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Zhaohao Li · Xiaoyu Wang · Junwei Shen · Yuning Zhang

Cavity Dynamics and Splashing Mechanism in Droplets

SpringerBriefs in Energy

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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

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024

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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

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Li et al., Cavity Dynamics and Splashing Mechanism in Droplets, SpringerBriefs in Energy, https://doi.org/10.1007/978-3-031-54246-6_1

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

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 Z. Li et al., Cavity Dynamics and Splashing Mechanism in Droplets, SpringerBriefs in Energy, https://doi.org/10.1007/978-3-031-54246-6_2

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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to the present time. But there are two points which I wish to explain concerning this hill of Kauiki. First: How it originated; secondly, the famous localities near to or connected with this hill. Let us therefore examine some of the erroneous ideas of the olden time.

: .

Olden people differed in this respect, four accounts of its origin being given 1. That this hill originated from the placenta of Hamoa,77 some claim that it originated [546]from the parents of Pele and her host; others, from Kaihuakala78 and Kahaule,79 his wife; still others, from Kalalawalu; and these are their accounts: Pele, Hiiaka and Puuhele were born of their parents; Kahinalii was their mother. However, Pele and Hiiaka were born with human bodies, while Puuhele was a bloody foetus when she was born. The elders despised this body, because when they beheld

mai ka wa kahiko mai a hiki i keia wa. Aka nae, elua a’u mea e hoakaka aku ai no keia puu o Kauiki; akahi, kona wahi i loaa mai ai; elua, na wahi kaulana e pili ana me keia puu. Nolaila, e nana ae kakou i ka manao kuhihewa o ka poe kahiko.

, .

Ua manao like ole ia e ka poe kahiko, eha wahi i loaa mai ai

keia puu. Ua loaa mai keia puu mai ka iewe mai o Hamoa, a o kekahi, mai na makua mai o Pele ma, [547]a o kekahi hoi, mai a Kaihuakala mai laua me Kahaule kana wahine, a o kekahi, mai a Kalalawalu mai; a penei na manao o lakou:

Ua hanauia mai o Pele ame

Hiiaka, ame Puuhele, e ko lakou mau makua, e Kahinalii ka makuahine o lakou; eia nae, o Pele ame Hiiaka ua hanau kino kanaka maoli ia mai laua, a o Puuhele hoi, ua hanau puu koko mai oia. No ka makemake ole

it it was not a human body, but only a foetus of blood; so the two conspired and said one to the other: “It were better for us to throw away our younger sister. How can we care for it? Of course we would care for it if it had a human body!” So they threw away the bloody foetus without the knowledge of the parents.

mai o na kaikuaana i keia kino, no ka mea, i ka nana ana iho, aole he kino maoli, aka, he puu koko wale no, nolaila, kuko iho la laua, a olelo aku la kekahi i kekahi: “E aho e kiloi kaua i ka pokii o kaua. Auhea hoi ka hoihoi aku, i ku no ka hoi ka hoihoi aku ke kino kanaka.” A o ko laua nei kiloi aku la la no ia i ua puu koko nei me ka ike ole o na makua i ko laua hoolei ana mai.

From that time onward this bloody foetus kept on traveling; it met Alenuihaha, that is the channel between Maui and Hawaii, and Alenuihaha asked it: “A journey! whither going?” The bloody foetus replied: “Just going to see what is ahead here; I was cast away by my elders and that is the cause of this journeying.”

A ma ia manawa mai, ua hoomaka keia puu koko e hele, a halawai mai oia me Alenuihaha, oia kela moana mawaena o Maui nei ame Hawaii, a ua ninau aku o Alenuihaha i ua puu koko nei: “He huakai! e hele mahea?” Olelo mai ua puu koko nei: “E hele ana ma keia wahi aku nei la, i kiloi ia mai au e o’u mau kaikuaana, a oia keia la e hele nei la.”

After their conversation the foetus passed on until it landed at Nuu;80 this place where it landed is at Kaupo; she walked on in the form of a human being.

A pau aela ka laua nei kamailio ana, o ka hoomaka mai la no ia o ua puu koko nei e hele mai, a pae ana keia i Nuu, aia ma Kaupo keia wahi ana i pae mai

When Nuu looked he beheld a most beautiful woman. She kept right on until she met Puuomaiai;81 she also was a good-looking woman. Puuomaiai asked the stranger: “To where are you journeying?” The other replied: “Going along to places ahead of here.” Puuomaiai requested that they be friends, and they became such. She was received; they ate, and after being satisfied they went along. While they were going Manawainui82 voiced a chant, thus:

A restlessness seizes me on account of thee, O stranger! It is causing numbness to my feelings.

Is it an antagonist? Or is it a traveling companion?

