⇦Chinese Physics Letters, 2019, Vol. 36, No. 8, Article code 084302 Oscillation and Migration of Bubbles within Ultrasonic Field * Wen-Hua Wu (吴文华), Peng-Fei Yang (杨鹏飞), Wei Zhai (翟薇)**, Bing-Bo Wei (魏炳波) Affiliations Department of Applied Physics, Northwestern Polytechnical University, Xi'an 710072 Received 1 April 2019, online 22 July 2019 *Supported by National Natural Science Foundation of China under Grant Nos 51327901, 51727803 and 51571164, the NPU Excellent Personnel Supporting Project, the Fundamental Research Fund of Northwestern Polytechnical University under No 3102018jcc039, and the Key Research Plan in Shanxi Province (2018GY-104).
**Corresponding author. Email: zhaiwei322@nwpu.edu.cn Citation Text: Wu W H, Yang P F, Di W and Wei B B 2019 Chin. Phys. Lett. 36 084302    Abstract The oscillation and migration of bubbles within an intensive ultrasonic field are important issues concerning acoustic cavitation in liquids. We establish a selection map of bubble oscillation mode related to initial bubble radius and driving sound pressure under 20 kHz ultrasound and analyze the individual-bubble migration induced by the combined effects of pressure gradient and acoustic streaming. Our results indicate that the pressure threshold of stable and transient cavitation decreases with the increasing initial bubble radius. At the pressure antinode, the Bjerknes force dominates the bubble migration, resulting in the large bubbles gathering toward antinode center, whereas small bubbles escape from antinode. By contrast, at the pressure node, the bubble migration is primarily controlled by acoustic streaming, which effectively weakens the bubble adhesion on the container walls, thereby enhancing the cavitation effect in the whole liquid. DOI:10.1088/0256-307X/36/8/084302 PACS:43.25.+y, 47.55.dd, 43.80.+p © 2019 Chinese Physics Society Article Text When an ultrasonic wave travels through liquids, the bubble dynamics including oscillation and migration is of great importance. This has aroused enormous attention in the fields of acoustics and fluid dynamics.[1–3] The former is related to the generation of ultrasonic cavitation, while the latter partially determines where the bubble cavitation could take place. When the driving pressure is lower than the cavitation threshold, the bubble only presents linear oscillation.[4] With the increase of driving pressure, the bubble undergoes stable cavitation and transient cavitation successively, which depends on whether bubble collapses in liquid.[5] All of these oscillation modes show important applications in different fields.[6–7] For sonochemical fabrication, transient cavitation can be used to produce instantaneous high pressure and temperature to synthesis materials with superior performance.[8–9] However, in the process of ultrasound-assisted drug transportation, most bubbles should keep linear oscillation or stable cavitation for delivering drug steadily.[10] To date, there has been no detailed report on how to predict the bubble oscillation mode under different conditions, especially in the cases of different driving sound pressures and initial bubble radii. For bubble migration in liquid, a vast number of works have been carried out by numerical simulation and experiment.[11–12] For example, Louisnard simulated the structure of cavitation bubble cloud induced by ultrasound in water based on Caflisch equation,[13] and Servantet studied the mechanism of Bjerknes (Bj) force on bubble cloud.[14] However, in most studies, bubbles are treated as an integrity, which cannot visually present the migration characteristics of individual bubbles in ultrasonic field. More importantly, acoustic streaming, which behaves as a kind of integral flow and inevitably affects the movement of bubbles,[15] is always ignored in previous investigations, which is not consistent with the actual situation. To accurately characterize the dynamics of individual bubble within intensive ultrasonic field, it is necessary to comprehensively study the mechanism of acoustic pressure and acoustic streaming on the oscillation and migration of individual bubbles. Hence, in this Letter, the oscillation characteristics of individual bubbles with different initial radii driven by increasing sound pressure are studied, based on which a selection map of bubble oscillation mode within 20 kHz ultrasonic field is established. Meanwhile, the bubble distribution profile is numerically and experimentally investigated by tracking the migration of individual bubbles, and the respective contribution of sound pressure gradient and acoustic streaming is revealed.
cpl-36-8-084302-fig1.png
Fig. 1. The schematic diagram of experimental setup (a) and modeling of numerical simulation (b).
