Study on the dynamics of a single cavitation bubble in a compressible spherical liquid confined by an elastic solid

Abstract

Cavitation bubbles within elastic solids are widespread phenomena in both natural environments and technological systems, yet their confinement-dependent dynamics remain not well understood. To systematically study the bubble behavior in these phenomena, we develop a coupled model to study the dynamical behavior of a single cavitation bubble within a spherically constrained compressible liquid domain, incorporating the first-order compressibility correction, within a finite-thickness elastic solid. Linear analysis reveals a fundamental inverse relationship between resonant frequency and inner radius of the elastic solid, which is agree with the experimental results obtained by Vincent when the outer radius of the elastic solid is greater than or equal to twice the inner radius. Linear analysis reveals that the bubble’s resonant frequency exhibits a non-monotonic response to inner radius of elastic solid: it initially decreases and then increases as the ambient bubble radius decreases under a certain inner radius of elastic solid. Nonlinear simulations demonstrate that the first-order compressibility correction significantly suppresses the amplitude of bubble rebounces and enhances oscillation stability. Parametric studies reveal that the maximum bubble radius is influenced by the geometric and material properties of the elastic solid: it grows proportionally with the solid’s inner radius, exhibits an inverse relationship with its outer radius, and increases as the bulk and shear moduli of the solid decrease. These findings analyze how geometric and material confinement parameters affect the dynamical behaviors of cavitation bubbles confined within elastic solids, thereby providing a theoretical foundation for enhancing ultrasound cavitation applications in confined environments, such as guiding the design of ultrasound contrast agent bubbles.

Highlights

• •Bubble model in compressible liquid with first-order correction within elastic solids
• •Analysis shows resonant frequency responds non-monotonically to solid’s inner radius
• •First-order compressibility correction reduces bubble rebounds and enhances stability
• •Effects of size and bulk and shear moduli of solid on bubble dynamics are studied

1. Introduction

The study of cavitation bubble dynamics has obtained significant attention due to the unique thermodynamic states — such as localized high temperatures up to several thousand degrees, high pressure up to several GPa, and shock-wave generation — produced during the rapid collapse of bubbles [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. These unique effects endow cavitation bubbles with broad applications, including biomedical engineering [12], sonochemical reactors [13], [14], [15], sonoprocessing of materials [16], [17], [18] and ultrasonic cleaning [19], [20]. While the dynamical behavior of cavitation bubbles in compressible spherical liquid entirely encapsulated within elastic solid have been attracting increasing interest recently.

Vincent [21], [22], [23] systematically investigated cavitation bubble dynamics within spherically confined liquid-filled micro-cavities embedded in elastic solids (stiff polymer hydrogels). Employing laser strobe photography, high-speed camera recordings, and light scattering, the experiments revealed that confinement dimensions, solid elasticity, and liquid compressibility collectively produced an order-of-magnitude elevation in bubble oscillation frequency and substantially augmented damping compared to the bubble in unbounded liquids. These findings emphasized the pivotal role of boundary constraints in governing cavitation behavior. Additionally, numerical comparisons with experimental data confirmed negligible contributions from interfacial surface tension, liquid viscosity, and gas inside the bubble effects under the studied conditions.

Wang [24] developed a theory to analyze bubble oscillations within liquid-filled cavities embedded in elastic solids. Results revealed that the confinement substantially reduced the maximum radius of the bubble and shortened its period. Additionally, the bubble required a longer duration to reach steady state and attained a larger equilibrium radius compared to bubbles in unbounded liquids. Zhang et al. [25] conducted numerical simulations of the radial oscillations and translational movements of cavitation bubbles within micro-cavities under confinement. The study demonstrated that cavity dimensions critically influence the bubble’s oscillation characteristics: specifically, smaller cavities correlate with shorter radial oscillation durations, while the equilibrium radius of gas-nucleated cavitation bubbles exhibits a direct proportional relationship with increasing cavity size. Zhang et al. [25] numerically calculated the radial pulsations and transnational movements of cavitation bubbles in micro-cavities confined within a confinement. The results show that the cavity dimensions critically influence the bubble’s oscillation characteristics: specifically, smaller cavities correlate with shorter radial oscillation durations, while the equilibrium radius of gas-nucleated cavitation bubbles exhibits a direct proportional relationship with increasing cavity size. Doinikov [26] proposed a theoretical model to characterize the dynamic behavior of a spherically symmetric bubble centrally embedded within a liquid-filled spherical cavity, which is itself encased in an infinitely extending elastic solid. Numerical simulations demonstrated that under conditions where the bubble radius is substantially smaller than the cavity dimensions, the gas content becomes a critical determinant of the system’s resonant characteristics. Marmottant et al. [27] proposed a refined model for ultrasound contrast agent microbubbles. The model predicted a distinct nonlinear dynamic response at large oscillation amplitudes, originating from the mechanical buckling of the lipid monolayer. This theoretical prediction was subsequently confirmed by their experimental recordings. van der Meer et al. [28] introduced a novel optical measurement technique employing ultrahigh-speed imaging to quantify the resonant frequency of a individual ultrasound contrast agent microbubble. The experimental observations demonstrated a critical influence of the encapsulating shell on bubble dynamics: specifically, shell elasticity elevates the resonance frequency compared to a uncoated bubbles, while shell viscosity emerges as a dominant damping mechanism. Liu [29] developed a specialized numerical framework employing boundary element method to characterize the dynamic behavior of bubbles within a fluid medium entirely enclosed by an elastic solid. The computational analyses demonstrated that such confinement induces rapid bubble oscillations, with jet formation occurring along the direction of eccentricity. Moreover, both the oscillation amplitude and period decreased monotonically with increasing elastic modulus of the solid and decreasing confinement size.