The foetus turned back and asked: “Why are you calling from behind?” Manawainui replied:

“Because I saw you very able in walking, that was why I called you.” “What have you to present that you should call?” “Because I saw that you were nothing but a bloody foetus cast away by your

ai, a hele kino kanaka mai la keia, a i ka ike ana’ku o Nuu, he keu a ka wahine maikai. O ko ianei hele mai la no ia a halawai me Puuomaiai, he wahine maikai no hoi keia. Olelo aku la o Puuomaiai i ka malihini: “E hele ana mahea kau huakai?” Olelo mai la kela: “E hele aku ana ma keia wahi aku.” A noi mai la o Puuomaiai e hoaikane, a hoaikane iho la laua nei, a hookipa ia iho la keia, ai laua a maona, a hele aku la laua nei. O ko laua nei hele ana, kani ana ka Manawainui mele, penei no ia:

Kau ka lia ia oe e ka malihini, Ke hele nei a maeele kuu manawa, He hoa paio paha, He hoa makaikai paha e?

Huli ae la ua puu koko nei i hope, a olelo aku: “Heaha kau o ke kahea ana mahope?” Pane mai o Manawainui: “No kuu ike ana ’ku nei ia oe, i ko mama i ka hele, nolaila kahea aku nei au ia oe.” “Heaha kau makana nui o ke kahea ana?” “No kuu ike ana aku nei ia oe, he puu koko wale

elders! and yet here you are walking! You had better be named Puuhele.”83 After their conversation the two came along until they arrived at Hana.

Puuomaiai said to Puuhele: “I had better return; we have come together; there is yet another time for meeting.” Puuhele consented, at the same time saying: “I know not how prosperous this journey is; I may see you again or I may not,” and Puuhele kissed her friend on the nose.

no i kiloi ia mai e kou mau

kaikuaana, nolaila, hele mai nei oe, e aho e kapa ia kou inoa o Puuhele.” A pau ka laua nei kamailio ana, o ko laua nei hele mai la no ia a hiki mai i Hana. Olelo aku a Puuomaiai ia

Puuhele: “E aho e hoi au, ua hele pu mai la no hoi kaua, eia aku no hoi ua wa e hui hou ai.”

O ka ae mai la no ia o Puuhele, me kae i mai: “Aole i akaka ko’u pono e hele nei, he ike hou paha ia oe, aole paha,” a o ka honi mai la no ia o ua Puuhele i ka ihu o ke aikane.

She no sooner commenced on her journey again when she met Kanahaha. Kanahaha first spied Puuhele, however, and she immediately became dead, her limbs extended and spread apart. When Puuhele came up, she found the other dead. This place Kanahaha is a hill from which gushes forth a spring of water to this day. Puuhele [548]passed right on until she met Lehoula. Lehoula said to her:

“What a beautiful woman you are!” Lehoula wished to follow her. Coming to Wananalua,84

O ko ianei hoomaka ana e hele, halawai aku la keia me Kanahaha, ike e mai la o Kanahaha, a o kona make koke iho la no ia, a waiho kahelahela iho la na uha o ua wahine nei. O ko ia nei hele aku no ia, a ike ua make. O keia wahi o Kanahaha, he [549]punawainui a hiki i keia wa. Hele loa aku aku la no o Puuhele a halawai me Lehoula. Olelo mai o Lehoula: “Nani hoi ka wahine maikai o oe,” a makemake o Lehoula e hele pu meia, a hiki ma Wananalua, hoohiki iho la o Puuhele ia wahi,

Puuhele made a vow to stay there, and she said to Lehoula: “I have a great desire for this place; I came along looking for a good place to locate in, and I have just located it. Therefore I will abide here until my death.” Lehoula returned to her usual home.

Puuhele lived here. When Kaihuakala saw her he came down and met her and spoke reprovingly. “By what right did you come here?” Puuhele humbly replied: “I came to see the country, and seeing this beautiful country, I vowed to stay.” Then Puuhele was killed, and buried. She used her divine power, the hill rose high.

Kaihuakala named the hill Kauiki, and that is the name by which it is known to the present time; that also is the cause for the song some people use, thus:

a olelo aku ia Lehoula: “Nani mai la kuu makemake i keia wahi; o i hele mai nei e imi i kahi kupono e noho ai, akahi no a loaa.

Nolaila, e noho au a hiki i ko’u wa e make ai,” a o ka hoi mai la no ia o Lehoula a hiki ma kona wahi mau.

Noho iho la no hoi o ua Puuhele; a ike mai la o Kaihuakala o ka iho mai la no ia o Kaihuakala a halawai me ia, a pane kiko e mai no o Kaihuakala: “Heaha kou kuleana nui o ka hiki ana mai o onei nei?” Olelo aku la o Puuhele me ka nawaliwali: “I hele mai nei hoi au i ka makaikai, a loaa keia wahi maikai, a hoohiki iho nei e noho.”

