**3. Results and discussion**

#### **3.1. Comparison of the bubble motion near a free boundary and a rigid boundary under an ultrasonic field**

**Figure 2** shows the motion characteristics of a bubble near a free boundary and a rigid boundary under an ultrasonic field in five sound cycles, for ultrasonic frequency of 20 kHz and acoustic amplitude of 0.2 MPa. The initial bubble radius is 10 μm, and the dimensionless distance from the bubble to the rigid boundary (*l*/*R*<sup>0</sup> ) is set to 1. The bubble motion near a rigid boundary is described by Eq. (3), and the bubble motion with a free boundary can be further obtained by ignoring the effect of the solid wall of Eq. (3), that is without the left third item in Eq. (3).

**Figure 2(a)** displays the variation of the bubble radius versus time. It can be seen that even though the bubble undergoes the dynamic process of growth, expansion, compression, collapse and rebound under the action of five sound cycles, there are obvious differences of the bubble radius in the two cases. Compared with the case under the free boundary, the bubble near the rigid boundary has a lower maximum radius and a longer collapse time. It illustrates that the process of bubble expansion and compression becomes slower because of the existence of the rigid boundary, that is, the rigid boundary plays a part in suppressing the bubble motion.

**Figure 2(b)** shows the variation of the bubble velocity versus time. As can be seen, the closer the bubble minimum radius, the greater is the bubble velocity. When the bubble radius is compressed to the minimum radius, the maximum velocity of the bubble can be obtained.

**Figure 2.** Motion characteristics of a bubble near a rigid boundary and a free boundary under ultrasound field: (a) bubble radius versus time and (b) bubble velocity versus time.

There is the damping of the sound wave in the liquid, and the oscillation of the bubble will become weaker and weaker, and thus the first sound cycle is taken an example to describe the bubble collapse approximately. Furthermore, for the free boundary, the bubble radius can be compressed to 0.1408 of the initial radius extremely, and the maximum velocity of the bubble can be up to 5422 m·s−1. However, for the rigid boundary, the bubble radius can merely be compressed to 0.1453 of the initial radius and the bubble velocity is 2661 m·s−1. Thus, compared with the free boundary, the compression ratio of the bubble under the rigid boundary is lower and the velocity of the bubble collapse is smaller. It also indicates that the rigid boundary has an inhibition effect for the bubble collapse.

#### **3.2. Effects of parameters on the velocity of the bubble collapse**

The collapse and rebound of the bubble near the rigid wall are closely related to the effects of microjets and shock waves of ultrasonic cavitation. To further study the effects of bubble collapse near the solid wall, the main parameters affecting the bubble collapse will be analyzed in the following aspects. In view of that, the theoretical and experimental research about acoustic cavitation are usually concerned about the size of the velocity of the bubble collapse [23], and the maximum value of the bubble velocity in an acoustic cycle is selected to record the velocity of the bubble collapse (*v*collapse).

#### *3.2.1. Effect of the bubble initial radius*

Eqs. (2), (3), and (5) into Eq. (7), the relationship between the velocity of the bubble collapse

**Figure 2** shows the motion characteristics of a bubble near a free boundary and a rigid boundary under an ultrasonic field in five sound cycles, for ultrasonic frequency of 20 kHz and acoustic amplitude of 0.2 MPa. The initial bubble radius is 10 μm, and the dimensionless

rigid boundary is described by Eq. (3), and the bubble motion with a free boundary can be further obtained by ignoring the effect of the solid wall of Eq. (3), that is without the left third

**Figure 2(a)** displays the variation of the bubble radius versus time. It can be seen that even though the bubble undergoes the dynamic process of growth, expansion, compression, collapse and rebound under the action of five sound cycles, there are obvious differences of the bubble radius in the two cases. Compared with the case under the free boundary, the bubble near the rigid boundary has a lower maximum radius and a longer collapse time. It illustrates that the process of bubble expansion and compression becomes slower because of the existence of the rigid boundary, that is, the rigid boundary plays a part in suppressing the

**Figure 2(b)** shows the variation of the bubble velocity versus time. As can be seen, the closer the bubble minimum radius, the greater is the bubble velocity. When the bubble radius is compressed to the minimum radius, the maximum velocity of the bubble can be obtained.

