Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 19, ICA 213 Montreal Montreal, Canada 2-7 June 213 Physical Acoustics Session 1pPAa: Acoustics in Microfluidics and for Particle Separation IIb: Bubbles and Drops 1pPAa3. Waves of acoustically induced transparency in bubbly liquids: theory and experiment Nail Gumerov*, Claus-Dieter Ohl, Iskander S. Akhatov, Sergei Sametov and Maxim Khasimullin *Corresponding author's address: Institute for Advanced Computer Studies, University of Maryland, College Park, Maryland 2742, The theory of self-organization of bubbles in acoustic fields predicts formation and propagation of waves of self-induced acoustic transparency. This is a strongly nonlinear effect, which is a result of a two-way coupling of the sound field with the bubble distribution. We are challenging the theory with an experiment. Here, a homogeneous distribution of gas bubbles is first generated and then an ultrasonic field (~1 khz) is switched on. The ultrasound leads to a rapidly propagating bubble sheet away from the transducer which leaves a liquid almost free of bubbles behind the front. The dynamics is observed with a high-speed camera and analyzed. A simplified theoretical model of this acousticallyinduced transparency is developed and numerical simulations for conditions of the experiment are performed. A comparison of the experimental data with the model shows a good agreement. The underlying physics of the problem is discussed. Published by the Acoustical Society of America through the American Institute of Physics 213 Acoustical Society of America [DOI: / ] Received 22 Jan 213; published 2 Jun 213 Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 1

2 INTRODUCTION There exists an extensive literature on single bubble dynamics in acoustic fields (e.g. [1]), as well on propagation of acoustic waves in bubbly liquids (e.g. [2]), which usually are considered somehow separately (e.g. [3]). However, an acoustic field acts on the bubbles and bubbles act on the field. Such two-way interaction triggers self-focusing of acoustic waves [4], self-induced acoustic transparency [4, 5], and structure formation [6, 7, 8]. All these phenomena can be described as self-organization of bubbles in acoustic fields [9, 1], or self-action of the acoustic field, which is an extremely rich physical phenomenon. Various theories are available in the mentioned references (see also [11]) as well as in a recent publication [12]. However, there is lack in comparisons of experiments and numerical modeling, which should benefit both progress in theory and practical applications of self-organization phenomena (surface chemistry, ultrasonic cleaning, etc.). In the present study, we are interested in a phenomenon of self-induced acoustic transparency of high frequency acoustic fields of the order of hundred kilohertz, which, in fact, appears to be not just acoustic transparency, but the effect of cleaning of liquid from bubbles. As a field is turned on it pushes bubbles away from the acoustic source and forms a zone free of bubbles. This leads to a propagating boundary between clear and bubbly liquid. The effect observed in experiments conducted at the Center for Micro- and Nanoscale Dynamics of Dispersed Systems in Ufa (Russia) is very strong and repeatable. In the paper we provide some details of this experiment. Such effect was observed previously in some cases of numerical simulation of bubble self-organization [12]. Having experimental data and a general theory, we developed a simplified quasi-one-dimensional model to study the phenomenon and compare the wave front dynamics obtained in computations and experiments. Such comparison shows good agreement, while there is a room for improvement of the theory, and experiments, which can be guided by the present model and insight obtained during the research. EXPERIMENTS The experimental setup includes a home-made transparent acrylic cuvette with inner dimensions of mm 3 and wall thickness of 5 mm (Fig.1). A piezo electric ceramic disc transducer (PZT) (STEMiNC) with resonance frequency of 1 MHz is glued to the bottom from outside the cuvette using degassed slowly curing epoxy. A sinusoidal voltage of adjustable frequency and amplitude is applied to