Numerical Simulation Study of Factors Influencing Ultrasonic Cavitation Bubble Evolution on Rock Surfaces during Ultrasonic-Assisted Rock Breaking

Abstract

With the increasing demand for deep oil and gas exploration and CCUS (Carbon Capture, Utilization, and Storage) engineering, improving rock-crushing efficiency stands as a pivotal technology. Ultrasonic vibration-assisted drilling has emerged as a novel rock-breaking technology. The high-frequency vibrations of ultrasonic waves impact rocks, inducing resonance and accelerating their fragmentation. At the same time, ultrasonic waves generate cavitation bubbles in the liquid near rock surfaces; the expansion and collapse of these bubbles further contribute to rock damage, thereby improving crushing efficiency. Therefore, investigating the dynamics and failure characteristics of cavitation bubbles near rock surfaces under ultrasonic influence is crucial for advancing ultrasonic-assisted rock-breaking technology. This study treats the liquid as compressible flow and investigates the movement and rupture of bubbles near rock surfaces under varying ultrasonic parameters, rock properties, characteristics of the circulating medium, and other relevant factors. The findings show that ultrasonic waves induce the oscillation, translation, collapse, and rebound of bubbles near rock surfaces. Higher ultrasonic frequencies correspond to larger collapse pressures and amplitudes near surrounding rocks, as well as longer expansion times and shorter collapse durations. In addition, bubble movement and collapse lead to rock material deformation, influenced by the rheological properties of the liquid medium. The study outcomes serve as a foundation for optimizing engineering parameters in ultrasonic-assisted rock breaking and provide theoretical support for the advancement of this technology.

1. Introduction

With the depletion of shallow, easily accessible resources and the ongoing development of CCUS projects, there is a growing demand for drilling deep and ultra-deep wells. The increased depth results in stronger rock formations and greater wear resistance, which significantly affects exploration efficiency and increases development costs for such wells. Researchers worldwide are therefore focusing on improving rock-breaking efficiency and drilling speed [1,2,3].

Ultrasonic waves are sound waves with frequencies exceeding 20 kHz, exhibiting high-frequency vibration, precise focusing, and strong penetrating capabilities. They find extensive application in mechanical manufacturing and biochemistry [4,5]. Ultrasonic-assisted rock crushing leverages the impact and resonance of high-frequency ultrasonic vibration on rocks, along with the destructive effect of cavitation bubble movement and collapse near rock surfaces, enhancing rock breakdown. In order to maximize the effect of ultrasonic cavitation on rock breaking, it is necessary to study the movement and collapse characteristics of bubbles near the rock wall under ultrasonic action. Ultrasonic cavitation involves dynamic processes in which bubbles, formed by gas accumulation during ultrasonic negative pressure phases, undergo activation during alternating positive and negative pressure cycles. This results in bubble movement and collapse near object surfaces, generating shock waves and micro-jets that induce surface deformation. Extensive theoretical, experimental, and simulation studies have been conducted on cavitation bubbles to investigate their dynamic characteristics and unveil the internal mechanisms of ultrasonic cavitation. Rayleigh [6] proposed the motion equation of spherical bubbles in an incompressible flow field based on fluid dynamics, deriving the Rayleigh equation describing bubble wall vibration velocity. Plesset [7] extended the Rayleigh equation by incorporating liquid viscosity and surface tension, resulting in the Rayleigh–Plesset (RP) equation. Gilmore [8] modified the RP equation to account for compressibility in liquid collapse stages, applying the Kirkwood–Bethe hypothesis. Benjamin et al. [9] introduced the Kelvin momentum concept to predict axial jets from bubble collapse. Zhang et al. [10] proposed a more rigorous expression for the acoustic damping constant based on Keller’s equation and considering the compressibility of the liquid for the forced radial oscillation of bubbles in liquids. The expression provided was also compared with that in published papers, and the results showed that it would improve the prediction of the total damping constant, especially for high frequencies and large bubbles. Sukwon Park et al. [11] studied the acoustic droplet vaporization(ADV) process and believed that the growth and collapse of cavitation bubbles depend on the pressure amplitude of the acoustic pulse. During the bubble collapse process, the surrounding liquid is compressed to high pressure. The compressibility of the liquid has a significant impact on the bubble behavior and the ADV threshold. A one-dimensional numerical model of volatile droplet ADV considering the compressibility of the liquid was proposed.