**Figure 2.** Motion characteristics of a bubble near a rigid boundary and a free boundary under ultrasound field: (a) bubble

) is set to 1. The bubble motion near a

**3.1. Comparison of the bubble motion near a free boundary and a rigid boundary** 

and microjet can be further obtained.

distance from the bubble to the rigid boundary (*l*/*R*<sup>0</sup>

radius versus time and (b) bubble velocity versus time.

**3. Results and discussion**

**under an ultrasonic field**

78 Cavitation - Selected Issues

item in Eq. (3).

bubble motion.

**Figure 3** shows the velocity of the bubble collapse versus the initial bubble radius for the ultrasonic frequency of 20 kHz, acoustic amplitude of 0.2 MPa and the dimensionless distance from the bubble to the rigid boundary of 1, for various initial bubble radius (10–100 μm). The

**Figure 3.** Velocity of the bubble collapse versus the initial bubble radius.

reason for the selection of that range of the bubble initial radius is that it is a common value for discussing cavitation and cavitation erosion [24]. As can be seen, the smaller the initial bubble radius, the higher is the velocity of the bubble collapse. With the increase of the initial bubble radius, the velocity of the bubble collapse decreases rapidly, which means the intensity of ultrasonic cavitation is weakened. This is mainly because the initial radius of the bubble used in the research is smaller than the resonance radius of the bubble (according to Minneart's theory, an ultrasonic wave with a frequency of 20 kHz has a resonance radius of several hundred micrometers [25]). Thus, for a bubble with a larger initial radius, it will begin to compress before it grows to the maximum. As a result, the expansion of the bubble is weakened, and the collapse time is prolonged, which results in the decrease of the velocity of the bubble collapse. For the same bubble initial radius, the velocity of the bubble collapse under the case of the rigid boundary is lower than that of the free boundary. In addition, in **Figure 3**, with the increase of the dimensionless distance from the bubble to the rigid boundary, the velocity of the bubble collapse gradually increases. The farther the distance from the bubble to the rigid boundary, the closer is the velocity of the bubble collapse under the rigid boundary to it under the free boundary. It indicates that compared with the free boundary, the rigid boundary suppresses the process of the bubble collapse among the discussed initial bubble radii.

#### *3.2.2. Effect of the distance from the bubble to the solid wall*

**Figure 4** shows the velocity of the bubble collapse versus the distance from the bubble to the solid wall for the ultrasonic frequency of 20 kHz, acoustic amplitude of 0.2 MPa and the initial bubble radius of 20 μm, for the dimensionless distance from the bubble to the rigid boundary (1*R*<sup>0</sup> –51*R*<sup>0</sup> ). It can be seen from **Figure 4**, the velocity of the bubble collapse under the free boundary can be up to 5569 m·s−1, and it is not related to the distance from the bubble to the solid wall. However, for the bubble near the rigid boundary, the distance from the bubble to the solid wall has a significant effect on the velocity of the bubble collapse. When the dimensionless distance between the bubble and the solid wall is relatively small, for instance, the bubble is just close to the solid wall, the velocity of the bubble collapse is 2756 m·s−1. With the increase of the distance from bubble to solid wall, the inhibitory action of the solid wall on the bubble motion is diminished and thus the velocity of the bubble collapse increases. Moreover, the greater the distance between the bubble and the solid wall, the slower is the increasing of the velocity of the bubble collapse. When the dimensionless distance from the bubble to the rigid boundary is 51, the velocity of the bubble collapse is 5422 m·s−1 which is very close to that of 5569 m·s−1 under the free boundary. It indicates that the farther the distance from the bubble to the solid wall, the smaller is the influence of the solid wall on the bubble. When the distance of the bubble away from the solid wall is up to a certain value, the effect of the solid wall on the bubble is almost negligible. In the situation, the bubble motion near the rigid boundary can be regarded as that under the free boundary.

