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Combined experimental and theoretical investigation of the gas bubble motion in an acoustic field
Accepted Manuscript Combined Experimental and Theoretical Investigation of the Gas Bubble Motion in an Acoustic Field Xiaojian Ma, Tianyu Xing, Qiuhe Li, Yifei Yang PII: DOI: Reference:
S1350-4177(17)30339-5 http://dx.doi.org/10.1016/j.ultsonch.2017.07.035 ULTSON 3790
To appear in:
Ultrasonics Sonochemistry
Received Date: Revised Date: Accepted Date:
16 February 2017 15 July 2017 24 July 2017
Please cite this article as: X. Ma, T. Xing, Q. Li, Y. Yang, Combined Experimental and Theoretical Investigation of the Gas Bubble Motion in an Acoustic Field, Ultrasonics Sonochemistry (2017), doi: http://dx.doi.org/10.1016/ j.ultsonch.2017.07.035
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Combined Experimental and Theoretical Investigation of the Gas Bubble Motion in an Acoustic Field Xiaojian Ma*1, 2, Tianyu Xing3, Qiuhe Li4, Yifei Yang5 1. School of Mechanical and Vehicular Engineering, Beijing Institute of Technology, Beijing 100081, China. 2. Department of Techniques Research and Development, Beijing Ferly Water Treatment Equipment Company Limited, Beijing 100022, China 3. School of Information Science & Technology, Beijing Forestry University, Beijing 100083, China. 4. Department of Mechanical and Electronic Engineering, Yingkou Institute of Technology, Liaoning 115014, China. 5. School of Journalism & Communication, Hebei Normal University, Hebei 050000, China.
Abstract The objective of this paper is to apply the combined experimental and theoretical method to investigate the various behaviors of gas bubbles in an acoustic field. In the experiments, high-speed video and ultrasonic processor are used to capture the transient evolution of gas bubble patterns, as well as velocity profiles. In the theoretical analysis, the theories of primary and secondary Bjerknes forces and buoyancy force are introduced to accurately demonstrate the variations of bubble volume and motion. Results are presented for gas bubbles with the radius of 1.4 mm under an acoustic field with a frequency of 18 kHz, for three cases, namely single bubble rising in a quiescent liquid, acoustic single bubble oscillation and two bubbles coalescence conditions. The results show that the fragments around the single gas
* Corresponding author: Xiaojian Ma E-mail: [email protected] Fax/Tel: + 86-010-68912395
bubble presents the periodical behaviors, namely, splitting, attraction, and secondary splitting motion. The centroid of the single gas bubble almost oscillates without motion upwards or downwards, because of the equilibrium of the primary Bjerknes force caused by acoustic waves and the effect of the buoyancy force. For the two coalescing bubbles, the resultant of buoyancy, primary and secondary Bjerknes forces acting on two bubbles are same in magnitude, but in opposite direction, which indicates that two gas bubbles attract each other and and coalesce into one. Keywords: gas bubble behaviors, acoustic fields, Bjerknes force, buoyancy force. Nomenclature Roman letters a A Bo
radius of the circular piston area of gas bubble on the images Bond number
R Re R0
cl
sound velocity in the liquid
R
instantaneous radius of a bubble Reynolds number initial radius of a bubble first time derivative of R
D
distance between the concerned two R bubbles diameter of the gas bubble t
er
unit vector
T
period of acoustic wave
f
frequency of acoustic field
V
instantaneous volume of a bubble
g
gravitational acceleration vector
V
second time derivative of V
Fb
buoyancy force
x
location vector of bubble
FB
primary or secondary Bjerknes force
y
vertical coordination
FB1
first Bjerknes force
Greek letters
secondary Bjerknes force acting on bubble 2 and caused by bubble 1 wave number vector of the acoustic wave mode number instantaneous acoustic pressure pressure amplitude of an acoustic wave
κ
polytrophic exponent
λ μ ρ
wavelength of the acoustic wave in the liquid viscosity of the fluid density of the liquid
σ
surface tension coefficient
d
F12
k n ps pa
second time derivative of R time
p
ambient pressure
ω
r
angular coordination
ωn
angular frequency of an acoustic wave natural frequency of bubble
1. Introduction Bubble dynamics in an acoustic field due to acoustic radiation force has been typical one of concerned multiphase fluidics. During the oscillation in an acoustic wave, gas bubbles present various behaviors, including high-speed jet [1], hot pot [2], high pressure [3] and sonoluminescence [4], etc. Those characteristics of oscillating gas bubbles have played a significantly important role in distinct industrial processes, such as environmental protection [5], water treatment [6], extraction [7] and chemical reaction [8]. A fundamental understanding of the acoustic gas bubble physics is particularly indispensable in many practical technologies for future works, ranging from the active control of medical ultrasound contrast agent to surface cleaning of precise instruments [9,10]. However, a comprehensive knowledge of dynamic behaviors of gas bubbles under the progressive acoustic wave fields is still limited, though extensive experimental investigation, theoretical solution and numerical simulation have been performed. Early experimental studies on bubble dynamics were reported in the works of Leighton [11], Brujan [12], Chen [13], and Ashokkumar [14]. They research on topic of the bubble dynamics included different boundary conditions, from a quiescent
liquid to an acoustic field and from infinite liquid to boundary wall. Results demonstrated that single bubble in a quiescent liquid presents oblate shape corresponding to the function of the buoyancy force [15]. When an acoustic wave is imposed on a liquid, single bubble is possible to trap in the acoustic field with various oscillations. Kim et al. [16] used the high-speed camera system to investigate the single bubble dynamics in an ultrasonic field and its destructive actions. They founded that single bubble behaviors have four different oscillations, namely volume, shape, splitting and chaotic oscillations, which depends on bubble size and acoustic pressure amplitude. Essentially, those different oscillations are generated due to the primary Bjerknes force, which cause by the pressure difference along the surface of bubble [17]. As for the coalescence of the bubbles, many researchers tended to create two bubbles with small distance in space, in order to investigate the interaction between the two bubbles. They founded that bubbles are attracted by the each other and migrate along the center line between two bubbles. Mettin et al. [18] demonstrated that bubbles with close distance tend to attract each other due to secondary Bjerknes force, which is caused by the oscillation of the bubble membranes. As a result, bubbles in an acoustic field may be acted by three kinds of forces, namely, the buoyancy force, primary and secondary Bjerknes forces. It worth noting that there exit significant differences between gas bubble and cavitation one. As known, the cavitation bubble is generated when the local pressure drops below the saturated one and the pressure of inner cavitation bubble is with saturated levels [19]. But the gas bubble is generated by the gathering of the gas, whose pressure depends on the environmental pressure. Furthermore, cavitation bubble is generally small in volume, and its buoyancy force can usually be neglected. As for the gas bubble, the buoyancy force is an importantly external force, which causes the deformation and motion of the bubbles. Due to the various mechanism of formation of bubbles, the surface tension coefficient and viscosity of the liquid may be both different [20]. For more details of the information about bubbles, readers are referred to systematic work by Hung el at. [21]. In present work, we concentrate our
attention on the transient behaviors of the gas bubbles, which are subjected to the acoustic travelling waves. To further investigate the mechanism of gas bubble dynamics in an acoustic field, theoretical solution and analysis has been also conducted to obtain the additional parameters. Due to highly nonlinearity of the compressible acoustic oscillation, it is difficult to capture the bubble motion and deformation [22]. Correct analysis of the bubble motion in an acoustic field depends upon the ability of theoretical methods. Over the years, the basic idea to overcome the problems is solving the bubble motion equation and the equations of primary and secondary Bjerknes force simultaneously. Many studies are performed to investigate the various forms of the equation to satisfy the different conditions [23,24]. Yasui et al. [25] used experimental and theoretical methods to investigate the direction of bubble clouds motion under an ultrasonic horn. In the theoretical methods, the interaction of the bubbles is considered and introduced. They founded the bubble clouds may migrate upwards the horn tip, which is strongly caused by the acoustic fields. But the interference comes from many other bubbles is impossible to eliminate, because the ultrasonic horn generates massive small bubbles, but not the one or two concerned bubble clouds, though this is considered in the theoretical mode of bubble motion equation. Although the researches on the dynamics bubbles have received much attention in the past years, the mechanisms of gas bubbles in an acoustic field are still not well understood, and hence additionally experimental and theoretical researches are still needed. The objectives of the present paper are (1) to observe and record three different gas bubble behaviors, namely rising, oscillation and coalescence; (2) to introduce and invalidate the theoretical mode to capture the oscillation of gas bubbles in an acoustic field; (3) to analyze the factors causing different behaviors of gas bubbles in an acoustic field, such as the buoyancy force, the primary and secondary Bjerknes force.