In numerical simulation, a 2D model was established, which is symmetric along the centerline ${\rm O}_{1}$O$_{2}$, as shown in Fig. 1(b). The 20 kHz ultrasound is introduced into a rectangle in size of $200 \times 150$ mm, where 2612 bubbles with an initial radius of $R_{0}=10\,µ$m are homogeneously distributed, and their migration processes in 1.5 ms (i.e. 30 acoustic wave periods) are calculated. Since the bubble radius $R$ is much smaller than the ultrasonic wavelength, the bubble shape is assumed to keep sphere during the oscillation. Due to the low volume fraction of bubbles in liquid, the individual bubble motion is unaffected by other bubbles. The ultrasonic horn with a radius of 10 mm is inserted into water by a depth of 15 mm, and its maximum displacement is 5 µm. Ultrasonic wave propagating in water is described by the nonlinear Helmholtz equation,[16] and the incompressible flow in water is solved by the standard $\kappa$–$\varepsilon$ model.[17] On the basis of Nyberg's streaming theory,[18] the bulk force ${\boldsymbol F} $ inducing the generation of acoustic streaming by a plane wave approximation is written as[19] $$\begin{align} F=\frac{2\alpha }{\rho c^{2}}\vert p\vert^{2}.~~ \tag {1} \end{align} $$ Here $\alpha$ is the sound absorption coefficient, and $p$ is the sound pressure; $\rho$ and $c$ represent water density and sound speed in water, respectively. The model details of the sound pressure and acoustic streaming are given in our previous study[20] and the work by Xu et al.[21] Furthermore, the individual bubble oscillation is described by the Keller-Mikisis equation[22] $$\begin{align} &\rho_{0} \Big[\Big(1-\frac{\dot{R}}{c_{0} }\Big)R\ddot{R} +\frac{3}{2}\dot{R}^{2}\Big(1-\frac{\dot{R}}{3c_{0} }\Big)\Big]\\ =\,&\Big(1+\frac{\dot{R}}{c_{0} }+\frac{R}{c_{0} }\frac{d}{dt}\Big)\Big(p_{\rm g} +p_{\rm v} -\frac{2\sigma }{R}-\frac{4\mu \dot{R}}{R}\\ &-p_{0} +p_{\rm a} \sin(2\pi ft)\Big),~~ \tag {2} \end{align} $$ where $\sigma$ and $f$ are the water surface tension and ultrasonic frequency, respectively, $p_{\rm v}$ is the saturated vapor pressure of water, $p_{\rm a}$ is the driving pressure amplitude, and $p_{\rm g}$ and $p_{0}$ are the pressure inside the bubble and static pressure, respectively.
Table 1. Physical parameters used in the calculation.
