Experimental study on attenuation effect of liquid viscosity on shockwaves of cavitation bubbles collapse

Highlights

• •Visualization of shockwaves from cavitation bubbles collapse in ultrasonic field.
• •Liquid viscosity can change development morphology of shockwave.
• •Liquid viscosity can effectively attenuate the peak pressure of shockwave.
• •Energy ratio during different stages of bubbles can be adjusted by liquid viscosity.

Abstract

How to precisely control and efficiently utilize the physical processes such as high temperature, high pressure, and shockwaves during the collapse of cavitation bubbles is a focal concern in the field of cavitation applications. The viscosity change of the liquid will affect the bubble dynamics in turn, and further affect the precise control of intensity of cavitation field. This study used high-speed photography technology and schlieren optical path system to observe the spatiotemporal evolution of shockwaves in liquid with different viscosities. It was found that as the viscosity of the liquid increased, the wave front of the collapse shockwave of the cavitation bubble gradually thickened. Furthermore, a high-frequency pressure testing system was used to quantitatively analyze the influence of viscosity on the intensity of the shockwave. It was found that the pressure peak of the shockwave in different viscous liquid was proportional to L^b (L represented the distance between the center of bubble and the sensor measuring point), and the larger the viscosity was, the smaller the value of b was. Through in-depth analysis, it was found that as the viscosity of the liquid increased, the proportion of the shockwave energy of first bubble collapse to the maximal mechanical energy of bubble gradually decreased. The proportion of the mechanical energy of rebounding bubble to the maximal mechanical energy of bubble gradually increased. These new findings have an important theoretical significance for the efficient utilization of ultrasonic cavitation.

Nomenclature

Symbols

Symbol Meaning Unit

R_eq

The equivalent radius of the cavitation bubble mm

R_max

The maximum radius of the cavitation bubble in the first cycle mm

R_r-max

The maximum radius of the cavitation bubble in the second cycle mm

r

The shockwave propagation radius mm

L

The distance between the center of bubble and the sensor measuring point mm

γ

The relative distance

η

The viscosity of the liquid mPa·s

T

The evolution time of the cavitation bubble shrinking from the maximum radius to the minimum radius in the first cycle ms

T_c

Rayleigh time ms

T_s

The impact time span of the collapse shockwave µs

P

The impact pressure of the collapse shockwave of the cavitation bubble MPa

E_s

The shockwave energy of first bubble collapse J

P_max

The pressure peak value of the shockwave MPa

E_i

The shockwave energy when γ was 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, respectively J

E_max

The maximal mechanical energy of bubble J

E_r-max

The mechanical energy of rebound bubble J

1. Introduction

Cavitation is common in fluid machinery and medical fields, such as ship propeller [1], [2], pump blade [3], [4], [5], ultrasonic cleaning [6], [7], [8], microbial degradation [9], [10], [11] and drug particle targeted delivery [12], [13]. The collapse of the cavitation bubble produces strong shockwave [14], [15], [16] and high-speed microjet [17], [18], which causes cavitation damage on the overflow surface. However, in some fields, the high temperature, high pressure and shockwave are efficient used, such as the cavitation field generated by ultrasound or the cavitation field generated by Venturi tubes to treat sewage [19], [20].

Since the cavitation phenomenon was discovered, many scientists have continued to study the cavitation dynamics. As early as 1917, Rayleigh [21] proposed an equation for the radial motion of a single spherical bubble in an incompressible fluid. This laid a foundation for the dynamics of a single bubble. Later, several scientists have conducted mechanistic studies on the dynamic characteristics of the bubble under different boundary influences in practical scenarios, such as the collapse behavior of the bubble near the wall [22], [23], [24], the collapse behavior of the bubble near the free surface [25], the influence of liquid physical properties on the bubble [26], and the interaction between spherical particles and the bubble [27], [28]. In the above cavitation scene, the physical properties of the liquid medium are the most common. In this field, Khavari et al. [26] not only discovered strong high-frequency pressure peak induced by bubble collapse, but also identified two obvious characteristics of shockwave, namely the increase in frequency peak within the MHz range (inherent) and the contribution to sub-harmonics (periodic). These research findings have significant implications for the application of cavitation. In recent years, with the application of high-speed photography technology in bubble dynamics research, the non-spherical collapse characteristics of bubble [29] have gradually attracted attention. One important feature was that non-spherical collapse of the bubble would generate multiple impacts related to different development processes of the bubble [30], [31]. Wang [32] proposed a weakly compressible theory for studying the asymmetric cavitation bubble dynamics of underwater explosions, and found that the energy of the cavitation bubble is most severely lost during the first collapse. Zhang et al. [33] proposed the dynamics theory of cavitation bubbles considered multiple physical factors, and found this theory could more accurately predict the migration of the cavitation bubble and the pressure pulse emitted by the cavitation bubble collapse under different conditions than previous theories. Moreover, due to the wide application of ultrasound cavitation in the fields of medicine and biodegradation, many scientists have also conducted research on the dynamics of sub-millimeter cavitation bubbles in ultrasound cavitation fields [34], [35], [36]. The evolution process of the cavitation bubble is microsecond or millisecond scale, and the accompanying physical phenomena bring great challenges to experimental research. Some scientists have studied bubble dynamics using a combination of numerical simulation and experimental methods. Based on potential flow theory and Boundary Element Method (BEM) discrete method, many scientists have reproduced the evolution process of cavitation bubble by numerical calculation [37], [38], [39], [40]. Wang et al. [41] solved the limitations of the BEM method in simulating the annular bubble by placing a vortex ring inside the bubble. Subsequently, Zhang et al. [42] introduced multiple vortex rings into the above model to simulate the splitting characteristics of the annular bubble. Faced with the interaction between the bubble and different boundaries in practical scenarios, BEM and finite element method were combined to reproduce the interaction process between the bubble and the wall [43], [44], [45]. Akhatov et al. [46] proposed a mathematical model and found that the calculated the radius of laser bubble and the intensity of shock wave generated during collapse were in good agreement with experimental results. Zhang et al. [47] proposed a method to predict the attenuation of acoustic waves in liquids with dilute bubbles under the condition that the external pressure outside the bubble is not uniform. However, there are differences in bubble dynamics under different boundary conditions, especially the emission of shockwave from the bubble collapse and the development of micro jets. These issues are the most concerning mechanism issues in the field of macroscopic applications of cavitation.