In addition to theoretical research, scholars have also carried out relevant experimental and numerical simulation research on this problem. Knapp et al. [12] pioneered the use of a high-speed camera on protrusions in 1948 to study cavitation, while Lauterborn [13] used holographic photography for bubble intensity and phase data. Lindau et al. [14] observed laser-induced cavitation bubble collapse and shock wave propagation using ultra-high-speed cameras. Zhang et al. [15,16] improved the low-voltage discharge device of Turangan et al. [17], using 220 V voltage and three 2200 μF capacitors in parallel to generate bubbles with a diameter of about 14 mm. The dynamic changes and mechanical processes of bubbles in different environments, such as free interface, horizontal rigid wall, inclined wall, curved wall, etc., were studied by filming with a high-speed camera with a maximum frame rate of 650,000 frames per second, and the effects of viscosity on bubble morphology, bubble wall velocity, and wall pressure were analyzed in detail. The study showed that the viscosity of the liquid not only affects the morphological changes of the bubbles, but also increases the viscosity, which slows down the speed of bubble collapse and reduces the pressure of the bubbles on the wall. Hasegawa et al. [18] designed a trumpet-shaped vibrator to enhance ultrasonic bubble transmission. Wang et al. [19] studied the dynamics of bubbles in compressible fluids. The study showed that the high-speed liquid bubble jet forms and is directed towards the node, impacting the opposite bubble surface and penetrating through the bubble to form a toroidal bubble. The bubble first rebounds in a toroidal form but re-combines into a singly connected bubble, expanding continuously and gradually returning to a near spherical shape, and these processes are repeated in the next oscillation. Zhang et al. [20], after considering the influence of liquid compressibility, conducted a numerical study on the radial motion law of a single bubble in an oscillating pressure field. The results showed that the cavitation motion exhibits strong nonlinear characteristics and that bifurcated chaotic phenomena occur under certain conditions. Zheng et al. [21], when studying the motion of micro-bubbles in blood vessels, derived a bubble model near the solid wall taking into account the bubble wall thickness, which was widely used in ultrasound contrast agents. Ma et al. [22] applied the CLSVOF method to study the dynamics of a single bubble in acoustic traveling waves. They proposed and verified the Naiver–Stokes equations taking into account the acoustic radiation force to capture the bubble behavior. The numerical results showed that bubbles exhibit different types of oscillations in acoustic traveling waves. The splitting oscillations, jet formation, bubble splitting, and sub-bubble rebound may lead to a significant increase in pressure fluctuations on the boundary. Wu et al. [23] studied the effects of surface tension on oscillating bubbles. Yasui, Brujan, and Sadighi et al. [24,25,26] investigated the effects of liquid properties on cavitation. Wu et al. [27] simulated micro-bubble collapse near walls, and Takahira et al. [28] analyzed bubble-induced damage in mercury using numerical methods.

The above overview reveals extensive international research in theoretical, experimental, and numerical studies on ultrasonic bubble movement and collapse. However, current studies often treat bubble-proximate walls as rigid, and do not consider the compressibility of the liquid, which is not consistent with reality, failing to accurately reflect wall–bubble interactions. Moreover, research on liquid properties and non-Newtonian fluid media in bubble dynamics remains sparse. In particular, the studies on bubble movement and collapse near rock walls under ultrasonic vibration are lacking. Therefore, further research is essential to advance and apply ultrasonic-assisted rock-breaking technology.

2. Methods

2.1. Selection of a Two-Phase Gas–Liquid Model

The VOF (volume of fluid) model simulates two or more immiscible liquids by solving separate momentum equations and tracking the volume fraction of each liquid flowing through the region. This study focuses on air bubbles and water and aims to observe a distinct gas–liquid interface and the bubble collapse process and to measure parameters such as the maximum velocity near the bubbles. Therefore, the VOF model is chosen to simulate bubble collapse near walls in an ultrasonic field.

