Translational instability of a spherical bubble in a standing ...

ubbles translating in standing waves, either towardsa pressure antinode, or towards a node. In thecomputations it is essential to simulate a continuouspressure sweep by keeping the previously calculatedbubble state as initial condition for the next parameterstep. It will be seen that in particular the regionof the (nonlinearly distorted) main resonanceexhibits an extended area of hysteresis, and that thiscan result in the absence of positional equilibria, i.e.,zeros of the primary Bjerknes force.Figure 1 presents the overviews in the P ex,a − R 0parameter plane for driving frequencies of 20 kHz,100 kHz, 300 kHz, and 1 MHz. The interesting regionis left of the linear resonance radius R res , which canbe found from the Minnaert frequency formula,[(2πf) 2 1=ρ 0 Rres2 3γP 0 + 2σ ](3γ − 1) , (11)R resand which is met by the lower right curve endings forzero excitation. The main picture is for all frequenciesthe same: a line running to the upper left, thusnarrowing the bubble size range that is attracted bythe pressure antinode for higher and higher driving.For medium high acoustic pressures, this border is“broadened” by bends, due to nonlinear resonances,and a “scattered” region appears where complicatedbubble oscillations occur. Very pronounced is a hysteresiszone of triangle shape, connected to the distortedmain (linear) resonance. The resonance bendstowards lower radii for larger driving because of thesoft spring character of the oscillating bubble [35],and its hysteresis zone is limited by saddle-node bifurcations[34]. This bifurcation creates (or annihilates)a pair of periodic solutions, one stable (attractor)and one unstable (repellor). As one stable solutionalready exists before the bifurcation appears,we find a coexistence of two different stable bubbleoscillations in such an area. It depends on the initialconditions, i.e., bubble radius and bubble wall velocityat the starting time, which of both possible stateswill be finally reached. It is now peculiar to note thatin wide parameter regions the different pulsationsresult also in different signs of the Bjerknes force!This is illustrated in Fig. 2, where a representativecase is shown. The different solutions manifest themselvesin non-unique values of the maximum bubbleradius, R max , which is depicted in the lower diagramof the figure. The vertical jumps are indications ofthe saddle-node bifurcations. The source of translationalinstability of a bubble with the indicated parametersis now the absence of a zero of f B with finiteslope: after crossing the bifurcation for upgoingpressure, a bubble is driven back to lower pressures,and a bubble for decreasing pressure falls on a valuedriving it back upwards. In a detailed simulation oftranslation, one would expect an oscillating translationalmotion in this case, and indeed this will bethe outcome (see Fig. 9c).Similar hysteresis scenarios repeat at nonlinearresonances of higher order, although they are lesspronounced. A prominent region is formed aroundthe second harmonic resonance (R 0 ≈ R res /2).Here, an overlay of further (ultrasubharmonic) resonances,period doubling, and chaotic solutionsresult in rather complicated multiattractor dynamics(compare [36]). In Fig. 1, scattered and irregularvalues of the Bjerknes force coefficient appear forsuch parameters. The example shown in Fig. 3 ischosen to pass this second harmonic parameter region.For a bubble of R 0 = 5.8 µm and otherwisethe same parameter conditions as in Fig. 2, f B andR max are given. The case is now more complicatedas additionally one stable and one unstable positionare present: at about 55 kPa and at 105 kPa theBjerknes coefficient crosses zero with positive slope(stable) and negative slope (unstable), respectively.This can be seen in the inset. Furthermore, beyond110 kPa, the upgoing branch shows chaotic bubblepulsations, visible by the broadened, i.e., multivaluedlines of f B and R max .From Figs. 2 and 3, the absolute strength of theBjerknes force on both branches, upgoing and downgoing,respectively, can be compared. It turns outthat the motion away from the pressure antinodeshould be much faster than the translation towardsit. This leads to asymmetric, sawtooth-like translationcurves of bubble position vs. time, and it is confirmedby results in the next Section. It is a consequenceof the larger bubble pulsations on the downgoingbranch of the hysteresis loop and should betypical for positional reciprocation of this type.4. Full simulation of translational motionIn this section, we propose a general classificationof the translational trajectories that are demonstratedby bubbles in the standing wave. Trajectoriesare modeled by numerically solving the systemof coupled equations that have been introduced inSection 2. They describe generally the instantaneousradial and translational motion of a spherical bubblein an ultrasound field, and are applied to thestanding wave case. The aim of this Section is both4