distance from the bubble to the rigid boundary of 1, for various acoustic pressure amplitude

on the velocity of the bubble collapse. When the acoustic pressure amplitude is very low, such as

the bubble collapse increases nearly in a linear manner. It illustrates that for a bubble in a free liquid, the increase of the acoustic pressure amplitude can significantly improve the severity of cavitation. Compared with the case under the free boundary, on the one hand, the velocity of the bubble collapse under the rigid boundary is lower because of the inhibitive action of the rigid boundary. On the other hand, the variation of the velocity of the bubble collapse under

pressure amplitude, the velocity of the bubble collapse presents the trend of increasing first and then decreasing. In addition, for the bubble near the rigid wall, it is noted that there is an optimal value of the acoustic pressure amplitude. Under the optimal value, the bubble collapse can be

of the bubble collapse can be up to the maximum value at the acoustic pressure amplitude of 3.5

, and it demonstrates there is the strongest cavitation effect. With the increase of the distance from the bubble to the solid wall, the optimal value of the acoustic pressure amplitude will gradually increase. When the bubble is far enough from the rigid boundary, the bubble motion

boundary, with the increase of the acoustic pressure amplitude, such as *pa* > 1*p*<sup>0</sup>

**Figure 4.** Velocity of the bubble collapse versus the distance from the bubble to the solid wall.

maximized. For instance, when the bubble is just close to the solid wall, that is *l* = *R*<sup>0</sup>

the rigid boundary is more special and complex. When *pa* > 1*p*<sup>0</sup>

). As can be seen from **Figure 5**, the acoustic pressure amplitude has a special influence

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, the velocity of the bubble collapse is almost close to zero for both free and rigid boundary. It is owing to the fact that the acoustic pressure amplitude is so lower that it is still unable to overcome the hydrostatic pressure of the liquid. Thus, in the case, the liquid has not yet caused cavitation. With the increase of the acoustic pressure amplitude, the velocity of the bubble collapse under the rigid boundary is different from the case under the free boundary. For the free

, the velocity of

, the velocity

, with the increase of the acoustic

(0*p*<sup>0</sup> –5*p*<sup>0</sup>

*p*0

*pa* ≤ 1*p*<sup>0</sup>

#### *3.2.3. Effect of acoustic pressure amplitude*

**Figure 5** shows the velocity of the bubble collapse versus the acoustic pressure amplitude for the ultrasonic frequency of 20 kHz, the initial bubble radius of 20 μm and the dimensionless

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**Figure 4.** Velocity of the bubble collapse versus the distance from the bubble to the solid wall.

reason for the selection of that range of the bubble initial radius is that it is a common value for discussing cavitation and cavitation erosion [24]. As can be seen, the smaller the initial bubble radius, the higher is the velocity of the bubble collapse. With the increase of the initial bubble radius, the velocity of the bubble collapse decreases rapidly, which means the intensity of ultrasonic cavitation is weakened. This is mainly because the initial radius of the bubble used in the research is smaller than the resonance radius of the bubble (according to Minneart's theory, an ultrasonic wave with a frequency of 20 kHz has a resonance radius of several hundred micrometers [25]). Thus, for a bubble with a larger initial radius, it will begin to compress before it grows to the maximum. As a result, the expansion of the bubble is weakened, and the collapse time is prolonged, which results in the decrease of the velocity of the bubble collapse. For the same bubble initial radius, the velocity of the bubble collapse under the case of the rigid boundary is lower than that of the free boundary. In addition, in **Figure 3**, with the increase of the dimensionless distance from the bubble to the rigid boundary, the velocity of the bubble collapse gradually increases. The farther the distance from the bubble to the rigid boundary, the closer is the velocity of the bubble collapse under the rigid boundary to it under the free boundary. It indicates that compared with the free boundary, the rigid boundary sup-

presses the process of the bubble collapse among the discussed initial bubble radii.