2.2. Governing Equations

In order to analyze the collapse process of micro-bubbles near the wall in an ultrasonic field, the Navier–Stokes equation combined with the VOF model is used to track the movement of the gas–liquid interface. The mass conservation equation, momentum conservation equation, and energy equation are, respectively, as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf{v})=0$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}(\rho \mathbf{v})+\nabla (\rho \mathbf{vv})=-\nabla p+\mu {\nabla }^2\mathbf{v}$$

(2)

$${\displaystyle \frac{\partial }{\partial t}}(\rho E)+\nabla \cdot (\mathbf{v}(\rho E+p))=0$$

(3)

The total energy of both gases and liquids follows the following formula:

$$E=h-{\displaystyle \frac{p}{\rho }}+{\displaystyle \frac{{\nu }^2}{2}}$$

(4)

$$h={\displaystyle \sum _j{Y}_j}{h}_j+{\displaystyle \frac{p}{\rho }}$$

(5)

The apparent enthalpy of compressible liquid is as follows:

$$h_j={\displaystyle {\int }_{T_{\mathrm{ref}}}^T{c}_{p,j}}dT$$

(6)

Furthermore, the density of a compressible liquid varies as a function of the pressure field, rather than remaining constant. To achieve a stable pressure solution in compressible flow calculations, an additional equation incorporating the speed of sound c must be introduced into the pressure correction equation:

$$c=\sqrt{{\displaystyle \frac{K}{\rho }}}$$

(7)

$$K=K_0+n\Delta p$$

(8)

$$\Delta p=p-p_0$$

(9)

Furthermore, the VOF model is employed to track the gas–liquid interface, where the volume fraction equation is as follows:

$${\displaystyle \frac{\partial {\alpha }_g}{\partial t}}+\left(\overrightarrow{\nu }\cdot \nabla \right){\alpha }_g=0$$

(10)

The volume fraction of the gas phase is set to 1 and that of the liquid phase to 0. The value of 0 < α < 1 represents the gas–liquid interface.

$${\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^n={\displaystyle \frac{K}{K_0}}$$

(11)

2.3. Solution

In order to improve the calculation accuracy while reducing workload, this study employs the PISO (Pressure Implicit with Splitting of Operators) algorithm available in ANSYS Fluent 2023 R1 to resolve the coupling between velocity and pressure. In contrast to the traditional SIMPLE algorithm, the PISO algorithm incorporates multi-step corrections per time step, making it particularly suitable for handling unsteady flows. In contrast, the SIMPLE algorithm often requires iterative calculations to ensure convergence of the mass conservation equation during unsteady flow simulations, significantly increasing computational burden. Additionally, the PISO algorithm includes a correction phase involving prediction and two correction steps, aiming to better satisfy both the momentum and continuity equations.

2.4. Physical Process and Calculation Assumptions

The specific physical process of the numerical simulation in this paper involves generating ultrasonic waves underwater, followed by introducing bubbles once the sound pressure field stabilizes. These bubbles collapse within one cycle of ultrasonic wave action, generating a jet directed towards the wall; the specific process is shown in Figure 1.

To simplify the calculation process and ensure the reliability of the numerical simulation results, this paper makes the following assumptions regarding the physical process:

(1)

Water and air are treated as compressible fluids satisfying the ideal gas state equation.

(2)

The flow process is assumed to be laminar due to the low Reynolds number throughout the process.

(3)

The bubbles are considered to have negligible mass and are unaffected by gravity at their initial moment.

(4)

Water and air are assumed to be immiscible, and mass transfer between them is neglected.

2.5. Geometric Model and Boundary Conditions

To improve computational efficiency, the numerical simulation uses a two-dimensional axisymmetric approach for geometric modeling. Figure 2 shows the schematic diagram of the computational domain, which is a rectangular area measuring 2 mm × 1 mm. Walls AB, BC, and CD are included, with the vibrating wall used to simulate ultrasonic vibration. The walls BC and CD are fixed, and AD serves as the axis of rotational symmetry. The temperature of the computational domain is set at room temperature (298.15 K), and the elastic modulus of water is 2.18 × 10⁶ kPa.