The seminal works by Leonov and Akhatov [30], [31], [32], [33] systematically investigated the complex nonlinear behavior of cavitation bubbles in an infinite elastic confinement. Their theoretical results revealed that volume confinement significantly affects both the classical Blake threshold criteria and bubble dynamic characteristics. Specifically, when liquid cell dimensions exceed a critical threshold, the bubble undergoes abrupt expansion until reaching its terminal equilibrium size under tension in the confined solid. Conversely, cavitation will be completely suppressed by the confinement. However, the assumption is made that the outer radius of the elastic confinement is infinite in their papers.

While existing studies demonstrate the essential role of elastic confinement in cavitation bubble dynamics, a fundamental limitation persists in current theoretical works: prior analyses of bubbles confined within elastic solids either neglect liquid compressibility effects, restrict consideration to zeroth-order approximations in compressibility analysis, or assume an infinitely thick elastic solid while disregarding thickness-dependent effects on bubble dynamics. However, these limitations become particularly pronounced under ultrasonic excitation with high-frequency ($\ge$ 1 MHz) or high-pressure amplitude ($\ge$ 5 MPa). In this paper, we relax the assumption of an infinite outer radius for the elastic solid and consider first-order compressibility corrections to derive a new model describing the dynamical behavior of a single bubble in a compressible spherical liquid entirely confined within an elastic solid. The sphericity of the entire system (cavitation bubble, compressible liquid, and elastic solid) is assumed throughout the dynamic process. Consequently, the liquid velocity field reduces to a purely radial function ${\mathbf{u}}_l=u_l(r,t)$, while the solid displacement ${\mathbf{u}}_s$ similarly retains solely a radial component $u_s(r,t)$. Additionally, mass transfer via evaporation/condensation and diffusion is neglected, i.e., $\text{d}m={\rho }_l\text{d}{V}_l+V_l\text{d}{\rho }_l\equiv 0$.

2. The bubble-liquid–solid system

In this section, the model of a single cavitation bubble in a spherical compressible liquid entirely confined within a finite-thickness elastic solid will be derived. The bubble-liquid–solid system, shown in Fig. 1, consists of the bubble, the compressible liquid, and the finite-thickness elastic solid surrounding the liquid. To enable tractable analytical solutions, the entire system is modeled as a set of concentric spheres throughout the whole dynamic process.

Fig. 1.

Bubble-liquid–solid system: a bubble in a compressible spherical liquid within an elastic solid (not to scale).

The ambient and dynamic bubble radii are symbolized as $R_{b0}$ and $R_b$. The ambient and dynamic radii of the liquid (the inner radius of the elastic solid) are symbolized as $R_{l0}$ and $R_l$, respectively. Similarly, the ambient and dynamic radii of the elastic solid (the outer radius of the elastic solid) are symbolized as $R_{s0}$ and $R_s$.

2.1. Liquid domain

The mathematical model for a compressible spherical liquid- confined cavitation bubble within a finite-thickness elastic solid is established through fundamental fluid dynamics principles. Governing equations derive from the Euler equation, continuity equation, and thermodynamic state equation in the liquid domain based on the assumptions made in section Introduction:

$$\frac{\partial u_l}{\partial t}+u_l\frac{\partial u_l}{\partial r}+\frac{1}{{\rho }_l}\frac{\partial p}{\partial r}=0,$$ (1)

$$\frac{1}{{\rho }_l}\frac{\text{d}{\rho }_l}{\text{d}t}+\nabla \cdot {\mathbf{u}}_l=0,$$ (2)

$$c^2={\left(\frac{\partial p}{\partial {\rho }_l}\right)}_s,$$ (3)

where ${\rho }_l$ is the liquid density, $c$ is the sound speed of liquid and $p$ is the pressure in liquid.

