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构建了弹性介质包裹的液体腔内的气泡振动模型, 并基于压力平衡关系得到了腔内液体中气核发展成为空化泡的Blake阈值以及液体腔临界半径表达式. 体积模量、泡核半径及表面张力系数等因素可影响Blake阈值压力和气泡大小, 形成触发或抑制空化发生的控制条件相关的参数空间. 基于拉格朗日方程推导了考虑腔外介质弹性影响的气泡动力学方程, 并基于此分析了激励声波频率、介质体积模量、腔内液体体积等因素对气泡振动行为的影响, 结果表明: 在声场驱动下, 泡核快速生长到新的平衡半径后振荡; 气泡平衡半径与超声无关, 但会影响气泡动力学行为. 当超声频率与气泡固有振荡频率相当时, 气泡在几个周期的剧烈振动后崩溃, 同时在腔内液体中形成明显的压力起伏变化. 高频超声驱动下气泡的响应相对较弱, 气泡主要表现为自由振荡.The cavitation effects occur in the ultrasound therapy technology. With the development of ultrasound therapy technology, cavitation effect in biological tissues has attracted more and more attention. The aim of the present study is to discuss the factors affecting cavitation nucleation and dynamics in tissues, and to provide a theoretical reference for the application of cavitation effects to ultrasound therapy. A model is developed for the cavitation inception in a spherical liquid cavity wrapped by an elastic medium. The Blake threshold value and the critical radius of the liquid cavity for the generation of spherical bubbles are obtained by the pressure equilibrium relationship. The effects of the excitation frequency, the volume modulus of the medium and the volume of the liquid in the cavity on the bubble vibration behavior are analyzed by deriving a bubble dynamic equation that consider the elastic effect of the medium outside the cavity using Lagrange equation. It is shown that the volume modulus, initial radius of bubble nucleus and surface tension can affect the Blake threshold pressure and bubble size, and those form a parameter reference for the control conditions that trigger or inhibit cavitation. The gas core can rapidly grow to a new equilibrium radius and oscillate under the action of an acoustic wave, and the bubble equilibrium radius is independent of the external field, but it can affect the bubble dynamic behavior. When the frequency of the ultrasonic signal is equal to the natural oscillation frequency of the bubble, the bubble collapses after several periods of intense vibration, and the pressure fluctuation in the liquid in the cavity is obvious. The response of bubbles under high frequency ultrasonic driving is relatively weak, and the oscillations of bubbles are dominated by free oscillation.
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Keywords:
- elastic medium /
- liquid cavity /
- Blake threshold /
- cavitation mechanics
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图 1 系统准静态示意图(空腔中气核在负压
$ {P_{{\text{l1}}}} $ 下生长) (a) 初始状态, 未受扰动; (b)液体腔在负压下被拉伸; (c) 气核生长Fig. 1. Configuration of the quasi-static motion of the system (schematically). The cavity contains a bubble nucleus, which grows under tension
$ {P_{{\text{l1}}}} $ : (a) Initial state, unperturbed; (b) the cavity stretched under tension; (c) a bubble nucleus grows
图 2 (a) 腔内液体体积变化曲线与无界液体中的平衡压力曲线对比; 刚性微腔与弹性介质中的球腔内液体空化模型对比, 其中图(b)是腔体体积变化量与气泡平衡半径的关系, 图(c)是气泡Blake阈值半径随腔体半径的变化关系
Fig. 2. (a) Comparison of the curve of the volume change of the liquid in the cavity and the equilibrium curve of the bubble in unbounded liquid. Comparison of cavitation models of liquid in spherical cavity between rigid microcavity and elastic medium: (b) The dependence of the relative change of liquid volume on the equilibrium bubble radius; (c) the dependence of the Blake threshold radius on the cavity radius.
图 3 体积模量对平衡半径的影响 (a) 腔外介质体积模量; (b) 液体体积模量.
Fig. 3. Influence of bulk modulus on equilibrium radius: (a) Influence of the bulk modulus of the medium outside the cavity; (b) influence of liquid bulk modulus.
图 4 不同初始半径及表面张力系数下气泡的平衡图 (a)腔体初始半径; (b)泡核半径; (c)表面张力系数.
Fig. 4. Equilibrium diagram of the bubble for the different Initial radius and surface tension: (a) Initial radius of cavity; (b) the radius of the bubble nucleus; (c) surface tension coefficient.