The initial pressure inside the bubble is given by the following:

$$P_{\mathrm{g}0}=P_0+{\displaystyle \frac{2\sigma }{R_0}}$$

(12)

This study simulates the active motion of the ultrasonic generator using a vibrating wall whose motion is controlled by a time-dependent velocity function. To accurately model this physical process, ANSYS Fluent 2023 R1 uses dynamic mesh technology. The AB wall serves as the boundary for the dynamic mesh, using a layering control mode for mesh updating with default parameters. The dynamic mesh motion is defined as rigid body motion, utilizing a User-Defined Function (UDF) to control rigid body velocity. The area of the dynamic mesh along the symmetry axis AD, connected to the AB wall and the boundary wall BC, is designated as the deformation area. In numerical simulations, the AB wall functions as a vibrating surface to simulate an ultrasonic transducer generating waves. The vibration mode is vertical sinusoidal resonance, and the vibration displacement follows this formula:

$$x(t)=-A\sin (2\pi ft)$$

(13)

2.6. Meshing

This study uses a two-dimensional structured grid to partition the computational domain. To conserve computing resources and accurately capture the evolution of bubbles, the grid resolution in the bubble region (0.3 mm × 0.15 mm) was refined. The Figure 3 shows the grid division diagram and the minimum grid size was set to 0.5 μm, with a refinement factor of 1.5.

The numerical simulation in this chapter was conducted by ANSYS Fluent 2023 R1 software. To solve the time-dependent mass conservation equation, a pressure-based transient solver with a first-order implicit transient equation was employed. The pressure–velocity coupling utilized the PISO algorithm. Spatial discretization employed the least squares method for the gradient term, PRESTO! format for pressure, and second-order upwind schemes for density, momentum, energy, and swirl velocity terms. The volume fraction term utilized the modified Geo-Reconstruct format. The simulations were performed on an HP-T7000 workstation equipped with a 1 TB hard disk, 8 GB RAM, and a 3.6 GHz CPU. Convergence criteria were set to 10⁻⁵ for the continuity equation, 10⁻⁶ for the momentum equation, and 10⁻⁷ for the energy equation to ensure satisfactory accuracy and convergence.

3. Results and Discussion

3.1. Model Validation

To verify the accuracy of the established model, experimental results from Wu et al. [27] were chosen as a benchmark. A dimensionless number λ is defined that describes the distance between the bubble and the wall and its initial radius. The equation is as follows:

$$\lambda ={\displaystyle \frac{L}{R_0}}$$

(14)

The experimental water temperature was maintained at 21 °C, and the bubbles produced were 35 µm in size. The dimensionless distance between the bubble center and the wall was λ = 1.85. An ultrasonic transducer connected to a horn was used to generate waves in the water, with the horn head positioned 10 mm from the bubble. The ultrasonic waves near the bubble had a frequency of 20.47 kHz and a sound pressure amplitude of 265 kPa. The wall elastic modulus was set to 10 GPa. Bubbles were initially generated at t = 0 when the ultrasonic generator was activated. For an accurate simulation resembling experimental conditions, a maximum time step of Δt = 2 × 10⁻⁸ s and a minimum time step of Δt = 1 × 10⁻¹⁰ s were chosen. To ensure calculation stability, each step was iterated 200 times.