Ia manawa, pepehi ia iho la o ua o Puuhele a make, kanuia iho la a nalo. E hao ae ana keia i kona mana, o ka pii ia o ua puu nei a kiekie, kapaia ka inoa o ua puu nei o Kaihuakala, o Puuiki, a mahope kapaia o Kauiki, a oia ka inoa e kapaia nei a hiki i keia wa, a oia iho la ke kumu o ke mele a kekahi poe; a penei no ia:

At Kauiki is the mail bag of the wind,

Being tossed about by the heavy black rain;

Facing is the surf of Anini to the windward,

Facing so as to head for the water of Punahoa.

But according to the idea of some people it was Lalawalu who brought it from Kahiki;85 she brought it as her foster child, but because she was vexed at the child for constantly nipping her breast, therefore the mother made up her mind to leave it. She brought it along to Koloa, Kauai, and there she wanted to cast it away, but the child did not fancy staying there. She persevered in carrying the child until they arrived at Kaena;86 again the child did not desire to be left there, so it was brought along until they landed at Kawaipapa,87 Hana, East Maui, and it was left with him; and there it stands until this day. That was the idea of some olden people.

Aia i Kauiki ka eke leta a ka makani, Kahi paialewa ia mai la e ka lau awa, Huli ka nalu o ke Anini i ka makani, I hoohuli no a huli i ka wai o Punahoa.

Aka, o ka manao o kekahi poe, na ka Lalawalu i lawe mai, mai Kahiki mai, i lawe keiki hanai ia mai, a no ka uluhua o ka

Lalawalu i ke nahu pinepine o ua keiki nei i ka waiu, nolaila, kupu ka manao iloko o ka makuahine e haalele ia ia, lawe mai la a pae ma Koloa, i Kauai. Manao ia e kiloi, aole nae he makemake o ua keiki nei e noho malaila. Hoomanawanui mai la no oia i ka hii ana, a hiki i Kaena ma Oahu. Aole no he makemake o ua keiki nei, nolaila, lawe loa ia mai a pae ma Kawaipapa ma

Hana i Maui Hikina, a hoonoho ia me ia a hiki i keia wa, a pela iho la kekahi manao o ka poe kahiko.

:

.

On this hill are many famous places; for instance, right on top of this hill was the house in which Peapea88 was consumed by fire, when he was burnt out by Liionaiwaa and others; thus the saying at the present time, “Consumed by fire is Peapea.” A little to the south of this hill is a famous landing place for canoes, called Kaihalulu (the roaring sea); concerning this place is the saying now quoted: “The roaring sea in the presence of Kauiki.” At the same place, too, are the coconuts of Kane; right makai of this place is a large rock in the sea which is called Mokuhano.

To the east of Kauiki is Pueokahi;89 this place was so named on account of an owl belonging to the chief, Peapea. When the bird saw there were plenty of people, it flew to the door of the chief, indicating a multitude. Afterwards it was killed, and that was why it was called Pueokahi.

:

Aia ma keia puu, he nui na wahi kaulana, oia hoi, maluna pono o keia puu ka hale o Peapea i pau ai i ke ahi, i puhiia ai e Liionaiwaa ma, a oia ka mea i oleloia: “Pau Peapea i ke ahi.”

Aia hoi ma ka hema iki o keia puu he awa pae waa keia, o Kaihalulu ka inoa, no ia wahi keia olelo e olelo ia nei, Kaihalulu i ke alo o Kauiki. Aia no hoi malaila na niu a Kane; aia aku makai ponoi o ia wahi he pohaku nui iloko o ke kai, ua kapaia ka inoa o ia pohaku o Mokuhano. Aia hoi ma ka hikina ponoi o Kauiki o ka Pueokahi, ka mea i kapaia ai ka inoa o ia wahi, he pueo no na ke alii na Peapea; aia ike ua pueo nei i ka nui o kanaka lele mai no ia a kau ma ke kikihi puka o ke alii, alaila, ua nui kanaka; a mahope pepehiia a make, a oia ka mea i kapaia ai o ka Pueokahi.