图 5 腔外介质体积模量对空化泡振动行为的影响 (a)空腔中气泡瞬态振荡; (b)超声作用下空腔中气泡振荡
Fig. 5. Influence of volume modulus of medium outside cavity on vibration behavior: (a) A transient bubble oscillating in a cavity; (b) a bubble in a confinement subject to an acoustic wave.
图 6 不同Rc0 /Rb0 取值对空化泡振动行为的影响 (a)空腔中气泡瞬态振荡的影响; (b)超声作用下空腔中气泡振荡的影响
Fig. 6. Influence of different values of Rc0 /Rb0 on the vibration behavior: (a) A transient bubble oscillating in a cavity; (b) a bubble in a confinement subject to an acoustic wave.
图 7 不同频率外加驱动信号下气泡半径与液体压力的变化 (a) f = 1 MHz; (b) f = 5 MHz; (c) f = 10 MHz; (d) f = 15 MHz
Fig. 7. Evolution of bubble radius and liquid pressure profiles at different driving frequencies: (a) f = 1 MHz; (b) f = 5 MHz; (c) f = 10 MHz; (d) f = 15 MHz.
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[1] Herbert E, Balibar S, Caupin F 2006 Phys. Rev. E 74 041603
Google Scholar
[2] Temesgen T, Bui T T, Han M, Kim T I, Park H 2017 Adv. Colloid. Interface. Sci. 246 40
Google Scholar
[3] Rooze J, Rebrov E V, Schouten J C, Keurentjes J T 2013 Ultrason. Sonochem 20 1
Google Scholar
[4] Brennen C E 2015 Interface. Focus 5 20150022
Google Scholar
[5] Bai W, Duan Q, Zhang Z 2016 Proceedings of the ASME 2016 Pressure Vessels and Piping Conference Vancouver, British Columbia, Canada, July 17–21, 2016 pV005T09A029
[6] Khokhlova V A, Fowlkes J B, Roberts W W, Schade G R, Xu Z, Khokhlova T D, Hall T L, Maxwell A D, Wang Y N, Cain C A 2015 Int. J. Hyperthermia 31 145
Google Scholar
[7] Roberts W W 2014 Curr. Opin. Urol 24 104
Google Scholar
[8] Bigelow T A, Thomas C L, Wu H, Itani K M F 2018 IEEE. Trans. Ultrason. Ferroelectr. Freq. Control 65 1017
Google Scholar
[9] 曹权, 胡玉刚, 周青, 郭瑞强 2020 医学研究杂志 49 126
Google Scholar
Cao Q, Hu YG, Zhou Q, Guo LQ 2020 J. Medical Res. 49 126
Google Scholar
[10] Mancia L, Vlaisavljevich E, Xu Z, Johnsen E 2017 Ultrasound. Med. Biol. 43 1421
Google Scholar
[11] Vlaisavljevich E, Maxwell A, Mancia L, Johnsen E, Cain C, Xu Z 2016 Ultrasound. Med. Biol. 42 2466
Google Scholar
[12] Keller J B, Miksis M, Acoust J 1980 Soc. Am. 68 628
Google Scholar
[13] Drysdale C, Doinikov A A, Marmottant P 2017 Phys. Rev. E 95 053104
Google Scholar
[14] Doinikov A A, Marmottant P 2018 J. Sound Vib. 420 61
Google Scholar
[15] Doinikov A A, Dollet B, Marmottant P 2018 Phys. Rev. E 97 013108
Google Scholar
[16] Wang Q X 2017 Phys. Fluids 29 072101
Google Scholar
[17] 张陶然, 莫润阳, 胡静, 陈时, 王成会, 郭建中 2020 69 234301
Google Scholar
Zhang T R, Mo R Y, Hu J, Chen S, Wang C H, Guo J Z 2020 Acta Phys. Sin. 69 234301
Google Scholar
[18] Vincent O, Marmottant P, Quinto-Su P A, Ohl C D 2012 Phys. Rev. Lett. 108 184502
Google Scholar
[19] Vincent O, Sessoms D A, Huber E J, Guioth J, Stroock A D 2014 Phys. Rev. Lett. 113 134501
Google Scholar
[20] Vincent O, Marmottant P 2017 J. Fluid Mech. 827 194
Google Scholar
[21] Leonov K, Akhatov I 2020 Int. J. Multiphase Flow 130 103369
Google Scholar
[22] Yang X M, Church C C 2005 J. Acoust. Soc. Am. 118 3595
Google Scholar
[23] Vincent O, Marmottant P, Gonzalez-Avila S R, Ando K, Ohl C D 2014 Soft Matter. 10 1455
Google Scholar
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