Single Spherical Bubble Research Articles

Data-driven acoustic control of a spherical bubble using a Koopman linear quadratic regulator.

Koopman operator theory has gained interest as a framework for transforming nonlinear dynamics on the state space into linear dynamics on abstract function spaces, which preserves the underlying nonlinear dynamics of the system. These spaces can be approximated through data-driven methodologies, which enables the application of classical linear control strategies to nonlinear systems. Here, a Koopman linear quadratic regulator (KLQR) was used to acoustically control the nonlinear dynamics of a single spherical bubble, as described by the well-known Rayleigh-Plesset equation, with several objectives: (1) simple harmonic oscillation at amplitudes large enough to incite nonlinearities, (2) stabilization of the bubble at a nonequilibrium radius, and (3) periodic and quasiperiodic oscillation with multiple frequency components of arbitrary amplitude. The results demonstrate that the KLQR controller can effectively drive a spherical bubble to radially oscillate according to prescribed trajectories using both broadband and single-frequency acoustic driving. This approach has several advantages over previous efforts to acoustically control bubbles, including the ability to track arbitrary trajectories, robustness, and the use of linear control methods, which do not depend on initial guesses.

The collapse of a sonoluminescent cavitation bubble imaged with X-ray free-electron laser pulses

Single bubble sonoluminescence (SBSL) is the phenomenon of synchronous light emission due to the violent collapse of a single spherical bubble in a liquid, driven by an ultrasonic field. During the bubble collapse, matter inside the bubble reaches extreme conditions of several gigapascals and temperatures on the order of 10000 K, leading to picosecond flashes of visible light. To this day, details regarding the energy focusing mechanism rely on simulations due to the fast dynamics of the bubble collapse and spatial scales below the optical resolution limit. In this work we present phase-contrast holographic imaging with single x-ray free-electron laser (XFEL) pulses of a SBSL cavitation bubble in water. X-rays probe the electron density structure and by that provide a uniquely new view on the bubble interior and its collapse dynamics. The involved fast time-scales are accessed by sub-100 fs XFEL pulses and a custom synchronization scheme for the bubble oscillator. We find that during the whole oscillation cycle the bubble’s density profile can be well described by a simple step-like structure, with the radius R following the dynamics of the Gilmore model. The quantitatively measured internal density and width of the boundary layer exhibit a large variance. Smallest reconstructed bubble sizes reach down to R≃0.8μm , and are consistent with spherical symmetry. While we here achieved a spatial resolution of a few 100 nm, the visibility of the bubble and its internal structure is limited by the total x-ray phase shift which can be scaled with experimental parameters.

Dynamics of Microbubbles Oscillating in Rheopectic Fluids Subject to Acoustic Pressure Field

In the present work, the dynamics of a single spherical gas bubble surrounded by a rheopectic fluid obeying the Quemada model is numerically investigated while the bubble undergoes oscillatory motion due to acoustic forcing. The generalized form of the Rayleigh–Plesset equation has been used for studying bubble dynamics in Quemada fluids. The integro-differential equation representing the dynamics of the bubble is solved numerically using the finite-element method (FEM) and also the Gauss–Laguerre quadrature (GLQ) method. The effect of rheopexy number (Rx) and viscosity ratio (ξ) are then investigated over a wide range of working parameters. Numerical results show that the rheopectic behavior of the fluid surrounding the bubble can dramatically affect the bubble dynamics. It is predicted that for highly anti-thixotropic fluids, harmonics are affected so much so that the bubble may exhibit chaotic behavior. For instance, at Rx = 0.001 and ξ = 1/81, a one-micron-sized bubble may attain a size almost 30 times of its initial size. The general conclusion is that, in sonography, microbubbles dispersed in rheopectic fluids may indeed be considered as a potent ultrasound contrast agent provided that the fluid is just moderately anti-thixotropic otherwise its chaotic response might damage the adjacent tissues.

