**2. Theoretical model**

uneven by the experiment for the first time. Brujan [2] measured the microjet released by the bubble collapse on a solid wall in water utilizing high-speed photography. It was shown that an ultrasonic wave with a frequency of 3.24 MHz has the capacity to generate a microjet of 80–130 m·s−1 when bubble's maximum radius is 150 μm. In addition, Brujan and Ikeda [3] demonstrated that the impact intensity of the microjet can be up to 1.3 ± 0.3 GP by capturing the bubble with a radius of 68 μm near a solid wall. However, not all bubbles near a solid wall can produce high-speed and large intensity microjets. Vignoli [4] proposed that the microjet would appear only if the velocity of the bubble collapse is higher than or even higher than that

The effect of microjets produced by cavitation bubbles under an ultrasound field is widely applied in ultrasonic medicine, ultrasonic chemistry, ultrasonic cleaning [5, 6] and so on. In recent years, the study of cavitation and cavitation erosion near a solid wall has also highly attracted in the field of ultrasonic vibration machining [7, 8]. On the one hand, the oscillating and collapsing bubble generated by ultrasonic cavitation can be used to clean the machining region. On the other hand, the microjet released by the bubble collapse near a solid wall can cause plastic deformation or even brittle fracture on the surface material. Nevertheless, cavitation mechanisms have not been revealed yet due to the complex relationship between

Vibration and collapse mechanisms of the cavitation bubble under the ultrasonic field can be described by motion equations of the bubble. Many scholars studied motion equations of the bubble under the ultrasonic field, some well-known models such as Rayleigh-Plesset equation [9], Gilmore equation [10], Keller-Miksis equation [11], and so on. Although these models are relatively reasonable to explore the dynamical behaviors of the cavitation bubble, they do not consider the action of a solid wall universally. It is certain to simplify calculation if ignoring the effect of a solid wall in analysis of the bubble motion inside a free boundary. However, due to the fact that there are always particle impurities and different types of structural walls in the actual liquid, theoretical models of the cavitation bubble are quite different from the actual environment. Thus, Doinikov [12] deduced a bubble model near a solid wall while exploring coated micro bubbles moving in the blood vessel in 2009. It took the wall thickness of the bubble into account and led to widespread application of ultrasound contrast agents [13, 14]. On the basis, the resonance frequency and vibration displacement of the bubble near a solid wall under an ultrasound field were derived by Qin [15]. It is noted that the solid wall can reduce the resonance frequency and increase the motion damping of the bubble. In order to deeply understand the motion and collapse characteristics of the bubble near a solid wall, the prediction and control strategies of microjets

In the research, based on the equation of the two bubbles under an ultrasonic field, a model for describing the growth and collapse of the bubble near the solid wall is established. The key parameters that affect the acoustic cavitation, the dynamics of bubble growth and collapse near the solid wall are discussed. The interaction of key parameters with the microjet is finally

of an acoustic wave propagating in a liquid.

74 Cavitation - Selected Issues

should be discussed theoretically.

investigated in detail.

the collapsing cavitation bubble and its microjet near a solid wall.

#### **2.1. Dynamical models of the bubble near a rigid wall under an ultrasonic field**

Refraction and reflection of acoustic waves will occur during the propagation of an ultrasonic field when it encounters rigid interfaces, for instance, planes, cylinders, or spheres. In the research, the physical process of ultrasound coming into contact with a rigid wall is assumed as total reflection, and the rigid wall is regarded as infinite. In order to reveal the influence of the rigid wall on the bubble motion, the two-bubble motion model of a free boundary under an ultrasound field is introduced at first. The model has assumptions as follows: (1) the bubble maintains a spherical shape during the process of expansion and contraction; (2) the radial motion of the bubble is taken into account, but the translational motion of the bubble is neglected; (3) the viscosity of the liquid, the surface tension, the vapor pressure, and the slight compressibility of the liquid are included; (4) the interaction between adjacent bubbles is also in view; and (5) heat exchange of the liquid, phase transitions of water vapor, gas mass exchange, and chemical reactions inside the bubble are not considered. Then, derived from the Doinikov equation, the dynamical model of two bubbles can be presented as follows [16]:

$$\begin{aligned} \left(\mathbf{R}\_i \mathbf{R}\_i + \frac{2}{D} \hat{\mathbf{R}}\_i^2 + \frac{1}{D} \frac{\mathbf{d}\left(\hat{\mathbf{R}}\_j^2 \hat{\mathbf{R}}\_i\right)}{\mathbf{d}t}\right) &= \frac{1}{\rho} \left(p\_{jl} + p\_v - \frac{2\sigma}{R\_i} - 4\eta \frac{\dot{R}\_i}{R\_i} - p\_o + p\_a \sin 2\pi ft\right) \\ + \frac{R\_i}{\rho c} \frac{\mathbf{d}t}{\mathbf{d}t} (p\_{jl} + p\_a \sin 2\pi ft) &\end{aligned} \tag{1}$$

where the subscript *i* and *j,* respectively, represent two different bubbles, *Ri* is the radius of the bubble *i* at any time, · indicates the derivative of time, *D* is the distance between two bubbles, *pgi* is the gas pressure within the bubble *i*, *pv* is the saturated vapor pressure inside the bubble, *ρ* is the density of the liquid, *σ* is the surface tension coefficient of the liquid, *η* is the viscosity coefficient of the liquid, *c* is the speed of sound in the liquid, *p*<sup>0</sup> is the hydrostatic pressure of the liquid, *pa* is the acoustic amplitude and *f* is the ultrasonic frequency.

In the research, the stage of the bubble collapse is the main focus of attention. Due to the fact that the bubble cannot be compressed indefinitely, the procedure of the gas changing inside the bubble is approximately treated as an adiabatic process. Then, the van der Waals gas is introduced to describe the bubble gas in the bubble *i* near a solid wall. The pressure *p*g*<sup>i</sup>* is described as follows [17]:

$$p\_{g\downarrow} = \left(p\_o + \frac{2\sigma}{R\_{bi}} - p\_v\right) \left(\frac{R\_{ii}^3 - h\_i^3}{R\_i^3 - h\_i^3}\right)^{\vee} \tag{2}$$

where *R*0*<sup>i</sup>* is the initial radius of the bubble *i*, *hi* is the van der Waals radius of the bubble *i*(for air, *Ri*<sup>0</sup> / *hi* = 8.54), *γ* is the multiparty index.

Thus, the bubble near the rigid wall is driven by the fluid pressure in the radial motion, and it can be affected by the action of the incident and reflected ultrasonic wave in particular. The reflection of ultrasonic wave is produced by the incident ultrasonic waves reflecting on the rigid wall. According to the principle of the mirror image, the action behavior of the bubble under the reflected wave near a rigid wall can be seen as that of a virtual mirror bubble under the incident wave. As a result, the motion of the bubble near a rigid wall can be regarded as a special case of the two bubbles system which consists of a bubble and its mirror image.

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

is introduced as follows [18]:

radius of the bubble.

*<sup>v</sup>*collapse <sup>≈</sup> \_\_2

expressed approximately as follows:

*t*

microjet can be expressed as follows:

*<sup>v</sup>*microjet <sup>≈</sup> <sup>2</sup> *<sup>R</sup>*\_\_\_\_\_ max

lapse time of the bubble can be expressed as:

*<sup>v</sup>*microjet <sup>=</sup> 2.677 <sup>√</sup>

**2.3. Numerical simulation and initial conditions**

The initial conditions for the simulation are when *t* = 0, *R* = *R*<sup>0</sup>

It is demonstrated that the microjet is caused by the uneven variation of the bubble wall near a solid wall. Since the bubble model is assumed as doing a spherical motion, we focus on the relationship between the velocity of the bubble collapse and microjet, in which the bubble is compressed to a minimum value. The nonspherical variation on the bubble wall is not included in the scope of this study. Thus, a simplified equation describing the bubble collapse

> 3 *p*<sup>∞</sup> − *p* \_\_\_\_\_\_*<sup>v</sup> ρ* (

where *p*∞ is the liquid pressure at infinity distance around the bubble, *R*max is the maximum

Blake and Gibson [19] indicated that the formation of the microjet is closely related to the bubble radius and the distance from the bubble to the solid wall through experiments. Ohl [20] and Tzanakis [21] used high-speed photography to record the relationship between the variation of the bubble wall and microjets near a solid wall. The results illustrated that the microjet produced by the bubble near the solid wall can be interpreted as the ratio of the maximum value of the bubble expansion to the collapse time of the bubble, which can be