To the north of Kahulili, with its foundation right under Kauiki, was what was known as the hair of Puuhele. Kaihuakala is mauka of Kauiki. Kaihuakala is not usually seen; when Maui is calm, then that locality is seen. Then Papahawahawa stands forth and brags, saying, “Here I have lived, and yet this is the first time I have [550]beheld the calmness of Maui; it is indeed clear, for Kaihuakala can be seen.” [One must behold] Kaihuakala mountainward and Kauiki seaward in order to complete one’s journey of sightseeing. Thus the saying by composers of chants:

Maui is calm, for Kaihuakala is seen; Kaihuakala mountainward, Kauiki to seaward. The surf breaks at Kaihalulu in the presence of Kauiki; Kauiki is nursing Mokuhano, Nursing the sand of Pueokahi, Nursing the cold wind from Kailiili, Nursing the surf of Keanini, Including Mokumana; A canoe from Hawaii

A ma ka akau ponoi no hoi o Kahulili, a malalo pono no o Kauiki kona kumu, ua kapaia oia na lauoho o Puuhele. Aia mauka o Kauiki o Kaihuakala. Aole e ike wale ia o Kaihuakala, aia a malie o Maui nei alaila, ike ia keia wahi. Ia wa no, ku mai la o Papahawahawa a akena iho la me ka i iho hoi, “He noho ae nei no hoi, akahi no a [551]ike ia ka malie a Maui, o ka malie ka ia ke ike ia aku la o Kaihuakala.” O Kaihuakala kai uka, o Kauiki ka i kai, alaila pau i ka makaikai ia na wahi a pau. A oia ka mea i olelo ia nei e ka poe haku mele, penei:

Malie o Maui, ua ikea o Kaihuakala, O Kaihuakala kai uka o Kauiki kai kai, I poia e ke kui, Kaihalulu i ke alo o Kauiki, Hii Kauiki ia Mokuhano, Hii ke one ka Pueokahi, Hii waikoloa i Kailiili, Hii ka nalu ia Keanini, Hoea ana i Mokumana, He waa Hawaii e, Waikaakihi ke awa,

[Enters] the harbor of Waikaakihi.

With the water at Punahoa, the sand at Pueokahi, [With] Honuaulu mauka, [and] Kahulili on top, Consumed indeed is Peapea by fire.

If some people are going to Kauiki [let them see] these places which have been mentioned. This is all of what has been told me by some one who knows the story of Kauiki; there is much more, but this which has been learned by heart is sufficient.

[M.]

Punahoa ka wai oneone ia ka Pueokahi, O Honuaula mauka o Kahulili ka iluna, Pau Peapea la i ke ahi.

A ina e hele ana kekahi poe ma Kauiki i keia mau wahi i hai ia ’ku la, a o keia iho la na mea e pili ana i ka hai ia ana mai ia’u e kekahi mea i ike i ka moolelo o Kauiki, he nui aku no, aka, ua lawa keia mau wahi mea i panaau no keia manawa.

[M.]

A STORY OF PUMAIA. HE MOOLELO NO PUMAIA.

Pumaia90 was born at Koloa, Kauai. Malaihi was the father and Kuhihewa was the mother.

Ua hanauia o Pumaia ma Koloa i Kauai, o Malaihi ka makuakane, o Kuhihewa ka makuahine. Oi

They lived together until the child had grown up, and his body had acquired strength, when Pumaia said to his father, “You will please let me go to see the club of my grandmother.” The father gave his consent. The son went, and arriving at the house of Kiha, his grandmother, he asked, “Where is the club?” “Here it is,” said the grandmother. He took it. The nets of calabashes91 were thereby dashed to the ground and broken to pieces. He shouldered the club and commenced his journey. He caught up with Wakaina, and the latter inquired, “Where are you journeying to?” “Going sightseeing,” replied Pumaia. “Please let me go with you.” Pumaia consented and they went along.

Puukolea stood forth and called out, “Whose offspring are you?”

“Our own,” replied Pumaia. “If you excel in strength, you are saved; if your strength is meager, I will kill you.” Then they commenced to fight. That man, however, had a dual body, while these had the ordinary living

noho lakou nei a nui ua keiki nei, a ikaika no hoi kona kino, olelo aku o Pumaia i ka makuakane: “E oluolu oe e hele au e ike i ka laau a ku’u kupunawahine,” o ka ae mai la no ia o ka makuakane. O ko ianei hele aku la no ia, a hiki ma ka hale o Kiha, ke kupunawahine, a olelo aku la: “Auhea la ka laau?” “Eia ae no,” wahi a ka luahine. O ko ianei lawe ae lo no ia. O ka paipu, ulupa ia iho la ilalo, a okaoka liilii. O ka hiu ae la no ia i ua laau nei a kau i ka a-i, a hoomaka e hele aku. Halawai mai la me ia o Wakaina, a pane mai o Wakaina: “He huakai e hele ana ihea?” “E hele ana i ka makaikai,” wahi a Pumaia. “E aho hoi ha owau kekahi me oe.” O ka ae mai la no ia; o ko laua hele aku la no ia.