A tractable mathematical model for rectified diffusion

A core unresolved challenge in rectified diffusion is the lack of predictive models, which allow accurate modelling of the growth of bubbles over millions of cycles of oscillation. The spherically symmetric problem involves two asymptotic regions in space for the convection and diffusion of the dissolved gas in the liquid, an inner region adjacent to the bubble and an outer region. It also involves three time scales: the shortest time scale associated with the bubble oscillation, the intermediate and the longest time scales for the mass transport across the bubble interface and the outer region, respectively. The problem is analysed systematically using matched asymptotic expansions in space and multi-scales in time, the large parameter being the Péclet number for mass transfer. A tractable mathematical model is formulated for rectified diffusion. It comprises the Rayleigh–Plesset equation for the spherical oscillations of the bubble on the shortest time scale, an ordinary differential equation for the mass of gas in the bubble on the intermediate time scale and a diffusion equation on the longest time scale and the long length scale. Predictions of this simplified model have excellent agreement with experimental results for a single spherical bubble in the bulk over millions of cycles of oscillation. A parametric study is undertaken with several new phenomena/features observed for acoustic frequencies in the range 12 to 62 kHz, acoustic field $0.12$ to $0.22$ bar and the initial gas concentration in the liquid being $0.95$ to $1.0$ of the saturation concentration.

Intrusive temperature measurement of the cavitation bubbles using cold wire

Cavitation is a phenomenon of rupturing a liquid when the liquid is subjected to a decrease in pressure at roughly constant temperature. Cavitation is the fundamental reason for the material erosion when the bubbles are collapsing closer to the surface. These bubbles also produce a very high temperature (thousands of degrees Celsius) for a very short time (in the order of nano or microseconds). Cavitation is also being used in the non-invasive cancer tumor ablation treatment using high intensity focused ultrasound (HIFU) called histotripsy. So, our research in measuring the temperature of the bubble collapse will help the histotripsy treatment to understand its thermal behavior. We have designed a cold wire sensor of the thickness of ∼10 to 50 micrometers. And, the sensing region is 2 millimeter. These cold wires are made with kevlar or nylon fibers, titanium, nickel, silver, and silicon dioxide. The single bubble or histotripsy cloud, is allowed to collapse on the sensor, and we measure the temperature based on its change in resistance. These changes in resistance are corresponded to the temperature using the mathematical function which is calibrated prior to the experiment. We measure it in three different experimental settings like Tube Arrest Method (TAM), LASER induced bubbles, and HIFU induced bubbles. The preliminary results have shown that the temperature of the single spherical bubble collapse (TAM) can rise to thousands of degrees celsius as mentioned in the literature based on the theoretical analysis.

A theoretical model for the growth of spherical bubbles by rectified diffusion

Rectified diffusion is a bubble growth phenomenon that occurs in acoustic fields. Despite the existence of a well-established spherically symmetric mathematical model, theoretical results have been unsuccessful in reproducing the bubble growth even in the case of a single spherical bubble in the bulk. In the latter case, the influence of surfactants and acoustic microstreaming have been speculated as the explanation for this disagreement. In this article, an exact solution for the leading-order concentration of gas in the liquid is determined. Using this exact solution, the well-established mathematical model is reduced to a system of two ordinary differential equations for the spherical bubble radius and the mass of gas in the bubble. This simplified model predicts the rapid bubble growth observed in experiments of a single spherical bubble in the bulk. The new results show that the bubble growth is asymptotically larger than at later times when the mass flux is limited by the slow diffusion of gas in a much larger region of the liquid surrounding the bubble. The new results also show that this bubble growth is relatively insensitive to a reduction in surface tension.

Stability analysis and sliding mode control of a single spherical bubble described by Keller–Miksis equation

This paper transforms the Keller–Miksis (KM) model, which is the main equation describing the bubble behavior, into a state space representation and presents a stability condition for the system using the linearized form of the KM equation. The dynamic of the cavitation bubble is analyzed for different gases, and its burst time is measured. Then, a sliding mode controller (SMC) is designed for the nonlinear system to regulate the radius of a single spherical bubble and prevent collapse occurrence, which has a great importance in some industrial applications. The system's robustness in the presence of uncertainties in bubble parameters is one of its most significant control objectives obtained in the controller designed in this paper. A comparison is also made between this controller and the controller previously designed for the simpler bubble model (Rayleigh–Plesset (RP) model). Also, the chattering created by SMC is eliminated by two methods. Simulation results are presented to show the effectiveness of the proposed method.