Figure 4 depicts the evolution of bubble shape under ultrasound. Initially, as shown in Figure 4a, the bubble is initially at a distance from the wall with a uniformly distributed pressure gradient in the surrounding liquid. Due to differential internal and external pressures, the bubble undergoes oscillatory expansion and contraction. In Figure 4b, it can be clearly seen that the bubble begins to move towards the wall, causing non-uniform pressure gradients to emerge between the wall-facing and the opposite ends of the bubble. Subsequently, the ultrasound field transitions from negative to positive pressure around the bubble over time. Figure 4c illustrates a rapid shrinkage of the bubble. The pressure differential at the end of the bubble furthest from the wall undergoes the most significant change during contraction due to wall proximity. Figure 4d shows the bubble assuming a nearly mushroom-like shape as liquid converges towards its tip, initiating inward sinking and collapse. Finally, Figure 4e demonstrates that once the far end of the bubble nears the wall and pierces it, a high-pressure, high-speed jet forms and impacts the rigid wall, resulting in lateral flow and vortex formation that act upon the toroidal bubble, initiating rebound.

In this numerical simulation, the bubble undergoes oscillation, translational motion, collapse, and rebound under ultrasound, consistent with the experimental findings of Wu et al. [27]. To validate the simulation results rigorously, this study analyzes the temporal variation in the bubble edge velocity preceding collapse and compares it with experimental data. For a direct comparison between simulation and experiment, this section non-dimensionalizes the time t experienced by the bubble across its four stages.

From Figure 5, it is evident that there are some differences in the curve trends of the bubble edge velocity in the simulation results and in the experimental results during the bubble oscillation and translation process, but they are within the acceptable error range, and their overall trends are basically consistent. However, both the collapse velocity simulated in the literature and our own simulation exceed the edge velocity measured in the experiment. This discrepancy can be attributed to the simplified assumptions regarding the flow and ultrasonic fields used in the simulation. Furthermore, while the experiment captures three-dimensional effects during image processing, the simulation is limited to two dimensions, contributing to significant measurement disparities. Additionally, the simulation overlooks complexities arising from the ultrasonic generator’s influence on water flow conditions.

The model successfully reproduces the four stages of oscillation, translational motion, collapse, and rebound observed in bubbles near a rigid wall under ultrasonic conditions. Numerical simulation data align closely with the fundamental trends observed in the experimental literature and demonstrate the feasibility of the model and computational approach.

3.2. Impact of Ultrasonic Parameters on Bubble Collapse

In ultrasonic-assisted rock breaking, optimizing the operational parameters of ultrasonic equipment is crucial. This is because ultrasonic waves influence various aspects such as bubble oscillation, collapse dynamics, and maximum velocity, all of which have a significant impact on the effectiveness of ultrasonic cavitation in rock crushing. Therefore, studying the influence of ultrasonic parameters on bubble collapse is essential for optimizing the efficiency of ultrasonic-assisted rock fragmentation.

3.2.1. Effect of Ultrasonic Frequency on Bubble Collapse

The amplitude of the ultrasonic generator is set to 1 µm, with ultrasonic wave frequencies of 20 kHz, 30 kHz, 40 kHz, and 50 kHz, while the wall elastic modulus is fixed at 10 GPa. In this section, the initial time is defined as the beginning of the eleventh cycle of the ultrasonic wave. Bubbles with initial radii of 30 µm, 40 µm, and 50 µm are individually positioned within the calculation area. The distance of the bubble near the wall, denoted as γ, is maintained at 1.6. The study investigates the expansion amplitude of the bubble, the collapse time, the collapse pressure, and the maximum jet velocity during the collapse process.

Figure 6 illustrates the relationship between pressure and frequency during bubble collapse. As ultrasonic frequency increases, the collapse pressure initially rises and then decreases for bubbles with initial radii of 30 µm and 40 µm, whereas for bubbles with an initial radius of 50 µm, the collapse pressure consistently increases, albeit at a decreasing rate. This trend indicates that, under constant amplitude, higher ultrasonic frequencies enhance wave energy. Furthermore, different initial bubble sizes have optimal frequencies for maximum collapse pressure. Therefore, adjusting the ultrasonic frequency according to the operating conditions can optimize collapse intensity effectively in practical applications.

Figure 7 depicts the relationship between ultrasonic frequency and bubble collapse time, revealing that the collapse time of bubbles decreases as ultrasonic frequency increases. Furthermore, bubbles with larger initial radii exhibit longer collapse times, suggesting that the contraction duration of the bubble increases with initial radius size.