According to the assumption made in Section Introduction $\text{d}m={\rho }_l\text{d}{V}_l+V_l\text{d}{\rho }_l\equiv 0$, thus one can rewrite Eq. (2) as follows

$$\nabla \cdot {\mathbf{u}}_l=\frac{1}{V_l}\frac{\text{d}{V}_l}{\text{d}t},$$ (4)

where $V_l$ is the liquid volume, $V_l=4/3\pi \left(R_l^3-R_b^3\right)$.

According to boundary conditions at the bubble-liquid interface $u_l(r=R_b)={\dot{R}}_b$ and $u_l(r=R_l)={\dot{R}}_l$, one can solve Eq. (4) and obtain [30], [33]

$$u_l\left(r,t\right)=\frac{R_l^2{\dot{R}}_l-R_b^2{\dot{R}}_b}{R_l^3-R_b^3}r+\frac{R_l^3{R}_b^2{\dot{R}}_b-R_b^3{R}_l^2{\dot{R}}_l}{\left(R_l^3-R_b^3\right)r^2}.$$ (5)

According to the assumption made in section Introduction, the liquid motion is irrotational and hence we can set $u_l=\frac{\partial \phi }{\partial r}$, where $\phi$ is the velocity potential. For a compressible liquid, $\phi$ satisfies [34]:

$${\nabla }^2\phi =\frac{1}{c^2}\frac{{\partial }^2\phi }{\partial t^2}.$$ (6)

In spherical coordinates, the above equation can be written as:

$$-\frac{1}{c^2}\frac{{\partial }^2\phi }{\partial t^2}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial \phi }{\partial r}\right)=0.$$ (7)

Due to the mathematical identity:

$$\frac{\partial }{\partial r}\left(r^2\frac{\partial \phi }{\partial r}\right)=r\frac{{\partial }^2}{\partial r^2}\left(r\phi \right).$$

One can rewrite Eq. (7):

$$-\frac{1}{c^2}\frac{{\partial }^2\phi }{\partial t^2}+\frac{1}{r}\frac{{\partial }^2}{\partial r^2}\left(r\phi \right)=0.$$

Multiplying through by $r$ gives the standard form of the spherical wave equation:

$$\left(\frac{{\partial }^2}{\partial r^2}-\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\right)\left(r\phi \right)=0.$$ (8)

This equation shows that $r\phi$ satisfies a one-dimensional wave equation, indicating disturbances propagate along $r$.

Factorizing the above equation gives:

$$\left(\frac{\partial }{\partial r}-\frac{1}{c}\frac{\partial }{\partial t}\right)\left(\frac{\partial }{\partial r}+\frac{1}{c}\frac{\partial }{\partial t}\right)\left(r\phi \right)=0,$$ (9)

This factorization separates the equation into outgoing and incoming wave components. Take a particular solution that propagates outwards and satisfies:

$$\phi +r\frac{\partial \phi }{\partial r}+\frac{r}{c}\frac{\partial \phi }{\partial t}=0.$$ (10)

Applying first-order time differentiation to Eq. (10) and using $u_l=\partial \phi /\partial r$ yields

$$\frac{\partial \phi }{\partial t}+r\frac{\partial u_l}{\partial t}+\frac{r}{c}\frac{{\partial }^2\phi }{\partial t^2}=0.$$ (11)

It should be noted that Eq. (11) is a reduced result of Eq. (7), representing the dynamic equation governing the spherically symmetric velocity potential after considering the first-order correction for liquid compressibility.

Next, we will process the dynamic equation Eq. (11). Integrating Eq. (1) from infinity to $r$:

$$\frac{\partial \phi }{\partial t}+\frac{1}{2}{u}_l^2+{\int }_{p_{\infty }}^p\frac{\text{d}p}{{\rho }_l}=0,$$ (12)

and then taking the partial derivative of Eq. (12) with respect to the time variable $t$,

$$\frac{{\partial }^2\phi }{\partial t^2}+u_l\frac{\partial u_l}{\partial t}+\frac{1}{{\rho }_l}\frac{\partial p}{\partial t}=0.$$ (13)