2. Experimental setup In present work, the experimental system mainly includes three parts, involving acoustic wave generator, gas bubble creator and high-speed video camera. The more information about each part, such as working function and parameter, is introduced in details as following. 2.1 The generation of acoustic waves The needed acoustic waves are generated by an acoustic processor (Institute of Acoustics, Chinese Academy of Science, China), which is with a constant frequency of 18 kHz. It notes that the electrical input power of the acoustic processor can be chosen in the range from 50 W to 250 W, which can be measured by a precise electrical power meter. The acoustic horn has a plane radiating surface, which is measured with 20 mm in diameter and oscillates with a simple harmonic motion. Fig.1(a) shows the schematic description of experimental setup for investigating the single bubble behaviors in acoustic waves. As observed, the tip of horn is merged into a water tank, which is made of transparent glass for photography and illumination. The water chamber is processed with a length of 320 mm, width of 240 mm, and height of 200 mm. The distance between the acoustic horn and the bottom wall of the water tank is about two times of the wave length in water in order to avoid the reflection of the acoustic waves. 2.2 The generation of gas bubbles Fig. 1(a) also shows that the bubble is generated by a motoring injection syringe with cylindrical shape, whose inner radius is about 1mm. The injection syringe, which is full of gas, can be driven by an electric motor, and its screw pitch is about 0.1 mm. By this application, the size and volume of the gas bubble can be precisely controlled with electric motor. A gas inlet is merged into the water tank and under the acoustic
horn. As shown in Fig. 1(b), the maximum distance between gas inlet and acoustic horn tip is about 8 mm. To create gas bubble at different position, the location of the gas inlet can be lifted. To increase the cavitation threshold, the water thank is the filled with sufficiently degassed water to the depth needed, and the temperature is about 25 ℃ [26]. Before the generation of a bubble, a ruler is placed in the same vertical plane with the acoustic horn and perpendicular to the camera lens, to be recorded as calibration for bubble size in length. As a result, spatial measurements are directly illustrated on the images. 2.3 The high-speed videos The temporal evolution of the bubble dynamics is recorded with a high-speed camera (HG-LE, by Redlake) at framing rates as high as 100,000 frames per second (fps). A lower recording speed of 3000 fps is adopted to obtain the desired spatial resolution in present investigation, and the exposure time of each frame is set as 30 μs, in order to measure the sharpness of the bubble outline. The light source is provided by a dysprosium lamp, and its maximum power is 1000 kW. To form a better light distribution in the background, a piece of ground glass is placed between the camera and the dysprosium lamp. Asynchronous system is controlled by a computer to realize the generation of gas bubble and capture of the high-speed camera at same time.
3. Experimental results 3.1 Bubble rising in a quiescent liquid To compare with the gas bubble behaviors in an acoustic wave, Fig. 2 shows the temporal evolution of the gas bubble rising in a quiescent liquid without any acoustic waves. The data from transient motion of gas bubble in Fig. 2 is treated as a control group in this paper. The typical images are chosen to illustrate the essential and pronounced features of gas bubble dynamics in temporal and spatial scales, at a recording rate of 3000 frames per second (fps). In this section, the radius of the
bubble is generated about 1.4 mm. As observed, the bubble presents almost the spherical shape in the rising process, due to the large surface tension caused by the small size of the radius. In our previous works about the regimes depending on the Reynolds and Bond numbers, the gas bubbles only remain spherical shape when they are rising in a quiescent liquid with low Reynolds or Bond numbers (Re < 150 or Bo < 1) [27]. Further increasing slightly in Reynolds number (Re = 300), the shape of gas bubbles still stay spherical for the lower Bond number, where the Reynolds and Bond numbers are defined as
Re =
ρg1/2 D3/2 ρgD ; Bo = μ σ
(1)
where ρ is the density of the liquid; D the gas bubble diameter; g the gravitational acceleration; μ the viscosity coefficient; and σ the surface tension coefficient. In order to further measure quantitatively the transient motion of the gas bubble in the quiescent liquid, the rising velocity of the gas bubble is investigated in present work. It is notable that the bubble presents a rising at a constant velocity for about 1.6 m/s. 3.2 Single bubble oscillation in an acoustic field Fig. 3 shows the temporal evolution of bubble oscillation in the acoustic field with frequency f = 18 kHz with the acoustic pressure amplitude of 5.40 bar. The images are captured at a recording rate of 12500 fps. Before the generation of the gas bubble, the acoustic horn is initialized to make sure the gas bubble located at an acoustic field. And this gas bubble is named as letter A. In present section, the size of gas bubble is same with the gas bubble in the quiescent liquid (1.4 mm), as discussed in Fig. 2. As observed, there are many fragments of gas bubble with small volume around the gas bubble during the process of oscillation. Fragments present the periodical motion that they splits from the gas bubble A, then be attracted, finally splits, repeatedly. It is worth noting that the centroid of the gas bubble almost has no movement in the original position. According to the Lamb’s expression for surface mode vibration [28], ωn = σ (n -1)(n +1)(n + 2) / ( ρR02 ) , where ωn is the natural
frequency of bubble, n is mode number, and R0 is the initial bubble radius, the experimental observations show the mode number of the gas bubble in present section is 1, which means gas bubble oscillates in a volumetric radial mode. However, its oscillation way for gas bubble in present investigation is so different with that reported with Kim et al. [16] that there are four kinds of oscillation, namely, shape, volume, splitting and chaotic oscillation for single cavitation bubbles. Especially, the phenomenon reported previous works of Wang et al. [29] is not founded that bubbles in a finite liquid domain can present high-speed liquid jet. To further broaden the various behaviors of the gas bubble, Fig. 4 shows the temporal evolution of two bubble coalescence in the acoustic field of f = 18 kHz with the acoustic pressure amplitude of 5.25 bar. Two gas bubbles both with initial radius of 1.4 mm, which are same with the cases of gas bubble rising in a quiescent liquid and oscillates in an acoustic wave, are created at the same initial time. The distance from bubble C to horn tip is about 2.5 mm, while that is about 5.5 mm for bubble D. To better distinguish the different bubbles, two gas bubbles are named with letters that upper bubble is named with C and lower bubble is D, respectively. The relative distance between gas bubble C and D is about 3 mm in the initial condition. Significant fragments of gas are founded abound the gas bubble C and D. The periodical behaviors, such as splitting, attraction, and splitting, are also founded near each gas bubble. As observed in Fig.4, gas bubble C oscillates (shrinks and expands a little) and migrates downwards, while gas bubble D oscillates and moves upwards. Finally, gas bubble C and D merge into one oscillating gas bubble E. To further quantitatively investigate the bubble behaviors in three different cases discussed above, Fig. 5 shows the comparisons of displacements of the gas bubble under three different cases, namely, single bubble rising in a quiescent liquid, acoustic single bubble oscillation and two acoustic bubbles coalescence. As discussed above, the radius of initial bubbles is all set as 1.4 mm. As observed, the gas bubble rises directly and upwards. Compared with that in a quiescent liquid, the centroid of single
gas bubble almost keeps constant in an acoustic field of f =18 kHz. However, when two gas bubbles are created with distance of 3 mm, they are attracted each other and coalesce into one oscillating gas bubble. The resultant forces caused by gravitational and acoustic field are the essential results of three different phenomena, namely, bubble rising in a quiescent liquid, single bubble oscillation and two bubbles interaction in an acoustic field. To have a clear interpretation about the gas bubble motion, the related theoretical models are presented and analyzed in the following sections.