Figure 8 illustrates the relationship between ultrasonic frequency and expansion amplitude. At a given frequency, the initial radius of the bubble affects the amplitude of expansion. Smaller initial bubble radii lead to larger expansion amplitudes. In particular, as ultrasonic frequency increases, the expansion amplitude of bubbles increases, with smaller bubbles exhibiting higher expansion speeds.

Figure 9 illustrates the relationship between ultrasonic frequency and bubble expansion time, indicating that the expansion time decreases as the frequency increases. At equivalent frequencies, bubbles of different initial radii exhibit comparable expansion times. The trend in the figure also reveals that bubble collapse time decreases with increasing ultrasonic frequency, while it increases with larger initial bubble radius.

If the contraction time is calculated as the difference between the expansion time and the collapse time at the same initial radius, it becomes evident that the contraction time decreases with increasing frequency.

3.2.2. Effect of Ultrasonic Amplitude on Bubble Collapse

Ultrasonic amplitude serves as a crucial parameter representing the magnitude of ultrasonic output energy. In this study, bubbles with initial radii of 30 µm, 40 µm, and 50 µm are positioned within the calculation area, maintaining a dimensionless distance γ = 1.6. The ultrasonic generator operates at a frequency of 20 kHz, with a wall elastic modulus set at 10 GPa. Vibration amplitudes of 1 µm, 2 µm, 3 µm, and 4 µm are applied. Since the frequency’s influence is not considered, bubbles are introduced during the first cycle of ultrasonic waves in this section. Parameters such as bubble expansion amplitude, collapse time, collapse pressure, and maximum velocity during the collapse process are investigated.

Figure 10 illustrates the relationship between ultrasonic amplitude and bubble collapse pressure. As ultrasonic amplitude increases, the collapse pressure of bubbles with different initial radii progressively increases. This trend arises because bubbles with smaller initial radii exhibit stronger surface tension, requiring greater energy absorption during collapse. Figure 11 depicts the relationship between ultrasonic amplitude and bubble collapse time, revealing that collapse time decreases with increasing ultrasonic amplitude. The propagation of ultrasonic waves involves transmitting vibrational energy, with amplitude serving as a critical measure of this energy. Higher ultrasonic amplitudes correspond to greater energy transmission, enabling bubbles to absorb more energy and collapse more quickly.

Figure 12 illustrates the relationship between ultrasonic amplitude and bubble expansion amplitude. For constant amplitude, the expansion amplitude decreases as the initial radius of the bubble increases, while for bubbles with the same initial radius, the expansion amplitude increases with amplitude. Figure 13 depicts the relationship between ultrasonic amplitude and bubble expansion time. For bubbles with the same initial radius, the expansion time decreases as the amplitude increases and stabilizes, eventually stabilizing as the initial bubble radius increases at a constant amplitude. The findings indicate that while bubbles with smaller radii exhibit greater surface tension, thereby potentially extending bubble expansion time, increased ultrasonic amplitude enhances energy input, diminishing the ability of surface tension to sustain prolonged expansion. Consequently, as amplitude increases, bubble expansion time tends to normalize.

3.3. Effect of Rock Material on Bubble Collapse

Rock crushing involves external forces causing surface deformation, detachment, and fragmentation. The high-speed micro-jet force generated during ultrasonic cavitation bubble collapse acts on the rock surface, contributing to its breakage. Therefore, when studying bubble dynamics on rock surfaces, it is important to treat the rock surface as an elastic wall. This section focuses on varying the elastic modulus, a key physical parameter of rocks, to explore its impact on bubble collapse.

This study only examined the effect of rock materials on bubble collapse. Therefore, the same as before, the liquid selected is water and the gas selected is air, and bubbles with an initial radius of 30 µm are positioned in the calculation area, maintaining a dimensionless distance γ = 1.6. The ultrasonic generator operates at a frequency of 20 kHz, with a vibration amplitude set to 1 µm. Considering the influence of frequency, bubbles are introduced at the start of the eleventh cycle of the ultrasonic wave. The investigation covers bubble expansion amplitude, collapse time, collapse pressure, and maximum velocity during the collapse process.

