Model for the dynamics of two interacting axisymmetric spherical bubbles undergoing small shape oscillations

Interaction between acoustically driven or laser-generated bubbles causes the bubble surfaces to deform. Dynamical equations describing the motion of two translating, nominally spherical bubbles undergoing small shape oscillations in a viscous liquid are derived using Lagrangian mechanics. Deformation of the bubble surfaces is taken into account by including quadrupole and octupole perturbations in the spherical-harmonic expansion of the boundary conditions on the bubbles. Quadratic terms in the quadrupole and octupole amplitudes are retained, and surface tension and shear viscosity are included in a consistent manner. A set of eight coupled second-order ordinary differential equations is obtained. Simulation results, obtained by numerical integration of the model equations, exhibit qualitative agreement with experimental observations by predicting the formation of liquid jets. Simulations also suggest that bubble-bubble interactions act to enhance surface mode instability.     

Numerical simulations of stable cavitation bubble generation and primary Bjerknes forces in a three-dimensional nonlinear phased array focused ultrasound field

We present a model developed for studying the generation of stable cavitation bubbles and their motion in a three-dimensional volume of liquid with axial symmetry under the effect of finite-amplitude phased array focused ultrasound. The density of bubbles per unit volume is determined by a nonlinear law which is a threshold-dependent function of the negative acoustic pressure reached in the liquid, in which nuclei are initially distributed. The nonlinear mutual interaction of ultrasound and bubble oscillations is modeled by a nonlinear coupled differential system formed by the wave and a Rayleigh-Plesset equations, for which both the pressure and the bubble oscillation variables are unknown. The system, which accounts for nonlinearity, dispersion, and attenuation due to the bubbles, is solved by numerical approximations. The nonlinear acoustic pressure field is then used to evaluate the primary Bjerknes force field and to predict the subsequent motion of bubbles. In order to illustrate the procedure, a medium-high and a low ultrasonic frequency configurations are assumed. Simulation results show where bubbles are generated, the nonlinear effects they have on ultrasound, and where they are relocated. Despite many physical restrictions and thanks to its particularities (two nonlinear coupled fields, bubble generation, bubble motion), the numerical model used in this work gives results that show qualitative coherence with data observed experimentally in the framework of stable cavitation and suggest their usefulness in some application contexts.     

A simple model of ultrasound propagation in a cavitating liquid. Part I: Theory, nonlinear attenuation and traveling wave generation

The bubbles involved in sonochemistry and other applications of cavitation oscillate inertially. A correct estimation of the wave attenuation in such bubbly media requires a realistic estimation of the power dissipated by the oscillation of each bubble, by thermal diffusion in the gas and viscous friction in the liquid. Both quantities and calculated numerically for a single inertial bubble driven at 20 kHz, and are found to be several orders of magnitude larger than the linear prediction. Viscous dissipation is found to be the predominant cause of energy loss for bubbles small enough. Then, the classical nonlinear Caflish equations describing the propagation of acoustic waves in a bubbly liquid are recast and simplified conveniently. The main harmonic part of the sound field is found to fulfill a nonlinear Helmholtz equation, where the imaginary part of the squared wave number is directly correlated with the energy lost by a single bubble. For low acoustic driving, linear theory is recovered, but for larger drivings, namely above the Blake threshold, the attenuation coefficient is found to be more than 3 orders of magnitude larger then the linear prediction. A huge attenuation of the wave is thus expected in regions where inertial bubbles are present, which is confirmed by numerical simulations of the nonlinear Helmholtz equation in a 1D standing wave configuration. The expected strong attenuation is not only observed but furthermore, the examination of the phase between the pressure field and its gradient clearly demonstrates that a traveling wave appears in the medium.     