> *t* collapse

> > \_\_\_\_\_\_ *<sup>ρ</sup>* \_\_\_\_\_\_ *p*<sup>∞</sup> − *pv*

\_\_\_\_\_\_\_\_\_\_\_ *v*collapse *R*max <sup>2</sup> *<sup>R</sup>* \_\_\_\_\_\_\_\_\_\_0 *R*max <sup>3</sup> − *R*<sup>0</sup>

where *t*collapse is the collapse time of the bubble. Based on the theory of Rayleigh [22], the col-

Combining Eqs. (4)–(6), the relationship between the velocity of the bubble collapse and

liquid temperature is 20°, and the main physical parameters are as follows: *ρ* = 1.0 × 10<sup>3</sup> kg·m−3, *σ* = 7.2 × 10−2 N·m−1, *p*<sup>v</sup> = 2.33 × 10<sup>3</sup> Pa, *c* = 1.5 × 10<sup>3</sup> m·s−1, *η* = 1.0 × 10−3 Pa·s, *γ* = 4/3. Therefore, the differential equation of Eqs. (2) and (3) for describing the dynamics of the bubble near a rigid wall can be calculated numerically by the Runge-Kutta fourth-order method. Taking

collapse <sup>≈</sup> 0.915 *<sup>R</sup>*<sup>0</sup> <sup>√</sup>

*R*max 3 \_\_\_\_ *R*0

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

<sup>3</sup> − 1) (4)

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

<sup>3</sup> (7)

, d*R*/d*t*=0. It is assumed that the

(5)

77

(6)

The coordinate system of the bubble near a rigid wall is established as shown in **Figure 1**, where *O*<sup>1</sup> and *O*<sup>2</sup> are the center coordinate of the bubble and its mirrored bubble, *l* is the distance between the center of the bubble and the rigid wall. There is the symmetric geometric relation of the bubble and the mirrored bubble in nature, that is *D* = 2 *l.* Therefore, ignoring the initial phase effect of the sound wave, the dynamical model of bubbles near a rigid wall under an ultrasonic field can be obtained as follows:

$$\begin{aligned} \text{Amount} &= \text{heat} \times \text{cm}^{2} \text{ to } \text{cm}^{2} \text{ to } \text{cm}^{2} \text{ to } \text{cm} \times \text{cm} \times \text{cm} \\\\ \text{R}\ddot{\text{R}} &+ \frac{3}{2}\dot{\text{R}}^{2} + \frac{1}{2l}\frac{\text{d}\left(\dot{\text{R}}^{2}\dot{\text{R}}\right)}{\text{d}t} = \frac{1}{\rho} \left(p\_{\text{g}} + p\_{v} - \frac{2\sigma}{R} - 4\eta\frac{\dot{\text{R}}}{R} - p\_{0} + p\_{\text{a}}\sin 2\pi\eta t\right) \\ &+ \frac{\text{R}}{\rho^{\text{ef}}}\frac{\text{d}}{\text{d}t} (p\_{\text{g}} + p\_{\text{a}}\sin 2\pi\eta t) \end{aligned} \tag{3}$$

Compared with the Doinikov model, Eq. (3) corrects the gas pressure inside the bubble *pg* and considers the weak compressibility in a liquid which can be seen in the second term on the right side of Eq. (3). In addition, the influence of the rigid wall on the bubble motion is especially included, which can satisfy the study of the bubble motion near the rigid wall.

**Figure 1.** Coordinate system of a bubble near a rigid wall.

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

Thus, the bubble near the rigid wall is driven by the fluid pressure in the radial motion, and it can be affected by the action of the incident and reflected ultrasonic wave in particular. The reflection of ultrasonic wave is produced by the incident ultrasonic waves reflecting on the rigid wall. According to the principle of the mirror image, the action behavior of the bubble under the reflected wave near a rigid wall can be seen as that of a virtual mirror bubble under the incident wave. As a result, the motion of the bubble near a rigid wall can be regarded as a special case of the two bubbles system which consists of

The coordinate system of the bubble near a rigid wall is established as shown in **Figure 1**,

distance between the center of the bubble and the rigid wall. There is the symmetric geometric relation of the bubble and the mirrored bubble in nature, that is *D* = 2 *l.* Therefore, ignoring the initial phase effect of the sound wave, the dynamical model of bubbles near a rigid wall under

*<sup>ρ</sup>*(*pg* <sup>+</sup> *pv* <sup>−</sup> \_\_\_ <sup>2</sup>*<sup>σ</sup>*