With the evolution of cavitation bubble, shockwaves phenomena would appear in different stages of the cavitation bubble development. Through experimental methods, some scientists have studied the shockwave mechanisms generated by bubble collapse under different conditions, including the erosion effect of the shockwave pressure on the surface of the specimen [48], the variation of the shockwave pressure when the cavitation bubble collapsed near the wall [49], and the influence of the interaction between the cavitation bubble and the air bubble on the shockwave in water gas two-phase flow [50]. Many scientists have conducted in-depth research on the shockwave intensity during the evolution of the bubble using advanced observation equipment. For example, the relationship between the pressure and propagation distance of expansion shockwave from a single laser bubble in water during its propagation [51], the differences in bubble behavior and acoustic emission in water, ethanol and glycerol [52], [53], [54], [55]. Among them, Tzanakis et al. [54] used a novel experimental design to directly observe cavitation and streaming, and found that the cavitation intensity in water and glycerol was in a similar range and higher than that in ethanol. On this basis, they found that water and liquid aluminum exhibited the closest cavitation behavior based on the measurement of cavitation intensity, and thus proposed for the first time that water can be used as a physical simulation body for the study of cavitation in liquid aluminum. Not only that, the shockwave evolution characteristics and shockwave intensity of bubble in some very large viscous liquids (2749.11 mPa·s, 5575.27 mPa·s) during their evolution have been extensively studied, which is of great help for efficient applications in biomedical fields [56], [57]. During the evolution of cavitation bubble, the shockwave not only emitted to the external water, but also induced secondary cavitation in the water body [58], [59].

With the gradual development and maturity of high-frequency measurement technology, the combination of laser-beam deflection probe and schlieren technology can also measured and observed the shockwave phenomenon emitted during bubble evolution from different angles [60]. High response frequency PVDF hydrophones were also gradually being applied in the quantitative research of shockwave [61], [62]. The shock waves emitted by multiple bubbles in the ultrasound field were observed and measured by Khavari et al. [63], and through spectral analysis, they first discovered a consistent resonance peak in a very narrow frequency range (3.27–3.43 MHz). This work profoundly elucidates the key characteristics and fundamental mechanisms of shock waves driven by cavitation, providing important value for the application of cavitation. With the emergence of ultra-high-speed photography technology, scientists have discovered that the shockwave from the bubble collapse not only appeared in the form of implosion, but also under the influence of external boundaries, before the bubble shrank to the minimum volume, the water hammer shockwave appeared before the implosion shockwave [64], [65].

From the above literature review, it can be seen that the physical properties of liquids have a significant impact on the spatiotemporal evolution and intensity of the shockwave during bubble evolution. These issues are significant for the efficient utilization of cavitation, such as ultrasonic cleaning [66] and ultrasonic degradation [67], etc. Therefore, this paper combined high-speed photography technology with schlieren optical path system, and observed and analyzed the evolution of shockwaves emitted by cavitation bubbles collapse in the ultrasonic field. To further quantify the evolution and development of the collapse shockwave from the cavitation bubble in different viscous liquids, a single cavitation bubble was created using the method of spark discharge, and the spatiotemporal evolution and intensity of the shockwave in different viscous liquids were observed. Finally, the variation of liquid viscosity on the shockwave attenuation was analyzed.

2. Experimental apparatus

In order to study the influence of the viscosity of liquid on the collapse shockwave of the cavitation bubble, ultrasonic cavitation equipment is used for the shockwave generated by the macroscopic cavitation field in this study, as shown in Fig. 1 (a). For the collapse shockwave of the single cavitation bubble in the mechanism study, the spark-induced cavitation bubble system, high-speed photography system, schlieren optical path system and transient pressure measurement system are used to collaborate and complete the observation and data acquisition synchronously, as shown in Fig. 1 (b). During the experiment, the liquid temperature was kept at (25 ± 1) °C and the ambient atmospheric pressure was about 96 kPa.