Ku ana o Puukolea, a olelo ana: “Nawai ke kupua o olua?” “Na maua no,” wahi a Pumaia. “A nui ka ikaika ola olua, hapa ka ikaika make olua ia’u.” O ko lakou nei hakaka iho la noia. O ke kino na’e o kela kanaka he kino papalua, o ko laua nei kino hoi he kanaka ola maoli no. Ua

body. Pumaia asked Wakaina, “Which of us shall be the first one to fight with this man?” The other replied, “It is better that I do it.” Pumaia consented and Wakaina went to fight Puukolea. Before they commenced fighting, however, he chanted a portion of a song, thus: [552]

ninau aku o Pumaia ia Wakaina: “Owai ka mea mua o kaua e hakaka me ianei?” Pane mai la kela: “E aho owau.” Ae aku la no hoi keia; o ko Wakaina hele aku la no ia e hakaka me Puukolea. Mamua nae o ko laua hoomaka ana e hakaka, hoopuka ae la oia i kekahi mau lalani mele, penei: [553]

Behold! behold! the mere lehua of Puuoni, Struggling with the clouds of the air, Now above, now below the rain clouds.

The other spoke up and said, “Why are you delaying? You are provoking me through your song!” Wakaina had not completed his song before they commenced to fight. From afternoon to near midnight [did they fight]. Wakaina said to Pumaia: “You simply look on and do not come to aid me; I am nearly dead.” Pumaia jumped in and fought with the other, while Wakaina went to seek food for them. They kept on fighting until Pumaia was weakened. He

Aia, la, aia la, o ka lehua wale o Puuoni,

Ke a uume inai la me opua i ka lewa

Maluna malalo ka wai opua.

O ka pane e mai la no ia o ua kanaka nei. “E hoololohi ana oe i ke aha? he mele hoonaukiuki kau no’u.” Aole nae i pau ke mele ana o Wakaina, o ka hakaka iho la no ia; mai ka auina la a kokoke aumoe. Kauoha aku la o Wakaina ia Pumaia: “O ka nana mai no ka kau, aole oe e kii mai ia’u la, ua kokoke au la e make.” O ka lele mai la no ia o Pumaia, a hakaka me iala; oi nei hoi, hele aku la e huli ai no laua nei. Oi nonoke aku no hoi laua nei i ka hakaka, a aneane

thought of his club; so when Wakaina came back and inquired, “How are you?” he answered, “I am weak; go and fetch my club; perhaps it can do something.” While they yet spoke the other man appeared in a different body. They fought on, and before Pumaia could reach for his club, he was killed by this man; Wakaina also was killed. Their spirits returned to their parents and were seen by them. (The parents of Wakaina had gone over and were living together with Pumaia’s parents.) While they were idling away their time a chant was voiced by the wandering spirits:

nawaliwali nae o Pumaia; noonoo keia i ka laau a ianei; a ku ana o Wakaina, a olelo mai, “Pehea oe?” Hai mai kela: “Ua nawaliwali au, e kii aku oe i kuu laau e lawe mai, malia paha he hana ka ia laau.” Ia laua nei e kamailio ana, ku ana ua kanaka nei me ke ano okoa. O ka hakaka iho la no ia; aole nae i lalau aku o Pumaia i ka laau a ia nei o ka make mai la no ia i ua kanaka nei; a make pu no hoi me Wakaina. O ka hoi iho la no ia o na uhane o laua nei a ku ana imua o na makua, a i ka ike ana mai o na makua o Wakaina a me Pumaia, no ka mea, ua hoi aku na makua o Wakaina a noho pu me na makua o Pumaia. Ia lakou nei e nanea ana, kani ana ke mele a ua mau uhane haukae nei, penei:

Fear is creeping over us, Coming for us to go; We can not; we are held by Waiauau.

Come to give us life! Life—indeed.

When these ghosts finished their chant the parents came out

Ko mai ana ke ano io maua nei, E kii mai ana ia maua e hele, Aole e hiki ua paa ia Waiauau, E kii mai ana ia maua e hoola— e E ola no—e.

A pau ke mele ana a ua mau eepa nei, o ka oili koke mai la no

immediately, but they saw nothing. They looked here and there, but they could not see them. Meanwhile the two had gone as spirits until they met Pupuilima. Some people saw these two coming, but Pupuilima said: “Those are not men; those are ghosts.” “How do you know those are ghosts?” said the others. So they laid wagers. Pupuilima then said to them, “I will spread an ape92 leaf on the ground; and if it breaks, then they are men, but if it does not, then they are spirits.” He spread it. Meanwhile Pumaia said to Wakaina, “There is our death being prepared.” “How shall we be saved?” Pumaia explained: “Where my feet tread there you tread; because I was raised from my young days until I died, and until I found you; for this process is like that done to Pamano when by the spreading of an ape leaf he was saved.” So when they proceeded Pumaia trod on the ape leaf and it broke through; the friend treaded after him. They were chased by the prophet until they were caught. That is what I have obtained.

ia o na makua; aole nae he ike iki. Oi nana wale lakou nei ma o a maanei, aole no he ike iki ia.

