A nonlinear analytic model describing the interaction of a spherical gas bubble and spherical rigid particle is presented. Both the bubble and particle are free to translate. The model is accurate to fifth order in terms of a nondimensional expansion parameter R/d, where R is a characteristic radius and d is the distance separating the bubble and particle. Numerical simulation results are presented to demonstrate the effects of key particle parameters and an external acoustic source.Applied Research Laboratorie

Hamilton, M. R.

Hay, T. A.

Ilinskii, Y. A.

Zabolotskaya, E. A.

English

Zabolotskaya, Evgenia A.

Texas ScholarWorks

Coupled pulsation and translation of a gas bubble and rigid particleT. A. Hay, M. F. Hamilton, Yu. A. Ilinskii, and E. A. Zabolotskaya  Citation: AIP Conference Proceedings 1022, 209 (2008); doi: 10.1063/1.2956188 View online: http://dx.doi.org/10.1063/1.2956188 View Table of Contents:http://scitation.aip.org/content/aip/proceeding/aipcp/1022?ver=pdfcov Published by the AIP Publishing  Articles you may be interested in Model of coupled pulsation and translation of a gas bubble and rigid particle J. Acoust. Soc. Am. 125, 1331 (2009); 10.1121/1.3077216  Numerical Study Of The Resonant Pressure Field Inside A Forced Rigid Cavity Filled With AHomogeneous Bubbly Liquid AIP Conf. Proc. 1022, 81 (2008); 10.1063/1.2956308  Hamiltonian Formalism in Bubble Dynamics AIP Conf. Proc. 1022, 58 (2008); 10.1063/1.2956294  The Bubble‐Bubble Interaction Under An Ultrasonic Horn AIP Conf. Proc. 1022, 245 (2008); 10.1063/1.2956199  Gas bubble pulsation in a semiconfined space subjected to ultrasound J. Acoust. Soc. Am. 109, 891 (2001); 10.1121/1.1346683   This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.Downloaded to  IP:  128.83.205.78 On: Fri, 20 Mar 2015 16:52:22Coupled pulsation and translation of a gasbubble and rigid particleT. A. Hay, M. F. Hamilton, Yu. A. Ilinskii and E. A. ZabolotskayaApplied Research Laboratories, The University of Texas at Austin, Austin, Texas 78713-8029, USAAbstract. A nonlinear analytic model describing the interaction of a spherical gas bubble andspherical rigid particle is presented. Both the bubble and particle are free to translate. The modelis accurate to ﬁfth order in terms of a nondimensional expansion parameter R/d, where R is acharacteristic radius and d is the distance separating the bubble and particle. Numerical simulationresults are presented to demonstrate the effects of key particle parameters and an external acousticsource.Keywords: bubble dynamics, particlesPACS: 43.25.YwINTRODUCTIONThe theoretical framework for the model of bubble-bubble interaction developed byIlinskii et al. [1] is applied to model the interaction between a spherical bubble and aspherical rigid particle. The investigation was motivated largely by the desire to accountfor the interaction of kidney stone fragments with cavitation clusters produced duringshock wave lithotripsy. Extension of the model to clusters of bubbles and particles willbe presented elsewhere. Here, we focus on the dynamics of a single bubble interactingwith a single rigid particle.MODEL EQUATIONSThe bubble and particle are positioned along the x axis in an incompressible liquid ofdensity ρ as shown in Fig. 1. Variables with subscript 1 correspond to the gas bubble,X1U1R˙1R1ρ2dX2U2R02xFIGURE 1. Notation and geometry for bubble and particle with collinear translational motion.and variables with subscript 2 correspond to the particle. Their positions are deﬁned byX1,2 and the separation distance d = |X2−X1|, while their instantaneous and equilibrium209CP1022, Nonlinear Acoustics—Fundamentals and Applications (ISNA 18), 18th International Symposium edited by B. O. Enflo, C. M. Hedberg, and L. Kari  © 2008 American Institute of Physics 978-0-7354-0544-8/08/$23.00 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.Downloaded to  IP:  128.83.205.78 On: Fri, 20 Mar 2015 16:52:22radii are given by R1 and R01,02, respectively. Their motions are described by the radialvelocity of the bubble, R˙1, and translational velocities U1,2 = X˙1,2, where dots overquantities indicate time derivatives. The equations of motion are obtained as expansionsin terms of R/d, where R is a characteristic radius, and they were derived followingthe formalism of Ilinskii et al. [1] The present model requires accuracy to order R5/d5,whereas the corresponding model for bubble-bubble interaction need only be accurateto order R2/d2. The radial equation of motion for the bubble isR1R¨1+32R˙21 =P1−P0ρ+14U21 +R3022d2U˙2−R3022d3U2 (U1 +2U2)− R3024d4(2R21R¨1+4R1R˙21)− R3022d5[R31U˙1 +4R21R˙1 (U1−U2)], (1)whereP1 =(P0+2σR01)(R01R1)3γ− 2σR1(2)is the liquid pressure at the bubble wall, γ is the ratio of speciﬁc heats and σ is surfacetension. The equations describing translation are M˙1 =−F and M˙2 = F , whereM1 =12ρV1U1− 32ρV02(R31d3U2− R51d5R˙1), (3)M2 =(12ρ +ρ2)V02U2− 32ρV02(R21d2R˙1 +R31d3U1), (4)F =3ρV02(R21d3R˙1U2 +R31d4U1U2− R41d5R˙21), (5)are the generalized momenta (M1, M2), and translational force (F) that acts equallyand oppositely on the bubble and particle. Quantities V1 = 43πR31 and V02 =43πR302are the volumes of the bubble and particle. Assuming small linear bubble pulsation,R1(t) = R01 + ξ0sinωt where