Parameter Unit Value
Density of water $\rho$ kg/m$^{3}$ $1.0\times 10^{3}$
Density of bubble $\rho_{\rm b}$ kg/m$^{3}$ 1.29
Sound velocity in water $c$ m/s $1.5\times 10^{3}$
Sound absorption coefficient $\alpha$ m$^{-1}$ 25
Water viscosity $\mu$ Pa$\cdot$s $1.8\times 10^{-3}$
Ultrasonic frequency $f$ Hz $2.0\times 10^{4}$
Vibration amplitude of horn $A_{0}$ m $5\times 10^{-6}$
Surface tension of water $\sigma$ N/m 0.0725
The Euler-Lagrange method is employed to track the migration of each individual bubble:[23] $$\begin{align} m_{\rm b} \frac{d\boldsymbol{v}_{\rm b}}{dt}=\boldsymbol{F}_{\rm B1} +\boldsymbol{F}_{\rm D} +\boldsymbol{F}_{\rm AM} +\boldsymbol{F}_{\rm V} +\boldsymbol{F}_{\rm G},~~ \tag {3} \end{align} $$ where $m_{\rm b}$ and ${\boldsymbol v}_{\rm b}$ are the mass and velocity of bubbles, respectively. ${\boldsymbol F}_{\rm B1}=-\langle V(t)\cdot\nabla p(t)\rangle$ is the primary Bjerknes force, which is determined by the bubble volume $V(t)$ and pressure gradient $\nabla p(t)$. In addition, ${\boldsymbol F}_{\rm D}=-m_{\rm b}({\boldsymbol v}_{\rm b}-{\boldsymbol v}_{\rm f})/\tau_{\rm b}$ is the drag force, where ${\boldsymbol v}_{\rm f}$ is the liquid velocity, and $\tau_{\rm b}$ is the relaxation time of bubbles, respectively. ${\boldsymbol F}_{\rm AM}=m_{\rm b} \rho (D{\boldsymbol v}_{\rm f}/Dt-d{\boldsymbol v}_{\rm b}/dt)/(2\rho_{\rm b})$ is the added mass force, where $\rho_{\rm b}$ is the bubble density. ${\boldsymbol F}_{\rm V}=\rho (dm_{\rm b}/dt)({\boldsymbol v}_{\rm f}-{\boldsymbol v}_{\rm b})/(2\rho_{\rm b})$ stands for the volume difference force, and ${\boldsymbol F}_{\rm G}=(1-\rho /\rho_{\rm b})m_{\rm b}{\boldsymbol g}$ is the gravitational force, which is much weaker than the other four forces. The walls AB, CD, EF, GH and AH are set as the absorption boundary, i.e. ${\boldsymbol v}_{\rm b}=0$. On the water surfaces BC and FG, bubbles are assumed to be disappeared, so the bubble number is always 0. The moving wall DE induced by ultrasonic transducer is set as resilient boundary, i.e. ${\boldsymbol v}_{\rm b}={\boldsymbol v}_{\rm c}-2({\boldsymbol n}\cdot{\boldsymbol v}_{\rm c})$, where ${\boldsymbol v}_{\rm c}$ is the velocity of the collision with the wall, and ${\boldsymbol n}$ is the unit normal vector. The physical parameters used in the calculation are listed in Table 1. A schematic diagram of the experimental setup is shown in Fig. 1(a). The size of the non-covered cubic glass container is $300\times 200\times 200$ mm, and the depth of the deionized water in the container is 150 mm. The initial pressure within water is 1 atm because of an open boundary on the container top, and the experimental temperature is 278 K. The experimental parameters, including the ultrasonic horn radius, vibration amplitude and immersion depth etc., are the same as those in the calculation. In our experiment, ultrasound at 20 kHz was generated by an ultrasonic transducer (SONICS VCX 1600). The container was illuminated by an LED light with a power of 5 W, and a high-speed camera (MigrationXtra HG-100k) with an image acquisition rate of 10000 frames per second (fps) was applied to track the migration of bubbles in water.
cpl-36-8-084302-fig2.png
Fig. 2. The oscillation characteristic of a single bubble: (a) $R_0=5 \,µ$m, (b) $p_{\rm a}=1.2\times 10^{5}$ Pa, (c) the distribution of oscillation mode.
The dependence of bubble oscillation on sound pressure is calculated for the bubbles with an initial radius of 5 µm. In Fig. 2(a$_{1}$), the bubble driven by low sound pressure of $1.2\times 10^{4}$ Pa behaves as a linearly harmonic oscillation, where higher order harmonic oscillation only appears at the initial stage of bubble oscillation. With the increase of driving sound pressure up to $1.2\times 10^{5}$ Pa, bubble oscillation exhibits distinct nonlinearity. In Fig. 2(a$_{2}$), the higher order harmonic oscillation increases sharply, and occupies the whole half period of positive pressure, even plus a small part of the negative pressure period. Afterwards, the bubble radius decays exponentially until it keeps stable, which is stable cavitation. If the driving pressure is further raised to $1.2\times 10^{6}$ Pa, then the bubble expands continuously to several hundred times of its original size, and then shrinks sharply (Fig. 2(a$_{3}))$. In this process, the bubble cannot keep stable oscillation and may collapse at any time (shown as the dotted line), namely transient cavitation.