The elastic modulus values for different rock materials are derived from results obtained by Zhang et al. [29], who used the three-point bending test method on various types of rocks and the detailed data are shown in

Table 1. Rock elastic modulus [29].

Serial Number	Lithology	Elastic Modulus (GPa)
1	Granite	42.3
2	Marble	20.2
3	Sandstone	8.8

.

Figure 14 illustrates the relationship between the elastic modulus of rock materials and the collapse time of bubbles near the wall. The figure demonstrates that as the Young’s modulus of the elastic wall increases, the collapse time of bubbles near the wall decreases. Specifically, the collapse times are 0.22 µs for granite, 0.39 µs for marble, and 0.52 µs for sandstone. These times exceed those reported for bubbles near rigid walls in the literature [30]. This discrepancy arises because the elastic wall undergoes deformation perpendicular to its surface when subjected to external forces, thereby altering the flow field dynamics between the bubble and the elastic wall. In contrast to bubbles near rigid walls, the fluid–solid coupling effect between the elastic wall and the surrounding flow field maintains a relatively small pressure difference at the bubble’s ends near the elastic wall. This effect slows down the bubble’s approach to the wall and prolongs its collapse time.

Figure 15 illustrates the displacement of the wall over time. As time progresses, the wall displacement gradually increases, aligning with basic physical principles where lower elastic modulus values for rock materials result in greater wall deformation.

Figure 16 depicts the relationship between the elastic modulus of rock material and the collapse time of the bubbles near the wall. Figure 16 illustrates that as the elastic modulus of the wall increases, the expansion amplitude of bubbles near the wall also increases. Conversely, when the elastic modulus of the elastic wall is small, the wall deformation is significant, resulting in less hindrance to bubble collapse, facilitating the easier collapse and larger expansion amplitudes of the bubbles.

3.4. Effect of Liquid Circulating Medium on Bubble Collapse

In practical drilling and rock-breaking operations, the presence of a circulating medium is essential. Ultrasonic cavitation occurs within this circulating medium, generating high-speed jets upon collapse that impact the rock surface, aiding in rock breaking. Besides water, various other drilling circulating media exist, necessitating a study of how different media affect bubble dynamics near rock walls under ultrasonic conditions.

Bubbles with an initial radius of 30 µm are positioned in the calculation area, maintaining a dimensionless distance γ = 1.6. The ultrasonic generator works with a frequency of 20 kHz with a vibration amplitude of 1 µm, and the wall elastic modulus is set to 10 GPa. This study begins with the onset of the eleventh cycle of the ultrasonic wave, investigating factors such as bubble expansion amplitude, collapse time, collapse pressure, and maximum velocity during the collapse process.

Initially, Newtonian fluids are examined. These fluids maintain constant viscosity regardless of shear rate, a characteristic referenced from the viscosity measurements of drilling circulating fluids in [30]. This section explores the collapse pressure of bubbles near the wall and the jet velocity generated upon their collapse under ultrasonic influence.

To thoroughly analyze the influence of viscosity characteristics on the dynamic process of bubble collapse, the maximum jet velocity generated during the initial collapse of bubbles in liquids with varying viscosities was examined. Figure 17 depicts the relationship between liquid dynamic viscosity and maximum jet velocity and the bubble collapse time. As the dynamic viscosity of the liquid increases, the jet velocity sharply decreases from 273 m/s to 234 m/s, accompanied by a prolonged collapse time. The kinematic viscosity directly influences the speed and duration of micro-jets produced by bubble collapse. This effect arises because higher liquid viscosity imposes greater resistance on bubble movement and leads to increased energy dissipation, thereby weakening the impact of bubbles on surfaces under ultrasonic action. These findings highlight the significant influence of liquid viscosity on the efficiency of ultrasonic cavitation applications.

Previously, this study focused on the influence of Newtonian fluid viscosity on ultrasonic cavitation bubbles. However, in practical drilling and rock-breaking operations, circulating media often exhibit non-Newtonian fluid characteristics. Therefore, it becomes imperative to investigate the behavior of cavitation bubbles within non-Newtonian fluid circulating media.