the piezo transducer through a waveform generator (WFG 33522A, Agilent Technologies) and a RF amplifier (AG 112, T&C Power Conversion). A needle hydrophone (HNR-1, Onda) and a storage oscilloscope (HRO 66Zi, LeCroy) are used to measure the pressure in the liquid filled tank. The hydrophone is attached to a xyz-stage that allows to position the needle tip precisely in all three dimensions. The bubble dynamics within the tank is recorded with a high speed camera (FASTCAM SA5, Photron) with a resolution pixels and at a frame rate of 2 fps. The camera is equipped with a macro objective lens (Micro-Nikkor 6mm f/2.8d, Nikon) that leads to a resolution of approximately 3 μm per pixel. To observe individual bubbles the camera was attached to the long-distance microscope (K2/SC, Infinity Photo-Optical Company) resulting in an increased resolution of approximately 3 μm per pixel. The image capturing is synchronized with the start of the voltage signal fed to the transducer. For all experiments deionized water is used, which is prepared from a ultrapure water purification system (Milli-Q Advantage A1, EMD Millipore). The bubbles in water are generated by using a home-made Venturi tube placed in a separate 5 ml reservoir. The bubbly liquid is prepared by running the Venturi tube for 5 minutes. Then the mixture is transferred to the tank for the observations. This procedure leads to bubbles with a mean radius of 16 μm and a void fraction of.3 -.5%. In all experiments the liquid volume is 2 ml. To avoid fast coalescence of the bubbles a salt (NaCl, concentration c=67g/l) was added to water. Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 2

3 signal generator tank filled by bubbly liquid water level is 23 mm 3 mm light source high speed camera 3 mm PZT amplifier FIGURE 1: A sketch of the experimental setup. The spatial distribution of the pressure is measured at a frequency 1 khz and at a low driving amplitude in the absence of bubbles to avoid a hydrophone damage. The magnitudes of the pressure is recorded at discrete points in the tank with a spatial resolution of 3 mm. Two maxima of the pressure are observed along the z-axis selected to be normal to the transducer surface. For sound speed in water 15 m/s and a frequency 1 khz the length of the standing wave is 15 mm which has been confirmed by our measurements. A small asymmetry of pressure distribution along the lateral axes is observed; this is likely due to some non-ideal acoustic properties of the tank and piezo attachment. This may also explain a slight asymmetry of the moving front observed and discussed below. To obtain the resonant frequencies the tank was filled with the salt water and the hydrophone was placed to the center of the cuvette. At a fixed signal amplitude the hydrophone response was recorded as a function of the frequency sweep from 2 khz to 22 khz. The most eminent resonant responses were found at frequencies 64.3 khz, 74 khz, 89 khz, 99.8 khz, and 29.2 khz. The maximum pressure, P max, in the salt water was determined as a function of the driving waveform signal amplitude (U wg ) at each frequency. For sufficiently small signal voltages the dependency of the pressure P max on U wg is linear. The experiment is conducted in the following way. First, the bubbly liquid is generated by running the Venturi tube in an external reservoir and then it is quickly transferred to the cuvette. Within time t on =6± 1 s the sound field is switched on at a preset value and together with that the high-speed recording started. To estimate the bubble concentration and size distribution at t on another tank with inner dimensions mm 3 was used to improve image contrast. Images of the bubbles, corresponding to time t on, are captured and analyzed with image processing routines consisting of thresholding, bubble contour detection, and fitting of ellipses to the contours. As a result we obtain a distribution of ellipses. It is from which we determine the center positions and geometry. The eccentricity is used to threshold between single and wrongly detected multiple bubbles. Figure 2 depicts the histogram which is not a Gaussian distribution. The mean bubble radius is 15.7 μm from the centroid positions of the bubbles and the average distance