Dynamics of bubbles under stochastic pressure forcing.

Several studies have investigated the dynamics of a single spherical bubble at rest under a nonstationary pressure forcing. However, attention has almost always been focused on periodic pressure oscillations, neglecting the case of stochastic forcing. This fact is quite surprising, as random pressure fluctuations are widespread in many applications involving bubbles (e.g., hydrodynamic cavitation in turbulent flows or bubble dynamics in acoustic cavitation), and noise, in general, is known to induce a variety of counterintuitive phenomena in nonlinear dynamical systems such as bubble oscillators. To shed light on this unexplored topic, here we study bubble dynamics as described by the Keller-Miksis equation, under a pressure forcing described by a Gaussian colored noise modeled as an Ornstein-Uhlenbeck process. Results indicate that, depending on noise intensity, bubbles display two peculiar behaviors: when intensity is low, the fluctuating pressure forcing mainly excites the free oscillations of the bubble, and the bubble's radius undergoes small amplitude oscillations with a rather regular periodicity. Differently, high noise intensity induces chaotic bubble dynamics, whereby nonlinear effects are exacerbated and the bubble behaves as an amplifier of the external random forcing.

Effect of non-condensable gas on a collapsing cavitation bubble near solid wall investigated by multicomponent thermal MRT-LBM* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11874140 and 11574072), the State Key Laboratory of Acoustics, Chinese Academy of Sciences (Grant No. SKLA201913), the National Key Research and Development Program of China (Grant No. 2016YFC0401600), the Primary Research

A multicomponent thermal multi-relaxation-time (MRT) lattice Boltzmann method (LBM) is presented to study collapsing cavitation bubble. The simulation results satisfy Laplace law and the adiabatic law, and are consistent with the numerical solution of the Rayleigh–Plesset equation. To study the effects of the non-condensable gas inside bubble on collapsing cavitation bubble, a numerical model of single spherical bubble near a solid wall is established. The temperature and pressure evolution of the two-component two-phase flow are well captured. In addition, the collapse process of the cavitation bubble is discussed elaborately by setting the volume fractions of the gas and vapor to be the only variables. The results show that the non-condensable gas in the bubble significantly affects the pressure field, temperature field evolution, collapse velocity, and profile of the bubble. The distinction of the pressure and temperature on the wall after the second collapse becomes more obvious as the non-condensable gas concentration increases.

A mass-preserving interface-correction level set/ghost fluid method for modeling of three-dimensional boiling flows

We present an efficient method for the direct numerical simulation of three-dimensional (3D) boiling flows. The liquid-vapor interface dynamics is captured using an interface-correction level set method, modified to account for the interface heat and mass transfer due to phase change. State-of-the-art computational techniques, such as the fast pressure-correction and ghost-fluid methods, are implemented to accurately solve the coupled thermo-fluid problem involving large density contrasts and jump conditions. The solver is thoroughly validated against four benchmark cases with increasing complexity, which show better mass conservation properties than traditional level set methods, thus allowing for coarser grid resolutions and lower computational costs. We further demonstrate our method by simulating two realistic 3D boiling flows in greater details. In the first case, a saturated film boiling of water vapor at near critical conditions over a horizontal hot flat plate is considered. The results are analyzed by comparing the transient evolution of the interface morphology, temperature distribution, space and time averaged Nusselt numbers obtained from numerical simulations with the semi-empirical correlation of Berenson and existing numerical literature. In the second case, we simulate the condensation and buoyancy-driven motion of a single spherical water vapor bubble at different subcooled liquid temperatures and saturation pressures. We find opposite trends of the condensation rate and the bubble rising velocity when the degree of subcooling is increased, and an increase of the condensation rate at lower saturation pressures, due to variation of the thermophysical properties.