4. Theoretical models In this section, three kinds of forces acting on a bubble in an acoustic field will be considered and analyzed, namely, the primary Bjerknes force, secondary Bjerknes force, and the buoyancy force [30]. According to the origin of the forces, the buoyancy force is generated due to the gravity field, as shown in Fig. 2. However, the primary Bjerknes force is the radiation force caused by the acoustic wave and even the ultrasound, while the secondary Bjerknes force is generated from the other oscillating bubbles. Essentially, the primary and secondary Bjerknes forces are initialized from the pressure difference along the bubble surface, which is distinctly from the buoyancy force. It is noting that viscous effects are of minor importance if the parameter Re 2
2 gd 3 3 105 ,where ρ is the density of the water; g is the 2
gravitational acceleration; d is the diameter of the gas bubble; and μ is the viscosity of the water. In present work, the effect of the viscosity is negligible. The time-averaged primary or secondary Bjerknes force is defined as [31]
FB = - V (t ) ps ( x, t )
T
(2)
where V(t) is the instantaneous volume of a bubble at time t; ps ( x, t ) is the instantaneous acoustic pressure at the location x , when time is t; the symbol
T
means the time-averaged value under one circle of an acoustic period T. As for the
instantaneous acoustic pressure emitted from the acoustic horn, it is defined as a progressive wave and the equation is [32]
ps ( x, t ) = - pa ( x)sin(ωt - k x)
(3)
pa ( x) is the pressure amplitude of an acoustic wave at the location x , ω is the
Here,
angular frequency of an acoustic wave and its function is ω = 2πf ; k is the wave number vector of the acoustic wave. The Eq. (4) of first Bjerknes force is obtained by substituting the Eq. (3) in to Eq. (2) as following FB1 = pa V (t )sin(ωt - k x)
T
- pa k V (t ) cos(ωt - k x)
T
(4)
Corresponding to the secondary Bjerknes force, the equation is defined as Eq. (5) and related process of derivation can be reviewed from the Ref. [33]
F12
V V 2 1 4 d 2
er
(5)
T
where F12 is the secondary Bjerknes force acting on bubble 2 and caused by bubble 1; parameter ρ is the density of the liquid; d is the distance between the concerned two bubbles; V1 is the volume of bubble 1, while V2 is the volume of bubble 2;
V1 d 2V1 / dt 2 ; er is the unit vector from bubble 1 to 2.
The buoyancy force acting on a bubble is defined as [34]
Fb g
4 3 R 3
(6) T
where g is the gravitational acceleration vector and R is a radius of a bubble. To close the equations discussed above, a classic equation of bubble motion is employed such as [35] R 3 R 1 R R 1 cl 2 3cl
2 P ( R, t ) R R dPdiff ( R, t ) R 1 diff cl cl dt
(7)
where 2 Pdiff ( R, t ) P R0
R0 3k 2 4 R P Ps ( ) R R R
(8)
Here, P is the ambient pressure; R dR / dt , R d 2 R / dt 2 ; σ is the surface tension coefficient; cl is the sound velocity in the liquid; k is the polytrophic exponent; Ps is the external acoustic wave pressure discussed above.
5. Theoretical results and discussion The spatial distribution of the pressure amplitude of acoustic wave emitted from a horn tip with the motion of circular piston is described as [36]
pa cl v0 2sin
x2 a2 x
(9)
Here, pa refers to the pressure amplitude on the symmetry axis of the horn; v0 is the velocity amplitude of the horn tip; a is the radius of the circular piston. In present work, the radius of the is 10 mm, and circular piston velocity amplitude of horn tip is about 0.5 m/s, when horn is vibrated at 18 kHz. Yasui et al. [25] considered that the presence of bubble layers under the ultrasonic horn significantly effects the distribution of the acoustic amplitude pressure. According to the theory, the acoustic radiation resistance deceases to 1/3 value when the presence of bubble layers, compared with that under no bubble layer case. Fig. 6 shows the predicted acoustic amplitude under an acoustic horn as a function of the distance from the acoustic horn tip on the symmetry axis in present work. As observed, there do not exit any bubble layers under the horn tip in present experiments, so the maximum acoustic amplitude at the origin of acoustic horn tip is about 5.74 bar. To validate the veracity of theoretical models in present work, Fig. 7 shows the comparisons of radius variations of gas bubble B along the time for theoretical and experimental results. To precisely obtain the bubble area from the experimental pictures, an example of a typical gas bubble pattern observed for bubble B is
presented in Fig. 7(a), which includes both original bubble visualization illuminated by high-speed camera and the schematic interpretation drawn by an in-house feature-recognition software package [37]. As observed in Fig. 7(b), red diamonds indicate the experimental results, which are obtained by software package to ensure the accuracy. The bubble radius is defined as R = A / π [38], where A is the area of gas bubble on the images. The black line shows the experimental result, which is obtained from the Eq. (7). The distance between gas bubble B and acoustic horn is about 3 mm. According to the Eq. (9), the maximum acoustic amplitude of pressure at bubble B is about 5.4 bar. It is notable that gas bubble B presents periodical oscillation, such as expansion and shrink. The period of gas bubble is 0.4 ms, and the theoretical model shows the best agreement with the measured results. Fig. 8 shows the predicted bubble radius as a function of time for three acoustic cycles when the distance between bubble and acoustic origin is 2.5 mm for gas bubble C and 5.5 mm for gas bubble D, at a frequency of 18 kHz. The acoustic amplitude in the bubble C and that in the bubble D have been assumed as 5.5 bar and 5.0 bar at a frequency of 18 kHz, respectively, assuming the spatial variation as Eq. (9). As observed, the black curve is for gas bubble C, while red one indicates the gas bubble D. It is worth noting that the tendency about oscillation way of bubble C and D is distinctly different from the single bubble B. The period of bubble C and D is obviously shorter than that of bubble B, which oscillates only acted by primary Bjerknes force. That’s because the interaction of two bubbles may contribute to the acceleration of the oscillation. Next, the phenomena will be quantitatively analyzed and discussed that bubble A migrates upwards, bubble B oscillates without motion, and bubble C and D coalesce into single one, as shown in Fig. 2, 3 and 4, respectively. The predicted results of the forces acting on each bubble have been summarized as follows. A. The forces acting on gas bubble A