between them can be estimated as μm. For these parameters the void fraction estimated using different models (a bubble in a spherical cell and a bubble in a cubic cell) should be in the range from.3% to.5%. After the sound field is switched on the initially homogeneous bubbly liquid quickly changes, with a remarkable redistribution of the bubble density. Initially, a reduction of the bubble density appears at the bottom of the tank. This region is separated by a thin layer with the high density of bubbles (front) from the homogeneous bubbly liquid in the upper part of the tank. The front shape has three maxima in the visualization plane, as shown in Fig. 3. Further the front was moving up to the top level of the liquid. The same picture was observed at all used resonant frequencies and amplitudes. The front velocity increased at the increasing waveform generator signal amplitude. A Matlab script was developed to estimate displacements of the bubble front. The script is based on detecting the front position by contrast variation. In the middle of the each frame the region of interest was cut along z-direction to get dependence of the intensity on the z-coordinate. After that the front Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 3

4 FIGURE 2: The measured bubble size (radius) distribution. position was determined to be at the local extrema of the obtained curve. Results showed high velocity of the front at the beginning of the process which reduced to almost constant speed at later times. Using the long-distance microscope observations near the bottom of the tank one can reveal that besides overall vertical motion upward, some bubbles move down picking up the neighboring bubbles which leads to cluster formation. The clusters move up and merge with the bubble front. The wave front looked like a densely packed bubble sheet. FIGURE 3: Propagation of the bubble front after switching on the acoustic field for two different frequencies: 89 khz (top) and 29.2 khz (bottom). MODEL We consider time harmonic acoustic fields of circular frequency ω in a bubbly liquid assuming harmonic bubble response p = p [1 + ɛre {Ae iωt}] [ {, a = a 1 + ɛre Λ(a ) Ae iωt}], (1) where p is the static pressure, ɛ the relative pressure amplitude, a and a are the bubble radius and its period average, A 1 is the normalized complex amplitude of pressure, and Λ(a ) is the bubble Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 4

5 response function. This function for a bubble in an infinite liquid can be found in [2], while a correction for finite void fraction, α, is provided in [13, 14], so a 2 ω Λ(a ) = a 2 r ( 1 α 1/3) a 2 iη, [ a2 r = 3γ + 2σ ] ( ) 3γ 1 a 2 ω p a a2 ω = p ω 2, ρ l (2) where a r and a ω are the resonance radius of a single bubble and the characteristic bubble length scale, σ and ρ l are the surface tension and the liquid density, and γ(a ) and η(a ) are the effective polytropic exponent and dissipation coefficient, η = [ a 2 4μl ω a 3 { ( iωa 2 ω + δ a p a 3 + Im Θ )}], γ = 13 { ( )} iωa 2 ω κ Θ, g κ g (3) Θ(ζ) = 3γ g ζ ( ) (ζ ζ + 3 γ g 1 cothζ 1/2 1 ), δ2 a = p ρ l C 2. l Here μ l and C l are the liquid viscosity and sound speed, while κ g and γ g are the gas thermal diffusivity and adiabatic exponent. If the nonlinearity plays some secondary role, the complex amplitude A should satisfy the Helmholtz equation with complex space-dependent wavenumber, k, which in one-dimensional case is 2 A z 2 + k2 A =, k 2 = (1 α) [ (1 α) k 2 l + ] k2 b, (4) where k l = ω/c l is the wavenumber for pure liquid and k b is a complex addition due to bubbles (see [14]), which for polydisperse systems has a form (e.g. [2]), k 2 b (z, t s) = 4πω2 ρ l N (ξ, z, t s )ξ 3 Λ(ξ) dξ, α(z, t s ) = 4 p 3 π N (ξ, z, t s )ξ 3 dξ. (5) Here N (ξ, z, t s ) is the bubble size distribution function, i.e. the probability density of the presence of bubble of size ξ at spatial point z at the moment of time t s, where t s is a slow time in which the period averaged characteristics of the bubble change. In the collisionless model we have d b N dt s + N v b z =, d b dt s = t s + v b z, (6) where v b = v b (ξ, z, t s ) is the period-averaged (drift) velocity of bubbles of fraction ξ, which can be found from the momentum