Fig. 1.

Experimental device diagram.

To observe the shape of the shockwave emitted by the cavitation bubble in the ultrasonic field with the viscosity of the liquid, the ultrasonic vibration cavitation system was selected in the experiment. The system consisted of ultrasonic generator, transducer, horn and water tank, as shown in Fig. 1 (a). The driving frequency was 12.5 kHz. The output power of the equipment was 110 W. In the experiment, the depth of the horn of the ultrasonic device in the liquid was 1.5 cm.

The spark-induced cavitation bubble system was composed of a charging circuit and a discharging circuit. In the experiment, the capacitor (4400 μF) was firstly charged at a voltage of 116 V. During the discharge process, the contact points of the two electrodes undergo electrolysis [68] and vaporization [69], generating gas that envelops the tiny gap between the electrodes, further intensifying the discharge and eventually inducing cavitation. The position and size of the cavitation bubble can be well controlled by using the method [70], which can improve the repeatability of the experiment. The cavitation bubble was repeated 5 – 8 times under each experimental condition, and the maximum radius was (11.5 ± 0.5) mm when it first expanded to the maximum. In order to reduce the influence of the electrodes on the cavitation dynamics as much as possible, the electrode with a diameter of 0.1 mm was used. The diameter of the electrode was less than 0.1 % of the maximum radius of the cavitation bubble [71]. Considering the reflection of shockwave on the water tank wall and free surface, a transparent glass tank with a size of 60 cm × 40 cm × 40 cm was used in this study. The cross point of electrodes was lapped at the center of the water body. The depth of the liquid level was 32 cm. The distance between electrodes overlap point and walls of the water tank, as well as between electrodes overlap point and the free surface, is greater than 10 times the maximum radius of the cavitation bubble [72].

The life cycle of spark-induced cavitation bubbles is very short (millisecond scale), and the evolution of microjet and shockwave during the cavitation bubble collapse is even shorter. Therefore, the collapse shockwave of the cavitation bubble must be observed by the high-speed photographic system. In this experiment, a high-speed camera (Fastcam SA-Z, Photron Inc., Japan) with a maximum shooting rate of 1,000,000 fps (Frame per second) was used for observation. Due to the millimeter scale of the cavitation bubble, the experiment needed to be filmed with a macro lens (NIKKOR 85 mm) combined with a high-speed camera. In the experiment, taking into account the image sharpness and shooting speed of high-speed photography, the shooting resolution was selected to be 512 pixel × 128 pixel. An exposure time of 0.25 µs was chosen to make the wave front of the shockwave clear. The propagation speed of the shockwave exceeded the sound speed in water [51]. When the shockwave propagated in the liquid, the density of the compressed liquid was higher than that of the uncompressed liquid [73]. When the light penetrated the compressed liquid, the transmittance was higher than that of the uncompressed liquid, which can be reflected in the imaging of high-speed photography. During the experiment, a point light source was used, and a concave mirror was placed on one side of the experimental tank to reflect the light beam to the water area where the cavitation bubble was located. When the light passed through the cavitation bubble and the surrounding water, it entered the lens of the high-speed camera, as shown in Fig. 1 (b).

The shockwave speed was very fast and the impact pressure was very high [16], [51]. When measuring the impact pressure from shockwave, it was necessary to consider the size, response frequency, rise time and maximum range of the sensor. In this experiment, a piezoresistive pressure sensor (Test Electronics Information Co. Ltd) was selected. The pressure sensing surface of the sensor was about 1 mm, the maximum measuring range was 20 MPa (the accuracy was 0.25 % of the maximum measuring range), and the response frequency was 0 kHz − 500 kHz. The rise time was 2 μs, and the sampling frequency during the experiment was 20 MHz. Before the experiment, the shockwave in various liquid was tested, and it was found that the time of a single waveform was about 4 μs − 40 μs, and the corresponding frequency was about 25 kHz − 250 kHz, so the selected pressure sensor was suitable for this study. The accuracy of the pressure sensor measurement used in this paper has been discussed in depth in the author's previous publication [50]. In order to more accurately measure the intensity variation of the collapse shockwave of the cavitation bubble when it developed outward, this experiment selected 2.5 times, 3.5 times, 4.5 times, 5.5 times, 6.5 times, 7.5 times, 8.5 times, 9.5 times of the maximum radius of the cavitation bubble to measure the shockwave intensity. On the one hand, if eight sensors were mounted at the same time, the development of the shockwave would be affected. On the other hand, the shooting light path of high-speed photography would be affected. Therefore, during the experiment, 8 measuring points were divided into 4 groups for measurement respectively (2.5 times and 6.5 times of the maximum cavitation bubble radius were measured at the same time, 3.5 times and 7.5 times of the maximum cavitation bubble radius were measured at the same time, 4.5 times and 8.5 times of the maximum cavitation bubble radius were measured at the same time, and 5.5 times and 9.5 times of the maximum cavitation bubble radius were measured at the same time), and finally the 4 groups of experiments were summarized together to clarify temporal and spatial variation of shockwaves.