cpl-36-8-084302-fig3.png
Fig. 3. Calculated and experimental results in 20 kHz ultrasound: (a) acoustic field (left) and flow field (right), (b) distribution of bubbles only driven by Bjerknes force at 1.5 ms, (c) distribution of bubbles by five forces at 1.5 ms, (d) the relationship between local pressure and bubble velocity at bubbles ${\rm B}_{1}$–${\rm B}_{6}$, (e) the comparison between bubble migration velocity $v_{\rm B}$ and local acoustic streaming $v_{\rm F}$ at the pressure nodes, (f) experimental and simulated bubble distribution underneath the ultrasonic horn.
Furthermore, the oscillation characteristics of bubbles with different radii ($R_{0}=0.5$, 5 and 50 µm) under the driving pressure of $1.2\times 10^{5}$ Pa are calculated by the Keller–Mikisis equation, as shown in Figs. 2(b$_{1}$)–(b$_{3}$). When the initial radius is 0.5 µm, the bubble eigen frequency is $4.1\times 10^{4}$ kHz. Note that the eigen frequency of bubble oscillation, $\omega^{2}=3\gamma p_{0}/(\rho_{0}R_{0}^{2})$, is obtained by applying the low amplitude approximation for Eq. (2). This is far higher than the ultrasonic frequency of 20 kHz, and there is low sound energy transferring to the bubble, leading to a linear oscillation, as shown in Fig. 2(b$_{1}$). In Fig. 2(b$_{2}$), the bubble with an initial radius of 5 µm can effectively absorb the sound energy, resulting in the formation of stable cavitation. When the initial radius increases to 50 µm in Fig. 2(b$_{3}$), the bubble eigen frequency is lower than the ultrasonic frequency, and the nonlinear effect is further strengthened. As a result, the amplitude of bubble radius becomes irregular; i.e., transient cavitation. The oscillation processes of mass of bubbles are calculated, and the intervals of bubble initial radius from 0.6 to 23.7 µm and driving sound pressure from $8.6\times 10^{4}$ to $6.01\times 10^{5}$ Pa are 0.1 µm and 1000 Pa, respectively. On the basis of calculated results, the selection map of oscillation mode in 20 kHz ultrasound with relevance to driving sound pressure and initial bubble radius is summarized in Fig. 2(c), in which the blue and red lines represent the sound pressure threshold of stable cavitation and transient cavitation, respectively. As the initial bubble radius increases, the pressure threshold of stable cavitation and transient cavitation is gradually reduced. If the initial radius is less than 0.8 µm or larger than 23 µm, the two pressure threshold curves almost coincide, which indicates that the stable cavitation is almost impossible to occur. One can predict the cavitation characteristics of bubbles with different initial radii and driving sound pressures from Fig. 2(c). To further investigate the migration of bubbles in liquid under 20 kHz ultrasound, the sound pressure distribution and flow field in water are calculated, and the results are shown in Fig. 3(a). The largest sound pressure is $5.03\times 10^{5}$ Pa, which is located very close to that at the bottom of ultrasonic horn (zone I). Additionally, there are other three pressure antinodes (zones II, III and IV) along the container centerline, as well as three pressure antinodes (zones V, VI and VII) distributed symmetrically on both sides of the container. Meanwhile, the flow is emitted from the ultrasonic horn center, which first increases and then decreases along the center axis. The average flow velocity is 0.15 m/s, and the maximum flow rate reaches 0.57 m/s. For quantitatively characterizing the bubble migration in different regions, nine bubbles marked as B$_1$–B$_9$ are extracted from the model, among which B$_1$–B$_6$ are initially located at pressure antinodes, and B$_7$–B$_9$ are from pressure nodes. For the sake of evaluating the effect of sound pressure, the bubble migration only driven by Bjerknes force is primarily calculated; i.e., there is only the term of Bjerknes force on the right side of Eq. (3). The calculated result is presented in Fig. 3(b), where every blue circle on the left represents a bubble, and the red arrow on the right depicts the migration direction of the corresponding bubble. It is obviously found that the effect of Bjerkens force makes bubbles with radii larger than 15 µm