Nonlinear dynamics of two coupled bubbles oscillating inside a liquid-filled cavity surrounded by an elastic medium

A theory is developed to model the nonlinear dynamics of two coupled bubbles inside a spherical liquid-filled cavity surrounded by an elastic medium. The aim is to study how the conditions of full confinement affect the coupled oscillations of the bubbles. To make the problem amenable to analytical consideration, the bubbles are assumed to be located on a diameter of the cavity, which makes the problem axisymmetric. Equations for the pulsation and translation motion of the bubbles are derived by the Lagrangian formalism. The derived equations are used in numerical simulations. The behavior of two bubbles in a cavity is compared with the behavior of the same bubbles in an unbounded liquid. It is found that both forced and free oscillations of two bubbles in a cavity occur differently than those in an unbounded liquid. In particular, it is shown that the eigenfrequencies of a two-bubble system in a cavity are different from those in an unbounded liquid.

Figure 1

Two bubbles inside a liquid-filled cavity.

Figure 2

Acoustic pulse used as excitation in Fig. 3. The mathematical definition is given by Eq. (32). The pulse parameters are $f=1.5$ MHz, $P_a=50$ kPa, and $N=5$.

Figure 3

Response of bubbles to the acoustic pulse shown in Fig. 2. Left panel: bubbles in a cavity. Right panel: bubbles in an unbounded liquid. $R_{10}=7.5\mu \mathrm{m}, R_{20}=5\mu \mathrm{m}, z_1\left(0\right)=-50\mu \mathrm{m}, z_2\left(0\right)=50\mu \mathrm{m}, R_{c0}=100\mu \mathrm{m}, R_s=500\mu \mathrm{m}$. The Minnaert frequencies of the bubbles are 468 and 724 kHz. The bubbles are excited above their natural frequencies.

Figure 4

Translational motion of two bubbles in a cavity and in an unbounded liquid. The parameters are as described in the caption of Fig. 3, except that the number of cycles in the excitation pulse is $N=500$. The bubbles are moving toward each other.

Figure 5

Response of bubbles to an acoustic pulse with the parameters $f=0.6$ MHz, $P_a=50$ kPa, $N=5$. Left panel: bubbles in a cavity. Right panel: bubbles in an unbounded liquid. $R_{10}=7.5\mu \mathrm{m}, R_{20}=5\mu \mathrm{m}, z_1\left(0\right)=-50\mu \mathrm{m}, z_2\left(0\right)=50\mu \mathrm{m}, R_{c0}=100\mu \mathrm{m}, R_s=500\mu \mathrm{m}$. The driving frequency lies between the natural frequencies of the bubbles.

Figure 6

Translational motion of two bubbles in a cavity and in an unbounded liquid. The parameters are as described in the caption of Fig. 5, except that $N=50$. The bubbles are moving away from each other.

Figure 7

Response of bubbles to an acoustic pulse with the parameters $f=0.3$ MHz, $P_a=50$ kPa, $N=5$. Left panel: bubbles in a cavity. Right panel: bubbles in an unbounded liquid. $R_{10}=7.5\mu \mathrm{m}, R_{20}=5\mu \mathrm{m}, z_1\left(0\right)=-50\mu \mathrm{m}, z_2\left(0\right)=50\mu \mathrm{m}, R_{c0}=100\mu \mathrm{m}, R_s=500\mu \mathrm{m}$. The bubbles are excited below their natural frequencies.

Figure 8

Translational motion of two bubbles in a cavity and in an unbounded liquid. The parameters are as described in the caption of Fig. 7, except that $N=1000$.

Figure 9

Free oscillations of two equal bubbles. Left panel: bubbles in a cavity. Right panel: bubbles in an unbounded liquid. $R_{10}=R_{20}=5\mu \mathrm{m}, z_1\left(0\right)=-50\mu \mathrm{m}, z_2\left(0\right)=50\mu \mathrm{m}, R_{c0}=100\mu \mathrm{m}, R_s=500\mu \mathrm{m}$. The Minnaert frequency of the bubbles is 724 kHz. The free oscillations are excited by a small initial deviation from the equilibrium bubble radii.

Figure 10

Free oscillations of two unequal bubbles. Left panel: bubbles in a cavity. Right panel: bubbles in an unbounded liquid. $R_{10}=7.5\mu \mathrm{m}, R_{20}=5\mu \mathrm{m}, z_1\left(0\right)=-50\mu \mathrm{m}, z_2\left(0\right)=50\mu \mathrm{m}, R_{c0}=100\mu \mathrm{m}, R_s=500\mu \mathrm{m}$. The Minnaert frequencies of the bubbles are 468 kHz and 724 kHz.

Figure 11

Growth and evolution of two bubbles in a rectangular microchannel. Images are taken after nucleation at times (a) 1.9 μs, (b) 15 μs, (c) 45 μs. The vertical scale is 100 μm.

Figure 12

Comparison of experimental data and theoretical predictions.