The fluids selected for this investigation are based on data from Qu et al. [31], involving two different concentrations of aqueous Carbopol solutions. A higher concentration solution conforms to the Herschel–Bulkley fluid, designated as liquid A, while a lower concentration solution conforms to the Power-Law fluid, designated as liquid B. The rheological curves and equations for these fluids are depicted in Figure 18.

Given the rheological dependence of non-Newtonian fluids on shear rate, the study of bubble dynamics in these fluids requires consideration of the frequency of ultrasonic vibrations. Accordingly, the frequencies of ultrasonic waves are set at 20 kHz, 30 kHz, 40 kHz, and 50 kHz, respectively. A user-defined function (UDF) program is developed to simulate the viscosity characteristics of the liquid medium, while other parameters remain consistent with the previous section. This investigation focuses on factors such as bubble collapse pressure, collapse time, maximum jet velocity during collapse, and other pertinent aspects of the bubble collapse process. To facilitate comparison with Newtonian fluids, the study also incorporates observations on bubble behavior in water under identical conditions.

Figure 19 illustrates the relationship between the ultrasonic frequencies of different liquids and bubble collapse pressures. It shows that the collapse pressure of bubbles in non-Newtonian fluids (liquid A and liquid B) gradually increases with higher vibration frequencies, contrasting with the behavior observed in a Newtonian fluid. Figure 20 displays the relationship between the ultrasonic frequencies of different liquids and bubble collapse times. In the Newtonian fluid water and in the non-Newtonian fluid liquid A, bubble collapse time decreases as ultrasonic frequency increases. However, bubble collapse time in the non-Newtonian fluid liquid A shows less sensitivity to changes in ultrasonic frequency, while in liquid B, the collapse time remains relatively unchanged despite variations in frequency. The change in vibration frequency is actually the change in the shear rate of a Newtonian fluid, as the rheological characteristics of a non-Newtonian fluid are different from those of a Newtonian fluid in terms of shear rate, which may affect the surface tension of the bubble, so that the bubble maintenance time is different, as is the phenomenon that the bubble collapse time does not change the vibration frequency significantly. Figure 21 indicates that the maximum velocity of bubble collapse jets shows poor correlation with changes in ultrasonic frequency for both Newtonian and non-Newtonian fluids. In these cases, the velocity does not significantly vary with changes in ultrasonic frequency.

4. Conclusions

This study employs numerical simulation techniques and treats the liquid as a compressible fluid and the rock surface as an elastic interface. It converts high-frequency ultrasonic vibrations into sinusoidal oscillations at the boundary to simulate ultrasonic stimulation in the fluid domain. The study investigates the movement and collapse of bubbles near the rock material surface under the influence of ultrasound. The research suggests that bubbles near the rock surface undergo approximately four stages of evolution: oscillation, translational movement, collapse, and rebound. Increasing the frequency and amplitude of ultrasound effectively enlarges the bubble expansion amplitude and reduces the collapse duration. The movement and collapse of bubbles can induce slight deformations on the rock surface, which in turn can affect the motion and collapse of bubbles. Therefore, the elastic modulus of the rock material surface can influence the dynamics of bubbles. Newtonian and non-Newtonian fluids exhibit different sensitivities to changes in ultrasound parameters due to their rheological properties. An increase in the viscosity of Newtonian fluids prolongs the bubble collapse time near the rock surface and reduces the maximum jet velocity at collapse. Non-Newtonian fluids, due to their complex rheological behavior, require specific analysis based on the fluid’s characteristics.

Ultrasonic-assisted rock breaking is an emerging and efficient technique for rock fragmentation. Due to limitations in theoretical understanding and the power of ultrasound equipment and its associated tools, it remains in the early stages of development. The movement and collapse of cavitation bubbles accelerate the rapid fragmentation of rock. The findings of this study can provide guidance for the rational selection of technical parameters in ultrasonic-assisted rock breaking, thereby promoting and improving drilling efficiency and speed.