conservation equation accounting for the added mass, Bjerknes, quasisteady viscous drag, and buoyancy forces, d b v b dt s = 1 [ { 3p ɛ 2 Re k a ρ l ΛA A z } + 2g βμ v b ], β μ = 18k μμ l ρ l a 2. (7) Here the bar denotes the complex conjugate, g is the gravity acceleration, and the added mass and viscous drag coefficients, k a and k μ, are k a = 1 + 2α 1 α, k μ = 1 1 α 5/ Re 3 (1 α) Re 1/2, Re= 2ρ l a v b. (8) μ l At large Reynolds numbers, Re, these coefficients can be found from the potential flow around a sphere moving in a cell of radius R c = α 1/3 a and the respective dissipation [15]. More accurate approximations of k μ (α,re) can be used as well. Given initial positions and velocities of the bubbles and boundary conditions for the acoustic field the model is closed. Assuming that the acoustic source is located at z = and its surface oscillates with amplitude Δ and defining parameter ɛ as ɛ = ωρ l C l Δ/p, we obtain A z = k l, A z=h =, (9) z= 1/2 Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 5

6 where H is the height of the tank filled by the bubbly liquid at which we have a free surface. Several approaches for numerical modeling, including finite difference, pseudospectral and Lagrangian methods were proposed and studied for computer simulations of self-organization phenomena [6, 8, 12]. In the present study we developed and tested a version of the particle-in-cell (PIC) method. To account for collisions a quasi three dimensional model is implemented, where the mixture characteristics are computed in one dimension, but bubbles are distributed in three dimensions. The entire computational domain of size h x h y H is subdivided into N grid = H/h z boxes. In this domain we randomly distribute N b bubbles of sizes with a given distribution N (ξ, z,) and uniform in x and y. Mixture characteristics α(z) and k 2 b (z) obtained via the particle-to-grid interpolation are used in the second order finite difference solver to find A (z). Then the grid-to-particle interpolation of α(z) and A (z) is used to compute dynamics of each bubble. Time marching is performed using the fourth order Runge-Kutta method, where the time step, h t, was selected to satisfy the Courant condition h t < Ch z, where C is the speed of the wave front, and be smaller than the viscous relaxation time, β 1 μ. In the collisionless model bubble distribution along x and y coordinates do not play any role. We also tried collision models with and without coalescene. RESULTS AND DISCUSSION To compare the results of simulations with experimental data we used the measured initial void fraction, α, the bubble size distribution, N (a ), and the amplitude of the acoustic field in a liquid without bubbles, from which ɛ and Δ can be found. Experiments at 89 khz and 29.2 khz were performed at the same U wg, and we assumed that this corresponds to the same Δ. This assumption provided average Δ =.355 ±.125 μm, and ɛ = 2.98 for 89 khz and ɛ = 7. for 29.2 khz. Computations were performed with the initial number of bubbles of the order of 1 3 and about 1 2 cells in the PIC method, and time step s. A simple model where each collision event results in coalescence showed effects, which are not observed in the experiments; namely, almost all bubbles near the front of the wave coalesce to one or a few large bubbles. A model of collisions without coalescence caused computational difficulties, as repacking in the dense bubbly region takes a substantial computational time. Perhaps, in the first report on a simple model revealing the basic physics there is no sense to resolve this issue which also involves physics of films and can be done in future. So, we report results obtained within the collisionless model. P a, kpa 1 5 t = 2 ms 29.2 khz α P a, kpa 1 5 t = 4 ms α P a, kpa 1 5 t = 1 ms α z, mm FIGURE 4: The void fraction (the blue lines) and the acoustic pressure amplitude (the red lines) at different instances of time or a wave of self-induced transparency at 29.2 khz and 7 kpa driving acoustic field. The intial void fraction is α =.4 and the bubble size distribution is shown in Fig. 2. It is seen in Fig. 4 that the region behind the wavefront is almost free of bubbles. In this domain an acoustic standing wave is formed rapidly as the size of this domain, z f (t), is much smaller than C l t. Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 6

