the distance between a point in the liquid and the center of the cavitation bubble is r , the pressure inside the liquid is p(r, t ), and the radial outward velocity is u(r, t ). According to the law of mass conservation, the velocity can be obtained as Eq. (2.12).
The Navier–Stokes equation for Newtonian fluids in the spherical coordinates can be expressed as Eq. (2.13).
Substituting Eq. (2.12) into Eq. (2.13), (2.14) can be obtained.
Through integrating from the radius of the cavitation bubble to the radius of the droplet, which is shown in Eqs. (2.15) and (2.16).
Substituting Eqs. (2.15) and (2.16) into Eq. (2.14), Eq. (2.17) can be obtained.
where “pout ” represents the gas pressure at the junction between droplet and outside air. “pin ” represents the gas pressure at the junction between droplet and cavitation bubble. “pd ” represents the liquid pressure at the junction between droplet and outside air. “pb ” represents the liquid pressure at the junction between droplet and cavitation bubble.
According to the law of mass conservation and the fact that the liquid is incompressible and does not evaporate or condense during bubble oscillation, the mathematical relationship can be obtained as Eq. (2.22).
Thus, the equation can be derived as Eq. (2.23).
The relationship between Rb and Rd can be described by Eq. (2.24).
2.2.2 Analytical Solution of Radial Motion Equation
The derivation of the radial motion equation is based on Ref. [1]. Equation (2.25)is the dynamic equation of a cavitation bubble inside a droplet considering the surface tension term.
The theoretical collapse time of the cavitation bubble is solved considering the surface tension. Multiplying Rb R 2 b on both the left and the right sides of Eqs. (2.25), (2.26) can be obtained as follows.
Then, by integrating the time, Eq. (2.27) can be obtained as follows.
When Rb = Rb,max , the expression of relevant parameters are shown in Eqs. (2.28) and (2.29).
Then, substituting the equations of the main parameters into Eq. (2.27), Eq. (2.30) can be obtained as follows.
When the radius of the bubble during the collapse reaches the maximum value, the collapse time can be obtained, which can be expressed as Eq. (2.31).
When it is subjected to external pressure, the cavitation bubble will oscillate. The variations of the bubble radius can be solved through the small perturbation method, which can be expressed as Eqs. (2.32) and (2.34).
where “ε ” represents the amplitude of small disturbance vibration. According to the aforementioned derivation, Eqs. (2.35–2.37) can be obtained as follows.
According to Eqs. (2.24), (2.38) can be obtained.
By retaining the first-order term, the simple harmonic oscillation equation considering damping can be obtained as Eq. (2.39).
Equation (2.40) is obtained according to the standard form of simple harmonic oscillation.
After simplification, Eq. (2.41) can be obtained.
The expressions of natural frequency and total damping are shown in Eqs. (2.42) and (2.43) respectively.
where “β tot ” represents the total damping constant term. “β ac ” represents the acoustic damping constant term. “β vis ” represents the viscous damping constant term. “β th ” represents the thermal damping constant term. Since only the viscous damping term is considered, the total damping can be obtained as Eq. (2.44).
Since x 1 and p0 ei(ω t +δ ) are of the same frequency, Eq. (2.45) can be obtained by considering the damping effect.
The complex form of the derivative of Eq. (2.45)isshown in Eq.(2.46).
Substituting Eqs. (2.41) and (2.46) into Eq. (2.40), Eq. (2.47) can be obtained as follows.
According to the complex number operational rules, there is a mathematical relationship as shown in Eq. (2.48).
The phase difference relation can be expressed by Eq. (2.49).
Substituting Eqs. (2.48) and (2.49) into Eq. (2.47), the numerical expression of the simple harmonic amplitude with the first-order accuracy is obtained, which can be expressed as Eq. (2.50).
The phase difference relation of the first-order caused by the viscosity term can be obtained as Eq. (2.51).
Similarly, the bubble dynamics equation considering the surface tension and the viscosity terms can be solved with the second-order accuracy, which can be expressed as Eq. (2.52).
Therefore, the standard form of harmonic oscillation with the second order can be expressed as Eq. (2.53).
The expressions of natural frequency and the total damping can be expressed as Eqs. (2.54) and (2.55).
Therefore, Eq. (2.56) can be obtained as follows.
Parameter ϕ 0 is employed to represent the right part of Eq. (2.52), and after performing a trigonometric functional transformation, Eq. (2.57) can be obtained.
According to the auxiliary angle equations, Eqs. (2.58) and (2.59) can be obtained.
Substituting Eq. (2.57) into Eq. (2.53), Eq. (2.60) can be obtained.
According to Eqs. (2.58) and (2.59), the components of Eq. (2.60) can be sorted out, and the results are shown in Eqs. (2.61) and (2.62).
Since x 2 and f [cos(2ω t )] are of the same frequency, Eq. (2.63) can be obtained.
After expressing Eq. (2.63) and its derivatives in the complex form, Eq. (2.64) can be obtained as follows.
Substituting Eqs. (2.64)into(2.60), (2.65) can be obtained.
Combined with the simple harmonic oscillation equation, the constant term needs to be consistent, which can be obtained as Eq. (2.66).
The phase difference relationship is shown in Eq. (2.67).
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