**Figure 4** shows the velocity of the bubble collapse versus the distance from the bubble to the solid wall for the ultrasonic frequency of 20 kHz, acoustic amplitude of 0.2 MPa and the initial bubble radius of 20 μm, for the dimensionless distance from the bubble to the

under the free boundary can be up to 5569 m·s−1, and it is not related to the distance from the bubble to the solid wall. However, for the bubble near the rigid boundary, the distance from the bubble to the solid wall has a significant effect on the velocity of the bubble collapse. When the dimensionless distance between the bubble and the solid wall is relatively small, for instance, the bubble is just close to the solid wall, the velocity of the bubble collapse is 2756 m·s−1. With the increase of the distance from bubble to solid wall, the inhibitory action of the solid wall on the bubble motion is diminished and thus the velocity of the bubble collapse increases. Moreover, the greater the distance between the bubble and the solid wall, the slower is the increasing of the velocity of the bubble collapse. When the dimensionless distance from the bubble to the rigid boundary is 51, the velocity of the bubble collapse is 5422 m·s−1 which is very close to that of 5569 m·s−1 under the free boundary. It indicates that the farther the distance from the bubble to the solid wall, the smaller is the influence of the solid wall on the bubble. When the distance of the bubble away from the solid wall is up to a certain value, the effect of the solid wall on the bubble is almost negligible. In the situation, the bubble motion near the rigid boundary can be regarded as that under the free boundary.

**Figure 5** shows the velocity of the bubble collapse versus the acoustic pressure amplitude for the ultrasonic frequency of 20 kHz, the initial bubble radius of 20 μm and the dimensionless

). It can be seen from **Figure 4**, the velocity of the bubble collapse

*3.2.2. Effect of the distance from the bubble to the solid wall*

–51*R*<sup>0</sup>

*3.2.3. Effect of acoustic pressure amplitude*

rigid boundary (1*R*<sup>0</sup>

80 Cavitation - Selected Issues

distance from the bubble to the rigid boundary of 1, for various acoustic pressure amplitude (0*p*<sup>0</sup> –5*p*<sup>0</sup> ). As can be seen from **Figure 5**, the acoustic pressure amplitude has a special influence on the velocity of the bubble collapse. When the acoustic pressure amplitude is very low, such as *pa* ≤ 1*p*<sup>0</sup> , the velocity of the bubble collapse is almost close to zero for both free and rigid boundary. It is owing to the fact that the acoustic pressure amplitude is so lower that it is still unable to overcome the hydrostatic pressure of the liquid. Thus, in the case, the liquid has not yet caused cavitation. With the increase of the acoustic pressure amplitude, the velocity of the bubble collapse under the rigid boundary is different from the case under the free boundary. For the free boundary, with the increase of the acoustic pressure amplitude, such as *pa* > 1*p*<sup>0</sup> , the velocity of the bubble collapse increases nearly in a linear manner. It illustrates that for a bubble in a free liquid, the increase of the acoustic pressure amplitude can significantly improve the severity of cavitation. Compared with the case under the free boundary, on the one hand, the velocity of the bubble collapse under the rigid boundary is lower because of the inhibitive action of the rigid boundary. On the other hand, the variation of the velocity of the bubble collapse under the rigid boundary is more special and complex. When *pa* > 1*p*<sup>0</sup> , with the increase of the acoustic pressure amplitude, the velocity of the bubble collapse presents the trend of increasing first and then decreasing. In addition, for the bubble near the rigid wall, it is noted that there is an optimal value of the acoustic pressure amplitude. Under the optimal value, the bubble collapse can be maximized. For instance, when the bubble is just close to the solid wall, that is *l* = *R*<sup>0</sup> , the velocity of the bubble collapse can be up to the maximum value at the acoustic pressure amplitude of 3.5 *p*0 , and it demonstrates there is the strongest cavitation effect. With the increase of the distance from the bubble to the solid wall, the optimal value of the acoustic pressure amplitude will gradually increase. When the bubble is far enough from the rigid boundary, the bubble motion

**Figure 5.** Velocity of the bubble collapse versus the acoustic pressure amplitude.

and collapse are equal to that under the free boundary, and then the optimal value of the acoustic pressure amplitude is not found easily.

collapse is getting higher. It presents that from the control point of view of the ultrasonic frequency, with the increase of ultrasonic frequency, the bubble under the rigid boundary is

