Interior non-uniformity of acoustically excited oscillating gas bubbles ＊

Yu-ning Zhang , Xiao-fei Li Zhong-yu Guo Yu-ning Zhang,

1. Key Laboratory of Condition Monitoring and Control for Power Plant Equipment (Ministry of Education),School of Energy, Power and Mechanical Engineering, North China Electric Power University, Beijing 102206, China 2. Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University, Chengdu 610039,China 3. College of Mechanical and Transportation Engineering, China University of Petroleum-Beijing, Beijing 102249, China 4. Beijing Key Laboratory of Process Fluid Filtration and Separation, China University of Petroleum-Beijing,Beijing 102249, China

Abstract: In the present paper, the interior non-uniformity of oscillating gas bubbles in liquids under an acoustic field is theoretically investigated, as well as the temperature non-uniformity in the surrounding liquid. The pressure, the density and the temperature inside the gas bubbles are quantitatively predicted for a wide range of bubble Mach numbers. It is shown that the bubble interior non-uniformity increases significantly with the increase of the bubble Mach number. In the case of a large Mach number, the values of aforementioned paramount parameters at the bubble center are significantly larger than those at the bubble interface. Except for a very thin layer close to the bubble interface, the effects of the temperature non-uniformity in the bulk liquid could be safely ignored.

Key words: Cavitation, gas bubbles, acoustic field, radial oscillation, non-uniformity, Mach number

Introduction

The bubble interior uniformity is generally an issue in the various kinds of cavitation. Five scenarios are responsible for the state of the bubble interior. The first scenario is when the bubble is excited by different types of external fields, e.g., the acoustic field(e.g., in the sonoluminescence, the surface cleaning, and the alloy enforcement), the electric field(e.g., in the enhanced bubble nucleation process,and the bubble detachment acceleration), the magnetic field (e.g., in biomedical applications for the theranostics[1-2]and in case of the bubble rising in the liquid metal[3-5]), the electromagnetic field (e.g., in the pressure measurement of bubble[6]) and the acousticmagnetic field (e.g., in the bubble manipulations[7]).For example, in the acoustic field, the mutual interactions between the bubble and the acoustic wave leads to the generation of the Bjerknes forces[8], which are the extra forces on the bubble interface. As a result,the bubble interior will see a strong non-uniformity with a non-spherical shape[9]. Comprehensive reviews of the acoustic bubble (e.g., the bifurcations[10], the harmonics and the subharmonics[11-12]), can be found in Refs. [11-14].

The second scenario is the bubble collapsing near the boundaries, e.g., the solid flat wall[15-20], the particles[21-22], the curved boundary[23], the corner[24-25],the crevice[26], the cell[27]or the combination of the above spots (e.g., the bubble-particle-wall system[28-29],and the droplet-bubble-wall system[30-31]). For this scenario, the existence of the boundaries could significantly alter the bubble collapsing behavior. For example, an elongation of the bubble is observed by the presence of the particle near the wall, leading to an anisotropic collapse.

The third scenario is the extreme condition inside the cavitation bubble[32-35]. As shown in the experimental data[32], the conditions inside the strongly collapsing bubbles are rather extreme, e.g. with a measured temperature exceeding 15 000 K (with a temperature rising rate over 1012K/s) and the pressure over 108Pa. Hence, it is not hard for us to imagine the great non-uniformities inside the bubbles.

The fourth scenario is the effects of shock wave[36-37]or complex flow on the bubbles. The shock wave could be generated by the surrounding flows or the well-designed devices (e.g., the extracorporeal shockwave lithotripters[37]). When the shock wave passes through the bubbly flow, the bubble will be greatly compressed (with great interior non-uniformity)and then rebound. A recent review of the bubbleshock wave interactions was made by Ranjan et al.[36].

The fifth scenario is the interactions between bubbles through various kinds of mechanisms[38]e.g.,the secondary Bjerknes force due to the radiation pressure. In the literature, the bulk uniform bubble cloud is of interest for a precise control of the bubble size non-uniformities in applications. However, the bubble-bubble interactions could significantly alter the bubble distributions. Specifically, this interaction strongly depends on the distance between the bubbles and the bubble oscillations. For a close distance (with a high void fraction), the bubble will deform due to the presence of other bubbles nearby and the bubble interior non-uniformity will be enhanced.