Inserting Eqs. (1), (12), (13) into Eq. (11) eliminates $\phi$ and links velocity and pressure directly:

$$u_l\frac{\partial u_l}{\partial t}+\frac{1}{{\rho }_l}\frac{\partial p}{\partial t}+c{u}_l\frac{\partial u_l}{\partial r}+\frac{c}{{\rho }_l}\frac{\partial p}{\partial r}+\frac{c}{2r}{u}_l^2+\frac{c}{r}{\int }_{p_{\infty }}^p\frac{\text{d}p}{{\rho }_l}=0.$$ (14)

Inserting Eqs. (1), (2), (3), (5) into Eq. (14) at $r=R_b$, one can obtain a ordinary differential equation involving the second-order temporal derivative of the cavitation bubble radius in a compressible spherical liquid entirely confined by a finite-thickness elastic solid:

$$\left(1-\frac{2{\dot{R}}_b}{c}\right)R_b{\ddot{R}}_b+\frac{3}{2}\left[1-\frac{4{\dot{R}}_b}{3c}+\frac{2R_b^3}{R_l^3-R_b^3}-\frac{2R_b^3{\dot{R}}_b}{c\left(R_l^3-R_b^3\right)}\right]{\dot{R}}_b^2=\frac{\mathrm{\Delta }P}{{\rho }_l}+\frac{3c{R}_b^3{\dot{R}}_b+3R_b{R}_l^2{\dot{R}}_b{\dot{R}}_l}{R_l^3-R_b^3}-\frac{3R_b{R}_l^2{\dot{R}}_b^2{\dot{R}}_l+3c^2{R}_b{R}_l^2{\dot{R}}_l}{c\left(R_l^3-R_b^3\right)}.$$ (15)

where $\mathrm{\Delta }P$ defines the interfacial pressure differential between the bubble wall and the liquid [28], [35]:

$$\mathrm{\Delta }P=\left(p_0+\frac{2\sigma }{R_{b0}}-p_v\right){\left(\frac{R_{b0}}{R_b}\right)}^{3\gamma }\left(1-\frac{3\gamma \dot{R_b}}{c}\right)+p_v-\frac{2\sigma }{R_b}-p_l-p_d\left(t\right),$$ (16)

where $p_d$ is the driving pressure of the acoustic wave acting on the bubble, and $p_l$ is the liquid pressure:

$$p_l=p_0-E_v\frac{R_l^3-R_b^3-R_{l0}^3+R_{b0}^3}{R_{l0}^3-R_{b0}^3}.$$ (17)

It was noted in Refs. [23], [24] that the validity of this approximation is contingent upon the condition that the radius of the liquid domain ($R_l$) exceeds twice the bubble radius ($R_b$).

2.2. Solid domain

The spatial variations of the displacement field $u_s$ and the radial component of the stress tensor within the elastic solid are analytically described by the following expressions [31], [36]:

$$u_s\left(r\right)=ar+\frac{b}{r^2},$$ (18)

$${\sigma }_{rr}^{\left(s\right)}\left(r\right)=3Ka-4G\frac{b}{r^3},$$ (19)

where $K$ and $G$ denote the elastic solid’s bulk and shear moduli, respectively. The coefficients $a$ and $b$ are governed by boundary conditions enforcing normal stress continuity at the liquid–solid interface ($R_l$) and the solid’s outer boundary ($R_s$):

$${\sigma }_{rr}^{\left(s\right)}\left(R_l\right)={\sigma }_{rr}^{\left(l\right)}\left(R_l,t\right)+p_0=-p_1,$$$${\sigma }_{rr}^{\left(s\right)}\left(R_s\right)=-\left(p_s-p_0\right)=-p_2,$$ (20)

where ${\sigma }_{rr}^{\left(l\right)}$ is the radial component of the stress tensor in the compressible liquid:

$${\sigma }_{rr}^{\left(l\right)}\left(r\right)=-p_l+2\mu \frac{\partial u_l}{\partial r}-\left(\frac{2}{3}\mu -\kappa \right)\nabla \cdot {\mathbf{u}}_l,$$ (21)

where $\mu$ and $\kappa$ denote the liquid’s shear and bulk viscosities, respectively. $p_l$ is the liquid pressure.