Coupled dynamics of translation and collapse of acoustically driven microbubbles

Pressure gradients drive the motion of microbubbles relative to liquids in which they are suspended. Examples include the hydrostatic pressure due to a gravitational field, and the pressure gradients in a sound field, useful for acoustic levitation. In this paper, the equations describing the coupled dynamics of radial oscillation and translation of a microbubble are given. The formulation is based on a recently derived expression for the hydrodynamic force on a bubble of changing size in an incompressible liquid [J. Magnaudet and D. Legendre, Phys. Fluids 10, 550-556 (1998)]. The complex interaction between radial and translation dynamics is best understood by examination of the added momentum associated with the liquid motion caused by the moving bubble. Translation is maximized when the bubble collapses violently. The new theory for coupled collapse and translation dynamics is compared to past experiments and to previous theories for decoupled translation dynamics. Special attention is paid to bubbles of relevance in biomedical applications.     

两个等径蛋白质气泡在Bingham流体中振动特性

利用黏弹性膜构成的蛋白质气泡有限变形方程,并考虑一个气泡在Bingham流体中振动产生的Bjerknes力对另一个气泡振动特性的影响,建立了两个等径蛋白质气泡在Bingham流体中振动的非线性方程.利用数值计算方法求解该方程,结果表明,增加Bingham流体的塑性黏度,蛋白质气泡振幅衰减速度加快,振动周期增加,频率减小;当两个气泡间的距离减小时,气泡振动频率会增加,振幅衰减速度加快;初始半径小的气泡振动频率高,振幅衰减快,而且振动的频率和振幅衰减的速率越大;与单个气泡相比,两个蛋白质气泡在Bingham流体中振动时,振动具有更高的振动频率,而且振幅衰减速度更快.     

Spatio-temporal dynamics of an encapsulated gas bubble in an ultrasound field

Coupled equations describing the radial and translational dynamics of an encapsulated gas bubble in an ultrasound field are derived by using the Lagrangian formalism. The equations generalize Church's theory by allowing for the translation motion of the bubble and radiation losses due to the compressibility of the surrounding liquid. The expression given by Church for the inner bubble radius corresponding to the unstrained state of the bubble shell is also refined, assuming that the shell can be of arbitrary thickness and impermeable to gas. Comparative linear analysis of the radial equation is carried out relative to Church's theory. It is shown that there are substantial departures from predictions of Church's theory. The proposed model is applied to evaluate radiation forces exerted on encapsulated bubbles and their translational displacements. It is shown that in the range of relatively high frequencies encapsulated bubbles are able to translate more efficiently than free bubbles of the equivalent size.     

Hydrodynamic approach to multibubble sonoluminescence

The velocity profile and radiation pressure field of a bubble cluster containing several thousand micro bubbles were obtained by solving the continuity and momentum equations for the bubbly mixture. In this study, the bubbles in the cluster are assumed to be generated and collapsed synchronously with an applied ultrasound. Numerical calculations describing the behavior of a micro bubble in a cluster included the effect of the radiation pressure field from the synchronizing motion of bubbles in the cluster. The radiation pressure generated from surrounding bubbles affects the bubble’s behavior by increasing the effective mass of the bubble so that the bubble expands slowly to a smaller maximum size. The light pulse width and spectral radiance from a bubble in a cluster subjected to ultrasound were calculated by adding a radiation pressure term to the Keller–Miksis equation, and the values were compared to experimental values of the multibubble sonoluminescence condition. There was close agreement between the calculated and observed values.     

非球形效应对强声场中次Bjerknes力的影响

考虑了非球形气泡在声场中的形状振动,推导了非球形气泡和球形气泡之间的次Bjerknes力方程,数值模拟了声场中非球形气泡和球形气泡之间的次Bjerknes力和两个球形气泡之间的次Bjerknes力,并对非球形气泡和球形气泡之间的次Bjerknes力的影响因素进行了分析讨论.研究结果表明:当驱动声压振幅大于非球形气泡的Black阈值且又能使得非球形气泡稳定振动时,在第一个声驱动周期内,非球形气泡和球形气泡之间的次Bjerknes力和两个球形气泡的次Bjerknes力方向差异较大,在大小上是两个球形气泡次Bjerkens力的数倍,且有着更长的作用距离.非球形气泡和球形气泡之间的次Bjerknes力取决于非球形气泡的形状模态、两个气泡初始半径的比值、驱动声压振幅、气泡间距和两个气泡的相对位置.     