Compared with the Doinikov model, Eq. (3) corrects the gas pressure inside the bubble *pg* and considers the weak compressibility in a liquid which can be seen in the second term on the right side of Eq. (3). In addition, the influence of the rigid wall on the bubble motion is especially included, which can satisfy the study of the bubble motion near the rigid wall.

are the center coordinate of the bubble and its mirrored bubble, *l* is the

*<sup>R</sup>* − 4*η R*̇ \_\_

<sup>d</sup>*t*(*pg* <sup>+</sup> *pa* sin <sup>2</sup>π*ft*) (3)

*<sup>R</sup>* <sup>−</sup> *<sup>p</sup>*<sup>0</sup> <sup>+</sup> *pa* sin <sup>2</sup>π*ft*)

a bubble and its mirror image.

an ultrasonic field can be obtained as follows:

2 *R*̇ 2 + \_\_<sup>1</sup> 2*l* d(*R*̇ 2 \_\_\_\_\_\_\_ *R*̇ )<sup>d</sup>*<sup>t</sup>* <sup>=</sup> \_\_<sup>1</sup>

**Figure 1.** Coordinate system of a bubble near a rigid wall.

+ \_\_*<sup>R</sup> c* \_\_d

and *O*<sup>2</sup>

*RR*¨ <sup>+</sup> \_\_3

where *O*<sup>1</sup>

76 Cavitation - Selected Issues

It is demonstrated that the microjet is caused by the uneven variation of the bubble wall near a solid wall. Since the bubble model is assumed as doing a spherical motion, we focus on the relationship between the velocity of the bubble collapse and microjet, in which the bubble is compressed to a minimum value. The nonspherical variation on the bubble wall is not included in the scope of this study. Thus, a simplified equation describing the bubble collapse is introduced as follows [18]:

$$
\sigma\_{\text{collapse}} \approx \frac{2}{3} \frac{p\_- - p\_v}{\rho} \left( \frac{R\_{\text{max}}^3}{R\_0^3} - 1 \right) \tag{4}
$$

where *p*∞ is the liquid pressure at infinity distance around the bubble, *R*max is the maximum radius of the bubble.

Blake and Gibson [19] indicated that the formation of the microjet is closely related to the bubble radius and the distance from the bubble to the solid wall through experiments. Ohl [20] and Tzanakis [21] used high-speed photography to record the relationship between the variation of the bubble wall and microjets near a solid wall. The results illustrated that the microjet produced by the bubble near the solid wall can be interpreted as the ratio of the maximum value of the bubble expansion to the collapse time of the bubble, which can be expressed approximately as follows:

$$
\upsilon\_{\text{metrojet}} \approx \frac{2 \, R\_{\text{max}}}{t\_{\text{collapor}}} \tag{5}
$$

where *t*collapse is the collapse time of the bubble. Based on the theory of Rayleigh [22], the collapse time of the bubble can be expressed as:

$$t\_{\text{collpas}} \approx 0.915 \, R\_0 \sqrt{\frac{\rho}{\overline{P\_{\text{un}} - \overline{P\_v}}}} \tag{6}$$

Combining Eqs. (4)–(6), the relationship between the velocity of the bubble collapse and microjet can be expressed as follows:

$$v\_{\text{microjet}} = 2.677 \sqrt{\frac{v\_{\text{collopos}} \, R\_{\text{max}}^2 \, R\_0}{R\_{\text{max}}^3 - R\_0^3}} \tag{7}$$

#### **2.3. Numerical simulation and initial conditions**

The initial conditions for the simulation are when *t* = 0, *R* = *R*<sup>0</sup> , d*R*/d*t*=0. It is assumed that the liquid temperature is 20°, and the main physical parameters are as follows: *ρ* = 1.0 × 10<sup>3</sup> kg·m−3, *σ* = 7.2 × 10−2 N·m−1, *p*<sup>v</sup> = 2.33 × 10<sup>3</sup> Pa, *c* = 1.5 × 10<sup>3</sup> m·s−1, *η* = 1.0 × 10−3 Pa·s, *γ* = 4/3. Therefore, the differential equation of Eqs. (2) and (3) for describing the dynamics of the bubble near a rigid wall can be calculated numerically by the Runge-Kutta fourth-order method. Taking Eqs. (2), (3), and (5) into Eq. (7), the relationship between the velocity of the bubble collapse and microjet can be further obtained.

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

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

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

79

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

**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

rigid boundary has an inhibition effect for the bubble collapse.

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

the velocity of the bubble collapse (*v*collapse).

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

*3.2.1. Effect of the bubble initial radius*