O ka uhaele uhane aku la no ia

o laua nei a halawai me Pupuilima. Ia wa ike mai na kanaka i ka hele aku o keia mau kanaka elua, ia wa olelo mai la o Pupuilima: “Aole kela he mau kanaka, he akua kela mau mea.”

“Mahea kahi i maopopo ai ia oe he akua?” wahi a ua poe nei. O ko lakou nei pili iho la no ia.

Olelo aku la o Pupuilima ia

lakou: “I halii auanei au i ka lau ape ma ke alanui nei a i nahae, alaila, he kanaka, aka hoi, ina nahae ole, alaila, he devil.” O ko ia nei halii iho la no ia. Ia wa, pane iho la o Pumaia ia

Wakaina, “Aia ko kaua make la, ke hana ia mai la.” “Pehea la auanei kaua e pakele ai?” Hai mai la o Pumaia: “Ma ko’u wawae e hehi ai, malaila mai no oe e hele ai, no ka mea, ua hanai ia wau mai kuu wa uuku a i ko’u wa i make ai, a loaa oe, no ka mea o keia hana ana, ua like me ko Pamano halii ia ana e ka lau ape, a pakele ke ola.” Ia laua nei i hele aku ai, o ka hehi iho la

While they were going along they were given chase. They came along until they caught up with Pueonuiokona.93 The owl, however, did not catch sight of them while they were coming. When they had passed ahead the prophet who was chasing them caught up with Pueonuiokona. The owl asked, “What is the cause of this heavy breathing and this perspiring?”

This one answered, “That you should be asking [554]‘what’? Spirits! and there they are! I am chasing them, but can not catch them; I have been wishing to get near them so that I can kill them, for I am possessed with great anger towards them.”

no ia o Pumaia i ka lau ape, a nahae, a mahope iho la no hoi ke keehi ana o ke aikane. O ko laua nei hahai ia aku ’la no ia e ua makaula nei a loaa i kahi i loaa ai.

Oia iho la no kahi mea i loaa ie mai nei.

Ia laua nei e hele ana, ua alualu ia mai la nae laua nei. O ko laua nei hele maila no ia a halawai me Pueonuiokona, aole nae he ike mai o ua Pueo nei i ka laua nei hele aku. A kaa laua nei mamua, ia manawa halawai mai la ka makaula e alualu nei ia laua me Pueonuiokona. Pane aku la ua Pueo nei: “Heaha ka mea i nui ai o ka hanu [555]a kahe hoi ka hou?” Hai aku la keia: “Heaha mai ka hoi kau, he mau uhane aia la, o ka’u ia e alualu nei aohe loaa iki; e ake ana hoi au o ka lihi launa aku, make la hoi ia’u, ua hele mai kuu ukiuki a nui ia laua.”

When the owl heard what the Ia lohe ana o Pueo i ka olelo a

prophet said, he said to him, “You are a prophet, and I am a prophet, still I did not see them; and now I hear you saying that if you catch them they die.” Where they were holding this conversation, however, was on the plain of Kamaomao. While the others prepared to come for the spirits, Pumaia said to his friend, “Here comes our death; but we will wait. If the new one gets ahead of the old one then we have hope for life.”

ka makaula, ia manawa oia i olelo aku ai i ua makaula nei: “He makaula oe, a he makaula wau, eia nae, aole wau i ike aku nei ia laua, a no kuu lohe ana mai nei i kau olelo, ke loaa aku ia oe make.” O kahi nae a laua e kamailio nei, aia ma ke kula o Kamaomao. Ia laua nei nae e hoomakaukau ana e kii i na uhane, olelo aku la o Pumaia i ke aikane: “Eia a’e ka make o kaua la, aka, i kali auanei kaua a i oi kela mea hou mamua o ka mea mua, alaila, manao ae ke ola.”

So they sat and watched the two prophets. When Pueo distanced the other, Pumaia said to the friend, “We are now saved; it were better that we go to our parents. It may be that we would be found there.” The friend consented. They came along Kealia, a large pond even to this day. These places above mentioned, the plain of Kamaomao and Kealia are at the eastern isthmus of Maui, connecting East and West Maui.