7 The amplitude of this wave depends on z f (t) and resonance phenomena are observed at front coordinates z f = (π/2 + πn)/k l, n =,1,... At such resonances the speed of the wave front at 89 khz increases, while a smooth behavior was obtained at 29.2 khz, which may be explained by different wave structures. In the region occupied by the bubbly liquid the amplitude of the wave attenuates rapidly in space. The mixture before the front almost does not feel the wave. One can also see some smaller amplitude short pressure waves just behind the wavefront. These waves are due to a relatively small amount of subresonant satellite bubbles, which form an acoustically transparent medium with a substantially smaller sound speed than in pure liquid. Figure 5 shows that the experimentally observed effect is quantitatively captured by the model khz 2 α =.3.5 Theory α =.4 z, mm khz 1 Experiment t, ms FIGURE 5: A comparison of the experimental data (the circles) and computations at different intial void fractions (the thin dashed lines) on the transparency wave position for two different frequencies. As the Bjerknes force here is the primary driving force, it is surprizing, why the subresonant and superresonant bubbles (the bubbles of sizes smaller and larger than the resonance value) are moving in the same direction? Indeed, it is well known that in a standing wave these two kind of bubbles behave differently, as the subresonant bubbles move towards pressure antinodes and the superresonant bubbles move towards pressure nodes. The answer comes from the analysis of the Bjerknes force acting on the bubbles in bubbly liquids. Indeed, we can assume that the complex sound amplitude in the bubbly liquid just before the wave front, z = z f,is A = A f e ik(z z f ), A f = A ( z f ), Im{k} >, (1) Here we have only the outgoing wave, as it attenuates to zero at some distance before the front. Considering for simplicity a monodisperse mixture we can write expression (7), (4), and (5) for the normalized Bjerknes force for the bubbles right on the front as F Bf = 3p ɛ 2 Re ρ l Λ A A z z=zf 3p 2 ɛ2 A f 2 Re { ik [ k 2 (1 α) 2 k 2 ] } l 4πω 2 ρ 2 = (1 α) l 3p ɛ 2 ρ l A f 2 Re { } ikλ 3p2 2 ɛ A f 2 [ k 2 + (1 α) 2 k 2 ] l 4πω 2 ρ 2 l (1 α) Im{k} >. The latter inequality holds due to Eq. (1), and does not depend on the real part of k 2, which is different for subresonant and superresonant bubbles. There are three basic mechanisms providing non-zero Im{k}. First, the dissipative mechanisms related to single bubble dynamics, mostly thermal dissipation in the gas. Second, in the case when the above effects are neglected, superresonant bubbles cause k 2 <, Im{k} >. Third, even when the mechanisms of the first kind are neglected the polydispersity causes the Landau damping [16]. In the absence of these mechanisms the effect should not exist. (11) Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 7

8 FIGURE 6: Visualization of the computed bubble structure near the wave front at frequencies 29.2 khz (on the left, t = 4 ms) and 89 khz (on the right, t = 12 ms) corresponding to the case α =.4 shown in Fig. 5. While the fact presented by Eq. (11) explains the essence of the phenomenon, experimental observations and numerical simulations show more complicated structure of the wave, as some bubbles may penetrate into the region behind the wave front, be trapped by the standing wave in the pure liquid or in the acoustically transparent bubbly layer, and show more complex behavior. However, since such standing wave is not stationary, its nodes and antinodes move as the front propagates, which cause motion of the trapped bubbles. Some single bubbles may still be trapped to the standing wave far behind the front. Figure 6 visualizes computational results near the front. Noteworthy that a good agreement of the theory and experiment is obtained for strong acoustic fields, while the theory neglects the nonlinearity of bubble oscillations. Explanation comes from the structure of the wave. Indeed, the presence of a few single bubbles behind the wave front, which behavior may not be properly described by the linear