to corroborate the theory presented in the precedingSection, and to demonstrate examples of trajectoriesthat are caused by the translational instabilitydue to the hysteresis effect.4.1. Translation stable bubblesBy translational stability, we mean that a bubblestays at a certain point of space or executes insignificanttranslational oscillations around a certain pointof space. Simulations show that there are three casesof translational stability, which will be referred toas “large” bubbles, “small” bubbles, and “intermediate”bubbles. They are described successively below.4.1.1. “Large” bubblesA “large” bubble is a bubble driven above resonance,i.e., its fundamental resonance frequency islower than the driving frequency of the imposed ultrasoundfield. In conformity with Eller’s formula[3], large bubbles move to the pressure node wheretheir radial oscillation dies down. Figure 4 showsan example of the translational trajectory of a largebubble for the driving frequency f = 300 kHz andthe acoustic pressure amplitude P a = 200 kPa. Theposition of the bubble in the ultrasound field is displayedin terms of a normalized distance from thepressure antinode. The distance is defined as x(t)/λ,where λ is the wavelength of sound in the surroundingliquid. The position 0 of the ordinate axis correspondsto the pressure antinode and 0.25 to thepressure node. The initial position of bubbles is setto be in the immediate vicinity of the pressure antinode.Doing so, we assume that bubbles arise due toacoustic cavitation in the area of high alternatingpressure. The horizontal line means that the bubblehas reached the node and is staying there.4.1.2. “Small” bubblesA “small” bubble is a bubble that moves to thepressure antinode and settles there, or executesinsignificant translational excursions around theantinode plane. Small bubbles are driven belowresonance and so, formally, their behavior obeysEller’s theory [3]. The trajectory of a small bubbleis exemplified in Fig. 5.4.1.3. “Intermediate” bubblesAn ”intermediate” bubble is a bubble that, inspite of the fact that it is driven below resonance,does not move towards the pressure antinode. Instead,it moves to an equilibrium point between theantinode and the node and remains there, undergoingstrong enough radial oscillations. Thus, intermediatebubbles no longer obey Eller’s linear theory.Examples of the trajectories of intermediate bubblesare shown in Fig. 6. The horizontal lines mean thatthe bubbles reached their equilibrium positions andare staying there. It can be said that intermediatebubbles are former small bubbles the Bjerknes forceon which changed the sigh and got directed awayfrom the pressure antinode because of high acousticpressure.The equilibrium position of an intermediate bubbleis dependent on the initial bubble radius, acousticpressure amplitude, and the driving frequency.Figure 7 shows the equilibrium position, x E , of anintermediate bubble as a function of R 0 at three valuesof acoustic pressure for f = 300 kHz and f = 1MHz. By way of example let us consider the 300 kHz200 kPa curve. The left end of the curve terminatesat about 2.3 µm, and the right end, at 6.25 µm.These values indicate the size range of intermediatebubbles for the specified values of frequency andpressure. Below 2.3 µm, the region of small bubbleslies. What happens to intermediate bubbles beyond6.25 µm is described in the following subsection.Turning back to Fig. 7, one can see that, as theacoustic pressure amplitude increases, increasinglysmaller bubbles turn to intermediate bubbles. Also,varying the acoustic pressure, one can change theposition of an intermediate bubble, moving it off ortowards the antinode. The gap in the 200 kPa curveat 1 MHz means that bubbles in this size range againbecome “small” and go to the antinode.The dependence of the equilibrium position of intermediatebubbles on the driving frequency is illustratedin Fig. 8. It is seen that, by increasing thedriving frequency, one can move increasingly smallerbubbles off the pressure antinode. However, the upperlimit of R 0 at which the equilibrium positioncurves terminate, decreases as well.4.2. Translation unstable (“traveling”) bubblesWhen R 0 exceeds a threshold value, bubbles haveno equilibrium space position. Such bubbles willbe called “traveling” bubbles hereinafter. Travelingbubbles execute translational oscillations in thespace between the antinode and the node. Examplesof the trajectories of traveling bubbles are given in5