By solving the Eqs. (19)–(21), one can find the solutions of a and b [36]:

$$a=\frac{p_1{R}_l^3-p_2{R}_s^3}{R_s^3-R_l^3}\frac{1}{3K},$$

$$b=\frac{R_l^3{R}_s^3\left(p_1-p_2\right)}{R_s^3-R_l^3}\frac{1}{4G}.$$

The change of the liquid volume can be approximately represented as [31]:

$$\mathrm{\Delta }{V}_l=\frac{4\pi }{3}\left(R_l^3-R_{l0}^3\right)=4\pi R_{l0}^{2}u\left(R_{l0}\right)=4\pi R_{l0}^2\left(a{R}_{l0}+\frac{b}{R_{l0}^2}\right),$$

Substituting $a,b,p_1,p_2$ and ignoring the effect of liquid’s shear and bulk viscosities, one can obtain:

$$\frac{R_l^3-R_{l0}^3}{R_{l0}^3}=\frac{1}{K}\frac{R_s^3\left(p_s-p_0\right)+\left(p_0-p_l\right)R_l^3}{-R_s^3+R_l^3}-\frac{3}{4G}\frac{R_s^3{R}_l^3\left(p_s-p_l\right)}{R_{l0}^3\left(R_s^3-R_l^3\right)}.$$ (22)

3. Linear analysis

In this section, the linear approximation of the model for the dynamic behavior of a single bubble embedded within a spherical compressible liquid surrounded by a confined solid is considered. We use the standard linearization procedure, where $R_b=R_{b0}+\varepsilon {\Re }_b\left(t\right),R_l=R_{l0}+\varepsilon {\Re }_l\left(t\right)$, $\varepsilon$ is a dimensionless small parameter. Inserting $R_b,R_l$ into Eq. (15), one can obtain the coefficient of $\varepsilon$:

$${\ddot{\Re }}_b+2\delta {\dot{\Re }}_b+{\omega }_0^2{\Re }_b=0,$$ (23)

where $\delta =\frac{2\mu }{{\rho }_l{R}_{b0}^2}-\frac{3c{R}_{b0}^2}{2(R_{l0}^3-R_{b0}^3)}$, and ${\omega }_0^2=\frac{3{\gamma }_1(p_0-p_v)}{R_{b0}^2\rho }+\frac{2\sigma (3{\gamma }_1-1)}{R_{b0}^3\rho }+\frac{36E_{v}GK{R}_{b0}(R_s^3-R_{l0}^3)}{\rho D}$, $D=8E_{v}G{R}_{l0}^6+6E_{v}K{R}_{l0}^6+9E_{v}K{R}_{l0}^3{R}_s^3+12GK(R_s^3-R_{l0}^3)(R_{l0}^3-R_{b0}^3)$. From the above equation, it is clear that the resonant frequency of a bubble oscillating in a liquid entirely confined within an elastic solid, $f_0={\omega }_0/\left(2\pi \right)$, is higher than that in an unbounded liquid. To validate the resonant frequency predicted by our theoretical model, we perform quantitative comparisons with experimental measurements by Vincent [22]. As shown in Fig. 2, the resonant frequency profiles correspond to four confinement conditions: the blue dash-dotted curve for $R_s=1.1R_{l0}$, the red dashed curve for $R_s=2R_{l0}$, the black solid curve for $R_s=5R_{l0}$, and the magenta dotted curve for a fixed $R_s=300\mathrm{\mu }\text{m}$. The detailed view in the figure illustrates $R_{l0}$ within the range of 130–180 $\mathrm{\mu }\text{m}$. Our analysis reveals a fundamental inverse relationship between resonant frequency ($f$) and elastic solid inner radius ($R_{l0}$), empirically described by $f=\mathcal{L}/R_{l0}$. The scaling coefficients $\mathcal{L}=\{51,110,121,122\pm 13\mathrm{m/s}\}$ correspond to the four confinement configurations in sequence. Notably, the theoretical result shows excellent agreement with Vincent’s experimental measurements and theoretical predictions ($\mathcal{L}=120\pm 10\mathrm{m/s}$) when $R_s\ge 2R_{l0}$.

Fig. 2.

Resonant frequency of a single bubble in a elastic solid. The parameters used are $K=1\mathrm{GPa},G=3K/4,R_{b0}=0.28R_{l0},E_v=2.2\mathrm{GPa}$[22].

To analyze the coupled dependence of bubble resonant frequency on the ambient radius $R_{b0}$ and the inner radius of the elastic solid $R_{l0}$, for the sake of simplicity, ${\dot{\Re }}_l$ can be ignored in the standard linearization procedure due to being much smaller than ${\dot{\Re }}_b$ and then one can obtain ${\omega }_0=\frac{1}{R_{b0}}\sqrt{\left[\frac{3\gamma }{{\rho }_l}\left(p_0+\frac{2\sigma }{R_{b0}}\right)-\frac{2\sigma }{{\rho }_l{R}_{b0}}+\frac{3E_v{R}_{b0}^3}{{\rho }_l\left(R_{l0}^3-R_{b0}^3\right)}\right]}$. The resonant frequency of a bubble within an elastic solid is presented in Fig. 3 as a function of the ambient radius of the bubble $R_{b0}$ (x axis) and the ambient radius of the liquid $R_{l0}$ (y axis). The parameters used here are: ${\rho }_l=998.2{\mathrm{kg/m}}^3,\gamma =1.33,p_0=1\mathrm{atm},\sigma =0.07275\mathrm{Pa}\mathrm{s},E_v=2.2\mathrm{GPa}$.

