PS17
音響場で浮遊する泡沫の浮遊安定性
Stability of levitated foam in acoustic field
大西一希 1, 長谷川浩司 2
Kazuki ONISHI1, Koji HASEGAWA212工学院大学大学院，Graduate School of Kogakuin University #1
工学院大学，Kougakuin University #2

1. Introduction
In recent years, non-contact fluid manipulation technologies without using containers have been attracting
great attention in various fields, including materials science, analytical chemistry, biology and medicine, andcombustion science1,2). One of the promising technology used in this study is the acoustic levitation method.
The acoustic levitation method utilizes ultrasound to levitate samples regardless the properties of the levitatedsample3). However, due to the principle of ultrasound-induced levitation, the levitated samples exhibit
nonlinear and complex behaviors, such as interfacial instability (interfacial deformation and atomization) andinternal/external acoustic streaming4). In particular, the interfacial behavior plays a vital role in the levitation
stability of the samples. While theoretical and numerical insights have been developed for linear regions inacoustics, to gain the insightful experimental understanding of the nonlinear and dynamic phenomena is thekey to succeed the stable sample levitation. Although many researchers investigated on the levitated droplet
and bubble in acoustic levitation, to the best of our knowledge, the levitated foam hasn’t been studied yet.
Therefore, this study aims to identify the interfacial stability of foams (sodium dodecyl sulfate (SDS) aqueoussolution) levitated in an acoustic field.
2. Experimental setup
Figure 1 shows a schematic diagram of the experimental apparatus used in this study. The experimental
procedure involves generating a sine wave signal with a function generator, which is then amplified by anamplifier and fed into an ultrasonic transducer. The sound waves emitted from the tip of the lower horn,connected to the ultrasonic transducer, are reflected by a reflector positioned at a predetermined distance. This
reflection forms an acoustic standing wave between the horn and the reflector. The foam was introduced near
the pressure nodes of the acoustic standing wave using a syringe for the levitation. The dynamic motion of thefoam was captured using a high-speed camera with a backlight method, and the resulting images wereprocessed on a computer.
Furthermore, the pressure gradient and changes in the pressure distribution caused by an increase inacoustic pressure within the test section of the experimental apparatus have a significant impact on thedynamic behavior of the levitated sample. Therefore, measuring the acoustic pressure and understanding its
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distribution within the test section are considered critical parameters when analyzing the dynamic behaviorof levitated droplets. To achieve this, a probe microphone was mounted on a traverse device and moved in 1mm increments. The acoustic pressure at each measurement point was recorded based on the waveform
displayed on the oscilloscope. Since the oscilloscope displays voltage waveforms, it is necessary to convert thevoltage values to acoustic pressure. The oscilloscope was calibrated so that a waveform of 1.0 mV corresponds
to 1.0 Pa. Additionally, the measurement region assumed axial symmetry of the acoustic standing wave, withthe horn center set as the origin (0), and measurements were conducted only on the right half of the horn.

The acoustic radiation force of the acoustic field enables the sample to be levitated by counteracting thegravitational force exerted on the sample. The levitation characteristics vary significantly depending on thecompetition between the acoustic field and the properties of the levitated fluid, such as surface tension. For
insufficient acoustic pressure, the sample cannot be levitated, while excessive acoustic pressure may lead toatomization due to the inability to maintain the interfacial shape.
Table 1 lists the experimental conditions used in this study. A reflector with a curvature radius of 36 mm(R36) was used to enhance the levitation stability of the sample in acoustic field. The frequency of input soundwave was approximately 19.3 kHz, resulting in a wavelength of the sound wave λ of approximately 17.9 mm.
The applied voltage was 750 mV. The levitated sample used was foam (sodium dodecyl sulfate (SDS) aqueoussolution). A 0.5% aqueous solution was prepared by mixing SDS with purified water, and air was injected into
the solution using a commercial hand soap dispenser.
In the acoustic levitation method, droplets levitated in the sound field become flattened and take on an
elliptical shape due to the relationship between acoustic radiation pressure and gravity. Therefore, in thisstudy, the equivalent diameter used for the foam is based on the volume-equivalent diameter, assuming theshape of a spheroid. The volume-equivalent diameter can be calculated using the following equation.
3𝑑 = √𝑎𝑏 2(1)Figure 1. Schematic diagram of the experimental apparatus: 1.function generator, 2. power amplifier, 3.
power meter, 4. acoustic levitator, 5. LED light, 6. high-speed video camera, and 7. PC
2 of 6Table 1. Experimental conditions
Reflector

Input

Input

Wave length

Temperature

Humidity

Test

Equivalent

type

frequency fvoltageof input

[℃]

[%]

samplediameter d

[kHz]

[V]

frequency λ[mm][mm]Foam,SDS

R36,
19.3

0.2

17.9

25±2

50±10

aqueous4.3-7.2

R∞
solution

3. Experimental result and discussion
3.1 Levitation stability of foam
Figure 2 shows the levitation range of the foam. In the figure, the horizontal axis represents the dimensionlessdiameter d, where d is the volume-equivalent diameter of the foam and λ is the wavelength of the acoustic
field. The vertical axis represents the sound pressure P in the test section. The blue line and orange triangle
plots represent for the stable foam levitation and unstable fell down due to insufficient applied acousticpressure. To determine the stable levitation conditions for the levitated samples, the acoustic pressure aroundthe foam was gradually reduced from a constant pressure when the foam was stably levitated until the foam
was no longer levitated and fell. The dimensionless diameter of the levitated foam was found to be within therange 0.24 < 𝑑 ⁄𝜆 < 0.4. Regarding the lower limit of levitation sound pressure, it was observed that foam often
fell around 0.6 kPa, and the minimum levitation threshold pressure being 0.34 kPa.
To compare with the dimensionless diameter of water, the dimensionless diameter will be calculated basedon the levitation conditions of water.