(1) The buoyant force:
g
4 3 R 3
1.13 104 N Towards the acoustic horn tip. T
(2) The sums of the forces:
1.13 104 N Towards the acoustic horn tip
B. The forces acting on gas bubble B (1) The buoyant force:
g
4 3 R 3
1.24 104 N Towards the acoustic horn tip. T
(2) The first Bjerknes force: pa V (t ) sin(ωt - k x)
T
- pa k V (t ) cos(ωt - k x)
= 0.84x10-4 ( N ) Away from the acoustic horn tip
T
= 0.42x10-4 ( N ) Away from the acoustic horn tip
(3) The sums of the forces:
0 N Oscillation without motion
C. The forces acting on gas bubble C (1) The buoyant force:
g
4 3 R 3
1.24 104 N Towards the acoustic horn tip. T
(2) The first Bjerknes force: pa V (t ) sin(ωt - k x) - pa k V (t ) cos(ωt - k x)
= 0.82x10-4 ( N ) Away from the acoustic horn tip
T
T
= 0.57x10-4 ( N ) Away from the acoustic horn tip
(3) The secondary Bjerknes force: V V C D 4 d 2
er 1.23 106 N Away from the acoustic horn tip T
(4) The sums of the forces:
0.20 104 N Away from the acoustic horn tip
D. The forces acting on gas bubble D (1) The buoyant force: g
4 3 R 3
1.24 104 N Towards the acoustic horn tip. T
(2) The first Bjerknes force: pa V (t ) sin(ωt - k x)
T
- pa k V (t ) cos(ωt - k x)
= 0.65x10-4 ( N ) Away from the acoustic horn tip
T
= 0.39x10-4 ( N ) Away from the acoustic horn tip
(3) The secondary Bjerknes force:
VC VD 2 4 d
er 1.23 106 N Towards the acoustic horn tip. T
(4) The sums of the forces:
0.21104 N Towards the acoustic horn tip.
Compared with the bubble A rising in the quiescent water due to the buoyant force, it is concluded that bubble B oscillates without motion upwards or downwards due to the equilibrium of the buoyant force and the first Bjerknes force. It is consistent with the experimental observation as shown in Fig. 3. Corresponding to the bubble C and D, the resultant of buoyancy, primary and secondary Bjerknes acting on two bubbles are same in magnitude, but in opposite direction, which indicates that two gas bubbles attract each other and coalesce into one. In present experiment, only the concerned bubbles are created to investigate the various forces acted on the bubbles, so the effect of interference factors (such as bubble layers under the horn tip) generated from the acoustic wave in reference is eliminated. The secondary Bjerknes force between bubbles from the external acoustic wave is negligible to ensure precision of the results.
6. Conclusions Experimental and theoretical studies are presented for gas bubbles in an acoustic field with the frequency of 18 kHz for three conditions, namely, gas bubble rising in a quiescent liquid, single bubble oscillation in acoustic, and two bubbles coalescence. High-speed videos of the evolution of the gas bubble dynamics is used to investigate the transient bubble patterns. Statistics of bubble velocity as well displacement are also presented to quantify the unsteady process. The theoretical model is introduced to quantitatively analyze the bubble behaviors in the matter of the buoyancy, secondary and primary Bjerknes force. The comparisons of the calculated results and the experimental results of the bubble behaviors have indicated that the gas bubble in an acoustic field presents the periodical behaviors, namely, expansion and shrink. The motion of gas bubbles in acoustic filed is closely similar with acoustic cavitation bubbles in finite liquid: the centroid of the gas bubble almost oscillates without motion upwards or downwards, because of the equilibrium of the primary Bjerknes force caused by acoustic waves and the effect of the buoyancy force. Furthermore, The resultant of buoyancy, primary and secondary Bjerknes forces acting on two bubbles are same in magnitude, but in opposite direction, which indicates that two gas bubbles attract each other and coalesce into one.
Acknowledgments The authors gratefully acknowledge the fund supported by Mr. Junhu Gao responsible for Beijing Ferly Water Treatment Equipment Co. Ltd. and Ferly Environmental Protection (Hebei) Equipment Co., Ltd.
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Legends Fig. 1 (a) The schematic description for experimental setup, and (b) the relative position about horn and gas inlet. Fig. 2 The temporal evolutions of bubble rising in a quiescent liquid without the acoustic wave. Time interval is 0.4 ms; bubble radius R0 = 1.4 mm; recording rate 2500 fps. Fig. 3 The temporal evolution of bubble oscillation without motion of bubble centroid in the acoustic field of f=18kHz.The distance between bubble and acoustic origin is 5 mm. Time interval is 0.08ms; bubble radius: R0 = 1.4 mm; the acoustic pressure amplitude is about 5.40 bar; recording rate: 12500 fps. Fig. 4 The temporal evolution of two gas bubble coalescence in the acoustic field of f=18 kHz. The initial distance between gas bubble C and D is about 3 mm. The distance from bubble C to horn tip is about 2.5 mm, while that is about 5.5 mm for bubble D. time interval t = 0.4ms; bubble radius of C and D: R0 = 1.4 mm; the acoustic pressure amplitude is about 5.25 bar; recording rate: 2500 fps. Fig. 5 The comparisons of displacements of the gas bubble under three different cases, namely, single bubble rising in a quiescent liquid, acoustic single bubble oscillation and two acoustic bubbles coalescence. Acoustic frequency: f =18 kHz; bubble radius: R0= 1.4 mm. Fig. 6 Predicted acoustic amplitude under an acoustic horn as a function of the distance from the acoustic horn tip on the symmetry axis. Fig. 7 (a) Typical bubble oscillation visualization and schematic interpretation, and (b) the comparisons of the theoretical and experimental bubble B radius as a function of the time for the initial condition: R0 = 1.4mm, f = 18 kHz, Pa = 5.4
bar. The distance from bubble C to horn tip is about 3 mm. The black line is the theoretical results and the red diamonds indicate the experimental one. And the red error bars represent the fluctuation span of the bubble radius. Fig. 8 Predicted bubble radius as a function of time for three acoustic cycles when maximum acoustic amplitude at bubble C and D is 5.5 bar and 5.0 bar at a frequency of 18 kHz, respectively. Initial bubble radius is R0 = 1.4 mm. The distance between bubble and acoustic origin is 2.5 mm for gas bubble C and 5.5 mm for gas bubble D. The black curve is for gas bubble C, while red one indicates the gas bubble D.
Highlights: 1.
Combined the experimental and theoretical methods are used to investigate the bubble behaviors in an acoustic field;
2.
Single gas bubble oscillates without motion due to the equilibrium of the primary Bjerknes and buoyancy force;
3.