Fig. 9. According to Fig. 7, for 300 kHz the travelingregime sets in at R 0 = 6.25 µm, and for 1 MHz,at R 0 = 2.09 µm. Combined diagrams for intermediateand traveling bubbles at P a = 200 kPa andf = 300 kHz and 1 MHz are presented in Fig. 10.The dashed lines show the far and near limits of thetranslational trajectories of traveling bubbles. For300 kHz, the dashed curves merge with the pressurenode at R 0 = 12 µm. From this value, the region oflarge bubbles begins. For 1 MHz, the dashed curvesmerge at R 0 = 3.59 µm, then the region of intermediatebubbles follows, and at R 0 = 3.74 µm theregion of large bubbles begins.The existence and the behavior of traveling bubblesare a result of the translational instability describedin Section 3. This is also confirmed by Fig. 11which shows a sharp jump in the amplitude of thebubble radial oscillation at the moment of the transitionfrom the intermediate to the traveling regime.4.3. Cumulative translation diagramFigure 12 shows the type of translational behaviorof a bubble depending on the equilibrium radiusand the acoustic pressure amplitude at 300 kHz and1 MHz. One can see that at f = 300 kHz there isan area denoted as “overnode bubbles”. It demonstratesthat at higher pressures, exceeding about 245kPa, some of intermediate bubbles go through thepressure node and settle between the node and thenext antinode, not between the node and the antinodefrom which they started. Such bubbles will becalled “overnode” bubbles. An example of the trajectoryof an overnode bubble is presented in Fig. 13.Figure 12 reveals that small bubbles can becometraveling bubbles skipping the intermediate regime.It is also interesting to note that at 1 MHz, for higherpressures, traveling bubbles are a dominant group.4.4. Effect of start positionAll the results presented above were obtained assumingthat bubbles start in the immediate vicinityof the pressure antinode. In many cases, the final positionof a bubble in space is independent of its initialposition. However, there are parameter regionswhere this is not the case. For instance, at 300 kHz,in the range of acoustic pressure from 104 kPa to 133kPa, a 5.8-µm-radius bubble can reach two equilibriumpositions depending on its start position. Thissituation is illustrated in Fig. 14. Figure 14a showsthat at 110 kPa the bubble goes to the antinode ifit starts at x(0)/λ < 0.052. If, however, the bubblestarts at x(0)/λ > 0.052, it goes to an equilibriumpoint with x E /λ ≈ 0.168. At 130 kPa, Fig. 14b,the boundary between the two types of trajectoriespasses at x(0)/λ = 0.1.Other examples for the start position dependenceare presented in Fig. 15. Figure 15a shows a situationopposite to that shown in Fig. 14. A 4.5-µm-radius bubble starting near the antinode at 110kPa has an equilibrium position situated at a distancefrom the antinode. If, however, the same bubblestarts closer to the node, it has the equilibriumposition at the very antinode. At 200 kPa, Fig. 15b,the same bubble has two equilibrium positions offthe antinode.Following the terminology of chaos physics [37],one can say that in the cases considered the bubblehas two coexisting attractors of the type “fixedpoint” in the translation state space, the basins ofwhich are divided at a certain value of the initialposition of the bubble x(0).5. ConclusionsThis paper shows that the translational behaviorof bubbles in standing ultrasound waves is ratherinvolved. At higher acoustic pressures, in additionto the equilibrium positions at the pressure antinodeand the pressure node which are know from thelinear theory for the primary Bjerknes force, spatialequilibria between the antinode and the nodeoccur. As a result, three groups of bubbles can bedistinguished: “large” bubbles going to the node,“small” bubbles going to the antinode, and “intermediate”bubbles having equilibrium positions betweenthe antinode and the node. Moreover, thereare translation unstable bubbles, which were namedhere “traveling” bubbles. These latter have no equilibriumspace position and have to execute translationalreciprocating oscillations between the antinodeand the node. This study discloses the physicalmechanism responsible for the behavior of travelingbubbles. It has been shown that the occurrenceof traveling bubbles is caused by a specific translationalinstability that results from the hysteresis inthe main bubble resonance. This study also revealedthat the translational behavior of bubbles possessesfeatures similar to those recognized for the nonlinearvolume bubble oscillation, such as bifurcations. Itwas shown that in certain parameter regions a bub-6

