Theoretical model for coupled radial and translational motion of two bubbles at arbitrary separation distances

A theoretical model is developed that describes nonlinear spherical pulsations and translational motions of two interacting bubbles at arbitrary separation distances between the bubbles. The derivation of the model is based on the multipole expansion of the bubble velocity potentials and the use of the Lagrangian formalism. The model consists of four coupled ordinary differential equations. Two of them are modified Rayleigh-Plesset equations for the radial oscillations of the bubbles and the other two describe the translational displacement of the bubble centers. The equations are not subject to the assumption that the distance between the bubbles is large compared to the bubble radii and hence make it possible to simulate the bubble dynamics starting from large separation distances up to contact between the bubbles providing that the deviation of the bubble shape from sphericity is negligible. Numerical simulations are carried out to demonstrate the capabilities of the developed model. It is shown that the correct modeling of the translational dynamics of the bubbles at small separation distances requires terms accurate up to ninth order in the inverse separation distance. Physical mechanisms are analyzed that lead to the change of the direction of the relative translational motion of the bubbles in finite-amplitude acoustic fields.

Figure 1

Two interacting bubbles: geometry of the system.

Figure 2

Time evolution of the distance between the bubble centers. Comparison of results obtained with different accuracy in $1/d$. The highest order in $1/d$ is indicated by the figures beside the curves for $d\left(t\right)$. The lower dotted curve is the sum of the instantaneous bubble radii.

Figure 3

Dynamics of bubbles with $R_{10}=1.5\mu \mathrm{m}$ and $R_{20}=2.0\mu \mathrm{m}$ at 1 kPa. (a) Resonance curves at separation distance equal to $4(R_{10}+R_{20})$. (b) Time evolution of the separation distance at $2.95$ and $2.15\mathrm{MHz}$.

Figure 4

Dynamics of bubbles with $R_{10}=1.5\mu \mathrm{m}$ and $R_{20}=2.0\mu \mathrm{m}$ at $40\mathrm{kPa}$. (a) Resonance curves at separation distance equal to $4(R_{10}+R_{20})$. (b) Time evolution of the separation distance at $2.85$ and $2.1\mathrm{MHz}$.

Figure 5

(a) Translational dynamics at different amplitudes of the driving acoustic pulse. (b) Resonance curves of the bubbles at separation distance $d_1$ for 40 and $100\mathrm{kPa}$. (c) Resonance curves at $100\mathrm{kPa}$ for separation distances $d_1$ and $d_2$. The vertical line with the arrow shows the position of the driving frequency.

Figure 6

Time evolution of the separation distance at $2.1\mathrm{MHz}$ for 40 and $100\mathrm{kPa}$. The other parameters are as in Fig. 5

Figure 7

Translational motion at increasing values of the acoustic pressure amplitude. Excitation is a 100-cycle, $1.1\mathrm{MHz}$ Gaussian pulse. The other parameters are as in Fig. 5

Figure 8

Fourier spectra of the radial oscillation of bubble 2 for the cases shown in Fig. 7