gather toward the center of pressure antinodes (zones I–IV) and with small bubbles (close 10 µm) tend to move to the pressure nodes. It should be also noted that there are many tiny bubbles sticking at the container walls. In contrast, when acoustic streaming is taken into account, the bubbles are driven by all the five forces described in Eq. (3), and the migration trail is plotted in Fig. 3(c). In zones I–VII, the large bubbles in the radius range of 17–32 µm move to the pressure antinodes, but small bubbles in the radius range of 7–10 µm escape from the pressure antinodes. This is in accordance with the former case, and indicates that Bjerknes force still dominates the bubble migration at the pressure antinode. To reveal the relationship among the velocity of bubbles and local pressure at the pressure antinodes, the migration characteristics of bubbles B$_1$–B$_6$ randomly chosen from zones II–VII are shown in Fig. 3(d). The highest velocities of bubbles occur in zones III and IV, where the local pressure above $4.2\times 10^{5}$ Pa is far higher than that in other regions. The velocities of bubbles B$_1$ and B$_2$ behaving transient cavitation are above 18 m/s, which is dozens of times higher than the maximum velocity of acoustic streaming. Even in zone VII with the lowest pressure, the migration velocity of bubble B$_4$ reaches 0.42 m/s, which is also higher than local flow rate. Hence, it can be summarized that the migration velocity of local bubble in high pressure region is relatively high. For the migration characteristic of bubbles at the pressure nodes, by comparing the bubbles trails (Fig. 3(c)) and the direction of acoustic streaming (Fig. 3(a)), the bubbles migration direction is consistent with that of acoustic streaming. The migration velocities of bubbles B$_7$–B$_9$ at the pressure nodes and local acoustic streaming are shown in Fig. 3(e), where the solid and dotted lines, respectively, represent the velocities of bubble migration and local acoustic streaming. During the whole calculation, the velocity magnitude of the three bubbles is almost equal to that of local acoustic streaming, which indicates that the bubble migration is primarily controlled by acoustic streaming at the pressure node. Moreover, there are fewer bubbles absorbing on the container walls in Fig. 3(c) than that in Fig. 3(b). Hence, acoustic streaming effectively inhibits the bubble adhesion on the wall, which leads to most of the bubbles entering the container inner. Figure 3(f) shows a comparison of the distribution of bubbles in zone I between the numerical simulation and the experimental observation. Their common characteristic is that lots of bubbles with relatively large radii gather underneath the ultrasonic horn. This is due to the existence of pressure antinode in this region, which is demonstrated in the numerical simulation result in Fig. 3(a), and the similar results have been drawn in other studies.[24] Moreover, bubble density decreases definitely away from the ultrasonic horn, leading to the formation of a conical bubble distribution, which is also in agreement with the experimental result. Hence, these results confirm the validity and accuracy of our simulation. In summary, the oscillation and migration of bubbles in water within 20 kHz ultrasonic field are investigated in detail. The selection diagram of oscillation mode with relevance to initial bubble radius and driving sound pressure is established, which indicates that the pressure threshold of stable and transient cavitation decreases with the increase of bubble radius in 20 kHz ultrasound. Meanwhile, the respective contribution of sound pressure gradient and acoustic streaming to bubble migration is revealed. At the pressure antinode, migration of bubbles is primarily controlled by Bjerknes force, leading to large bubbles gathering toward the center of pressure antinode and small bubbles escaping from the pressure antinode. By contrast, the bubble migration at the pressure node is dominated by acoustic streaming, which effectively weakens the bubble adhesion on container walls, thereby enhancing the cavitation effect in the whole liquid. 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