le can have two equilibrium space positions whichare realized depending on the start position of thebubble in space.From the given theory, there are no limitationson driving frequencies and the size of bubbles forwhich the translational phenomena described inthis paper can occur. Practically, one should expectthat parameter change and further physical effectscan have a crucial influence. We have checked thatour findings are robust against a change of the polytropicexponent to isothermal bubble oscillations,and against a slow bubble growth/dissolution byrectified diffusion. Nevertheless, spherical surfaceinstability might limit the observability of travelingbubbles to certain parameter ranges. Preliminarycalculations of parametric instabilities confirmedthe expectation that the described phenomena willbe observed more readily for higher frequenciesand micron-sized bubbles. Under such conditionsthe bubbles are more resistant to surface instabilitiesand can retain the spherical form even at highenough acoustic pressures. This fact contributes tothe importance of the findings, since MHz frequenciesand micron-sized bubbles just form a regionof currently growing interest of various engineeringand medical ultrasound applications.AcknowledgementsThe authors like to thank F. Holsteyns and A.Lippert for stimulating discussions. R.M. thanksthe members of the cavitation group at DrittesPhysikalisches Institut, Göttingen University, forcontinuous support and fruitful joined theoreticaland experimental work.References[1] Plesset MS, Prosperetti A. Bubble dynamics andcavitation. Ann Rev Fluid Mech 1977;9:145-185.[2] Feng ZC, Leal LG. Nonlinear bubble dynamics. AnnRev Fluid Mech 1997;29:201-43.[3] Eller A. Force on a bubble in a standing acoustic wave.J Acoust Soc Am 1968;43(1):170-1.[4] Gaines N. Magnetostriction oscillator producing intenseaudible sound and some effects obtained. Physics1932;3:209-29.[5] Kornfeld M, Suvorov L. On the destructive action ofcavitation. J Appl Phys 1944;15:495-506.[6] Benjamin TB, Strasberg M. Excitation of oscillationsin the shape of pulsating gas bubbles; theoretical work(abstract). J Acoust Soc Am 1958;30(7):697.[7] Strasberg M, Benjamin TB. Excitation of oscillations inthe shape of pulsating gas bubbles; experimental work(abstract). J Acoust Soc Am 1958;30(7):697.[8] Benjamin TB. Surface effects in non-spherical motionsof small cavities. In: Davies R, editor. Cavitation in realliquids. Elsevier; 1964. p. 164-180.[9] Eller AI, Crum LA. Instability of the motion of apulsating bubble in a sound field. J Acoust Soc Am1970;47(3):762-7.[10] Benjamin TB, Ellis AT.Self-propulsion of asymmetrically vibrating bubbles. JFluid Mech 1990;212:65-80.[11] Mei CC, Zhou X. Parametric resonance of a sphericalbubble. J Fluid Mech 1991;229:29-50.[12] Feng ZC, Leal LG. Translational instability of a bubbleundergoing shape oscillations. Phys Fluids 1995;7:1325-36.[13] Doinikov A. Translational motion of a bubbleundergoing shape oscillations. J Fluid Mech 2004;501:1-24.[14] Miller DL. Stable arrays of resonant bubbles in a 1-MHz standing-wave acoustic field. J Acoust Soc Am1977;62(1):12-19.[15] Khanna S, Amso NN, Paynter SJ, Coakley WT.Contrast agent bubble and erythrocyte behavior in a 1.5MHz standing ultrasound wave. Ultrasound Med Biol2003;29:1463-70.[16] Kuznetsova LA, Khanna S, Amso NN, Coakley TW,Doinikov AA. Cavitation bubble-driven cell and particlebehavior in an ultrasound standing wave. J Acoust SocAm 2005;117(1):104-12.[17] Watanabe T, Kukita Y. Translational and radialmotions of a bubble in an acoustic standing wave field.Phys Fluids A 1993;5(11):2682-88.[18] Doinikov AA. Translational motion of a spherical bubblein an acoustic standing wave of high intensity. PhysFluids 2002;14(4):1420-25.[19] Leighton TG. The Acoustic Bubble. London: AcademicPress; 1994.[20] Keller JB, Miksis M. Bubble oscillations of largeamplitude. J Acoust Soc Am 1980;68(2):628-33.[21] Matula TJ. Bubble levitation and translation undersingle-bubble sonoluminescence conditions. J AcoustSoc Am 2003;114(2):775-81.[22] Mettin R, Luther S, Ohl C-D, Lauterborn W. Acousticcavitation structures and simulations by a particlemodel. Ultrason Sonochem 1999;6:25-29.[23] Appel J, Koch P, Mettin R, Krefting D, Lauterborn W.Stereoscopic high-speed recording of bubble filaments.Ultrason Sonochem 2004;11:39-42.[24] Luther S, Mettin R, Lauterborn W. Modeling acousticcavitation by a Lagrangian approach. In: LauterbornW, Kurz T, editors. Nonlinear acoustics at the turn ofthe millennium - Proceedings of the 15th internationalsymposium on nonlinear acoustics. AIP ConferenceProceedings; 2000. Vol. 524. p. 351-54.7