theory should not affect the driving acoustic field significantly. The bubbles at some distance before the front also do not feel the acoustic field at all. Finally, the bubbles close to the front are placed either in a substantially reduced acoustic field (due to its strong attenuation), or/and in substantially constrained conditions due to relatively high volume concentrations preventing their high amplitude oscillations. So, in the entire domain the effect of the bubble dynamics nonlinearity may not be very strong, while more detailed studies are required. CONCLUSION We conducted experiments demonstrating strong manifestation of acoustically induced transparency in bubbly liquid and developed a simplified theory which agrees well with the observations. It was found that the pressure and bubble concentration wave structures are correlated. The entire domain can be subdivided into an almost unperturbed bubbly liquid region before the acoustically induced transparency front, a densely packed bubble sheet representing this front, and a region of liquid almost free of bubbles behind the front. Such structure appears as a result of the two-way interaction of bubbles and acoustic field. Dissipation of the acoustic wave energy by the bubbles is found to be a major mechanism of formation and propagation of the acoustically induced transparency front. More work is needed to reveal the detailed structure and internal dynamics of this front and to understand the role of the neglected effects. Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 8

9 ACKNOWLEDGMENTS This study is supported by the Grant of Ministry of Education and Science of the Russian Federation (11.G ). REFERENCES [1] P. Koch, T. Kurz, U. Parltz, and W. Lauterborn, Bubble dynamics in a standing sound field: The bubble habitat, J. Acoust. Soc. Am. 13, (211). [2] K.W. Commander and A. Prosperetti, Linear pressure waves in bubbly liquids: Comparison between theory and experiments, J. Acoust. Soc. Am. 85, (1989). [3] X. Xi, F. Cegla, R. Mettin, F. Holsteyns, and A. Lippert, Collective bubble dynamics near a surface in a weak acoustic standing wave field, J. Acoust. Soc. Am. 132, (212). [4] Yu.A. Kobelev and L.A. Ostrovsky, Nonlinear acoustic phenomena due to bubble drift in a gas-liquid mixture, J. Acoust. Soc. Am. 85, (1989). [5] N.A. Gumerov, On waves of the self-induced acoustic transparency in mixtures of liquid and vapor bubbles, In: S. Morioka and L. van Wijngaarden (eds), IUTAM Symposium on Waves in Liquid/Gas and Liquid/Vapour Two-Phase Systems, Kyoto, Japan, Kluwer, Netherlands, [6] I. Akhatov, U. Parlitz, and W. Lauterborn, Pattern formation in acoustic cavitation, J. Acoust. Soc. Am. 96, (1994). [7] U. Parlitz, C. Scheffczyk, I. Akhatov, and W. Lauterborn, Structure formation in cavitation bubble fields, Chaos, Solitons and Fractals, 5, (1995). [8] R. Mettin, S. Luther, C. Ohl, and W. Lauterborn, Acoustic cavitation structures and simulations by a particle model, Ultrason. Sonochem, 6, 25-3 (1999). [9] I. Akhatov, U. Parlitz, and W. Lauterborn, Towards a theory of self-organization phenomena in bubble-liquid mixtures, Phys. Rev. E, 54, (1996). [1] N.A. Gumerov, On self-organization of voids in acoustic cavitation, Proceedings of the Third International Conference on Multiphase Flow, ICMF 98, paper p512, Lyon, France, [11] A.A. Doinikov, Mathematical model for collective bubble dynamics in strong ultrasound fields, J. Acoust. Soc. Am. 116, (24). [12] N.A. Gumerov and I.S. Akhatov, Numerical simulation of 3D self-organization of bubbles in acoustic fields, In: C.-D. Ohl, E. Klaseboer, S.W. Ohl, S.W. Gong, and B.C. Khoo (eds), Proceedings of the 8th International Symposium on Cavitation, Singapore, 212. [13] W. Lauterborn, T. Kurz, and I. Akhatov, Nonlinear acoustics of fluids, in T.D. Rossing (ed), Springer Handbook of Acoustics, Springer, Berlin, 27. [14] R.I. Nigmatulin, Dynamics of Multiphase Media, Vol. 1, 2. Hemisphere, Washington, D.C., [15] V.G. Levich, Physico-Chemical Hydrodynamics, Fizmatgiz, Moscow, [16] D.D. Ryutov, Analog of the Landau damping in the problem of sound wave propagation in a bubbly liquid, Pis ma v JETF, 22, (1975) (in Russian). Proceedings of Meetings on Acoustics, Vol. 19, 4512 (213) Page 9