**1. Introduction**

The dynamical behavior of the bubble near a solid wall has crucial and practical significance for exploring the industrial application of ultrasonic cavitation. In 1966, Benjamin and Ellis [1] found out that it may lead to a high-speed microjet impinging on a solid wall through the bubble when the pressure on the upper and lower wall of the bubble near a rigid wall was

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 of an acoustic wave propagating in a liquid.

**2. Theoretical model**

bubbles can be presented as follows [16]:

*i* + \_\_3 2 *R*̇ *i* 2 + \_\_<sup>1</sup> *D* <sup>d</sup>(*R*̇ *j* 2 *R*̇ *<sup>j</sup>*) \_\_\_\_\_\_\_\_ <sup>d</sup>*<sup>t</sup>* <sup>=</sup> \_\_<sup>1</sup>

*pgi* is the gas pressure within the bubble *i*, *pv*

*pgi* <sup>=</sup> (*p*<sup>0</sup> <sup>+</sup> \_\_\_ <sup>2</sup>*<sup>σ</sup>*

is the initial radius of the bubble *i*, *hi*

/ *hi* = 8.54), *γ* is the multiparty index.

+ *R*\_\_*i c* \_\_d

*Ri <sup>R</sup>*¨

the liquid, *pa*

where *R*0*<sup>i</sup>*

air, *Ri*<sup>0</sup>

described as follows [17]:

**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

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

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

<sup>d</sup>*t*(*pgi* <sup>+</sup> *pa* sin <sup>2</sup>π*ft*) (1)

bubble *i* at any time, · indicates the derivative of time, *D* is the distance between two bubbles,

*ρ* is the density of the liquid, *σ* is the surface tension coefficient of the liquid, *η* is the viscosity

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>*

*R*0*i*

<sup>−</sup> *pv*) (

*R*0*i* <sup>3</sup> − *hi* 3 \_\_\_\_\_ *Ri* <sup>3</sup> − *hi* <sup>3</sup>) *γ*

is the acoustic amplitude and *f* is the ultrasonic frequency.

where the subscript *i* and *j,* respectively, represent two different bubbles, *Ri*

coefficient of the liquid, *c* is the speed of sound in the liquid, *p*<sup>0</sup>

*Ri* − 4*η R*̇ \_\_\_*i Ri*

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

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

75

is the saturated vapor pressure inside the bubble,

is the van der Waals radius of the bubble *i*(for

is the radius of the

is

(2)

is the hydrostatic pressure of

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

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 should be discussed theoretically.

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 investigated in detail.