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**Figure 7** shows the relationship between the velocity of the bubble collapse and the microjet under the action of one sound cycle, for the ultrasonic frequency of 20 kHz, the initial radius of the bubble of 20 μm and the dimensionless distance from the bubble to the rigid wall of 1. It can be seen from **Figure 7**, the velocity of the microjet responds to changes in the velocity of the bubble collapse, with the increase of the acoustic pressure amplitude. From the above analysis of **Figure 5**, the velocity of the bubble collapse can be up to the maximum

the acoustic pressure amplitude. However, in **Figure 7**, the velocity of the microjet reaches a

be treated as another optimum value of acoustic pressure amplitude to improve the microjet effect. It can be seen that the optimum value of the acoustic pressure amplitude of the microjet is lower than that of the velocity of the bubble collapse. In addition, the dotted line in **Figure 7** represents the position where the velocity of the bubble collapse is 1500 m·s−1, and the acoustic

. When *pa* ≤ 1.6 *p*<sup>0</sup>

less than the propagation velocity of an ultrasonic wave in water (1500 m·s−1), in which there is no microjet appearing near the solid wall. Thus, it can be determined that the analysis for the velocity of the bubble collapse is contributed to seek the optimal value of the microjet and to distinguish the range of the variation of the microjet. Based on the earlier analysis, the velocity

maximum (67.9 m·s−1), corresponding to the acoustic pressure amplitude of 3.1 *p*<sup>0</sup>

, which is the optimal value of

, the velocity of the bubble collapse is

, which can

more easily to collapse than that under the rigid boundary.

**Figure 6.** Velocity of the bubble collapse versus the ultrasonic frequency.

value (5488 m·s−1) at the acoustic pressure amplitude of 3.5 *p*<sup>0</sup>

pressure amplitude is relevant to 1.6 *p*<sup>0</sup>

**3.3. Relationship between the velocity of the bubble collapse and microjet**

#### *3.2.4. Effect of ultrasound frequency*

**Figure 6** shows the velocity of the bubble collapse versus the ultrasonic frequency for the acoustic pressure amplitude of 0.2 MPa, the initial bubble radius of 20 μm and the dimensionless distance from the bubble to the rigid boundary of 1, for various ultrasonic frequency (18–30 kHz). As can be seen from **Figure 6**, when the ultrasonic frequency is low, the velocity of the bubble collapse is high. As the ultrasonic frequency increases, the velocity of the bubble collapse gradually decreases. It means that a weaker effect of the cavitation will be obtained with the increase of the ultrasonic frequency. It is mainly due to the fact that with the increase of ultrasonic frequency, the cycles of the bubble expansion and compression are getting faster and faster. Thus, the bubble may not have enough time to grow to produce the cavitation effect or the bubble may not be compressed enough to collapse. These may result in reducing the growth and collapse of the bubble and further reducing acoustic cavitation effect. Especially for the higher-frequency ultrasound, the bubble does not have enough time to store the ultrasonic energy and begins to collapse. Therefore, when the ultrasonic frequency is increasing, the velocity of the bubble collapse will continue to decrease, and eventually it will tend to be stable. It can also be found in **Figure 6**, the velocity of the bubble collapse under the rigid boundary is lower than that under the free boundary, at the same ultrasonic frequency. When the ultrasonic frequency varies from 18 to 30 kHz, the velocity of the bubble collapse is reduced by 48.84 and 53.94% under the rigid and free boundary, respectively. Moreover, as the increase of the distance from the bubble to the solid wall, the velocity of the bubble The Relationship between the Collapsing Cavitation Bubble and Its Microjet near a Rigid Wall… http://dx.doi.org/10.5772/intechopen.79129 83

**Figure 6.** Velocity of the bubble collapse versus the ultrasonic frequency.

and collapse are equal to that under the free boundary, and then the optimal value of the acous-