Fig. 3.

Resonant frequency of a single bubble in an elastic solid. The $x$ axis is the ambient radius of the bubble ($R_{b0})$ and the $y$ axis is the ambient radius of the liquid ($R_{l0}$).

Fig. 3 reveals that the bubble’s resonant frequency exhibits an increasing trend as the ambient inner radius of the solid $R_{l0}$ decreases for a certain ambient bubble radius $R_{b0}$. However, Fig. 3 also shows that the resonant frequency of the bubble decreases and then increases with the increase of the ambient radius of the bubble $R_{b0}$ for a certain ambient radius of the liquid $R_{l0}$ (inner radius of the elastic solid). The red line depicts the stationary points of the resonance frequency with respect to $R_{b0}$ within the specific range $10\mathrm{\mu }\mathrm{m}\le R_{l0}\le 40\mathrm{\mu }\mathrm{m}$. This non-monotonic behavior indicates that surface tension and gas compressibility dominate when $R_{b0}$ falls below the critical value (denoted by the red line), whereas liquid compressibility prevails when $R_{b0}$ exceeds this threshold. These findings establish a theoretical foundation for optimizing cavitation reactor design in confined environments.

In the case where the bubble undergoes a small-amplitude oscillation, it is evident that $f_0={\omega }_0/2\pi$ will reduces to the Minnaert frequency of the single bubble in an unbounded liquid as $R_{l0}\to \infty$, given by

$$f_0=\frac{{\omega }_0}{2\pi }=\frac{1}{2\pi R_{b0}}\sqrt{\left[\frac{3\gamma }{{\rho }_l}\left(p_0+\frac{2\sigma }{R_{b0}}\right)-\frac{2\sigma }{{\rho }_l{R}_{b0}}\right]}.$$ (24)

4. Nonlinear bubble dynamics

In this section, the dynamics of a single cavitation bubble in a spherical compressible liquid within an elastic solid under driving pressure is studied to investigate the first-order liquid compressibility correction. The liquid (water) parameters are set to: atmospheric pressure $p_0=1\mathrm{atm}$, density $\rho =1000{\mathrm{kg/m}}^3$, sound speed $c=1500\mathrm{m/s}$, surface tension $\sigma =0.07286\mathrm{N/m}$, viscosity $\mu =0.001\mathrm{Pa\; s}$, and heat capacity ratio of gas inside the bubble $\gamma =1.33$ [25], [31]. The elastic solid parameters are bulk modulus $K=4.5\mathrm{GPa}$ and shear modulus $G=0.74\mathrm{GPa}$ [31]. The ambient bubble radius is $0.5\mathrm{\mu }$m. The ambient inner and outer radii of the elastic solid are $R_{l0}=40\mathrm{\mu }$m and $R_{s0}=60\mathrm{\mu }$m, respectively. The driving pressure follows $p_d\left(t\right)=p_{\mathrm{ac}}sin\left(2\pi ft\right)$ with amplitude $p_{\mathrm{ac}}=10p_0$ and frequency $f=1\mathrm{MHz}$.

The numerical solutions for bubble radius $R_b$ and the inner radius of the elastic solid $R_l$ (Eqs. (15), (22)) are shown in Fig. 4 and Fig. 5. The red solid lines stand for $R_b$ and $R_l$ with considering the first-order correction for liquid compressibility, and the black dashed lines stand for $R_b$ and $R_l$ without considering the first-order correction for liquid compressibility. From Fig. 4, one can clearly see that the first-order correction significantly damps the amplitude of bubble rebounds from 9.7 $R_{b0}$ (the first rebound of black dashed line) to 1.4 $R_{b0}$ (the first rebound of red solid line), which is consistent with the result in an unbound liquid [37], [38]. Both figures further reveal that the first-order correction effects improve cavitation bubble stability through enhanced damping mechanisms.

Fig. 4.

The radial pulsations of the bubble a function of time with or without considering the first-order correction for liquid compressibility.

Fig. 5.

