Figure 1

Scheme of the experimental setup. (a) Cylindrical Pyrex resonator. The main resonance frequency was experimentally determined to be about 30.33 kHz, which corresponds to the lowest oscillation mode of the resonator with a pressure antinode in the geometrical center of the flask. (b) Upper view of the acoustic chamber. The Figure details the setup used for the bubble radius temporal evolution measurements via standard Mie scattering technique. An Oriel 77340 phototube was used to acquire the scattered light collected with a Newport KPX142 lens (focal length $=$ 50.2 mm).Reuse & Permissions

Figure 2

Bubble radius temporal evolution obtained from Mie scattering for argon bubbles in SA85 [Green (gray) circles; the SL light pulses has been removed from the experimental data]. The red (black) curve represents the best numerical fit achieved (minimizing χ${}^2$) and the light gray line the absolute error. The bubbles were driven using a biharmonic signal with $f_0$ $=$ 30358.1 Hz and the third harmonic ($N$ $=$ 3). In the presented data the calculated $c_{\infty }/c_0$ was $\sim 7\times {10}^{-3}$ for $P_0$ $=$ 925 mbar and the bubbles were spatially fixed occupying almost the same position in the resonator (∼7 mm away from the geometrical center of the cylinder). The experimental data is the average of more than 30 traces of the scattered light. (a) The static absolute pressure in the fluid for this case was (700 ± 15) mbar. The fitted parameters were: bubble ambient radius $R_0$ $=$ 10.4 μm, low-frequency acoustic pressure $P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 1.24 bar, high-frequency acoustic pressure $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 1.35 bar, and relative phase between the driving frequencies φ $=$ 5.3 μs. The computed maximum temperature at the instant of collapse was $T_{\max }$ $=$ 22 kK. (b) In this case $P_0$ was (925 ± 15) mbar. $R_0$ $=$ 9.8 μm, $P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 1.50 bar, $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 1.75 bar, and φ $=$ −5.1 μs. $T_{\max }$ was 28 kK. (c) In this measurement $P_0$ was (1200 ± 15) mbar. $R_0$ $=$ 9.1 μm, $P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 1.86 bar, $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 2.10 bar, and φ $=$ $-$4.5 μs. $T_{\max }$ was 36 kK. (d) $P_0$ was (1600 ± 15) mbar. $R_0$ $=$ 9.4 μm, $P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 2.48 bar, $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 1.90 bar, and φ $=$ −3.7 μs. $T_{\max }$ was 43 kK.Reuse & Permissions

Figure 3

Bubble radius temporal evolution for argon bubbles in SA85 for three distinct $P_0$. The green (dark gray), cyan (gray), and light gray dots correspond to the acquired experimental data for 0.5 bar, 1.2 bar, and 1.6 bar respectively. The red (black) curves represent the best numerical fit achieved for each case (minimizing χ${}^2$). The bubbles were driven using a biharmonic signal with $f_0$ ≈ 30.4 kHz and the second harmonic ($N$ $=$ 2). The low-frequency applied voltage signal had different amplitudes in each case, being of 18.5 $V_{\mathrm{Rms}}$ for 0.5 bar, 25 $V_{\mathrm{Rms}}$ for 1.2 bar, and 58 $V_{\mathrm{Rms}}$ for 1.6 bar; the calculated $c_{\infty }/c_0$ parameter was $\sim 15\times {10}^{-3}$ for $P_0$ $=$ 925 mbar and the bubbles were spatially fixed. Each measurement is the result of averaging 20 traces of the scattered light. The fitted parameters for each case were: [Green (dark gray) - $P_0$ $=$ 0.5 bar] Bubble ambient radius $R_0$ $=$ 14.2 μm, low-frequency acoustic pressure $P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 0.96 bar, high-frequency acoustic pressure $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 1.20 bar, and relative phase between the driving frequencies φ $=$ 7.0 μs. The computed maximum temperature was 24 kK. [Cyan (gray) - P${}_0$ $=$ 1.2 bar] $R_0$ $=$ 9.6 μm, $P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 2.26 bar, $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 1.30 bar, and φ $=$ 8.0 μs. $T_{\max }$ was 41 kK. (Light gray - P${}_0$ $=$ 1.6 bar) $R_0$ $=$ 6.6 μm, $P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 3.14 bar, $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 0.25 bar, and φ $=$ 4.6 μs. $T_{\max }$ was 72 kK.Reuse & Permissions