In the experiment, in order to achieve changes in fluid viscosity while also maintaining consistency in density and surface tension as much as possible, the liquid with varying viscosity is prepared by mixing water and glycerol in specific proportions. The viscosity of liquid was measured by a digital display viscometer (Shanghai Lichen-BX Instrument Technology Co., Ltd.). The physical parameters of the liquid were shown in Table 1.

Table 1.

Physical parameters of liquid (25 ± 1) ℃.

Liquid	Deionized water	20 % Gl solution	30 % Gl solution	40 % Gl solution	52 % Gl solution	60 % Gl solution
Viscosity（mPa·s）	0.88	1.69	2.29	3.68	6.55	10.32
Density （g / cm3）	1.00	1.05	1.07	1.10	1.12	1.16
sound velocity (m/s, [74])	1482.90	1579.10	1588.70	1631.40	1684.18	1723.50

3. Collapse shockwaves of cavitation bubbles in ultrasonic cavitation field

Fig. 2 was high-speed photographic images of the shockwaves emitted during the evolution of cavitation bubbles in the ultrasonic cavitation field. The time marked in the lower left corner of the images started from the frame before the shockwaves generated by cavitation bubbles. The black rectangular area at the top of the image was the horn of the ultrasonic instrument. In Fig. 2 (a-f), the viscosity η of the liquid was 0.88 mPa·s, 1.69 mPa·s, 2.29 mPa·s, 3.68 mPa·s, 6.55 mPa·s, and 10.32 mPa·s, respectively.

Fig. 2.

Shockwaves from collapsing of cavitation bubbles in ultrasonic cavitation field. (shooting rate: 180000 fps; exposure time: 0.25 µs; (a) η = 0.88 mPa·s; (b) η = 1.69 mPa·s; (c) η = 2.29 mPa·s; (d) η = 3.68 mPa·s; (e) η = 6.55 mPa·s; (f) η = 10.32 mPa·s).

In Fig. 2 (a₁-f₁), the red arrow indicated cavitation bubbles induced by the ultrasonic. In Fig. 2 (a₃-f₃), the red arrow indicated the shockwaves emitted by cavitation bubbles in the evolution process of the ultrasonic field. In Fig. 2 (a-f), cavitation bubbles shrank to the minimum volume at t = 0.000 ms, and emitted implosion shockwaves when it rebounded at t = 0.006 ms.

Fig. 2 (a₃) showed the multiple clear shockwaves emitted by cavitation bubbles during rebound. Fig. 2 (b₃) showed multiple weaker shockwaves emitted from rebound of cavitation bubbles. Fig. 2 (c₃-f₃) showed the shockwaves with successively weakened states emitted by cavitation bubbles during rebound. Therefore, it could be seen from Fig. 2 (a₃-f₃) that as the viscosity of the liquid increased, implosion shockwaves form of emission during the cavitation bubbles rebound gradually weakened.

Due to the high frequency vibration of the horn in the ultrasonic cavitation field, thousands of cavitation bubbles appeared under the horn. The continuous expansion and collapse of these cavitation bubbles caused multiple shockwaves [63]. In addition, the cavitation bubble generated by ultrasonic field or hydrodynamic effect often appears in the form of group cavitation bubbles or cavitation cloud, which brings great challenges to the study on collapse shockwave of the cavitation bubble caused by the change of viscosity of liquid. Therefore, this study adopted the technology of spark-induced cavitation bubble, and analyzed the influence of liquid viscosity on the collapse shockwave of the cavitation bubble.

4. Influence of viscosity of liquid on shockwave evolution

To visually analyze the influence of liquid viscosity on the evolution of cavitation bubble collapse shockwave, this section would present a visual representation of the evolution of shockwave.

4.1. Effect of viscosity of liquid on the evolution of shockwave

Fig. 3 is high-speed photographic images of the evolution of shockwave emitted by the cavitation bubble collapse in different liquid. In Fig. 3 (a-f), η was 0.88 mPa·s, 1.69 mPa·s, 2.29 mPa·s, 3.68 mPa·s, 6.55 mPa·s and 10.32 mPa·s, respectively, and the maximum radius R_max of the corresponding cavitation bubble was 11.20 mm, 11.44 mm, 11.33 mm, 11.17 mm, 11.39 mm and 11.12 mm, respectively. The process of rebound regeneration occurred when the cavitation bubble collapsed for the first time to the minimum [75]. The radius R_r-max of the rebound cavitation bubble when it expanded again to the maximum was 4.10 mm, 6.57 mm, 6.85 mm, 6.92 mm, 7.37 mm, 7.92 mm, respectively. In Fig. 3 (a₅-f₅), the two red arrows in each image indicated the wave front of the collapse shockwave of the cavitation bubble.

Fig. 3.

Shockwave morphology in different viscosity liquid. (shooting rate: 180000 fps; exposure time: 0.25 µs; (a) η = 0.88 mPa·s; (b) η = 1.69 mPa·s; (c) η = 2.29 mPa·s; (d) η = 3.68 mPa·s; (e) η = 6.55 mPa·s; (f) η = 10.32 mPa·s).