A new pressure formulation for gas-compressibility dampening in bubble dynamics models

We formulated a pressure equation for bubbles performing nonlinear radial oscillations under ultrasonic high pressure amplitudes. The proposed equation corrects the gas pressure at the gas–liquid interface on inertial bubbles. This pressure formulation, expressed in terms of gas-Mach number, accounts for dampening due to gas compressibility during the violent collapse of cavitation bubbles and during subsequent rebounds. We refer to this as inhomogeneous pressure, where the gas pressure at the gas–liquid interface can differ to the pressure at the centre of the bubble, in contrast to homogenous pressure formulations that consider that pressure inside the bubble is spatially uniform from the wall to the centre. The pressure correction was applied to two bubble dynamic models: the incompressible Rayleigh–Plesset equation and the compressible Keller and Miksis equation. This improved the predictions of the nonlinear radial motion of the bubble vs time obtained with both models. Those simulations were also compared with other bubble dynamics models that account for liquid and gas compressibility effects. It was found that our corrected models are in closer agreement with experimental data than alternative models. It was concluded that the Rayleigh–Plesset family of equations improve accuracy by using our proposed pressure correction.     

Numerical simulation of bubble dynamics in a Phan-Thien–Tanner liquid: Non-linear shape and size oscillatory response under periodic pressure

We study non-linear bubble oscillations driven by an acoustic pressure with the bubble being immersed in a viscoelastic, Phan-Thien–Tanner liquid. Solution is provided numerically through a method which is based on a finite element discretization of the Navier–Stokes flow equations. The proposed computational approach does not rely on the solution of the simplified Rayleigh–Plesset equation, is not limited in studying only spherically symmetric bubbles and provides coupled solutions for the velocity, stress fields and bubble interface. We present solutions for non-spherical bubbles, with asphericity being addressed by means of Legendre polynomials or associated Legendre functions. A parametric investigation of the bubble dynamical oscillatory response as a function of the fluid rheological properties shows that the amplitude of bubble oscillations drastically increases as liquid elasticity (quantified by the Deborah number) increases or as liquid viscosity decreases (quantified by the Reynolds number). Extensive numerical calculations demonstrate that increasing elasticity and/or viscosity of the surrounding liquid tend to stabilize the shape anisotropy of an initially non-spherical bubble. Results are shown for pressure amplitudes 0.2–2 MPa and Deborah, Reynolds numbers in the intervals of 1–8 and 0.094–1.256, respectively.     

Numerical simulation of 3D bubbles rising in viscous liquids using a front tracking method

The rise of bubbles in viscous liquids is not only a very common process in many industrial applications, but also an important fundamental problem in fluid physics. An improved numerical algorithm based on the front tracking method, originally proposed by Tryggvason and his co-workers, has been validated against experiments over a wide range of intermediate Reynolds and Bond numbers using an axisymmetric model [J. Hua, J. Lou, Numerical simulation of bubble rising in viscous liquid, J. Comput. Phys. 22 (2007) 769–795]. In the current paper, this numerical algorithm is further extended to simulate 3D bubbles rising in viscous liquids with high Reynolds and Bond numbers and with large density and viscosity ratios representative of the common air–water two-phase flow system. To facilitate the 3D front tracking simulation, mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh. On the latter mesh, the governing Navier–Stokes equations for incompressible, Newtonian flow are solved in a moving reference frame attached to the rising bubble. Specifically, the equations are solved using a finite volume scheme based on the Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm, and it appears to be robust even for high Reynolds numbers and high density and viscosity ratios. The 3D bubble surface is tracked explicitly using an adaptive, unstructured triangular mesh. The numerical model is integrated with the software package PARAMESH, a block-based adaptive mesh refinement (AMR) tool developed for parallel computing. PARAMESH allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer. Further, Peskin distribution function is applied to interpolate the variable values between the front and the background meshes. Detailed sensitivity analysis about the numerical modeling algorithm has been performed. The current model has also been applied to simulate a number of cases of 3D gas bubbles rising in viscous liquids, e.g. air bubbles rising in water. Simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities. In addition, we applied this model to simulate the interaction between two bubbles rising in a liquid, which illustrated the model’s capability in predicting the interaction dynamics of rising bubbles.     

