Path suppression of strongly collapsing bubbles at finite and low Reynolds numbers

We study, numerically and experimentally, three different methods to suppress the trajectories of strongly collapsing and sonoluminescent bubbles in a highly viscous sulfuric acid solution. A new numerical scheme based on the window method is proposed to account for the history force acting on a spherical bubble with variable radius. We could quantify the history force, which is not negligible in comparison with the primary Bjerknes force in this type of problem, and results are in agreement with the classical primary Bjerknes force trapping threshold analysis. Moreover, the present numerical implementation reproduces the spatial behavior associated with the positional and path instability of sonoluminescent argon bubbles in strongly gassed and highly degassed sulfuric acid solutions. Finally, the model allows us to demonstrate that spatially stationary bubbles driven by biharmonic excitation could be obtained with a different mode from the one used in previous reported experiments.

Figure 1

(a)–(g) Trajectories of an argon bubble with $R_0=9.0\mu$m in SA85 driving at $f_0=30$ kHz under different pressure amplitudes at the resonator center ($p_a^0$). (a) $p_a^0=1.30$ bar. (b) $p_a^0=1.32$ bar. (c) $p_a^0=1.40$ bar. (d) $p_a^0=1.50$ bar. (e) $p_a^0=1.60$ bar. (f) $p_a^0=1.70$ bar. The black dot indicates the origin of coordinates. (g) $p_a^0=1.65$ bar, $p_a^0=1.70$ bar, $p_a^0=1.80$ bar, $p_a^0=1.90$ bar, $p_a^0=2.00$ bar, $p_a^0=2.10$ bar, $p_a^0=2.20$ bar, $p_a^0=2.30$ bar, $p_a^0=2.40$ bar, $p_a^0=2.50$ bar, $p_a^0=2.60$ bar, and $p_a^0=2.70$ bar. (h) The inset shows for the case with $p_a^0=1.65$ bar, the direction of the $F_{\mathrm{history}}$ (red) and $F_{\mathrm{Bjerknes}}$ (green) for the first 1500 radial cycles.

Figure 2

Mean levitation distance as function of $p_a^0$ using the ambient radius $R_0$ as a parameter for argon bubbles in SA85 solution driven at $f_0=30$ kHz. The gray bars indicate the approximate maximum amplitude of the pseudo-orbits for the particular case of $R_0=9\mu$m. For driving pressures below $p_a^0<1.31$ bar no pseudo-orbits appeared. The onset of the path instability at $p_a^0=1.32$ bar as well as the onset of the positional instability at $p_a^0=1.60$ bar, is indicated with gray dotted line. The inset shows in detail the transition between the non-moving bubble and the moving-bubble for the particular case of $R_0=9\mu$m.

Figure 3

Ratio $\frac{c_{\infty }}{c_0}$ as a function of $p_a^0$ for different $R_0$ for argon bubbles in SA85 solution driven at $f_0=30$ kHz. For $p_a^0>1.70$ bar, a plateau is reached in all cases studied.

Figure 4

Primary Bjerknes force module averaged in each radial cycle using the ambient radius as a parameter for an argon bubble in SA85 driven at 30 kHz.

Figure 5

History force module averaged in each radial cycle using the ambient radius as a parameter for an argon bubble in SA85 driven at 30 kHz.

Figure 6

Added mass peak force module using the ambient radius as a parameter for an argon bubble in SA85 driven at 30 kHz.

Figure 7

Phase diagram of an argon bubble driving at $f_0=29.2$ kHz in SA85. (green line) Indicates the trapping threshold due to primary Bjerknes force for bubbles that remain spatially fixed in the pressure field driven by $p_a^{b\mathrm{LF}}$. In green is shown the region where the primary Bjerknes force is able to trap the bubble near the central antinode of pressure, while in white is shown the region where the bubble is repealed from the central antinode of pressure. (Solid black line) Indicates different diffusive constant contour levels ($\frac{c_{\infty }}{c_0}=\text{cte}$). (Dotted black line) The contour $\frac{c_{\infty }}{c_0}=0.010$ indicates the threshold between low and high gas concentration dissolved in the liquid. (dotted blue line) Indicates the parametric shape instability boundary calculated using the vorticity boundary layer approximation [29]. (Red dotted line) Indicates the Blake threshold, which marks the onset of nonlinear bubble response to the acoustic pressure. (Red dots) Indicate moving bubbles. (Yellow dots) Indicate nonmoving bubbles. (Inset) Shows the accumulative points. When $p_a^0$ rises, the bubble shifts away from the center and lower pressures are applied on the bubble wall up to an asymptotic value. For $\frac{c_{\infty }}{c_0}=0.017$ the asymptotic value is $p_a^{b\mathrm{LF}}=1.59$ bar, while for $\frac{c_{\infty }}{c_0}=0.006$, $p_a^{b\mathrm{LF}}=1.69$ bar.