In Fig. 3 (a), the cavitation bubble expanded to the maximum volume when t = 1.161 ms and shrank to the minimum volume when t = 2.222 ms. In Fig. 3 (a₄), the shockwave completely separated from the cavitation bubble and rapidly developed to the external water body. The wave front of the collapse shockwave in the high-speed photographic image was very clear. The same phenomenon also reported in Beig [76]. It could be measured that the thickness of shockwave was about 3.171 mm. With the rebound of the cavitation bubble, its volume gradually increased, and the surface gradually became not smooth, as shown in Fig. 2 (a₇-a₁₀).

In Fig. 3 (b-f), the cavitation bubble expanded to its maximum volume at t = 1.167 ms, 1.222 ms, 1.133 ms, 1.200 ms, 1.222 ms, respectively. The cavitation bubbles shrank to the minimum volume at t = 2.250 ms, 2.328 ms, 2.200 ms, 2.344 ms, 2.328 ms, respectively. The wave front gradually weakened from a clear form to a fuzzy form, as shown in Fig. 3 (b₄-f₄). The corresponding thickness of wave front of the shockwave was about 3.500 mm, 3.784 mm, 4.419 mm, 4.750 mm, and 5.117 mm, respectively. In the above experiments, when the shockwave of the cavitation bubble collapse for the first time propagated to the wall or free surface of the water tank during the process of the cavitation bubble rebound and regeneration, reflected waves would appear. Many waves would appear when these reflected waves propagated to the viewing field of high-speed photography, as shown in Fig. 3 (a₁₀). An interesting phenomenon was that as the viscosity of the liquid increased, this phenomenon of reflected waves gradually weakened, as shown in Fig. 3 (b₁₀-f₁₀).

According to the high-speed photographic observation of the collapse shockwave of the cavitation bubble of similar size under different viscous liquid conditions in Fig. 3, it could be found that with the increase of viscosity, the wave front of the shockwave gradually thickened, and the maximum expansion radius of the rebound cavitation bubble gradually increased. A scatter plot of the change of the cavitation bubble normalization time T / T_c with η was drawn through the repetition experiment (5–8 times) of the cavitation bubble under various viscous liquid conditions, as shown in Fig. 4. T was the cavitation bubble shrinking from the maximum radius to the minimum radius in the first cycle. T_c was Rayleigh time. T_c was calculated using the following Equation (1) [21]:

$$T_c=0.915R_{\max }\sqrt{\rho /(p_{\infty }-p_v)}$$ (1)

where $\rho$ is the liquid density,$p_{\infty }$ is the hydrostatic pressure, taking 96 kPa, When calculating Rayleigh time in this paper, $p_{\infty }$ is selected as atmospheric pressure in Liu et al. [52], $p_v$ is the vapor pressure, taking 0 kPa.

Fig. 4.

Variation of the cavitation bubble normalization time with liquid viscosity.

In Fig. 4, when η was 0.88 mPa·s, 1.69 mPa·s, 2.29 mPa·s, 3.68 mPa·s, 6.55 mPa·s and 10.32 mPa·s, the mean value of T / T_c was 0.996, 0.999, 1.004, 1. 006, 1. 007 and 1.015, respectively. It could be seen that T / T_c showed a monotonically increasing trend with the increase of viscosity η. Meanwhile, the values T / T_c in the literature [52] were also shown in Fig. 4. The standardized time in the literature and the standardized time of the bubble in different viscous liquids in this paper were shown in Table 2.

Table 2.

The standardized time T / T_c of bubble in different liquid.

Viscosity（mPa·s）	Present data (25℃)						Liu et al. [52] (20℃)
	0.88	1.69	2.29	3.68	6.55	10.32	1.00	1.78	3.85	9.46	60.07	1499
Radius (mm)	11.5 ± 0.5						1.14	1.00	0.88	0.79	0.68	0.52
T / Tc	0.996	0.999	1.004	1.006	1.007	1.015	0.91	1.02	1.30	1.51	1.79	2.44

4.2. Effect of viscosity of liquid on impact time span of the shockwave

In order to further study the influence of the viscosity of liquid on the thickness of wave front of shockwave, the impact time span T_s of collapse shockwave in different viscosity liquids is compared. T_s was extracted from the pressure profile line collected by the pressure sensor. The extraction principle of T_s was to select the time corresponding to the impact pressure being 20 % of the pressure peak before and after the peak value of the shockwave.

Fig. 5 showed the influence of viscosity of liquid on T_s. T_s gradually increased with the increase of η, as shown in Fig. 5. In deionized water, the mean value of T_s was about 3.015 µs, and when η increased to 10.32 mPa·s, the corresponding mean value of T_s increased to about 20.823 µs.

Fig. 5.

Impact time span of the cavitation bubble collapse shockwave in different viscosity liquid. (Error bar of gray square is hidden in gray square. The high-speed photography of collapse shockwave and the pressure profile line of impact time span are shown in the upper left corner).