Evaluation of Thermal and Acoustic Energy during Collapse of Cavitation Bubbles

The results of a numerical study of the thermal and acoustic energies released during the collapse of a single spherical cavitation bubble in water at a pressure of 10 bar and a temperature of 20°С are given. In the model used, we take into account the thermal conductivity of the vapor in the bubble and the surrounding liquid, heat transfer, evaporation/condensation on the surface of the bubble, and the fluid compressibility. The conversion of mechanical energy into heat due to the fluid viscosity is not accounted for. When the bubble collapses, the energy of acoustic radiation due to radial pulsations of the bubble is shown to be approximately nine times greater than the energy spent on heating the liquid. The value of this energy is proportional to the cube of the initial bubble radius.

Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique.

A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

Ultrasound-enhanced mass transfer during single-bubble diffusive growth

The effect of ultrasound on the diffusive growth of a single spherical bubble growing on a substrate in supersaturated carbonated water is quantified. It is found that the diffusive growth of surface bubbles can be easily enhanced by 2 orders of magnitude during volumetric resonance. This convective growth is shown to be caused by vigorous acoustic microstreaming arising from the nonspherical bubble oscillations.

High-performance GPU computations in nonlinear dynamics: an efficient tool for new discoveries

The main aim of this paper is to demonstrate the benefit of the application of high-performance computing techniques in the field of non-linear science through two kinds of dynamical systems as test models.It is shown that high-resolution, multi-dimensional parameter scans (in the order of millions of parameter combinations) via an initial value problem solver are an efficient tool to discover new features of dynamical systems that are hard to find by other means. The employed initial value problem solver is an in-house code written in C++ and CUDA C software environments, which can exploit the high processing power of professional graphics cards (GPUs). The first test model is the Keller–Miksis equation, a non-linear oscillator describing the dynamics of a driven single spherical gas bubble placed in an infinite domain of liquid. This equation is important in the field of cavitation and sonochemistry. Here, the high-resolution parameter scans gave us the opportunity to lay down the basis of a non-feedback technique to control multi-stability in which direct selection of the desired attractor is possible. The second test model is related to a pressure relief valve that can exhibit a special kind of impact dynamics called grazing impact. A fine scan of the initial conditions revealed a second focal point of the grazing lines in the initial-condition space that was hidden in previous studies.

GPU accelerated study of a dual-frequency driven single bubble in a 6-dimensional parameter space: The active cavitation threshold.

The active cavitation threshold of a dual-frequency driven single spherical gas bubble is studied numerically. This threshold is defined as the minimum intensity required to generate a given relative expansion (Rmax-RE)/RE, where RE is the equilibrium size of the bubble and Rmax is the maximum bubble radius during its oscillation. The model employed is the Keller-Miksis equation that is a second order ordinary differential equation. The parameter space investigated is composed by the pressure amplitudes, excitation frequencies, phase shift between the two harmonic components and by the equilibrium bubble radius (bubble size). Due to the large 6-dimensional parameter space, the number of the parameter combinations investigated is approximately two billion. Therefore, the high performance of graphics processing units is exploited; our in-house code is written in C++ and CUDA C software environments. The results show that for (Rmax-RE)/RE=2, the best choice of the frequency pairs depends on the bubble size. For small bubbles, below 3μm, the best option is to use just a single frequency of a low value in the giant response region. For medium sized bubbles, between 3μm and 6μm, the optimal choice is the mixture of low frequency (giant response) and main resonance frequency. For large bubbles, above 6μm, the main resonance dominates the active cavitation threshold. Increasing the prescribed relative expansion value to (Rmax-RE)/RE=3, the optimal choice is always single frequency driving with the lowest value (20kHz here). Thus, in this case, the giant response always dominates the active cavitation threshold. The phase shift between the harmonic components of the dual-frequency driving (different frequency values) has no effect on the threshold.

Liquid Heating During the Collapse of a Single Cavitation Bubble

This paper presents the results of a numerical study of the liquid heating process during collapse of a single spherical cavitation bubble in water at a liquid pressure of 10 bar, temperature of 20°C, and an initial bubble radius of 500 μm. The simulation of this phenomenon took into account the thermal conductivity of the vapor in the bubble and the surrounding liquid, heat transfer and evaporation/condensation on the surface of the bubble, and the effects of viscosity and compressibility of the liquid. It is shown that, as a result of bubble collapse, the liquid heats up in a region of about 60 μm radius. The temperature in the center of this region is about 50°C higher than in the surrounding liquid. The thermal energy expended on heating the liquid in this region is approximately equal to 25 μJ.