Water, with its higher density compared to foam, is considered a

representative levitation sample in acoustic levitation methods. The levitation conditions for water include theminimum levitation sound pressure Pmin5) as described by King and the maximum levitation sound pressurePmax6) as described by Danilov et al., as shown below.
𝑃min = √1.6𝜌1 𝜌0 𝑔𝑐 2𝑃max = √(2)

𝑘6.8𝜎𝜌0 𝑐 2𝑑(3)The diameter at which the minimum and maximum levitation sound pressures coincide, as derived fromthese theoretical formulas, represents the theoretical maximum levitation diameter. Upon calculation, themaximum diameter was found to be d=5.5 mm, and the maximum dimensionless diameter was 𝑑 ⁄𝜆=0.31.
Since a typical water droplet has a dimensionless diameter greater than 0.31, it can be inferred that the diameterof the levitated foam is approximately 1.3 times larger compared to that of the water droplet.

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1.2
1Upper limit for water droplet

P [kPa]

0.8
0.6
0.4

stable levitation

0.2

falling down

00.2

0.25

0.3

0.35

0.4

0.45

d/λ [-]
Figure 2. Levitation characteristics for foam in acoustic fields.

3.2 Foam collapse
Figure 3(a) shows the collapse process of foam levitated in an acoustic field. Snapshots was captured fromabove in Fig. 3(a). The start time of levitation was set to 0 seconds. At 100 seconds, it was observed that thebubbles in the foam had expanded. By 150 seconds, the expanded bubbles began to concentrate radially within
the foam. Following this, the foam rapidly decreased in volume due to bubble rupture, and this change became
more gradual by 250 seconds. At 270 seconds, expanded bubbles similar to those seen at 100 seconds wereobserved again. By 290 seconds, a rapid decrease in volume was noted, followed by the foam eventually falling.
Foam levitated in an acoustic field undergoes evaporation and drainage over time, causing the liquid film tothin.7) This leads to the coalescence of bubbles, resulting in an increase in bubble size. As evaporation anddrainage continue, the enlarged bubbles eventually lose their pressure equilibrium in the acoustic field andburst. It is believed that the secondary droplets generated during the burst impact other bubbles, causing achain reaction of consecutive collapse. Fig. 3(b) shows the spatiotemporal diagram of the foam collapse

process. The spatiotemporal diagram was reconstructed by extracting the slice image from a specified
row in each captured frame and aligning them over time to form a single image. This diagram allows us
to observe the interfacial deformation before collapse．Additionally, as time progresses, enlarged bubbles
become visible as dark regions at the edges of the bubble radius, where the background light istransmitted through the bubbles.

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(a) Snapshots of the levitated foam(b) Time-space diagram

Figure 3. Collapse process of levitated foam in acoustic levitation
The drainage process during the foam collapse can be observed through the reduction in foam volume
captured in the recorded images.
Figure 4 shows the results of the temporal change in foam area. The horizontal axis represents the time withthe start of foam levitation set as 0 seconds, and the vertical axis represents the dimensionless area obtainedby dividing the area S from the images by the initial area S0. The comparison was made with samples having
initial diameters of d=12.08, d =8.38 [mm]. A gradual decrease is observed from 0 to around 100 seconds. Arapid decrease in area begins around 130 seconds. After approximately 230 seconds, the change becomes
gradual again, with another rapid decrease observed around 270 seconds. The cause of the rapid decrease inarea is attributed to the continuous collapse of bubbles. The bubbles that make up the foam vary in volume,
resulting in differences in internal pressure. This pressure difference causes the liquid films to merge. The
merged bubbles eventually burst due to external pressure, and the secondary droplets ejected during thisrupture can penetrate other bubbles, causing a chain reaction of continuous collapse.
1.21

d=8.4 [mm]S/S0 [-]

0.8
0.60.4d=12.1 [mm]

0.2
0050

100

150200Time [s]250300350

Figure 4. The evolution of foam area
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4. Conclusion
In this study, we aimed to understand the interfacial instability phenomena of levitated foam and achieve
high-precision stable control. We investigated the levitation conditions of the foam, visualized the collapse
process, and measured the area changes during collapse.
The levitation conditions for the foam showed that it remained suspended within the dimensionlessdiameter range of 0.24< 𝑑 ⁄𝜆 < 0.4. The visualization of the foam collapse process revealed that, over time,expanded bubbles appeared in parts of the foam. There was a rapid decrease in area at a certain point, andthis process repeated several times. The temporal change in the area of the levitated foam indicated a gradualdecrease up to around 100 seconds, followed by a rapid decrease starting at approximately 130 seconds. Thechange became gradual again after 230 seconds, with another rapid decrease observed around 270 seconds.
The cause of the rapid area decrease is attributed to the continuous collapse of bubbles.

References1)

4)

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6)7)Danilov, S. D. and Mironov, M. A.: Breakup of a droplet in a high‐intensity sound field, The Journal of the AcousticalSociety of America, Vol. 92, No. 5 (1992) pp. 2747-2755.
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