Nonlinear oscillations of gas bubbles in viscoelastic fluids

Nonlinear oscillations of gas bubbles in viscoelastic fluids of a three-constant Oldroyd model are theoretically investigated. The equation of motion for a bubble in a viscoelastic fluid subject to a pulsating pressure, and the pressure equation at the bubble wall are obtained. By numerical calculations on these equations, the effects of relaxation time and retardation time on frequency response curves, and the relation between the maximum pressure at the bubble wall and the initial radius of the bubble are clarified.     

导电流体中气泡径向振动行为的磁场控制

液态金属中气泡行为是磁流体力学的重要方面。为对磁场条件下导电流体中气泡动力学行为作全面理解,基于磁流体动力学方法建立了磁场条件下导电流体中气泡径向振动的无量纲化动力学方程,数值研究了磁场对导电流体中气泡径向非线性振动稳定性、泡内温度、泡内气压及液体空化阈值的影响。结果显示:磁场增强了气泡非线性振动的稳定性,随着磁场增强且当作用在泡上的电磁力与惯性力数量级可比时,气泡运动为稳定的周期性振动;同时,磁场引起泡内温度、泡内压力及液体空化阈值变化。研究表明,可用磁场调节和控制液态金属中气泡的运动使其满足工程应用需求。      

Mathematical simulation of the compression process of a vapor bubble at a pressure increase in the surrounding liquid

The thermophysical and hydrodynamic processes in a spherical vapor bubble and the surrounding liquid at increasing external pressure are investigated by using a numerical simulation method. The investigation is performed on the basis of a new mathematical model belonging to the class of models of homobaric bubbles (the pressure in the bubble is homogeneous at nonhomogeneous temperature and density). The model takes into account the following main physical effects: the viscosity of the liquid, the heat conductivity of the liquid and vapor, the surface tension, and the phase transitions at the bubble surface. An energy equation taking into account convective heat transfer and viscous dissipation in the liquid is used to calculate the temperature fields in the liquid and vapor. The model also takes into account the dependence of the thermophysical properties on the temperature. A distinctive feature of the proposed model is that the integral conservation law of the system’s total energy (including the kinetic energy of the liquid, the surface energy, and the internal energy of the liquid and vapor) is exactly satisfied (without allowance for the kinetic energy of the vapor). As a result of the numerical simulation of the compression of vapor bubbles in water, we obtained data for the major characteristics of the process at considerable degrees of compression. It is shown that the heat and mass transfer between the vapor in a bubble and the surrounding liquid considerably slow down the temperature increase in the bubble.     

Calculation of shock wave propagation in water containing reactive gas bubbles

The entry of a shock wave from air into water containing reactive gas (stoichiometric acetylene–oxygen mixture) bubbles uniformly distributed over the volume of the liquid has been numerically investigated using equations describing two-phase compressible viscous reactive flow. It has been demonstrated that a steady-state supersonic self-sustaining reaction front with rapid and complete fuel burnout in the leading shock wave can propagate in this bubbly medium. This reaction front can be treated as a detonation-like front or “bubble detonation.” The calculated and measured velocities of the bubble detonation wave have been compared at initial gas volume fraction of 2 to 6%. The observed and calculated data are in satisfactory qualitative and quantitative agreement. The structure of the bubble detonation wave has been numerically studied. In this wave, the gas volume fraction behind the leading front is approximately 3–4 times higher than in the pressure wave that propagates in water with air bubbles when the other initial conditions are the same. The bubble detonation wave can form after the penetration of the shock wave to a small depth (~300 mm) into the column of the bubbly medium. The model suggested here can be used to find optimum conditions for maximizing the efficiency of momentum transfer from the pressure wave to the bubbly medium in promising hydrojet pulse detonation engines.     