Figure 4

Fitted parameters of the experimental data as a function of $P_0$ for four distinct data series. The symbols in the graphs were coded as follows. The squares, circles, stars, and rhombus represent each data series with $c_{\infty }/c_0\sim 7\times {10}^{-3}$, $c_{\infty }/c_0\sim 8\times {10}^{-3}$, $c_{\infty }/c_0\sim 15\times {10}^{-3}$, and $c_{\infty }/c_0\sim 4\times {10}^{-3}$, respectively, taken at $P_0$ $=$ 925 mbar. In the figures, the filled markers indicate measurements made with the second harmonic of the fundamental driving frequency ($f_0\sim 30356\mathrm{Hz}$), the half-filled markers correspond to measurements made applying the third harmonic ($N=3$), and the unfilled markers are associated with the sixth harmonic ($N$ $=$ 6). The dotted gray lines connecting the star markers are drawn to ease the visualization of the data measured using $c_{\infty }/c_0\sim 15\times {10}^{-3}$. The uncertainty in $P_0$ was Δ$P_0$ $=$ 15 mbar. The relative error of the other physical quantities shown are: $\Delta P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 1$\%$, $\Delta P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 3$\%$, Δ(dR/dt) $=$ 2.8$\%$, Δ($R_{\max }/R_0$) $=$ 1.5$\%$ and $\Delta \left(P_{\mathrm{Ac}}^{\mathrm{LF}}{\left(R_{\max }/R_0\right)}^3\right)$ $=$ 2.5$\%$. (a) Acoustic pressures ($P_{\mathrm{Ac}}^{\mathrm{LF}}$ and $P_{\mathrm{Ac}}^{\mathrm{HF}}$) acting on the bubbles as a function of $P_0$. The solid line is a linear fit of the filled markers ($P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 1.98$P_0$ - 0.04; $r^2$ $=$ 0.966) and the dashed line is a fit of the measurements taken using harmonics $N$ $=$ 3 and $N$ $=$ 6 ($P_{\mathrm{Ac}}^{\mathrm{LF}}$ $=$ 1.40$P_0$ $+$ 0.17; $r^2$ $=$ 0.938). Here, the light gray markers signify the high-frequency acoustic pressure ($P_{\mathrm{Ac}}^{\mathrm{HF}}$) associated with the data points with the same marker in black. (b) Maximum bubble wall velocity dR/dt vs. $P_0$. It grows rapidly with an increment of $P_0$ as can be seen from the plot. (c) $R_{\max }/R_0$ ratio vs. $P_0$. This parameter, related with the strength of collapse, grows with $P_0$ in a quadratic fashion doubling its value in a raise of 1 bar (0.5 to 1.5 bar). (d) Mechanical energy density for distinct $P_0$. According to the nonlinear evolution of the expansion ratio, this quantity can be enhanced in a factor 25 augmenting $P_0$ in 1 bar (0.5 to 1.5 bar).Reuse & Permissions