[25] Doinikov AA. Equations of coupled radial andtranslational motions of a bubble in a weaklycompressible liquid. Phys Fluids 2005;17(12):128101.[26] Doinikov AA. Lagrangian formalism in bubbledynamics. Far East J Appl Math 2006;25(2):159-77.[27] Doinikov AA, Dayton PA. Spatio-temporal dynamics ofan encapsulated gas bubble in an ultrasound field. JAcoust Soc Am 2006;120(2):661-69.[28] Mei R, Klausner JF, Lawrence CJ. A note on the historyforce on a spherical bubble at finite Reynolds number.Phys Fluids 1994;6:418-420.[29] Magnaudet J, Eames I. The motion of high-Reynoldsnumberbubbles in inhomogeneous flows. Ann Rev FluidMech 2000;32:659-708.[30] Lauterborn W, Kurz T, Mettin R, Ohl CD.Experimental and theoretical bubble dynamics. In:Prigogine I, Rice SA, editors. Advances in ChemicalPhysics. Vol. 110. New York: John Wiley and Sons; 1999.p. 295-380.[31] Mettin R. From a single bubble to bubble structures inacoustic cavitation. In: Kurz T, Parlitz U, Kaatze U,editors. Oscillations, Waves and Interactions. Göttingen:Universitätsverlag Göttingen; 2007. p. 171-198.[32] Guckenheimer J, Holmes PJ. Nonlinear Oscillations,Dynamical Systems and Bifurcations of Vector Fields.New York: Springer; 1983.[33] Thompson JMT, Stewart HB. Nonlinear Dynamics andChaos. New York: John Wiley and Sons; 1986.[34] Parlitz U, Englisch V, Scheffczyk C, Lauterborn W.Bifurcation structure of bubble oscillators. J. Acoust.Soc. Am. 1990;88(2):1061-77.[35] Lauterborn W. Numerical investigation of nonlinearoscillations of gas bubbles in liquids. J. Acoust. Soc.Am. 1976;59(2):283-93.[36] Lauterborn W, Mettin R. Nonlinear Bubble Dynamics- Response Curves and More. In: Crum LA, MasonTJ, Reisse JL, Suslick KS, editors. Sonochemistryand Sonoluminescence. Dordrecht: Kluwer AcademicPublishers; 1999. p. 63-72.[37] Lauterborn W, Parlitz U. Methods of chaos physicsand their application to acoustics. J Acoust Soc Am1988;84(6):1975-93.8