The resultant of forces acting on two bubbles are same in magnitude, but in opposite direction, leading to the coalescence of two gas bubbles.
S1350-4177(17)30339-5 http://dx.doi.org/10.1016/j.ultsonch.2017.07.035 ULTSON 3790
To appear in:
Ultrasonics Sonochemistry
Received Date: Revised Date: Accepted Date:
16 February 2017 15 July 2017 24 July 2017
Please cite this article as: X. Ma, T. Xing, Q. Li, Y. Yang, Combined Experimental and Theoretical Investigation of the Gas Bubble Motion in an Acoustic Field, Ultrasonics Sonochemistry (2017), doi: http://dx.doi.org/10.1016/ j.ultsonch.2017.07.035
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Combined Experimental and Theoretical Investigation of the Gas Bubble Motion in an Acoustic Field Xiaojian Ma*1, 2, Tianyu Xing3, Qiuhe Li4, Yifei Yang5 1. School of Mechanical and Vehicular Engineering, Beijing Institute of Technology, Beijing 100081, China. 2. Department of Techniques Research and Development, Beijing Ferly Water Treatment Equipment Company Limited, Beijing 100022, China 3. School of Information Science & Technology, Beijing Forestry University, Beijing 100083, China. 4. Department of Mechanical and Electronic Engineering, Yingkou Institute of Technology, Liaoning 115014, China. 5. School of Journalism & Communication, Hebei Normal University, Hebei 050000, China.
Abstract The objective of this paper is to apply the combined experimental and theoretical method to investigate the various behaviors of gas bubbles in an acoustic field. In the experiments, high-speed video and ultrasonic processor are used to capture the transient evolution of gas bubble patterns, as well as velocity profiles. In the theoretical analysis, the theories of primary and secondary Bjerknes forces and buoyancy force are introduced to accurately demonstrate the variations of bubble volume and motion. Results are presented for gas bubbles with the radius of 1.4 mm under an acoustic field with a frequency of 18 kHz, for three cases, namely single bubble rising in a quiescent liquid, acoustic single bubble oscillation and two bubbles coalescence conditions. The results show that the fragments around the single gas
* Corresponding author: Xiaojian Ma E-mail: [email protected] Fax/Tel: + 86-010-68912395
bubble presents the periodical behaviors, namely, splitting, attraction, and secondary splitting motion. The centroid of the single gas bubble almost oscillates without motion upwards or downwards, because of the equilibrium of the primary Bjerknes force caused by acoustic waves and the effect of the buoyancy force. For the two coalescing bubbles, the resultant of buoyancy, primary and secondary Bjerknes forces acting on two bubbles are same in magnitude, but in opposite direction, which indicates that two gas bubbles attract each other and and coalesce into one. Keywords: gas bubble behaviors, acoustic fields, Bjerknes force, buoyancy force. Nomenclature Roman letters a A Bo
radius of the circular piston area of gas bubble on the images Bond number
R Re R0
cl
sound velocity in the liquid
R
instantaneous radius of a bubble Reynolds number initial radius of a bubble first time derivative of R
D
distance between the concerned two R bubbles diameter of the gas bubble t
er
unit vector
T
period of acoustic wave
f
frequency of acoustic field
V
instantaneous volume of a bubble
g
gravitational acceleration vector
V
second time derivative of V
Fb
buoyancy force
x
location vector of bubble
FB
primary or secondary Bjerknes force
y
vertical coordination
FB1
first Bjerknes force
Greek letters
secondary Bjerknes force acting on bubble 2 and caused by bubble 1 wave number vector of the acoustic wave mode number instantaneous acoustic pressure pressure amplitude of an acoustic wave
κ
polytrophic exponent
λ μ ρ
wavelength of the acoustic wave in the liquid viscosity of the fluid density of the liquid
σ
surface tension coefficient
d
F12
k n ps pa
second time derivative of R time
p
ambient pressure
ω
r
angular coordination
ωn
angular frequency of an acoustic wave natural frequency of bubble
1. Introduction Bubble dynamics in an acoustic field due to acoustic radiation force has been typical one of concerned multiphase fluidics. During the oscillation in an acoustic wave, gas bubbles present various behaviors, including high-speed jet [1], hot pot [2], high pressure [3] and sonoluminescence [4], etc. Those characteristics of oscillating gas bubbles have played a significantly important role in distinct industrial processes, such as environmental protection [5], water treatment [6], extraction [7] and chemical reaction [8]. A fundamental understanding of the acoustic gas bubble physics is particularly indispensable in many practical technologies for future works, ranging from the active control of medical ultrasound contrast agent to surface cleaning of precise instruments [9,10]. However, a comprehensive knowledge of dynamic behaviors of gas bubbles under the progressive acoustic wave fields is still limited, though extensive experimental investigation, theoretical solution and numerical simulation have been performed. Early experimental studies on bubble dynamics were reported in the works of Leighton [11], Brujan [12], Chen [13], and Ashokkumar [14]. They research on topic of the bubble dynamics included different boundary conditions, from a quiescent
liquid to an acoustic field and from infinite liquid to boundary wall. Results demonstrated that single bubble in a quiescent liquid presents oblate shape corresponding to the function of the buoyancy force [15]. When an acoustic wave is imposed on a liquid, single bubble is possible to trap in the acoustic field with various oscillations. Kim et al. [16] used the high-speed camera system to investigate the single bubble dynamics in an ultrasonic field and its destructive actions. They founded that single bubble behaviors have four different oscillations, namely volume, shape, splitting and chaotic oscillations, which depends on bubble size and acoustic pressure amplitude. Essentially, those different oscillations are generated due to the primary Bjerknes force, which cause by the pressure difference along the surface of bubble [17]. As for the coalescence of the bubbles, many researchers tended to create two bubbles with small distance in space, in order to investigate the interaction between the two bubbles. They founded that bubbles are attracted by the each other and migrate along the center line between two bubbles. Mettin et al. [18] demonstrated that bubbles with close distance tend to attract each other due to secondary Bjerknes force, which is caused by the oscillation of the bubble membranes. As a result, bubbles in an acoustic field may be acted by three kinds of forces, namely, the buoyancy force, primary and secondary Bjerknes forces. It worth noting that there exit significant differences between gas bubble and cavitation one. As known, the cavitation bubble is generated when the local pressure drops below the saturated one and the pressure of inner cavitation bubble is with saturated levels [19]. But the gas bubble is generated by the gathering of the gas, whose pressure depends on the environmental pressure. Furthermore, cavitation bubble is generally small in volume, and its buoyancy force can usually be neglected. As for the gas bubble, the buoyancy force is an importantly external force, which causes the deformation and motion of the bubbles. Due to the various mechanism of formation of bubbles, the surface tension coefficient and viscosity of the liquid may be both different [20]. For more details of the information about bubbles, readers are referred to systematic work by Hung el at. [21]. In present work, we concentrate our
attention on the transient behaviors of the gas bubbles, which are subjected to the acoustic travelling waves. To further investigate the mechanism of gas bubble dynamics in an acoustic field, theoretical solution and analysis has been also conducted to obtain the additional parameters. Due to highly nonlinearity of the compressible acoustic oscillation, it is difficult to capture the bubble motion and deformation [22]. Correct analysis of the bubble motion in an acoustic field depends upon the ability of theoretical methods. Over the years, the basic idea to overcome the problems is solving the bubble motion equation and the equations of primary and secondary Bjerknes force simultaneously. Many studies are performed to investigate the various forms of the equation to satisfy the different conditions [23,24]. Yasui et al. [25] used experimental and theoretical methods to investigate the direction of bubble clouds motion under an ultrasonic horn. In the theoretical methods, the interaction of the bubbles is considered and introduced. They founded the bubble clouds may migrate upwards the horn tip, which is strongly caused by the acoustic fields. But the interference comes from many other bubbles is impossible to eliminate, because the ultrasonic horn generates massive small bubbles, but not the one or two concerned bubble clouds, though this is considered in the theoretical mode of bubble motion equation. Although the researches on the dynamics bubbles have received much attention in the past years, the mechanisms of gas bubbles in an acoustic field are still not well understood, and hence additionally experimental and theoretical researches are still needed. The objectives of the present paper are (1) to observe and record three different gas bubble behaviors, namely rising, oscillation and coalescence; (2) to introduce and invalidate the theoretical mode to capture the oscillation of gas bubbles in an acoustic field; (3) to analyze the factors causing different behaviors of gas bubbles in an acoustic field, such as the buoyancy force, the primary and secondary Bjerknes force.