**Figure 6** shows the velocity of the bubble collapse versus the ultrasonic frequency for the acoustic pressure amplitude of 0.2 MPa, the initial bubble radius of 20 μm and the dimensionless distance from the bubble to the rigid boundary of 1, for various ultrasonic frequency (18–30 kHz). As can be seen from **Figure 6**, when the ultrasonic frequency is low, the velocity of the bubble collapse is high. As the ultrasonic frequency increases, the velocity of the bubble collapse gradually decreases. It means that a weaker effect of the cavitation will be obtained with the increase of the ultrasonic frequency. It is mainly due to the fact that with the increase of ultrasonic frequency, the cycles of the bubble expansion and compression are getting faster and faster. Thus, the bubble may not have enough time to grow to produce the cavitation effect or the bubble may not be compressed enough to collapse. These may result in reducing the growth and collapse of the bubble and further reducing acoustic cavitation effect. Especially for the higher-frequency ultrasound, the bubble does not have enough time to store the ultrasonic energy and begins to collapse. Therefore, when the ultrasonic frequency is increasing, the velocity of the bubble collapse will continue to decrease, and eventually it will tend to be stable. It can also be found in **Figure 6**, the velocity of the bubble collapse under the rigid boundary is lower than that under the free boundary, at the same ultrasonic frequency. When the ultrasonic frequency varies from 18 to 30 kHz, the velocity of the bubble collapse is reduced by 48.84 and 53.94% under the rigid and free boundary, respectively. Moreover, as the increase of the distance from the bubble to the solid wall, the velocity of the bubble

tic pressure amplitude is not found easily.

**Figure 5.** Velocity of the bubble collapse versus the acoustic pressure amplitude.

*3.2.4. Effect of ultrasound frequency*

82 Cavitation - Selected Issues

collapse is getting higher. It presents that from the control point of view of the ultrasonic frequency, with the increase of ultrasonic frequency, the bubble under the rigid boundary is more easily to collapse than that under the rigid boundary.

#### **3.3. Relationship between the velocity of the bubble collapse and microjet**

**Figure 7** shows the relationship between the velocity of the bubble collapse and the microjet under the action of one sound cycle, for the ultrasonic frequency of 20 kHz, the initial radius of the bubble of 20 μm and the dimensionless distance from the bubble to the rigid wall of 1. It can be seen from **Figure 7**, the velocity of the microjet responds to changes in the velocity of the bubble collapse, with the increase of the acoustic pressure amplitude. From the above analysis of **Figure 5**, the velocity of the bubble collapse can be up to the maximum value (5488 m·s−1) at the acoustic pressure amplitude of 3.5 *p*<sup>0</sup> , which is the optimal value of the acoustic pressure amplitude. However, in **Figure 7**, the velocity of the microjet reaches a maximum (67.9 m·s−1), corresponding to the acoustic pressure amplitude of 3.1 *p*<sup>0</sup> , which can be treated as another optimum value of acoustic pressure amplitude to improve the microjet effect. It can be seen that the optimum value of the acoustic pressure amplitude of the microjet is lower than that of the velocity of the bubble collapse. In addition, the dotted line in **Figure 7** represents the position where the velocity of the bubble collapse is 1500 m·s−1, and the acoustic pressure amplitude is relevant to 1.6 *p*<sup>0</sup> . When *pa* ≤ 1.6 *p*<sup>0</sup> , the velocity of the bubble collapse is less than the propagation velocity of an ultrasonic wave in water (1500 m·s−1), in which there is no microjet appearing near the solid wall. Thus, it can be determined that the analysis for the velocity of the bubble collapse is contributed to seek the optimal value of the microjet and to distinguish the range of the variation of the microjet. Based on the earlier analysis, the velocity

**Figure 7.** Relationship between the collapse velocity of a bubble and microjet under acoustic pressure.

of the microjet depends on the velocity of the bubble collapse and changes with the velocity of the bubble collapse. Control and utilization of the velocity of the bubble collapse can be used indirectly to achieve the control of the microjet.

**Figure 8** shows the relationship between the velocity of the bubble collapse and the microjet under various ultrasonic frequencies. It can be seen that the variation of the velocity of the microjet has the corresponding law to the velocity of the bubble collapse under different ultrasonic frequency. Furthermore, it can be known from Eq. (7) that the velocity of the micro-jet is also directly affected by the radius parameters of the bubble such as *R*<sup>0</sup> and *R*max. Thus, in **Figure 8**, the velocity of the microjet presents a different variation from the velocity of the bubble collapse occasionally, such as a turning point on the curve of the microjet with the ultrasonic frequency of 28 kHz.