ξ0  R01, and averaging Eq. (5) over one acoustic cycleyields the time averaged force〈F〉= 3ρV02(ρ −ρ02ρ +2ρ02)R401d5ω2ξ 20 +O(ξ30 ). (6)Equation (6) is the expression derived by Coakley and Nyborg [2] by an entirely differentapproach. It indicates that for a particle having density greater than that of the liquid,ρ02 >ρ , the time-averaged interaction force is attractive while for a less dense particle,ρ02<ρ , the bubble and particle repel. The result also demonstrates why it is essential toretain terms through order R5/d5.FREE RESPONSEThe free response of the system is investigated by setting the initial bubble radius to anon-equilibrium value R1(0)>R01 = 100μm and releasing it from rest. Parameters γ210 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.Downloaded to  IP:  128.83.205.78 On: Fri, 20 Mar 2015 16:52:22ρpart/ρliq = 2(d)XbubXpart010495500Position[μm]0 200 400 600 800Time [μs]ρpart/ρliq = 1XbubXpart(c)010495500Position[μm]ρpart/ρliq = 0.5XbubXpart(b)−50495500Position[μm]0 200 400 600 800Time [μs](a)90100110120Rbub[μm]FIGURE 2. Transient response of bubble and rigid particle for different particle densities.and σ were chosen to be 1.4 and 0.072 N/m, respectively. The particle is initially at restin its equilibrium state and has radius R02 = 2R01 = 200μm. The initial positions ofthe bubble and particle are X1(0) = 0 and X2(0) = 2.5R02 = 500μm when the bubble isreleased from its initial radius R1(0) = 120μm. Figure 2 shows results for three differentparticle densities: ρ2 = 500, 1000, and 2000kg/m3 (ρ2/ρ = 0.5 ,1, and 2). To aid thereader, the indices i = 1 and i = 2 on parameters shown in the ﬁgures are replaced withsubscripts “bub” and “part”, respectively, and the density of the liquid is denoted ρliq.The radial response of the bubble, which is independent of the density ratio ρpart/ρliq,is shown in Fig. 2(a). Parts (b)–(d) of Fig. 2 display the positions of the bubble andparticle for the three density ratios (note the split vertical axes). Consistent with theapproximate analytical result for the time-averaged interaction force given by Eq. (6),the bubble and particle repel for ρpart/ρliq = 0.5, they attract for ρpart/ρliq = 2, and thereis virtually no translation for ρpart/ρliq = 1.FORCED RESPONSEThe system may also be driven by an acoustic source as was included in the modelof Ilinskii et al. [1] The acoustic pressure perturbation was deﬁned to be a sinusoidalplane wave with amplitude 1 kPa. Figure 3 shows the positions and separation distancesbetween the bubble and particle for the three density ratios (ρpart/ρliq = 0.5, 1, 2) for asource located to the left [Fig. 3(a-c)] or right [Fig. 3(d-f)] of the bubble and particle.At this small separation distance the direction of bubble and particle translation[Fig. 3(a,b,d,e)] is determined by the particle density. However, the source radiationpressure (primary Bjerknes force) on the bubble has a non-zero time average. There-fore the direction of bubble motion [Fig. 3(a,d)] is also inﬂuenced by the propagationdirection of the acoustic excitation. This effect is most apparent for a neutrally buoy-ant particle, where the bubble-particle interaction force is negligible. In the case of a211 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.Downloaded to  IP:  128.83.205.78 On: Fri, 20 Mar 2015 16:52:22ρpart/ρliq = 0.5ρpart/ρliq = 1ρpart/ρliq = 2(c)495500505510|Xpart−Xbub|[μm]0 1000 2000Time [μs]ρpart/ρliq = 0.5ρpart/ρliq = 1ρpart/ρliq = 2(b)498500502Xpart[μm]Right-travelling waveρpart/ρliq = 0.5ρpart/ρliq = 1ρpart/ρliq = 2(a)−505Xbub[μm]ρpart/ρliq = 0.5ρpart/ρliq = 1ρpart/ρliq = 2(f)0 1000 2000Time [μs]ρpart/ρliq = 0.5ρpart/ρliq = 1ρpart/ρliq = 2(e)Left-travelling waveρpart/ρliq = 0.5ρpart/ρliq = 1ρpart/ρliq = 2(d)FIGURE 3. Bubble and particle translation for right- and left-traveling plane waves.right-traveling wave the radiation pressure tends to push the bubble towards the parti-cle, causing the separation distance to decrease [Fig. 3(c)]. In the case of a left-travelingwave, the radiation pressure tends to push the bubble away from the particle. If the ini-tial separation distance were doubled the source radiation pressure would dominate thebubble-particle interaction force.SUMMARYEquations describing the interaction of a gas bubble and a rigid particle were presented.The equations are accurate to order R5/d5, the minimum required accuracy to accountfor the interaction force. The system dynamics show dependence on the density and sizeof the particle as well as acoustic source parameters.REFERENCES1. Yu. A. Ilinskii, M. F. Hamilton, and E. A. Zabolotskaya, “Bubble interaction dynamics in Lagrangianand Hamiltonian mechanics,” J. Acoust. Soc. Am. 121, 786 (2007).2. W. T. Coakley, and W. L. Nyborg, “Cavitation Dynamics of Gas Bubbles,” in Applications in Ultra-sound: Its Applications in Medicine and Biology, edited by F. J. Fry, Elsevier, New York, 1978.212 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions.Downloaded to  IP:  128.83.205.78 On: Fri, 20 Mar 2015 16:52:22

Coupled Pulsation And Translation Of A Gas Bubble And Rigid Particle

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Coupled Pulsation And Translation Of A Gas Bubble And Rigid Particle

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