2. Experimental setup In present work, the experimental system mainly includes three parts, involving acoustic wave generator, gas bubble creator and high-speed video camera. The more information about each part, such as working function and parameter, is introduced in details as following. 2.1 The generation of acoustic waves The needed acoustic waves are generated by an acoustic processor (Institute of Acoustics, Chinese Academy of Science, China), which is with a constant frequency of 18 kHz. It notes that the electrical input power of the acoustic processor can be chosen in the range from 50 W to 250 W, which can be measured by a precise electrical power meter. The acoustic horn has a plane radiating surface, which is measured with 20 mm in diameter and oscillates with a simple harmonic motion. Fig.1(a) shows the schematic description of experimental setup for investigating the single bubble behaviors in acoustic waves. As observed, the tip of horn is merged into a water tank, which is made of transparent glass for photography and illumination. The water chamber is processed with a length of 320 mm, width of 240 mm, and height of 200 mm. The distance between the acoustic horn and the bottom wall of the water tank is about two times of the wave length in water in order to avoid the reflection of the acoustic waves. 2.2 The generation of gas bubbles Fig. 1(a) also shows that the bubble is generated by a motoring injection syringe with cylindrical shape, whose inner radius is about 1mm. The injection syringe, which is full of gas, can be driven by an electric motor, and its screw pitch is about 0.1 mm. By this application, the size and volume of the gas bubble can be precisely controlled with electric motor. A gas inlet is merged into the water tank and under the acoustic
horn. As shown in Fig. 1(b), the maximum distance between gas inlet and acoustic horn tip is about 8 mm. To create gas bubble at different position, the location of the gas inlet can be lifted. To increase the cavitation threshold, the water thank is the filled with sufficiently degassed water to the depth needed, and the temperature is about 25 ℃ [26]. Before the generation of a bubble, a ruler is placed in the same vertical plane with the acoustic horn and perpendicular to the camera lens, to be recorded as calibration for bubble size in length. As a result, spatial measurements are directly illustrated on the images. 2.3 The high-speed videos The temporal evolution of the bubble dynamics is recorded with a high-speed camera (HG-LE, by Redlake) at framing rates as high as 100,000 frames per second (fps). A lower recording speed of 3000 fps is adopted to obtain the desired spatial resolution in present investigation, and the exposure time of each frame is set as 30 μs, in order to measure the sharpness of the bubble outline. The light source is provided by a dysprosium lamp, and its maximum power is 1000 kW. To form a better light distribution in the background, a piece of ground glass is placed between the camera and the dysprosium lamp. Asynchronous system is controlled by a computer to realize the generation of gas bubble and capture of the high-speed camera at same time.
3. Experimental results 3.1 Bubble rising in a quiescent liquid To compare with the gas bubble behaviors in an acoustic wave, Fig. 2 shows the temporal evolution of the gas bubble rising in a quiescent liquid without any acoustic waves. The data from transient motion of gas bubble in Fig. 2 is treated as a control group in this paper. The typical images are chosen to illustrate the essential and pronounced features of gas bubble dynamics in temporal and spatial scales, at a recording rate of 3000 frames per second (fps). In this section, the radius of the
bubble is generated about 1.4 mm. As observed, the bubble presents almost the spherical shape in the rising process, due to the large surface tension caused by the small size of the radius. In our previous works about the regimes depending on the Reynolds and Bond numbers, the gas bubbles only remain spherical shape when they are rising in a quiescent liquid with low Reynolds or Bond numbers (Re < 150 or Bo < 1) [27]. Further increasing slightly in Reynolds number (Re = 300), the shape of gas bubbles still stay spherical for the lower Bond number, where the Reynolds and Bond numbers are defined as
Re =
ρg1/2 D3/2 ρgD ; Bo = μ σ
(1)
where ρ is the density of the liquid; D the gas bubble diameter; g the gravitational acceleration; μ the viscosity coefficient; and σ the surface tension coefficient. In order to further measure quantitatively the transient motion of the gas bubble in the quiescent liquid, the rising velocity of the gas bubble is investigated in present work. It is notable that the bubble presents a rising at a constant velocity for about 1.6 m/s. 3.2 Single bubble oscillation in an acoustic field Fig. 3 shows the temporal evolution of bubble oscillation in the acoustic field with frequency f = 18 kHz with the acoustic pressure amplitude of 5.40 bar. The images are captured at a recording rate of 12500 fps. Before the generation of the gas bubble, the acoustic horn is initialized to make sure the gas bubble located at an acoustic field. And this gas bubble is named as letter A. In present section, the size of gas bubble is same with the gas bubble in the quiescent liquid (1.4 mm), as discussed in Fig. 2. As observed, there are many fragments of gas bubble with small volume around the gas bubble during the process of oscillation. Fragments present the periodical motion that they splits from the gas bubble A, then be attracted, finally splits, repeatedly. It is worth noting that the centroid of the gas bubble almost has no movement in the original position. According to the Lamb’s expression for surface mode vibration [28], ωn = σ (n -1)(n +1)(n + 2) / ( ρR02 ) , where ωn is the natural
frequency of bubble, n is mode number, and R0 is the initial bubble radius, the experimental observations show the mode number of the gas bubble in present section is 1, which means gas bubble oscillates in a volumetric radial mode. However, its oscillation way for gas bubble in present investigation is so different with that reported with Kim et al. [16] that there are four kinds of oscillation, namely, shape, volume, splitting and chaotic oscillation for single cavitation bubbles. Especially, the phenomenon reported previous works of Wang et al. [29] is not founded that bubbles in a finite liquid domain can present high-speed liquid jet. To further broaden the various behaviors of the gas bubble, Fig. 4 shows the temporal evolution of two bubble coalescence in the acoustic field of f = 18 kHz with the acoustic pressure amplitude of 5.25 bar. Two gas bubbles both with initial radius of 1.4 mm, which are same with the cases of gas bubble rising in a quiescent liquid and oscillates in an acoustic wave, are created at the same initial time. The distance from bubble C to horn tip is about 2.5 mm, while that is about 5.5 mm for bubble D. To better distinguish the different bubbles, two gas bubbles are named with letters that upper bubble is named with C and lower bubble is D, respectively. The relative distance between gas bubble C and D is about 3 mm in the initial condition. Significant fragments of gas are founded abound the gas bubble C and D. The periodical behaviors, such as splitting, attraction, and splitting, are also founded near each gas bubble. As observed in Fig.4, gas bubble C oscillates (shrinks and expands a little) and migrates downwards, while gas bubble D oscillates and moves upwards. Finally, gas bubble C and D merge into one oscillating gas bubble E. To further quantitatively investigate the bubble behaviors in three different cases discussed above, Fig. 5 shows the comparisons of displacements of the gas bubble under three different cases, namely, single bubble rising in a quiescent liquid, acoustic single bubble oscillation and two acoustic bubbles coalescence. As discussed above, the radius of initial bubbles is all set as 1.4 mm. As observed, the gas bubble rises directly and upwards. Compared with that in a quiescent liquid, the centroid of single
gas bubble almost keeps constant in an acoustic field of f =18 kHz. However, when two gas bubbles are created with distance of 3 mm, they are attracted each other and coalesce into one oscillating gas bubble. The resultant forces caused by gravitational and acoustic field are the essential results of three different phenomena, namely, bubble rising in a quiescent liquid, single bubble oscillation and two bubbles interaction in an acoustic field. To have a clear interpretation about the gas bubble motion, the related theoretical models are presented and analyzed in the following sections.
