Dynamics of an externally driven cavitation bubble in an elastic microconfinement

The subject of the present study is the dynamics of a single cavitation bubble in a spherical liquid cell surrounded by an infinite elastic solid. It is shown that volume confinement strongly affects the manifestation of the classical cavitation Blake threshold. In particular, at liquid cell sizes smaller than some critical size, the cavitation is completely suppressed by volumetric confinement. The system of equations for the dynamics of a confined bubble, accounting for the mass of gas in the bubble, surface tension, liquid compressibility, solid elasticity, and damping due to viscosity in the liquid cell, is derived. The pressure in the solid far away from the bubble is used as an external driving force. Linear analysis of the bubble dynamics, including consideration of the natural frequency and amplitude-phase frequency response of the bubble-in-cell system, is conducted. In the nonlinear case, bifurcation diagrams are considered to determine the dynamic response of small and large bubbles in the states below and above the cavitation threshold, respectively.

Figure 1

The sequence of quasistatic states of the system. The elastic solid contains a spherical liquid cell with a gas–vapor bubble nucleus inside, which grows under tension $p_{\infty }$.

Figure 2

The dependence of the nondimensionalized tension in a solid on the (a) bubble radius, (b) relative liquid cell radius, (c) pressure in the liquid cell, and (d) gas pressure in the bubble. The dotted horizontal line $\overline{p_{\infty }}=1$ corresponds to the reference state; $R_{b0}=300\mathrm{nm}, R_{l0}=40\mu \mathrm{m}$.

Figure 3

Equilibrium diagram of a bubble in microconfinement for different initial radii of the liquid cell; $R_{b0}=300\mathrm{nm}$.

Figure 4

Ratio of the modified Blake threshold to the classical one as a function of the initial radius of the liquid cell [Eq. (29), solid line]. The full circles denote the Blake threshold numerically calculated from system (24). The double arrow shows the difference ${\mathrm{\Delta }}^{**}$ between the modified Blake threshold and the numerically calculated one for the critical initial radius of the liquid cell $R_{l0}^{\mathrm{crit}}\simeq 21.5\mu \mathrm{m}, R_{b0}=300\mathrm{nm}$.

Figure 5

The natural frequency of bubble oscillations as a function of the (a) bubble radius, (b) tension in the elastic solid, and (c) relative liquid cell radius in hertz ($f={\omega }_0/\phantom{{\omega }_{0}2\pi }2\pi$) for different initial radii of the liquid cell: $R_{l0}=40\mu \mathrm{m}$ (solid line), $R_{l0}^{\mathrm{crit}}\simeq 21.5\mu \mathrm{m}$ (dashed line), $R_{l0}=10\mu \mathrm{m}$ (dotted-dashed line). The long-dashed line in (a) corresponds to the natural frequency in the bulk liquid [Eq. (37)], while that in (b) to the natural frequency in the case of viscous damping [Eq. (38)]; $R_{b0}=300\mathrm{nm}$.

Figure 6

The quality factor of the bubble-in-cell system as a function of the tension in the elastic solid; $R_{b0}=300\mathrm{nm}, R_{l0}=40\mu \mathrm{m}$.

Figure 7

(a) The equilibrium diagram of a bubble in microconfinement; the amplitude-phase frequency response of a small bubble at point A (b), (c) and a large bubble at point N (d), (e); $\overline{p_{\infty ,eq}^A}=-1.67, \overline{p_{\infty ,eq}^N}=-2.4, R_{b0}=300\mathrm{nm}, R_{l0}=40\mu \mathrm{m}$.

Figure 8

The relative contribution of the three terms in the square brackets of Eq. (19) and the term with viscosity from the generalized Rayleigh-Plesset equation (5). The dotted line is the liquid pressure, the solid line is the term with viscosity in Eq. (19), the dotted-dashed line is constant tension ($p_{\infty }=-0.24\mathrm{MPa}$), and the dashed line is the term with viscosity from the generalized Rayleigh-Plesset equation.

Figure 9

(a) Equilibrium diagram of a bubble in a confined liquid. The solution to the system of Eqs. (20) and (5) for a small bubble [(b), (c)] and a large bubble [(d), (e)]. The solid line is the bubble radius, the dashed line is the pressure in the liquid cell, and the dotted-dashed line is the tension in the elastic solid; $R_{b0}=300\mathrm{nm}, R_{l0}=40\mu \mathrm{m}$.

Figure 10

Bifurcation diagram of the bubble oscillations in a liquid cell around the state at point A [see Fig. 9] in the driving frequency range $f_{\mathrm{ext}}=0.1-30\mathrm{MHz}$; $R_{b0}=300\mathrm{nm}, R_{l0}=40\mu \mathrm{m}$.

Figure 11

(a) Evolution of the bubble radius at a driving frequency $f_{\mathrm{ext}}=1.1\mathrm{MHz}$. (b) Bifurcation diagram of the bubble oscillations in a liquid cell around the state at point N [see Fig. 9] in the driving frequency range $f_{\mathrm{ext}}=0.1-5\mathrm{MHz}$. (c), (d) Enlarged views of (b); $R_{b0}=300\mathrm{nm}, R_{l0}=40\mu \mathrm{m}$.