Figure 8

(a) SL argon bubble in SA85 driven only with a frequency equivalent to the first eigenvalue of the resonator ($f_0=29.2$ kHz). The experiment was made with a high gas concentration diluted in the liquid ($\frac{c_{\infty }}{c_0}=0.017$). $p_a^0$ was gradually increased in a quasistatic manner [32]. (b) Numerical simulations keeping constant $\frac{c_{\infty }}{c_0}$ as in the real experiment. For the selected concentration ($\frac{c_{\infty }}{c_0}=0.017$), the numerical model reproduces the “path instability” as well as the “positional instability” when $p_a^0$ rises. (c) (Black line) Temporal evolution of the bubble radius ($R_0=9.3\mu$m; $p_a^0=1.50$ bar), positional stable and path unstable case. (Solid green line) Module of the history force acting on the bubble in the radial time scale. (Dotted green line) Module of the primary Bjerknes force. (d) (Black line) Temporal evolution of the bubble radius ($R_0=11.5\mu$m; $p_a^0=1.70$ bar), positional unstable and path unstable case. (Solid green line) Module of the history force acting on the bubble in the radial time scale. (Dotted green line) Module of the primary Bjerknes force. (e) Module of the primary Bjerknes force in the time scale of spatial trajectories. The simulations were made during 10 000 radial cycles. (Blue line) Case ($R_0=9.3\mu$m; $p_a^0=1.50$ bar), positional stable and path unstable case. (Red line) Case ($R_0=11.5\mu$m; $p_a^0=1.70$ bar), positional unstable and path unstable case. (f) Module of the history force in the time scale of spatial trajectories. The simulations were made during 10 000 radial cycles. (Blue line) Case ($R_0=9.3\mu$m; $p_a^0=1.50$ bar), positional stable and path unstable case. (Red line) Case ($R_0=11.5\mu$m; $p_a^0=1.70$ bar), positional unstable and path unstable case.

Figure 9

(a) SL argon bubble in SA85 driven only with a frequency equivalent to the first eigenvalue of the resonator ($f_0=29.2$ kHz). The experiment was made with a low gas concentration diluted in the liquid ($\frac{c_{\infty }}{c_0}=0.006$). $p_a^0$ was gradually increased in a quasistatic manner [32]. (b) Numerical simulations keeping constant $\frac{c_{\infty }}{c_0}$ as in the real experiment. For the selected concentration ($\frac{c_{\infty }}{c_0}=0.006$), the numerical model reproduces the mean levitation position shifting away from the center when $p_a^0$ rises (“positional instability”). No “path instability” could be observed for bubbles in a highly degassed SA85. (c) (Black line) Temporal evolution of the radius of the bubble ($R_0=2.5\mu$m; $p_a^0=1.45$ bar). (Solid green line) Module of the history force acting on the bubble in the radial time scale. (Dotted green line) Module of the primary Bjerknes force. (d) (Black line) Temporal evolution of the radius of the bubble ($R_0=7.6\mu$m; $p_a^0=2.52$ bar). (Solid green line) Module of the history force acting on the bubble in the radial time scale. (Dotted green line) Module of the primary Bjerknes force. (e) Module of the primary Bjerknes force in a time scale of spatial trajectories. The simulations were made during 10 000 radial cycles. (Blue line) Case ($R_0=2.5\mu$m; $p_a^0=1.45$ bar). (Red line) Case ($R_0=7.6\mu$m; $p_a^0=2.52$ bar). (f) Module of the history force in the time scale of spatial trajectories. The simulations were made during 10 000 radial cycles. (Blue line) Case ($R_0=2.5\mu$m; $p_a^0=1.45$ bar). (Red line) Case ($R_0=7.6\mu$m; $p_a^0=2.52$ bar).

Figure 10

(gray) Bubble radius temporal evolution measured with Mie scattering technique of an argon bubble in SA85 solution driving under bi-harmonic excitation ($f_0=29081$ Hz and $f_7=7f_0=203567$ Hz). (Black) Numerical fitting with the present model. Fitted parameters: $R_0=(10.7\pm 0.1)\mu$m, $p_a^{b\mathrm{LF}}=(1.34\pm 0.02)$ bar, $p_a^{b\mathrm{HF}}=(1.19\pm 0.02)$ bar, $\varphi =(1.80\pm 0.01)$ rad. The concentration of gas in the liquid is equivalent to $\frac{c_{\infty }}{c_0}=0.030$. The time reference $t=0$ is the zero crossing of the acoustic pressure in the fundamental mode with positive slope.