From the high-speed photographic images and data of the above experiments, it could be found that the liquid viscosity has a significant influence on the evolution of the cavitation bubble and the morphology of the collapse shockwave. Specifically, the thickness wave front of the collapse shockwave from the cavitation bubble gradually increased with the increase of η, until it became blurred when η was very large, and the corresponding T_s also lengthened. There were two main spatiotemporal characteristics of collapse shockwave of the cavitation bubble in different viscosities. On the one hand, when the implosion shockwave generated, the viscous liquid had an obvious blocking effect on the shockwave. Moreover, the blocking effect was also reflected in the spatial structure of the shockwave. That was, during the propagation of the shockwave, there was a large pressure gradient in the normal direction of the wave front, the wave front of the shockwave was thickened. On the other hand, viscosity also had a significant blocking effect on the collapse stage of the cavitation bubble. That was, the collapse time of the cavitation bubble increased with the increase of viscosity, which led to the reduction of the intensity of the implosion shockwave (this content would be discussed in depth in Section 5). This finding was consistent with the previous conclusions on cavitation dynamics in viscous liquids [52], [55], [57], [77].

5. Effect of liquid viscosity on the shockwave pressure

In order to further explore the influence of the viscosity of liquid on the shockwave, this section would analyze the changing of pressure profile and pressure peak of shockwaves.

5.1. Influence of viscosity of liquid on pressure profile of the shockwave

Fig. 6 (a-h) was the pressure profile of eight different measuring points in the space when η changed, the correspondingly relative distance γ was 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5 and 9.5 respectively. γ was the relative distance which the distance between the center of the bubble and the sensor measuring point divided by the maximum radius of the bubble. L was the distance between the center of the bubble and the sensor measuring point. r was the shockwave propagation radius. P was the impact pressure of the collapse shockwave of the cavitation bubble.

Fig. 6.

Pressure profile of shockwave of cavitation bubble collapse in different liquid. ((a) γ = 2.5; (b) γ = 3.5; (c) γ = 4.5; (d) γ = 5.5; (e) γ = 6.5; (f) γ = 7.5; (g) γ = 8.5; (h) γ = 9.5).

In Fig. 6 (a), when γ = 2.5, the impact pressure of the pressure profile decreased to different degrees with the increase of η. The higher the viscosity is, the more obvious the impact pressure would decrease compared with deionized water. In Fig. 6 (b-h), the variation of the impact pressure of the pressure profile is the similar as that in Fig. 6 (a). In Fig. 6 (a-h), when η is the same, the impact pressure of the pressure profile decreased to different degrees with the increase of γ.

As can be seen from Fig. 6 (a-h), when η = 0.88 mPa·s, the negative pressure generated by the collapse shockwave at the sensor wall gradually decreased with the increase of γ. In addition, the pressure peak of the shockwave also showed varying degrees of attenuation. In the next section, the attenuation amplitude of the pressure peak of the collapse shockwave in different viscous liquids will be analyzed in depth.

5.2. Attenuation rate of viscosity of liquid to pressure peak of shockwave

The pressure peak is an important index to evaluate the potential damage to the wall caused by the shockwave or the efficiency of ultrasonic cleaning. The relationship between the pressure peak of the shockwave and the distance and the relative distance and the viscosity of liquid is shown in Fig. 7. Fig. 7 (a) showed the attenuation of the pressure peak of shockwave in space (Logarithmic coordinates are used in the figure to facilitate comparison with the results from Reference [51]). Fig. 7 (b) comprehensively analyzed the relationship between the pressure peak of the shockwave and its relative distance, as well as the viscosity of the liquid, with the purpose of predicting the attenuation of the shock wave pressure peak from the cavitation bubble collapse in a determined solution.

Fig. 7.

Relation between the pressure peak of shockwave, the distance, the relative distance and the viscosity of liquid. ((a) The relationship between the pressure peak of shockwave and the distance; (b) The change of pressure peak of shockwave with viscosity and relative distance; The experimental data points in (a) and (b) have error bars, among which the data points cited by Vogel et al. [51] do not have error bars).

As can be seen from Fig. 7 (a), in the same liquid viscosity, the pressure peak P_max of the shockwave showed a monotonically decreasing trend as L gradually increased. Vogel et al. [51] obtained the spatial variation of the intensity of the expansion shockwave of the laser-induced cavitation bubble. This was almost consistent with the spatial variation of the pressure peak of the collapse shockwave of spark-induced cavitation bubble in this paper. The difference was that the cavitation bubble in Reference [51] was induced by laser focusing, and the maximum radius of the cavitation bubble was about 1 mm.

In Fig. 7 (a), It can be observed that there is a relationship between P_max and L as follows:

$$P_{\mathrm{m}\mathrm{a}\mathrm{x}}\propto a{L}^b$$ (2)

As the viscosity of the liquid changed, the attenuation of the pressure peak of the collapse shockwave of the cavitation bubble showed a similar trend in space. The difference was that when each viscosity coefficient changed, the value of a and the value of b in Relation (2) were different, and the approximate coefficients were shown in Table 3.

Table 3.

Coefficients of Relation (2) in different viscous liquid.