4. Theoretical models In this section, three kinds of forces acting on a bubble in an acoustic field will be considered and analyzed, namely, the primary Bjerknes force, secondary Bjerknes force, and the buoyancy force [30]. According to the origin of the forces, the buoyancy force is generated due to the gravity field, as shown in Fig. 2. However, the primary Bjerknes force is the radiation force caused by the acoustic wave and even the ultrasound, while the secondary Bjerknes force is generated from the other oscillating bubbles. Essentially, the primary and secondary Bjerknes forces are initialized from the pressure difference along the bubble surface, which is distinctly from the buoyancy force. It is noting that viscous effects are of minor importance if the parameter Re 2
2 gd 3 3 105 ,where ρ is the density of the water; g is the 2
gravitational acceleration; d is the diameter of the gas bubble; and μ is the viscosity of the water. In present work, the effect of the viscosity is negligible. The time-averaged primary or secondary Bjerknes force is defined as [31]
FB = - V (t ) ps ( x, t )
T
(2)
where V(t) is the instantaneous volume of a bubble at time t; ps ( x, t ) is the instantaneous acoustic pressure at the location x , when time is t; the symbol
T
means the time-averaged value under one circle of an acoustic period T. As for the
instantaneous acoustic pressure emitted from the acoustic horn, it is defined as a progressive wave and the equation is [32]
ps ( x, t ) = - pa ( x)sin(ωt - k x)
(3)
pa ( x) is the pressure amplitude of an acoustic wave at the location x , ω is the
Here,
angular frequency of an acoustic wave and its function is ω = 2πf ; k is the wave number vector of the acoustic wave. The Eq. (4) of first Bjerknes force is obtained by substituting the Eq. (3) in to Eq. (2) as following FB1 = pa V (t )sin(ωt - k x)
T
- pa k V (t ) cos(ωt - k x)
T
(4)
Corresponding to the secondary Bjerknes force, the equation is defined as Eq. (5) and related process of derivation can be reviewed from the Ref. [33]
F12
V V 2 1 4 d 2
er
(5)
T
where F12 is the secondary Bjerknes force acting on bubble 2 and caused by bubble 1; parameter ρ is the density of the liquid; d is the distance between the concerned two bubbles; V1 is the volume of bubble 1, while V2 is the volume of bubble 2;
V1 d 2V1 / dt 2 ; er is the unit vector from bubble 1 to 2.
The buoyancy force acting on a bubble is defined as [34]
Fb g
4 3 R 3
(6) T
where g is the gravitational acceleration vector and R is a radius of a bubble. To close the equations discussed above, a classic equation of bubble motion is employed such as [35] R 3 R 1 R R 1 cl 2 3cl
2 P ( R, t ) R R dPdiff ( R, t ) R 1 diff cl cl dt
(7)
where 2 Pdiff ( R, t ) P R0
R0 3k 2 4 R P Ps ( ) R R R
(8)
Here, P is the ambient pressure; R dR / dt , R d 2 R / dt 2 ; σ is the surface tension coefficient; cl is the sound velocity in the liquid; k is the polytrophic exponent; Ps is the external acoustic wave pressure discussed above.
5. Theoretical results and discussion The spatial distribution of the pressure amplitude of acoustic wave emitted from a horn tip with the motion of circular piston is described as [36]
pa cl v0 2sin
x2 a2 x
(9)
Here, pa refers to the pressure amplitude on the symmetry axis of the horn; v0 is the velocity amplitude of the horn tip; a is the radius of the circular piston. In present work, the radius of the is 10 mm, and circular piston velocity amplitude of horn tip is about 0.5 m/s, when horn is vibrated at 18 kHz. Yasui et al. [25] considered that the presence of bubble layers under the ultrasonic horn significantly effects the distribution of the acoustic amplitude pressure. According to the theory, the acoustic radiation resistance deceases to 1/3 value when the presence of bubble layers, compared with that under no bubble layer case. Fig. 6 shows the predicted acoustic amplitude under an acoustic horn as a function of the distance from the acoustic horn tip on the symmetry axis in present work. As observed, there do not exit any bubble layers under the horn tip in present experiments, so the maximum acoustic amplitude at the origin of acoustic horn tip is about 5.74 bar. To validate the veracity of theoretical models in present work, Fig. 7 shows the comparisons of radius variations of gas bubble B along the time for theoretical and experimental results. To precisely obtain the bubble area from the experimental pictures, an example of a typical gas bubble pattern observed for bubble B is
presented in Fig. 7(a), which includes both original bubble visualization illuminated by high-speed camera and the schematic interpretation drawn by an in-house feature-recognition software package [37]. As observed in Fig. 7(b), red diamonds indicate the experimental results, which are obtained by software package to ensure the accuracy. The bubble radius is defined as R = A / π [38], where A is the area of gas bubble on the images. The black line shows the experimental result, which is obtained from the Eq. (7). The distance between gas bubble B and acoustic horn is about 3 mm. According to the Eq. (9), the maximum acoustic amplitude of pressure at bubble B is about 5.4 bar. It is notable that gas bubble B presents periodical oscillation, such as expansion and shrink. The period of gas bubble is 0.4 ms, and the theoretical model shows the best agreement with the measured results. Fig. 8 shows the predicted bubble radius as a function of time for three acoustic cycles when the distance between bubble and acoustic origin is 2.5 mm for gas bubble C and 5.5 mm for gas bubble D, at a frequency of 18 kHz. The acoustic amplitude in the bubble C and that in the bubble D have been assumed as 5.5 bar and 5.0 bar at a frequency of 18 kHz, respectively, assuming the spatial variation as Eq. (9). As observed, the black curve is for gas bubble C, while red one indicates the gas bubble D. It is worth noting that the tendency about oscillation way of bubble C and D is distinctly different from the single bubble B. The period of bubble C and D is obviously shorter than that of bubble B, which oscillates only acted by primary Bjerknes force. That’s because the interaction of two bubbles may contribute to the acceleration of the oscillation. Next, the phenomena will be quantitatively analyzed and discussed that bubble A migrates upwards, bubble B oscillates without motion, and bubble C and D coalesce into single one, as shown in Fig. 2, 3 and 4, respectively. The predicted results of the forces acting on each bubble have been summarized as follows. A. The forces acting on gas bubble A
(1) The buoyant force:
g
4 3 R 3