200updown250updown150200P ex,a [kPa]100P ex,a [kPa]1501005020 kHz50100 kHz00 20 40 60 80 100 120 140 160 180R 0 [µm]00 5 10 15 20 25 30 35R 0 [µm]300250updown400350updown300P ex,a [kPa]200150100P ex,a [kPa]25020015050300 kHz100501 MHz00 2 4 6 8 10 12R 0 [µm]00 0.5 1 1.5 2 2.5 3 3.5 4R 0 [µm]Fig. 1. Locations of zero Bjerknes coefficient, f B = 0, for the acoustic frequencies 20 kHz, 100 kHz, 300 kHz, and 1 MHz.Zero crossings for increasing (“up”) and for decreasing (“down”) driving pressure are indicated by solid and dashed lines,respectively. The vertical lines in the 300 kHz plot at 8 and 5.8 µm mark the parameter scans of Figs. 2 and 3.9

12001000800R 0 = 8 µm600500400300100-10f B [µm 3 ]600400f B [µm 3 ]2001000 20 40 60 80 100 1202000R max [µm]0updown-2000 25 50 75 100 125 150P ex,a [kPa]25201510R 0 = 8 µmupdown2015510500 1 per. 2 3 400 25 50 75 100 125 150P ex,a [kPa]R(t) [µm]Fig. 2. Hysteresis in the Bjerknes force coefficient f B (top)and the maximum radius R max (bottom) for variable drivingpressure P ex,a and f = 300 kHz, R 0 = 8 µm. Dynamicsresulting from upgoing (downgoing) driving pressure is givenby solid (dashed) lines. The inset shows the bubble radiusR(t) vs. time in driving periods for the coexisting solutionsat P ex,a = 50 kPa.R max [µm]-100R 0 = 5.8 µmupdown-2000 25 50 75 100 125 150P ex,a [kPa]252015105R(t) [µm]201510500 1 2 3 4periodsR 0 = 5.8 µmupdown00 25 50 75 100 125 150P ex,a [kPa]Fig. 3. Hysteresis in f B (top) and R max (bottom) for variableP ex,a and f = 300 kHz, R 0 = 5.8 µm. Dynamics resultingfrom upgoing (downgoing) driving pressure is given by solid(dashed) lines. The top inset magnifies part of the graph,showing two driving pressures with zero Bjerknes force. Thebottom inset shows the bubble radius R(t) vs. time in drivingperiods for the coexisting solutions at P ex,a = 125 kPa, oneoscillation being chaotic.10

x ( t ) / 0.250.20.150.10.05Pressure nodef = 300 kHzPa= 200 kPaR = 12 m0Pressure antinode200 400 600 800 1000Acoustic cyclesFig. 4. Translational trajectory of a large bubble in a planestanding wave.0.00001x ( t ) / f = 300 kHzPa= 200 kPaR = 1.5 m00.0500 1000 1500 2000Acoustic cyclesFig. 5. Translational trajectory of a small bubble in a planestanding wave.0.120.10R 0= 3.5 m3.2 mx ( t ) / 0.080.060.043.0 m2.8 mR 0= 2.5 m0.02500 1000 1500 2000 2500Acoustic cyclesFig. 6. Translational trajectories of intermediate bubbles forf = 300 kHz and P a = 200 kPa.11