η（mPa·s）	1.04 (22℃) [51]	0.88	1.69	2.29	3.68	6.55	10.32
a	12.94	187.19	212.01	131.4	131.2	128.94	107.0
b	−1.06	−1.03	−1.16	−1.18	−1.21	−1.23	−1.28
R2	0.997	0.987	0.994	0.994	0.968	0.880	0.987

It could be seen from Table 3 that the pressure peak P_max was related to the distance L^b. The changing of the value of b in Table 3 was consistent with the results in the literature [51]. In addition, under the condition of increasing viscosity in the experiment in this paper, the value of b in Relation (2) gradually decreased.

In Fig. 7 (b), the parameter (1 / γ)(1 / η) was comprehensively reflected the variation of shockwave pressure peak with viscosity and relative distance. As can be seen from Fig. 7 (b), within the viscosity range of 0.88 mPa·s to 10.32 mPa·s, the peak pressure of the shock wave roughly followed a linear variation pattern with (1 / γ)(1 / η).

Through the above analysis, there were two main reasons for the influence of the change of liquid viscosity on the impact pressure peak. On the one hand, in the stage of the cavitation bubble contraction and collapse, the viscosity of the liquid inhibited the contraction speed of the cavitation bubble and delayed the process of the cavitation bubble contraction and collapse. At the same time, more mechanical energy of the cavitation bubble was dissipated in the stage of the cavitation bubble contraction, resulting in the decrease of the cavitation bubble's ability to form implosion shockwave. On the other hand, when the shockwave propagated to the outside liquid after breaking away from the cavitation bubble [51], the viscosity of the liquid further dissipated the energy of the shockwave in space, and this dissipation effect gradually intensified with the increase of the distance from the cavitation bubble center. The similar results also appeared in Vogel et al. [51], Brujan and Williams [78] and Brujan et al. [79].

6. Attenuation and redistribution of shockwave energy due to viscosity of liquid

Evaluation of shock wave energy is also an important aspect of shock intensity research. Shockwave energy could be obtained by pressure profile [50], [80], [81].

$$E=\frac{4\pi L^2}{{\rho }_0{c}_0}\int P^{2}dt$$ (3)

where L is the distance from the center of the cavitation bubble to the pressure sensor, ${\rho }_0$ is the density of liquid, and $c_0$ is the sound speed in liquid. The specific parameters of ${\rho }_0$ and $c_0$ were shown in Table 1. The integral bounds includes the complete waveform of the shockwave.

6.1. Attenuation rate of viscosity of liquid to shockwave energy

Fig. 8 showed the relationship between the ratio of the absolute value of difference which shockwave energy E_i subtracted the cavitation bubble maximal mechanical energy E_max in the same viscous liquid to the cavitation bubble maximal mechanical energy E_max with the variation of liquid viscosity. E_i was the shockwave energy when the parameter γ was 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, respectively.

Fig. 8.

Attenuation of shockwave energy of the cavitation bubble collapse due to liquid viscosity. (all data points in the Fig. 8 have error bars)

As can be seen from Fig. 8, under the condition of the same η, the attenuation rate (|E_i − E_max| / E_max) of shockwave energy gradually increased as the γ gradually increased. In Fig. 8, when γ was the same, the attenuation rate (|E_i − E_max| / E_max) of shockwave energy gradually increased with the increase of η.

In order to further quantitatively analyze the attenuation rate of the shockwave energy of the cavitation bubble collapse, the attenuation rate of shockwave energy and the viscosity coefficient of liquid were fitted, as shown in the red dashed line in Fig. 8. The fitting formula was shown next to the red dashed line in Fig. 8. The value of s was −0.3. The value of w was 0.72. The corresponding R² was 0.8.

According to the fitting analysis in Fig. 8, there was a trend of exponential function change between the attenuation rate of the shockwave energy from the cavitation bubble collapse and liquid viscosity. When the viscosity of liquid gradually increased, the attenuation rate of shockwave energy gradually approached 1. When the viscosity of the liquid was 0, that was, the ideal liquid, the attenuation rate of shockwave energy in the liquid was almost 0.

6.2. Influence of viscosity of liquid on the energy of the rebound bubble

The energy distribution of the cavitation bubble during collapse is an important index to reflect the dynamic characteristics of the cavitation bubble. We further discussed the relationship between the energy E_s of the collapse shockwave of the cavitation bubble, the maximal mechanical energy E_max and the mechanical energy E_r-max when the rebound cavitation bubble expanded to the maximum volume. The calculation formulas for E_max and E_r-max were as follows [30], [78], [81], [82].

$$E_{\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{4\pi }{3}{R}_{\mathrm{m}\mathrm{a}\mathrm{x}}^3(p_{\infty }-p_{\mathrm{v}})$$ (4)

$$E_{\mathrm{r}-\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{4\pi }{3}{R}_{\mathrm{r}-\mathrm{m}\mathrm{a}\mathrm{x}}^3(p_{\infty }-p_{\mathrm{v}})$$ (5)

where $p_{\infty }$ is the hydrostatic pressure, taking 96 kPa; $p_v$ is the saturated vapor pressure of the cavitation bubble, taking 0 kPa [82].