Figure 5

Fitted parameters of the experimental data as a function of the maximum temperature achieved in the instant of collapse ($T_{\max }$). The symbols in the graphs are coded in the same fashion explained in the legend of Fig. 4. The uncertainty in $P_0$ was Δ$P_0$ $=$ 15 mbar. The relative error of the other physical quantities shown are: Δ($c_{\infty }$/c${}_0$) $=$ 4$\%$, Δ($R_{\max }/R_0$) $=$ 1.5$\%$, $\Delta \left(P_{\mathrm{Ac}}^{\mathrm{LF}}{\left(R_{\max }/R_0\right)}^3\right)$ $=$ 2.5$\%$, and Δ($T_{\max }$) $=$ 3$\%$. (a) Static pressure vs. $T_{\max }$. The dotted gray lines connecting the star markers are drawn to ease the visualization of the data measured using $c_{\infty }/c_0\sim 15\times {10}^{-3}$. The second harmonic of $f_0$ is the most satisfactory to achieve high-energy focusing (in the context of this particular experiment). (b) $c_{\infty }/c_0$ related with the maximum temperature. This figure shows that large $T_{\max }$ could be obtained using low dissolved gas concentration and also shows the decay occurring in $c_{\infty }/c_0$ with augmented $P_0$ (or the growth of $c_0$ according to Henry's Law). (c) $R_{\max }/R_0$ as a function of $T_{\max }$. The relation between these two variables can be approximated in a linear fashion with different slopes for distinct harmonics. (d) Relation between the mechanical energy density for $P_{\mathrm{Ac}}^{\mathrm{LF}}$ and the $T_{\max }$ obtained by means of gas kinetic theory. This plot shows that for $N$ $=$ 6, the mechanical energy that was actually transformed into heat was found to be less than the one for lower harmonics ($N$ $=$ 2, 3).Reuse & Permissions

Figure 6

Computed $R_0$-$P_{\mathrm{Ac}}^{\mathrm{LF}}$ parameter map for the Ar-SA85 system for the case of a bifrequency driving ($f_0\approx 30.36$ kHz, $N$ $=$ 3) and different static pressures. The solid blue (black) line is the Bjerknes stability threshold ($F_{\mathrm{Bj}}$ $=$ 0). This line delimits the positionally stable region of the map [shaded in light blue (gray)] and the unstable region (uncolored). In each subplot, the parameters for the calculations has been set in order to match the fitted parameters of the experimental data points (•) shown in Fig. 2. The label between parenthesis correlates with the $R$($t$) in Fig. 2. The thick dash-doted red (black) line is the Blake threshold. The solid line in light gray is the Rayleigh-Taylor shape instability for the mode $n$ $=$ 2. The thick dashed line (black) corresponds to the parametric shape instability threshold. In the latter two curves, the stable region is always below the lines, or outside the closed curves for the “instability islands” in the plots. The green (gray) solid curves are the contours of constant $T_{\max }$ (in units of kK) and the thin dotted curves (black) are the contours of constant $c_{\infty }/c_0$. The graphs demonstrate that there is an expansion of the bubble habitat as the static pressure of the system is increased.Reuse & Permissions

Figure 7

Computed Bjerknes stability threshold for different static pressures. In all the presented data, the parameters used in the numerical model corresponded to the case discussed in the legend of Fig. 2d ($f_0$ ≈ 30.36 kHz, $N$ $=$ 3, $P_{\mathrm{Ac}}^{\mathrm{HF}}$ $=$ 1.9 bar, and φ $=$ −3.7 μs). The dash-dotted blue (black) lines are the Bjerknes stability threshold ($F_{\mathrm{Bj}}$ $=$ 0) with a label indicating the static pressure used for the computation of each curve in units of mbar. The stable Bjerknes zone is always to the left of each curve. The light gray solid line is the Rayleigh-Taylor shape instability for the mode $n$ $=$ 2 and an initial perturbation of 1 nm for the case with $P_0$ $=$ 5 bar. The thick dashed line (black) corresponds to the parametric shape instability threshold for the case with $P_0$ $=$ 5 bar. In the latter two curves, the stable region is always below the lines. The green (gray) solid curves are the contours of $T_{\max }$ (in units of kK) and the thin dotted curves (black) are the contours of constant $c_{\infty }/c_0$ for the case with $P_0$ $=$ 5 bar. As the static pressure is increased, there was a shift of both Bjerknes threshold and Blake threshold (not shown) to regions of higher acoustic pressures. For the maximum computed static pressure in the plot, the numerical model predicted that a bubble with ambient radius of ∼4 μm could achieve a maximum temperature of approximately 170 kK.Reuse & Permissions