Robust versus optimal control for the radius of spherical bubble in a perfect incompressible liquid, LMI optimization approach

Control of the bubble radius is essential in many medical and industrial applications. In this paper, the nonlinear governing dynamic of single spherical bubble in a perfect incompressible liquid is introduced. In continuation, the state space form of the nonlinear dynamic is presented and then observability and controllability are checked by Lie derivative concepts and finally the linearized state space is obtained around its equilibrium point. Second part of this paper is devoted to designing control strategies for the nonlinear governing dynamics of bubble, where the ultrasonic pressure plays the role of control input and the output is radius of bubble which is intended to follow desired radius in order to prevent from bubble collapse phenomenon. A novel robust control is proposed to ensure the desired level of performance in the presence of parametric uncertainty. In this method, the stability of the closed loop system is guaranteed by Lyapunov theorem and the stabilizing gain is achieved by Linear Matrix Inequality (LMI) optimization approach. For comparison, Linear Quadratic Regulator method is selected which is a very classical approach. The optimal feedback gain is also achieved through LMI approach. Simulation results indicate the effectiveness and efficiency of the robust over optimal controller.

Route to shrimps: Dissipation driven formation of shrimp-shaped domains

Abstract In this paper, two scenarios for the formation of shrimp-shaped domains [1] are presented. The employed test model is the Keller–Miksis equation that is a second order, harmonically forced nonlinear oscillator describing the dynamics of a single spherical gas bubble placed in a liquid domain. The results have shown that with an increasing dissipation rate (liquid viscosity), shrimp-shaped domains are evolved within the complex structure of each subharmonic resonances in the amplitude-frequency parameter plane of the external forcing. The mechanism is the coalescence and interaction of two pairs of a period-doubling and a saddle-node codimension-two bifurcation curves.

Experimental and theoretical investigation of CO2 and air bubble rising velocity through kerosene and distilled water in bubble column

This paper concerns with developing of parameters which influence terminal velocities of air and CO2 bubbles in distilled water and kerosene pools. The objective of this study is to validate and correct the formulas that were developed by previous investigators for prediction of terminal velocities. The investigation revealed that the terminal velocity of a single rigid spherical bubble in Newtonian fluids can be developed by balancing of mechanical forces acting on the bubble. However, for large bubbles, because of deforming of the bubble which is a result of interfacial tension, the effect of surface tension should be considered in the terminal velocity prediction formula. By using PSO algorithm and plotting experimental data of terminal velocity against the size of gas bubbles, the suitable equation for each of systems was chosen. Results showed that Jamialahmadi model is more practical for terminal velocity prediction. Jamialahmadi model requires a modification to be utilized for air-kerosene, CO2-kerosene, air- distilled water and CO2-distilled water systems. The developed PSO algorithm model is accurate for prediction of experimental data with an average R2 value of 0.9722.

Estimation of liquid‐side mass transfer coefficient and liquid film thickness in a bubble column using single spherical bubble model

AbstractA macroscopic mass transfer model based on the unsteady‐state liquid film mass transfer mechanism for a single spherical bubble was formulated. Analytical solution of the model equation was obtained in Laplace transform using surface renewal rates based on Danckwerts' surface age distribution function. The mass transfer coefficient, kL, in a slurry bubble column under different operating conditions of temperature, pressure, gas flow rate, and solid concentration has been simulated using a program code “BUBBLESIM” in MATLAB®, developed by the authors. The proposed model has been validated using secondary data for a slurry system under a wide range of operating conditions. The predicted values of kL show very good agreement with the experimental data within an average deviation of ±2%. The results show that the mass transfer coefficient, kL, increased with increasing superficial gas velocity and temperature and decreased with increase in slurry concentration, while it changed slightly with pressure. Based on the present work, empirical correlations have been proposed for the prediction of δ in terms of dimensionless groups for H2–, CO–, and CO2–paraffin–quartz sand systems under elevated temperatures (298–423 K) and elevated pressures (1–3 MPa) in a slurry bubble column.