At present, many scholars have used the high-speed photography technique to observe and track the ultrasonic cavitation effect and obtain the same variational laws of microjets near the solid wall. However, because of the instability of the bubble collapse near the solid wall of different targets, the accuracy of the measuring instruments and human errors and so on, the quantitative measurement of the velocity of microjets has not been fixed. In order to verify the rationality of the theoretical model, the bubble model and its relationship with the microjet will be examined under different acoustic cavitation test conditions near the rigid wall.

the bubble will oscillate for many sound cycles before it begins to collapse. It leads to the inconvenience of the reasonable selection of initial parameters such as the maximum radius of the bubble and the velocity of the bubble collapse, and thus the instability of the numerical calculation of microjets will increase. However, to sum up, the calculated values of the microjet produced by the ultrasonic cavitation in the research approximately equal to the literature values, and it is to be in the range of an order of magnitude error. Therefore, the bubble model and its relationship with the microjet have certain rationality in the theoretical prediction for

**Literature Model**

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**Figure 8.** Relationship between the velocity of the bubble collapse and microjet under ultrasonic frequency.

 80–130 [2] 19.98 25–30 [4] 24.57 15–20 [24] 123 33.8 [23] 49.43

**Table 1.** Velocity comparison between the literature and the model of the microjet.

In the research, the dynamical model of the bubble near the rigid boundary is established using the principle of the mirror image, and the growth and collapse characteristics of the bubble are

the microjet generated by the ultrasonic cavitation under the rigid boundary.

**4. Conclusion**

**Number Microjet/(m·s−1)**

**Table 1** shows the velocity comparison between the literature and the model of the microjet. Numbers 1–3 in **Table 1** are the experiment results of the microjet, and Number 4 is the numerical simulation results of the microjet. It can be seen from **Table 1**, at a higher ultrasonic frequency or a larger acoustic amplitude, the error between the model and the literature on the value of the microjet is relatively great, which can be illustrated by numbers 1 and 3. It is because that when the ultrasonic frequency is high or the acoustic amplitude is large, The Relationship between the Collapsing Cavitation Bubble and Its Microjet near a Rigid Wall… http://dx.doi.org/10.5772/intechopen.79129 85

**Figure 8.** Relationship between the velocity of the bubble collapse and microjet under ultrasonic frequency.


**Table 1.** Velocity comparison between the literature and the model of the microjet.

the bubble will oscillate for many sound cycles before it begins to collapse. It leads to the inconvenience of the reasonable selection of initial parameters such as the maximum radius of the bubble and the velocity of the bubble collapse, and thus the instability of the numerical calculation of microjets will increase. However, to sum up, the calculated values of the microjet produced by the ultrasonic cavitation in the research approximately equal to the literature values, and it is to be in the range of an order of magnitude error. Therefore, the bubble model and its relationship with the microjet have certain rationality in the theoretical prediction for the microjet generated by the ultrasonic cavitation under the rigid boundary.

#### **4. Conclusion**

of the microjet depends on the velocity of the bubble collapse and changes with the velocity of the bubble collapse. Control and utilization of the velocity of the bubble collapse can be used

**Figure 7.** Relationship between the collapse velocity of a bubble and microjet under acoustic pressure.

**Figure 8** shows the relationship between the velocity of the bubble collapse and the microjet under various ultrasonic frequencies. It can be seen that the variation of the velocity of the microjet has the corresponding law to the velocity of the bubble collapse under different ultrasonic frequency. Furthermore, it can be known from Eq. (7) that the velocity of the micro-jet

**Figure 8**, the velocity of the microjet presents a different variation from the velocity of the bubble collapse occasionally, such as a turning point on the curve of the microjet with the

At present, many scholars have used the high-speed photography technique to observe and track the ultrasonic cavitation effect and obtain the same variational laws of microjets near the solid wall. However, because of the instability of the bubble collapse near the solid wall of different targets, the accuracy of the measuring instruments and human errors and so on, the quantitative measurement of the velocity of microjets has not been fixed. In order to verify the rationality of the theoretical model, the bubble model and its relationship with the microjet will be examined under different acoustic cavitation test conditions near the rigid wall.