Fig. 9 showed E_s / E_max (solid scatter point in Fig. 9) and E_r-max / E_max (the hollow scatter point in Fig. 9) varied with η, respectively. As can be seen from Fig. 9, E_s / E_max gradually decreased, while E_r-max / E_max gradually increased, as η increased. In addition, in the same η, E_s / E_max decreased gradually with the increase of γ between the measuring point and the cavitation bubble center.

Fig. 9.

Effect of liquid viscosity on the distribution of shockwave energy in the first collapse of the cavitation bubble and mechanical energy when the rebound cavitation bubble expanded to the maximum. (The black dashed line fitted the variation relationship between E_s / E_max and η. The red dashed line fitted the variation relationship between E_r-max / E_max and η.)

In order to quantitatively analyze the influence of η on E_s / E_max and E_r-max / E_max, we respectively made a fitting analysis of the variation of E_s/E_max and E_r-max/E_max with η. The fitting formulas were written in Fig. 9. In Fig. 9, the corresponding intersection point was obtained by fitting two curves, with a viscosity of 2.82 mPa·s and a value of 0.23 where E_s / E_max was equaled to E_r-max / E_max.

First of all, in Fig. 9, it could be seen that as η increased, E_s / E_max gradually decreased. When η was very small, the value of E_s / E_max was approximately 95.6 %. When the viscosity of the medium increased, a small portion of the energy of the bubble is released in the form of the shockwave, while most of the energy is dissipated by the liquid and transferred to the subsequent evolution process of the bubble.

Secondly, in Fig. 9, it could be seen that with the increase of η, E_r-max / E_max gradually increased. When η was very small, E_r-max / E_max was about 3.9 %. With η increasing to 12 mPa·s, E_r-max / E_max was closed to 32 %.

Finally, it could also be found from Fig. 9 that when η = 2.82 mPa·s, E_s was approximately equal to E_r-max. This meant that when η was less than 2.82 mPa·s, E_s was higher than E_r-max. When η was higher than 2.82 mPa·s, E_s was less than E_r-max. This changing of energy configuration between the viscosity of liquid and the evolution of the cavitation bubble could provide a theoretical basis for ultrasonic cleaning in liquids with different viscosities to make rational use of their respective regions. For example, in the liquid with low viscosity, the strength of cavitation should be controlled to the maximum extent and the shockwave capacity emitted by the cavitation bubble during the first collapse should be exerted as much as possible. In the liquid with high viscosity, the first life cycle of the cavitation bubble should be controlled as much as possible to make the rebound cavitation bubble obtain more energy.

7. Conclusion

In order to explore the differences of cavitation dynamics in liquids with different viscosities, this study used the high-speed photography and the high-frequency pressure measurement system to analyze attenuation effect of liquid viscosity on the shockwave of the cavitation bubble collapse. The preliminary conclusions are as follows:

• (1)The viscosity of liquid has a significant influence on the spatiotemporal evolution of the shockwave of the cavitation bubble collapse. With the increase of the viscosity of the liquid, the wave front of the shockwave from the cavitation bubble collapse becomes thicker gradually, and the corresponding shockwave impact time span also increases. In addition, because the viscosity delays the first collapse period of the cavitation bubble, the cavitation bubble in the viscous liquid obtains a large energy during the rebound, and eventually the expansion radius of the rebound cavitation bubble increases correspondingly.
• (2)The viscosity of liquid has a significant influence on the pressure peak of the shockwave of the cavitation bubble collapse. In the same viscous liquid, the pressure peak value of the shockwave decreases gradually as the shockwave is transmitted to the outside liquid. In different viscous liquid, the pressure peak value of shockwave at the same distance from the cavitation bubble center in space decreases with the increase of viscosity of the liquid. Moreover, there is a relationship between the pressure peak P_max of the shockwave and the distance L^b from the center point of the cavitation bubble, and the higher the viscosity is, the smaller the value of b is.
• (3)The viscosity of liquid has a significant influence on the energy configuration of the cavitation bubble collapse shockwave and rebound cavitation bubble. In different viscous liquids, the energy of shockwave shows different degrees of attenuation when it propagates to the outside liquid. And the higher the viscosity is, the higher the attenuation degree is. Moreover, with the increase of liquid viscosity η, the ratio of the shockwave energy E_s to the maximal mechanical energy E_max gradually decreases, while the ratio of the mechanical energy E_r-max to the maximal mechanical energy E_max increases gradually.

The above new findings have an important theoretical significance for the efficient utilization of cavitation in industrial production. For example, in the liquid with low viscosity, the intensity of cavitation should be controlled to the maximum extent, and the shockwave energy emitted by the cavitation bubble when it first collapses should be used as much as possible. While in the liquid with high viscosity, the first life cycle of the cavitation bubble should be controlled as much as possible, so that the rebound cavitation bubble can obtain greater energy to achieve the maximum effect.

CRediT authorship contribution statement

Jing Luo: Methodology, Investigation, Formal analysis. Guihua Fu: Investigation, Data curation. Weilin Xu: Conceptualization. Yanwei Zhai: Investigation. Lixin Bai: Methodology. Jie Li: Investigation. Tong Qu: Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.