II. Recirculation flow induced by a bubble stream rising in a viscous liquid

The recirculation flow induced by the rising motion of a bubble stream in a viscous fluid within an open-top rectangular enclosure is studied. The three-dimensional volume averaged conservation equations are solved by a control-volume method using a hybrid finite differencing scheme to describe the liquid phase hydrodynamics. The momentum exhange between the bubbles and the liquid phase is modeled with a source term equals to the volumetric buoyancy force acting on the gas in the bubble stream. The volumetric buoyancy force accounts for in line interactions between bubbles through the average gas volume fraction in the gas liquid column which depends on the size and the rising velocity of bubbles. The fluid flow within an open-top rectangular enclosure is further investigated by particle image velocimetry for a bubble stream rising in a water-glycerol solution. The measured fluid velocities in a vertical plane are compared with the predictions of the numerical model over a wide range of fluid viscosity (43 mPa s-800 mPa s) and gas flow rates. Finally, the recirculation flows resulting from the interaction of two neighbouring vertical bubble streams are studied. Received: 23 July 1997 / Revised: 19 December 1997 / Accepted: 11 May 1998     

The characteristics of sonoluminescence

Cavitation luminescence is light emission from gases that are compressed to high temperature and high pressure inside a bubble or group of bubbles. The numerical simulation in this study indicates that if the temperature and pressure inside a bubble are not high enough, then dim and spectral line emission dominates. However, if the temperature and pressure inside the bubble are very high, then the light is bright and a continuum spectrum will be generated. Calculations of the spectrum using modified equations of bubble motion can simulate the spectral profile well. However, pulse width calculations using these equations only partly agree with the experimental results.     

Bubble interaction dynamics in Lagrangian and Hamiltonian mechanics

Two models of interacting bubble dynamics are presented, a coupled system of second-order differential equations based on Lagrangian mechanics, and a first-order system based on Hamiltonian mechanics. Both account for pulsation and translation of an arbitrary number of spherical bubbles. For large numbers of interacting bubbles, numerical solution of the Hamiltonian equations provides greater stability. The presence of external acoustic sources is taken into account explicitly in the derivation of both sets of equations. In addition to the acoustic pressure and its gradient, it is found that the particle velocity associated with external sources appears in the dynamical equations.     

Dynamics of a bubble cluster in an acoustic field

A mathematical model describing the dynamics of clustered gas bubbles under the effect of an acoustic field is presented. The proposed model is used as the basis for an analytical study of small bubble oscillations in monodisperse and polydisperse clusters and for a numerical study of nonlinear bubble oscillations under high-amplitude external pressures. The following effects are found to occur in a polydisperse cluster: a synchronization of the collapse phases of bubbles with different radii and a collapse intensification for bubbles of one size in the presence of bubbles of another size. These effects are explained by the interaction between bubbles of different radii in the cluster.     

含气泡液体中气泡振动的研究

研究了含气泡液体中单个气泡在驱动声场一定情况下的振动过程. 让每次驱动声场作用的时间特别短, 使气泡半径发生微小变化后再将其变化反馈到气泡群对驱动声场的散射作用中去, 从而可以得到某单个气泡周围受气泡散射影响后的声场, 接着再让气泡在该声场作用下做短时振动, 如此反复. 通过这样的方法, 研究了液体中单个气泡的振动情况并对其半径变化进行了数值模拟, 结果发现, 在液体中含有大量气泡的情况下, 某单个气泡的振动过程明显区别于液体中只有一个气泡的情况. 由于大量气泡和驱动声场的相互作用, 使气泡半径的变化存在多种不同的振动情况, 在不同的气泡大小和含量的情况下, 半径变化过程分别表现为: 在平衡位置附近振荡的过程; 周期性的空化过程; 一次空化过程后保持某一大小振荡的过程; 增长后维持某一大小振荡的过程等. 所以, 对于含气泡液体中气泡振动的研究, 在驱动声场一定的情况下, 必须考虑气泡含量的因素. 关键词： 含气泡液体 超声空化 散射 数值模拟