In the various kinds of models (e.g., the polytropic model for linking the pressure and the bubble volume), a uniform state is usually assumed. In the present paper, the bubble interior non-uniformity is theoretically investigated with the focus on the distributions of the pressure and the temperature.

1. Theoretical model

In this section, the basic equations of oscillating bubbles in liquids are presented together with the boundary conditions, solutions and parameter definitions under the following assumptions:

(1) The bubbles in radial oscillations are spherical rather than non-spherical.

(2) The acoustic pressure amplitude is assumed to be limited and hence the complex resonances are avoided.

(3) A single bubble oscillating in the acoustic field is investigated. For the polydisperse bubble, the cases will be much more complex.

(4) The non-uniformities of parameters (e.g., the temperature and the pressure) inside the bubble interior are allowed together with the temperature non-uniformity in the surrounding liquid.

1.1 Parameter definitions

The parameters involved are assumed to be deviated from the equilibrium values as follows[39]

whereρ,P,Tg,lTandRrepresent the dynamic values of the density, the pressure, the temperature of the gas inside the bubble, the temperature in the liquid and the bubble radius, respectively,η,p,gθ,lθandxrepresent the nondimensional deviations of the aforementioned parameters from the equilibrium values,gρis the density of the gas,0Pis the ambient pressure,T∞is the ambient temperature,R0is the bubble radius in equilibrium,σis the surface tension coefficient of the surrounding liquid.

Based on Eqs. (1)-(5) and the ideal gas law, the relationship between the bubble interior parameters can be written as[39]

For the convenience of discussions, the parameters are expressed as

whereAis a given parameter and could beη,p,gθ,lθorx, the subscriptrrefers to the amplitude of the oscillations of the given parameter,εis the non-dimensional acoustic pressure amplitude,ωis the angular frequency of the acoustic wave,δis the phase and the subscriptArepresents the given parameter. Hence,Aris the (real) amplitude of the non-dimensional deviation of the related parameter.

Asxris independent of the radial coordinate(r), we further define

The Mach number (Ma) is defined as follows

1.2 Basic equations

The governing equations of an oscillating gas bubble in the bulk liquid are[39]

with the following boundary conditions[39]

whereuis the velocity of the gas inside the bubble,ris the radial coordinate,kgis the gas thermal conductivity,Dg,vis the gas thermal diffusivity with a fixed volume,Dlis the liquid thermal diffusivity with a constant volume.

1.3 Solutions

The solutions of aforementioned equations could be expressed as follows[39]

In Eqs. (19)-(23), the detailed expressions for some parameters (e.g.A1) are not given because they are too lengthy[39]. As shown in Eqs. (19), (20), the solutions have a singular point at the bubble center(r/R0=0). Hence, in the following sections, the positions very close to the bubble center (r/R0=0.01)are used to represent the bubble center.

To reflect the non-uniformity, several new parameters (p**,η**andθ**) are defined as:

Here, the above parameters reflect the ratios between the values at the bubble center and those at the bubble interface.

2. General features of phenomenon

In this section, the general features of the acoustically excited bubble interior non-uniformity are discussed with demonstrating examples (air bubbles in water) for a wide range of Mach numbers.

Figures 1, 2 show the amplitudes of the non-dimensional deviations of the important parameters inside the bubble, including the amplitudes of the pressure (pr), the density (ηr) and the temperature(θgr) with theMabeing 0.1, 0.7. For the low Mach number (Fig. 1), the variations of the parameters are limited and they can be safely treated as pure constants (i.e., the uniform distributions not varied with the radial coordinate). However, for the high Mach number (Fig. 2), the parameters vary significantly with a relatively large value at the bubble center. Hence, Figs. 1, 2 reflect the cases of the bubble interior being uniform and non-uniform, respectively.As shown in Fig. 2, with the increase ofr/R0, the parameters will decrease with the minimum at the bubble interface (r/R0=1.0). Generally speaking,forMa＞0.15, the effects of bubble interior non-uniformity should be considered.