O ko laua nei noho iho la no ia nana no laua nei i ua mau makaula nei. A oi no o ua o Pueo mamua, olelo aku o Pumaia i ke aikane: “Akahi kaua a pakele, e aho e uhaele kaua a kahi o na makua o kaua; malia paha, o loaa ae kaua ilaila.” Ae mai la ke aikane. O ko laua nei hoomaka mai la no ia e hele ma Kealia, he loko nui no hoi a hiki i keia wa. O keia mau wahi nae i hai ia a’e nei, no kula o Kamaomao a me Kealia, aia no ma ka puali hikina o Maui nei,

Let us leave the ghosts and speak of the prophets. While the prophets were following, and because Pueonuiokona was the faster, an idea occurred to one of them, and he said to himself, “If I catch up to that prophet I shall kill him, for he is simply going after my enemies to save them.” The owl went along slowly until his fellow prophet thundered after him and gave his hand a jerk. Pueo asked, “What wrong has a fellow committed that you should tug at the arm from behind and thereby nearly tripping me?” The other replied with blazing eyes, “Who told you to go after my enemies with the intention of saving them? Why don’t you look for your own, and on them practice saving?” The owl replied: “If you excel in strength you can obtain them, but if you are weak you can not have them; because I, even I, am an old resident of this plain. When spirits from the dead arrive I cause them to revive. Also no prophet comes to this

alaila pau o Maui Hikina, pau o Maui Komohana.

E waiho ae kakou no ua mau

akua lapu nei, a e kamailio

kakou no na makaula. Aia i ua mau makaula e alualu nei, a no ka mama o Pueonuiokona, kupu mai la ka manao iloko o kekahi o laua, me ka nune iho iloko ona: “Ina e loaa ana ia’u kela makaula e make ana oia ia’u, no ka mea he kii wale i ko’u mau

enemi e hoola.” Naue malie aku la nae ua Pueo nei, a halulu ana ua makaula nei, a huki mai la i ka lima o Pueo. A ninau mai la o

Pueo: “Heaha iho la ka hewa o ke kanaka o kau huki ana i ka lima mahope, a mai hina wau?” Olelo mai kela me ka inoino o na maka: “Nawai oe i i aku e kii i ko’u mau enemi, me ko olelo iho e hoola? Aole no hoi oe e huli i mau mea okoa aku nau, ilaila oe e hoola ai.” Pane mai la ua Pueo nei: “Aia a nui ko ikaika, lilo ia oe, aka hoi, hapa ko ikaika, aole e loaa ia oe; no ka mea, owau nei la, he kamaaina wau no keia kula. Hele mai na uhane make, na’u no e hoola aku, aole no hoi he makaula hiki ma keia kula,

plain; this is the first time that I have seen a prophet on this plain, and that is yourself. Because you complained of my desire to give life to those spirits, so be it! By a test of strength will [we know] whether you obtain them or I do.”

They immediately commenced to fight. Where they fought was at Kalepolepo,94 near this isthmus. The prophet was killed and his entrails were disemboweled by Pueonuiokona and placed on the akolea. 95 That shrub used to be plentiful at that place, but it is destroyed now on account of numerous animals.

August 8, 1872. K.

akahi wale no wau a ike o oe hookahi ka makaula hiki maanei. A no ko olelo ana mai nei ia’u no kuu kii e hoola i kela mau mea; heaha la ka hana, aia mamuli o ka ikaika e lilo aku ai oe, a ia’u paha laua.”

O ko laua nei hoomaka koke iho la no ia i ka hakaka. O ko laua wahi i hakaka ai, ma Kalepolepo, kokoke no ma keia puali. A make ua makaula nei, kua’i ia ae la ka opu o ua makaula nei e Pueonuiokona, a kauhihi ia ae la maluna o ka akolea. He laau nui keia ma kela wahi, a ua lilo nae i mea ole i keia manawa, no ka nui o na holoholona.

August 8, 1872. K. A STORY OF PUUPEHE.

Puupehe was the child of O Puupehe he keiki ia na

Kapokoholua and Kapoiliili. On the island of Kaululaau96 lived the subject of this story, the champion of that past age. Let us glance briefly at the occupation of the parents. While they were living together the woman [556]realized that she would obtain a beautiful fruit from her womb. The husband was a fisherman who used to go fishing for them, and he passed long periods at his occupation. One day Kapokoholua sailed out on the ocean in a canoe to fish. When he had gone Puupehe commenced a conversation from inside the mother’s womb. When Puupehe spoke the mother was startled from her sleep, and she wondered at this strange sound which she heard. She went to sleep again; the live one called out, “Say, my dear mother, do thou awake; father is dead.” The mother then realized that it was a human voice, but she could see no human being. She ran quickly and stood at the door of the house and looked round for the owner of the voice, but she could see no one; she thought, however, that it was her