The inner radius of the elastic solid $R_l$ as a function of time with or without considering the first-order correction for liquid compressibility.

In order to compare the present model (Eqs. Eq. (15) and Eq. (22)) with the model presented in Ref. [31], which does not consider the first-order correction for liquid compressibility (Eqs. (5), (20) therein), we set the driving pressure acting on the bubble-cell system as follows [31]:

$$p_s=\left\{\begin{array}{c}\hfill p_{instant}\hfill \\ \hfill p_{instant}-\Delta psin\left(2\pi f\left(t-t_{act}\right)\right)\hfill \end{array}\right)\begin{array}{c}\hfill 0\le t<t_{act}\hfill \\ \hfill t\ge t_{act}\hfill \end{array}$$ (25)

Fig. 6 shows the responses of a bubble in an elastic solid to a driving pressure with the parameters $f=10\mathrm{kHz},p_{\text{instant}}=-1.67p_0$, $\mathrm{\Delta }p=0.73p_0,t_{\text{tact}}=0.4\mathrm{\mu }s$. The red solid line and the blue dotted-dashed line represent the radius of the bubble based on the present model with the outer radius of the elastic solid $R_{s0}=60\mathrm{\mu }\mathrm{m}$ and $R_{s0}\to \infty$ respectively; the black dashed line represents the bubble radius based on Eqs. (5), (22) in Ref. [31]. Fig. 6(a) illustrates the results of the first 100 $\mathrm{\mu }s$. By comparing the red solid line and the blue dotted-dashed line, one can observe that as the outer radius of the elastic solid increases, the oscillation duration of the cavitation bubble becomes shorter, and its oscillation amplitude decreases significantly. Furthermore, comparing the blue dotted-dashed line with the black dashed line, we find that incorporating the first-order correction for liquid compressibility also results in a reduction in the oscillation duration of the cavitation bubble, while the oscillation amplitude significantly increases during the rebound phase. Fig. 6(b) shows the bubble radius between 0.0784 and 0.08 s. From this subfigure, we can see that the radius of the bubble based on Leonov model is unstable and collapses during the 797th period; in contrast, the bubble radius considering first-order correction for liquid compressibility remains stable. This demonstrates that the present model exhibits significantly greater stability compared to the one without considering the first-order correction for liquid compressibility.

Fig. 6.

Responses of a bubble in an elastic solid to a driving pressure with the parameters $f=10\mathrm{kHz},p_{instant}=-1.67p_0$, $\mathrm{\Delta }p=0.73p_0,t_{tact}=0.4\mathrm{\mu }s$. The black line is the bubble radius based on Eqs. (5), (22) in Ref. [31]; the red line and the dotted-dashed line are based on the present model with $R_{s0}=60\mathrm{\mu }\mathrm{m}$ and $R_{s0}\to \infty$, respectively. (a) stable part; (b) unstable part.

5. The effect of the parameters of elastic solid

In this section, numerical simulations are conducted to investigate the influence of an elastic solid on the dynamics of cavitation bubbles. From Eqs. (15), (22), it can be observed that the factors related to the elastic solid that influence the pulsation of cavitation bubbles include the inner and outer radii of the elastic solid, as well as its bulk modulus and shear modulus. Unless otherwise specified, all parameters are set to be consistent with those used in Section 4.

5.1. The effect of size of the elastic solid $R_{l0}$ and $R_{s0}$

In this subsection, we firstly investigate the effect of size of the elastic solid $R_{l0}$ and $R_{s0}$ on the dynamic behavior of a bubble within an elastic solid. The results corresponding to variation in $R_{l0}$ with a fixed $R_{s0}=100\mathrm{\mu }\mathrm{m}$ are presented in Fig. 7. The results corresponding to variation in $R_{s0}$ with a fixed $R_{l0}=40\mathrm{\mu }\mathrm{m}$ are presented in Fig. 8.

Fig. 7.

The effect of inner radius of the elastic solid $R_{l0}$ on the dynamic behavior of a single bubble in an elastic solid. (a) The radius of bubble with $R_{l0}=40,60,80\mathrm{\mu }\mathrm{m}$, respectively. (b) The variation of maximum radius ratio of the bubble ($max\left(R_b\right)/R_{b0}$) as $R_{l0}$ ranges from 40 to 80 $\mathrm{\mu }\mathrm{m}$.

Fig. 8.

The effect of outer radius of the elastic solid $R_{s0}$ on the dynamic behavior of a single bubble in an elastic solid. (a) The radius of bubble with $R_{s0}=60,80,100\mathrm{\mu }\mathrm{m}$, respectively. (b) The variation of maximum radius ratio of the bubble ($max\left(R_b\right)/R_{b0}$) ranges from 50 to 300 $\mathrm{\mu }\mathrm{m}$.