**Table 1** shows the velocity comparison between the literature and the model of the microjet. Numbers 1–3 in **Table 1** are the experiment results of the microjet, and Number 4 is the numerical simulation results of the microjet. It can be seen from **Table 1**, at a higher ultrasonic frequency or a larger acoustic amplitude, the error between the model and the literature on the value of the microjet is relatively great, which can be illustrated by numbers 1 and 3. It is because that when the ultrasonic frequency is high or the acoustic amplitude is large,

and *R*max. Thus, in

is also directly affected by the radius parameters of the bubble such as *R*<sup>0</sup>

indirectly to achieve the control of the microjet.

ultrasonic frequency of 28 kHz.

84 Cavitation - Selected Issues

In the research, the dynamical model of the bubble near the rigid boundary is established using the principle of the mirror image, and the growth and collapse characteristics of the bubble are analyzed. The results of numerical analysis illustrate that the bubble under the rigid boundary has a lower maximum radius and a longer collapse time than the bubble under the free boundary, which indicates that the rigid boundary has an inhibition effect for ultrasonic cavitation. The velocity of the bubble collapse decreases with the increase of the initial radius of the bubble, and it rises with the increase of the dimensionless distance from the bubble to the solid wall. Especially when the bubble reaches a certain value away from the solid wall, the bubble motion near the solid boundary can be approximated as the bubble motion under the free boundary. Whatever for the solid boundary and the free boundary, when the acoustic pressure amplitude is less than 1 *p*<sup>0</sup> , the ultrasonic cavitation cannot form in the liquid. For the free boundary, the velocity of the bubble collapse rises approximately linearly, as the acoustic pressure amplitude is greater than 1 *p*<sup>0</sup> . However, the velocity of the bubble collapse under the rigid boundary can increase first and then decrease. Thus, the optimal acoustic pressure amplitude can be obtained, at which the velocity of the bubble collapse can be up to maximum and cavitation effect is most violent. In addition, the velocity of the bubble collapse under the free boundary decreases faster than that under the rigid boundary, and then it can decrease as the ultrasonic frequency increases. Based on that, the relationship between the velocity of the bubble collapse and the microjet is established. It can be determined that the analysis for the collapse velocity of the bubble is contributed to seek the optimal value of the microjet and furthermore to achieve the purpose of indirect judgment and control microjets. Moreover, the velocity of the microjet obtained in the research is in the range of tens of micrometers, which is nearly the same magnitude with the experiments measured by Brujan and other scholars. Therefore, it can be considered that the bubble model and its relationship with the microjet have a certain reference value in theory, which provides an implication for further understanding the dynamics of cavitation bubbles on the solid wall induced by the ultrasonic field.

*h* van der Waals radius of the bubble

*pa* acoustic pressure amplitude

*pg* gas pressure within the bubble

*p*<sup>0</sup> hydrostatic pressure of the liquid

*R*max maximum radius of the bubble

*R*<sup>0</sup> initial radius of the bubble

collapse collapse time of the bubble

*v*collapse velocity of the bubble collapse

*η* viscosity coefficient of the liquid

Address all correspondence to: guoce1027@163.com

College of Mechanical Engineering, Taiyuan University of Technology, Taiyuan, China

London. 1966;**260**(1110):221-240. DOI: 10.1098/rsta.1966.0046

[1] Benjamin TB, Ellis AT. The collapse of cavitation bubbles and the pressures thereby produced against solid boundaries. Philosophical Transactions of the Royal Society of

*σ* surface tension coefficient of the liquid

*v*microjet velocity of the microjet

*i*, *j* different bubbles

*γ* multiparty index

*ρ* density of the liquid

**Author details**

Ce Guo

**References**

*R* radius of the bubble

*t*

*pv* saturated vapor pressure inside the bubble

*p*<sup>∞</sup> liquid pressure at infinity distance around the bubble

*l* distance between the center of the bubble and the rigid wall

The Relationship between the Collapsing Cavitation Bubble and Its Microjet near a Rigid Wall…

http://dx.doi.org/10.5772/intechopen.79129

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