Kapokoholua ame Kapoiliilii. Aia ma ka mokupuni o Kaululaau ka mea nona keia moolelo, ka eueu hoi oia kau i hala. E huli iki ae kakou no na hana a kona mau makua. Aia i ka wa a laua e noho nei, ua hoomaka mai kana [557]wahine e loaa nei mea he hua maikai no kona puhaka. O ka hana hoi a ke kane o ka hele i ka lawaia na laua, a hala kekahi manawa loihi, o ka hoomau ana o kana kane i ka lawaia. I kekahi la, ua holo aku ua Kapokoholua nei i ka moana maluna o ka waa i ka lawaia, a i kona kaawale ana aku ua hoomaka mai o Puupehe e kamailio ae iloko o ka opu o kona makuahine. Ia wa a Puupehe e kamailio nei, ua puiwa ae la kona makua mai kona hiamoe ana, a hoomaopopo iho la keia no keia mea kupanaha ana i lohe ai. Hoi hou iho keia moe no, kani ana kahea a ua eueu nei: “E! ko’u mama aloha, e ala mai oe ua make o papa.” Ia manawa hoomaopopo loa iho la keia he leo kanaka, koe nae kona ike ana aku i ke kino maoli. Holo wikiwiki aku la keia a ku ana ma ka puka o ka hale, nana aku la

husband, but she was disappointed. She returned and sat in the house. The child voiced a chant thus:

How I desire the moisture of the dew,

That water proudly hanging on the tree; When the rain falls on the precipice

It fills up the streams. That is the benefit my eyes behold!

My eyes to behold, my hands to rustle.

When the child finished chanting his mother became possessed and was greatly troubled. I had better explain shortly about his chanting and falsely stating that his father was dead. It was not true as he chanted. He had gone to watch his father fishing, and he had sent for a great number of fish to come and bite the hook. He saw that his father had caught a great many fish, but he needed the second [requisite], the awa root. He knew his

hoi o ka ike aku i ka mea nona keia leo; aole nae he ike ia, ua manao nae ia o kana kane, aka ua hoka nae ia. Hoi mai la no keia a noho iloko o ka hale. Kani ana ke oli a ua keiki nei, penei:

Aloha au o kahi wai a ke kehau, Oia wai haaheo mai iluna o ka laau.

A ua mai ka ua i na pali

E hoopiha ana i na kahawai. O ka waiwai no ia a kuu maka o ka ike aku.

A ike kuu maka a nehe kuu lima.

A pau ke oli ana a ua keiki nei, ua ano e mai la ka makuahine, a ku a pilikia maoli ia. E pono e wehewehe iki aku wau i ke ano o kana oli ana a me ke kamailio hoopunipuni ana ua make ka makuakane; aole he oiaio o ke ano o ke oli ana. Ua hele ia e nana i ka lawaia ana o kona makuakane, a ua kii aku ia i na i’a he nui loa e hele e ai i ka makau. No ka ike ana ua nui na i’a i loaa i kona papa, ua koe hoi ka lua, oia ka awa. Ua ike ia

parents had none; that was why he voiced the few lines of song above written.

Let us drop what the child did for some later time and turn and talk of the father. While his father was fishing he became very much interested because he caught so many. When he glanced shoreward he could not see land, because Puupehe had covered it completely with fog. He thought to himself, “What can this wonderful thing be? There is now no wind to bring the fog on to the land!” He had a premonition, however, concerning his wife, so he commenced to pull in his line. When it was near the top his line was held by a shark. The name of this shark was Puaiki.

aohe awa a kona mau makua, nolaila oia i puana ae ai i keia mau lalani mele e kau ae la maluna.

E waiho iki kakou i na hana a ke keiki a mahope aku. E huli ae kakou a kamailio no ka makuakane. A i kona makua e lawaia nei, ua nanea loa ia i ka nui o na i’a i loaa iaia, a i kona nana ana mai iuka aohe ike ia aku o uka, no ka mea ua uhi iho la o Puupehe i ka ohu a nalowale ka aina. I iho la keia iloko ona: “Heaha la hoi keia mea kupanaha, nokamea, aole hoi he makani nana e lawe mai ka ohu a kau iluna o ka aina.” Ua halialia wale mai nae na ano o kana wahine, hoomaka iho la ia e huki mai i ka aho a kokoke e pau mai iluna, paa ana ke aho a ianei i ka mano, o ka inoa o keia mano o Puaiki.

Let me say a few words concerning this despicable fish. When he realized that it was a shark holding on he jerked the line; the fish took him to a point near the cape of Papawai here on Maui; from there he again

E kamailio iki ae au no keia i’a ino. Ia ike ana oia nei he mano ka mea nana e paa mai la, huki mai la ia me ka ikaika, e hao aku ana ua i’a nei, kokoke keia mawaho ae o ka lae o Papawai ma Maui nei, malaila aku hiki

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