From Fig. 7, one can observe that as $R_{l0}$ increases, the maximum radius of the bubble also increases, and the collapsing time (the instantaneous moment at which the bubble attains its minimum radius) significantly delays. Conversely, from Fig. 8, it is evident that as $R_{s0}$ increases, the maximum radius of the bubble decreases, and the collapsing time shortens as well. These findings suggest that increasing the inner radius of the elastic solid or reducing its outer radius — effectively enlarging the liquid volume surrounding the cavitation bubbles or decreasing the thickness of the elastic solid — can enhance the pulsation amplitude of the cavitation bubbles.

5.2. The effect of bulk and shear moduli of elastic solid

In this subsection, we subsequently investigate the effect of the properties of the elastic solid (the bulk modulus $K$ and the shear modulus $G$) on the dynamic behavior of a bubble within an elastic solid. The results corresponding to variation in $K$ with a fixed $G=0.74\mathrm{GPa}$ are presented in Fig. 9. The results corresponding to variation in $G$ with a fixed $K=4.5\mathrm{GPa}$ are presented in Fig. 10. From both Fig. 9, Fig. 10, one can observe that as the bulk modulus $K$ or the shear modulus $G$ of the elastic solid increases, the maximum radius of the bubble decreases. This inverse correlation clearly indicates that higher material stiffness (via increased K or G) significantly suppresses the expansion of the cavitation bubble.

Fig. 9.

The effect of bulk modulus of the elastic solid $K$ on the dynamic behavior of a bubble in an elastic solid. (a) The radius of bubble with $K=1,4.5,8\mathrm{GPa}$, respectively. (b) The variation of maximum radius ratio of the bubble ($max\left(R_b\right)/R_{b0}$) as bulk modulus of elastic solid $K$ ranges from 1 to 100 $\mathrm{GPa}$.

Fig. 10.

The effect of shear modulus of the elastic solid $G$ on the dynamic behavior of a bubble in an elastic solid. (a) The radius of bubble with $G=0.4,0.74,1\mathrm{GPa}$, respectively. (b) The variation of maximum radius ratio of the bubble ($max\left(R_b\right)/R_{b0}$) as shear modulus $G$ ranges from 0.2 to 10 $\mathrm{GPa}$.

6. Conclusion

In this paper, we develop a coupled model to simulate the dynamics of a single cavitation bubble within a compressible spherical liquid, incorporating the first-order compressibility correction, entirely confined by a finite-thickness elastic solid. Linear analysis reveals a fundamental inverse relationship between resonant frequency and inner radius of the elastic solid $f=\mathcal{L}/R_{l0}$. The scaling coefficient $\mathcal{L}\approx 120\mathrm{m/s}$ agrees with the experimental results obtained by Vincent when $R_s\ge 2R_{l0}$. Moreover, linear analysis reveals a non-monotonic relationship between resonant frequency of the bubble and inner radius of the elastic solid: as the ambient bubble radius decreases, the frequency initially decreases and then increases under a specific inner radius of the elastic solid. The first-order correction for liquid compressibility plays an crucial role in the rebounce phase of the bubble within the compressible liquid entirely confined by an elastic solid.

Additionally, the numerical calculations are conducted to investigate the effect of the size and the properties of elastic solid on the bubble behavior. The results indicate that the maximum radius ratio of the bubble ($max\left(R_b\right)/R_{b0}$) increases with an increase of the inner radius of elastic solid, a decrease of the outer radius of the elastic solid, and a decrease of bulk and shear moduli of the elastic solid. Furthermore, an increase of the inner radius significantly prolongs the collapsing time of the confined bubble. These findings provide a foundational work for predicting and controlling cavitation dynamics confined in an elastic solid.

However, the model has several limitations. First, we assumed the bubble-liquid–solid system maintains spherical symmetry throughout, which may not hold under high driving pressures. We also ignored mass transfer effects and did not account for shock wave impacts on the elastic solid. Future studies will target these aspects along with non-spherical modeling.

CRediT authorship contribution statement

Yang Shen: Writing – original draft, Validation, Supervision, Methodology, Investigation, Funding acquisition, Conceptualization. Yisa Wang: Writing – original draft, Investigation. Long Xu: Supervision, Methodology, Conceptualization.

Declaration of competing interest

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

Appendix A. Supplementary data

The following is the Supplementary material related to this article.

MMC S1

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