1.13 104 N Towards the acoustic horn tip. T
(2) The sums of the forces:
1.13 104 N Towards the acoustic horn tip
B. The forces acting on gas bubble B (1) The buoyant force:
g
4 3 R 3
1.24 104 N Towards the acoustic horn tip. T
(2) The first Bjerknes force: pa V (t ) sin(ωt - k x)
T
- pa k V (t ) cos(ωt - k x)
= 0.84x10-4 ( N ) Away from the acoustic horn tip
T
= 0.42x10-4 ( N ) Away from the acoustic horn tip
(3) The sums of the forces:
0 N Oscillation without motion
C. The forces acting on gas bubble C (1) The buoyant force:
g
4 3 R 3
1.24 104 N Towards the acoustic horn tip. T
(2) The first Bjerknes force: pa V (t ) sin(ωt - k x) - pa k V (t ) cos(ωt - k x)
= 0.82x10-4 ( N ) Away from the acoustic horn tip
T
T
= 0.57x10-4 ( N ) Away from the acoustic horn tip
(3) The secondary Bjerknes force: V V C D 4 d 2
er 1.23 106 N Away from the acoustic horn tip T
(4) The sums of the forces:
0.20 104 N Away from the acoustic horn tip
D. The forces acting on gas bubble D (1) The buoyant force: g
4 3 R 3
1.24 104 N Towards the acoustic horn tip. T
(2) The first Bjerknes force: pa V (t ) sin(ωt - k x)
T
- pa k V (t ) cos(ωt - k x)
= 0.65x10-4 ( N ) Away from the acoustic horn tip
T
= 0.39x10-4 ( N ) Away from the acoustic horn tip
(3) The secondary Bjerknes force:
VC VD 2 4 d
er 1.23 106 N Towards the acoustic horn tip. T
(4) The sums of the forces:
0.21104 N Towards the acoustic horn tip.
Compared with the bubble A rising in the quiescent water due to the buoyant force, it is concluded that bubble B oscillates without motion upwards or downwards due to the equilibrium of the buoyant force and the first Bjerknes force. It is consistent with the experimental observation as shown in Fig. 3. Corresponding to the bubble C and D, the resultant of buoyancy, primary and secondary Bjerknes acting on two bubbles are same in magnitude, but in opposite direction, which indicates that two gas bubbles attract each other and coalesce into one. In present experiment, only the concerned bubbles are created to investigate the various forces acted on the bubbles, so the effect of interference factors (such as bubble layers under the horn tip) generated from the acoustic wave in reference is eliminated. The secondary Bjerknes force between bubbles from the external acoustic wave is negligible to ensure precision of the results.
6. Conclusions Experimental and theoretical studies are presented for gas bubbles in an acoustic field with the frequency of 18 kHz for three conditions, namely, gas bubble rising in a quiescent liquid, single bubble oscillation in acoustic, and two bubbles coalescence. High-speed videos of the evolution of the gas bubble dynamics is used to investigate the transient bubble patterns. Statistics of bubble velocity as well displacement are also presented to quantify the unsteady process. The theoretical model is introduced to quantitatively analyze the bubble behaviors in the matter of the buoyancy, secondary and primary Bjerknes force. The comparisons of the calculated results and the experimental results of the bubble behaviors have indicated that the gas bubble in an acoustic field presents the periodical behaviors, namely, expansion and shrink. The motion of gas bubbles in acoustic filed is closely similar with acoustic cavitation bubbles in finite liquid: the centroid of the gas bubble almost oscillates without motion upwards or downwards, because of the equilibrium of the primary Bjerknes force caused by acoustic waves and the effect of the buoyancy force. Furthermore, The resultant of buoyancy, primary and secondary Bjerknes forces acting on two bubbles are same in magnitude, but in opposite direction, which indicates that two gas bubbles attract each other and coalesce into one.
Acknowledgments The authors gratefully acknowledge the fund supported by Mr. Junhu Gao responsible for Beijing Ferly Water Treatment Equipment Co. Ltd. and Ferly Environmental Protection (Hebei) Equipment Co., Ltd.
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Legends Fig. 1 (a) The schematic description for experimental setup, and (b) the relative position about horn and gas inlet. Fig. 2 The temporal evolutions of bubble rising in a quiescent liquid without the acoustic wave. Time interval is 0.4 ms; bubble radius R0 = 1.4 mm; recording rate 2500 fps. Fig. 3 The temporal evolution of bubble oscillation without motion of bubble centroid in the acoustic field of f=18kHz.The distance between bubble and acoustic origin is 5 mm. Time interval is 0.08ms; bubble radius: R0 = 1.4 mm; the acoustic pressure amplitude is about 5.40 bar; recording rate: 12500 fps. Fig. 4 The temporal evolution of two gas bubble coalescence in the acoustic field of f=18 kHz. The initial distance between gas bubble C and D is about 3 mm. The distance from bubble C to horn tip is about 2.5 mm, while that is about 5.5 mm for bubble D. time interval t = 0.4ms; bubble radius of C and D: R0 = 1.4 mm; the acoustic pressure amplitude is about 5.25 bar; recording rate: 2500 fps. Fig. 5 The comparisons of displacements of the gas bubble under three different cases, namely, single bubble rising in a quiescent liquid, acoustic single bubble oscillation and two acoustic bubbles coalescence. Acoustic frequency: f =18 kHz; bubble radius: R0= 1.4 mm. Fig. 6 Predicted acoustic amplitude under an acoustic horn as a function of the distance from the acoustic horn tip on the symmetry axis. Fig. 7 (a) Typical bubble oscillation visualization and schematic interpretation, and (b) the comparisons of the theoretical and experimental bubble B radius as a function of the time for the initial condition: R0 = 1.4mm, f = 18 kHz, Pa = 5.4
bar. The distance from bubble C to horn tip is about 3 mm. The black line is the theoretical results and the red diamonds indicate the experimental one. And the red error bars represent the fluctuation span of the bubble radius. Fig. 8 Predicted bubble radius as a function of time for three acoustic cycles when maximum acoustic amplitude at bubble C and D is 5.5 bar and 5.0 bar at a frequency of 18 kHz, respectively. Initial bubble radius is R0 = 1.4 mm. The distance between bubble and acoustic origin is 2.5 mm for gas bubble C and 5.5 mm for gas bubble D. The black curve is for gas bubble C, while red one indicates the gas bubble D.
Highlights: 1.
Combined the experimental and theoretical methods are used to investigate the bubble behaviors in an acoustic field;
2.
Single gas bubble oscillates without motion due to the equilibrium of the primary Bjerknes and buoyancy force;
3.
The resultant of forces acting on two bubbles are same in magnitude, but in opposite direction, leading to the coalescence of two gas bubbles.