0.250.20Pressure nodef = 300 kHz0.250.20Pressure nodef = 1 MHzx E/ 0.150.10225 kPax E / 0.150.10300 kPa200 kPa0.05200 kPa175 kPaPressure antinode0.05250 kPaPressure antinode200 kPa2 3 4 5 6R 0[ m]0.6 0.8 1 1.2 1.4 1.6 1.8 2R 0[ m]Fig. 7. Equilibrium position of intermediate bubbles at different acoustic pressures.12

x E / 0.250.200.150.10750 kHzPressure nodeP a= 200 kPa500 kHz300 kHz0.05Pressure antinode2 3 4 5 6R 0[ m]Fig. 8. Equilibrium position of intermediate bubbles at differentdriving frequencies.13

0.25(a)0.25(b)x ( t ) / 0.20.150.10.05x ( t ) / R 0 = 6.3 mR 0 = 7.0 m0.20.150.10.05500 1000 1500 2000Acoustic cycles500 1000 1500 2000Acoustic cycles0.25(c)0.25(d)0.20.2x ( t ) / 0.150.1R 0= 8.0mx ( t ) / 0.150.1R 0= 9.0m0.050.05500 1000 1500 2000Acoustic cycles500 1000 1500 2000Acoustic cycles(e)(f)0.250.250.20.2x ( t ) / 0.150.1R 0 = 10.0mx ( t ) / 0.150.1R 0 = 10.5m0.050.05500 1000 1500 2000Acoustic cycles500 1000 1500 2000Acoustic cyclesFig. 9. Translational trajectories of traveling bubbles for f = 300 kHz and P a = 200 kPa.14

x E / 0.250.200.150.100.05Pressure nodePressure antinode2 4 6 8 10 12R 0 [ m]f = 300 kHzP = 200 kPaax E / 0.250.200.150.100.050Pressure nodeR 0[ m]f = 1 MHzP = 200 kPaaPressure antinode1.5 2 2.5 3 3.5Fig. 10. Equilibrium position of intermediate bubbles (curved solid lines) and the far and near limits of the translationaltrajectories of traveling bubbles (dashed lines). The two horizontal solid lines in the 1 MHz plot display the equilibrium positionof small bubbles.15

2520f = 300 kHzP = 200 kPaaR max [ m]15105IntermediatebubblesTraveling bubbles4 6 8 10 12R 0[ m]Fig. 11. The maximum radius of a bubble versus its equilibriumradius: Transition from the intermediate to the travelingregime.Fig. 12. Cumulative translation diagrams for f = 300 kHzand 1 MHz.16

0.3x ( t ) / 0.250.20.1Pressure nodeR 0= 4.0 mPressure antinode200 400 600 800 1000Acoustic cyclesFig. 13. Example of the trajectory of an overnode bubble atf = 300 kHz and P a = 275 kPa.17

0.25(a)0.25(b)0.200.20x ( t ) / 0.150.100.05f = 300 kHzPa= 110 kPaR = 5.8 m0x ( t ) / 0.150.100.05f = 300 kHzPa= 130 kPaR = 5.8 m0500 1000 1500 2000 2500 3000Acoustic cycles500 1000 1500 2000Acoustic cyclesFig. 14. Example of a bubble with two equilibrium space positions depending on the start position of the bubble.0.25(a)0.25(b)x ( t ) / 0.200.150.100.05f = 300 kHzPa= 110 kPaR = 4.5 m0x ( t ) / 0.200.150.100.05f = 300 kHzPa= 200 kPaR = 4.5 m0500 1000 1500 2000 2500 3000Acoustic cycles500 1000 1500 2000Acoustic cyclesFig. 15. Examples of the start position dependence of bubble trajectories.18