Fig. 1 The variations of pr, ηr and θgr versus r/ R0.Ma=0.1

Fig. 2 The variations of pr, ηr and θgr versusr/ R0.Ma=0.7

Fig. 3 The variations of pr, ηr and θgr versus r/ R0.Ma=0.2

Fig. 4 The variations of pr, ηr and θgr versus r/ R0.Ma=0.3

3. Influences of Mach number on the bubble interior non-uniformity

In this section, the influences of the Mach number on the bubble interior non-uniformity are discussed. The primary parameters of interest are the oscillation amplitudes of the pressure, the density and the temperature inside the bubble.

Fig. 5 The variations of pr, ηr and θgr versus r/ R0.Ma=0.4

Fig. 6 The variations of pr, ηr and θgr versus r/ R0.Ma=0.5

Figures 3-7 show the oscillation amplitudes of three non-dimensional parameters (pr,rηandθgr)against the Mach numbers (Ma=0.2, 0.3, 0.4, 0.5 and 0.6, respectively), which reflect different degrees of the bubble interior non-uniformity. In those figures,the same vertical coordinates are employed for each plot for the convenience of comparisons. For a fixed Mach number, the oscillation amplitude of the non-dimensional pressure (pr) is the largest among the three parameters. And, the oscillation amplitude of the non-dimensional density (ηr) is similar toprbut with a little lower value. The oscillation amplitude of the non-dimensional temperature (θgr) is the smallest, indicating that the change of the temperature is less violent in the bubble interior.

Fig. 7 The variations of pr, ηr and θgr versus r/ R0.Ma=0.6

Fig. 8 (Color online) The variations of parameter pr* against Mach numbers

Fig. 9 (Color online) The variations of parameter against Mach numbers

Fig. 10 (Color online) The variations of parameter against Mach numbers

Fig. 11 (Color online) The variations of p**, η** and θ**versus Mach numbers

In Figures 8-10, the values of,andversusr/R0are further compared for seven different Mach numbers ranging from 0.10 to 0.70. Based on Figs. 8-10, one can clearly find that the bubble interior non-uniformity increases gradually with the increase of the Mach number.

Figure 11 shows the variations ofp**,η**andθ**versus the Mach numbers. Figure 11 reveals that the bubble interior non-uniformity increases dramatically with the increase of the bubble Mach number.Furthermore, the above three parameters are nearly identical in the given cases, indicating that the bubble interior non-uniformity is largely affected by the Mach number.

4. The temperature non-uniformity in the liquid

In this section, the temperature non-uniformity in the surrounding liquid is evaluated. Figure 12 shows the oscillation amplitudes of the non-dimensional temperature (θlr) in the liquid versusr/R0with a fixed Mach number. At the bubble interface (r/R0=1),θlris of the order of 10-3. With the increase ofr/R0, the value ofθlrdecreases dramatically and atr/R0=1.1,θlrdrops to the order of 10-6. Hence, in most practical cases, the variations of the temperature in the liquid could be safely ignored except for very thin layers near the bubble interface.

Fig. 12 The variations of parameter lrθ versus r/ R0 in the liquid

5. Conclusions

In the present paper, the radial oscillations of the gas bubbles in liquids are theoretically analyzed with the focus on the bubble interior non-uniformity.Typical parameters (the pressure, the density and the temperature) inside the gas bubbles are discussed and demonstrated with plenty of examples. For the completeness, the temperature non-uniformity in the liquid is also considered. It is found that the bubble interior non-uniformity increases significantly with the increase of the bubble Mach number. For a large Mach number, the values of related parameters at the bubble center are higher than those at the bubble interface. The effects of the temperature non-uniformity in the liquid are only prominent within a very thin layer near the bubble interface.

In the present paper, our discussions are within limited values of the bubble Mach number (up to 0.7).For larger Mach numbers, the bubble interface will no longer be in the spherical shape, leading to prominent vortex generations, which could be further captured through employing the advanced vortex identification methods. For even larger Mach numbers, the bubble will break up, leading to a much complex scenario.

Acknowledgement

This work was supported by the Open Research Fund Program of Key Laboratory of Fluid and Power Machinery, Ministry of Education, Xihua University(Grant Nos. szjj2019-002; szjj2019-004;szjj-2017-100-1-002).