Figure 11

Simulated trajectories for the experimental case shown in Fig. 10. The initial conditions for the system of equations Eqs. (1) are as follows: $R_i=R_0$, ${\dot{R}}_i=0.0\frac{\mathrm{m}}{\mathrm{s}}$, $T_b^i=T_{\mathrm{liq}}$, $x_i=0.10$ mm, ${\dot{x}}_i=0.0\frac{\mathrm{m}}{\mathrm{s}}$, $y_i=-0.15$ mm, ${\dot{y}}_i=0.0\frac{\mathrm{m}}{\mathrm{s}}$,$z_i=0.0$ mm, ${\dot{z}}_i=0.0\frac{\mathrm{m}}{\mathrm{s}}$. (a) Bubble driving by the fundamental mode only, $p_a^7=0.00$ bar. (b) $p_a^7=0.15$ bar. (c) $p_a^7=0.30$ bar. (d) $p_a^7=0.45$ bar. (e) $p_a^7=0.50$ bar. (f) $p_a^7=0.55$ bar. (g) $p_a^7=0.60$ bar. (h) $p_a^7=1.05$ bar. (i) $p_a^7=1.19$ bar. Spatially fixed bubble by biharmonic excitation within $6\mu$m. This size of trajectories is of the order of the bubble ambient radius and this movement is considered negligible from the experimental point of view.

Figure 12

(Green) Module of the forces acting on an argon bubble in SA85 in the time scale imposed by ultrasound. (Blue) Buoyant force. (Black) Temporal evolution of the radius of the bubble.

Figure 13

Forces acting on an argon bubble in SA85 during 10 000 radial cycles ($\sim$0.344 s). (Black line) Bubble driving by only the fundamental frequency $f_0$. Moving bubble according to case (a) of Figure 11. (Gray line) Bubble under biharmonic excitation with $f_0$ and $7f_0$. Nonmoving bubble according to the case (i) of Fig. 11.

Figure 14

Solution of the history force kernel model made by Magnaudet and Legendre [13] and the approximation proposed by Ren and MacKenzie [18].

Figure 15

(upper panel) The solid black line is the radius of the bubble at previous times calculated with the numerical model considering the coupling between the radial and translational dynamics but without the feedback of the history force. The red dotted line is the history force in the $x$ direction on the bubble at maximum expansion ($t_{\mathrm{back}}=0$) calculated with different numbers of points backwards. The parameters of the argon bubble in SA85 are: $R_0=9.0\mu$m and $p_a^0\simeq p_a^b=1.65$ bar at $f_0=30.0$ kHz. (Lower panel) The solid black line indicates the value of the kernel approximated by Ren-Mackenzie form in logarithmic scale. With red dotted line is shown the absolute value of $\frac{d\left[R\left(t\right){\vec{U}}_x(r,t)\right]}{dt}$ in logarithmic scale. For the time interval defined by 400 points backwards (equivalent to $t_{\mathrm{win}}$), indicated with gray dotted line, the variations of $\frac{d\left[R\left(t\right){\vec{U}}_x(r,t)\right]}{dt}$ are rather small compared with the variation during main collapse. The variations are lower than $0.03\%$ from the mean value indicated with black dotted line. During this time interval the “window model” used to calculate the history force is valid according to Ref. [20]. This analysis is similar for the $y$ direction and $z$ direction.

Figure 16

(Upper panel) Temporal evolution of the radius of an argon bubble in water at a temperature of $6{}^{\circ }$C. The noncondensible gas, the liquid and driving parameters are the same that the ones reported in Ref. [31] ($f_0=19570$ Hz, $p_a^b=1.28$ bar, $p_{\infty }=0.95$ bar, $c_w=1430$ m/s, ${\mu }_w=0.0015\mathrm{Pa}\mathrm{s}$, ${\sigma }_w=0.075$ N/m, ${\rho }_w=999.94$ kg/m${}^3$). (Solid black line) Simulation (1) with an ambient radius $R_0=5.0\mu$m. (Red dotted line) Simulation (2) with an ambient radius $R_0=5.5\mu$m. (Black dots) Indicate the instant $t=-1$ ns. At $t=0$ ns the minimum radius is reached by the bubble. (Lower panel) Radial velocity of the bubble interface. (Thin dotted black line) Speed of sound in the liquid.

Figure 17

(a) Temporal evolution of the radius of an argon bubble in water, with the liquid and driving parameters reported in Ref. [31]. (Solid black line) Simulation (1) with an ambient radius $R_0=5.0\mu$m. (Dotted red line) Simulation with an ambient radius $R_0=5.5\mu$m. (Black dots) Indicate the instant $t=-1$ ns. (b) Radial velocity of the bubble interface. (c) Modulus of the added mass force on the bubble around main collapse. For simulation (1) the maximum value is $F_{\mathrm{am}}^{\max }=140\mu \mathrm{N}$ while simulation (2) $F_{\mathrm{am}}^{\max }=130\mu \mathrm{N}$. (d) Modulus of the drag force on the bubble around main collapse. For simulation (1) the maximum value is $F_{\mathrm{Drag}}^{\max }=1.18\mu \mathrm{N}$ while simulation (2) $F_{\mathrm{Drag}}^{\max }=1.09\mu \mathrm{N}$.