Spherical Bubble Research Articles

Application of Koopman operator theory to the control of nonlinear bubble dynamics

Microbubbles are widely used in biomedicine for ultrasound contrast imaging and are a promising vehicle for targeted drug and gene delivery. The ability to control the oscillations of bubbles through the applied ultrasound can improve the effectiveness of these treatments, for example, by enhancing the acoustic echo or inciting bubble rupture at precise locations. Koopman operator theory has gained interest in the past decade as a framework for rigorously 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 purely through machine learning and data-driven methodologies, which then enables the application of classical linear control strategies to strongly nonlinear systems. Here, we use a Koopman linear quadratic regulator (LQR) to control the nonlinear dynamics of spherical bubbles, as described by the well-known Rayleigh-Plesset equation, with two novel objectives: (1) stabilization of the bubble at a nonequilibrium radius and (2) simple harmonic oscillation at amplitudes large enough to incite nonlinearities. Control is implemented through both broadband and single-frequency transducers that are modulated by the Koopman LQR controller. We repeat these results using Koopman model predictive control (MPC), which allows for the implementation of constraints. This work is a step towards controlling nonspherical shape modes of encapsulated microbubbles.

Investigation of unsteady cryogenic cavitating flow and induced noise around a three-dimensional hydrofoil

In this paper, the Large Eddy Simulation (LES) combined with the Schnerr–Sauer cavitation model and the permeable Ffowcs Williams–Hawkings (FW-Hpds) acoustic analogy approach are introduced to study the unsteady cavitation behaviors and the radiated noise characteristics of the transient liquid nitrogen (LN2) cavitating flow around a NACA66 (National Advisory Committee for Aeronautics) hydrofoil. Satisfactory agreement is obtained between the numerical predictions and experimental measurements. The cavitation noise is predicted based on the sound radiation theory for spherical bubbles and compared with the sound pressure levels of non-cavitating flow from the FW-Hpds equation. It is found that the cavity volume acceleration is directly responsible for driving the generation of cavitation noise, and the sound pressure caused by the development of LN2 cavitation is shown to vary with the periodic pulsing cavity volume evolution, indicating a strong link between cavity evolutions and radiated noises. The transient cavitation structures of the sheet and cloud cavitation are well captured, and the evolution features of the cavities and vortex structures are analyzed in detail. The collapse of the detached small cloud cavity downstream is the main mechanism for generating intense acoustic impulses for both sheet and cloud cavitation. While the strong interaction between the re-entrant jet and the main flow results in violent pressure fluctuations, and thus produces instantaneous extreme dipole noise, which accounts for another distinctive mechanism to induce intense acoustic impulses for cloud cavitation, the presented study provides a deep understanding of the nature of cavitation-dominated noise for cryogenic cavitating flow.

BCS-BEC crossover of atomic Fermi superfluid in a spherical bubble trap

We present a theory of a two-component atomic Fermi gas with tunable attractive contact interactions on a spherical shell going through the Bardeen-Cooper-Schrieffer (BCS) - Bose Einstein condensation (BEC) crossover, inspired by the realizations of spherical bubble traps for ultracold atoms in microgravity. The derivation follows the BCS-Leggett theory to obtain the gap and number equations. The BCS-BEC crossover can be induced by tuning the interaction, and the properly normalized gap and chemical potential exhibit universal behavior regardless of the planar or spherical geometry. Nevertheless, the spherical-shell geometry introduces another way of inducing the crossover by the curvature. The curvature-induced BCS-BEC crossover is made possible by fixing the particle number and interaction strength while shrinking the sphere, causing a reduction to the ratio of the pairing and kinetic energies and pushing the system towards the BCS limit. The saturation of the superfluid density further confirms the ground state is a Fermi superfluid.

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.

Motion of nanoparticles near rising and dissolving microbubbles

This paper presents a computational study of the motion of inertia-less particles in the vicinity of freely rising bubbles of carbon dioxide that dissolve into the suspension. The purpose is to assess the extent of particle motion due to diffusiophoresis that is caused by ion concentration gradients, in this case protons and bicarbonate ions. Calculations are reported for spherical bubbles with diameters from 10 to 100μm. Particle trajectories are computed for bubbles with slip and no slip surfaces, which (respectively) represent the extremes of a high purity system and a contaminated system in which surfactants fully immobilise the surface. The extent of particle motion towards or away from the bubbles (depending on the sign of the diffusiophoretic mobility) is quantified. Particle motion due to removal of dissolved carbon dioxide from a suspension into air bubbles is also considered. In the cases where particles are attracted to the bubbles, some particle trajectories reach the surface of the bubble. The results suggest that this mechanism could therefore be exploited to extend the use of flotation to separate smaller particles.

Propagation of waves in a hydroelastic shell‐viscous liquid system, in the presence of gas bubbles

In this article, a thin‐walled isotropic infinitely long‐circular cylindrical elastic tube and the pulsed flow of an aerohydroelastic system within a compressible bubble fluid have been presented in 3D. The equation has been solved for an axisymmetric waves propagation in the liquid–shell system. The form and frequency of the specific oscillations formed in the shell–fluid dynamic system have been determined. The results obtained for the case when the material of thin‐walled pipes is made from various modifications of high‐strength steel and non‐classical polymeric materials, and water containing spherical air bubbles is taken as a two‐phase liquid. The fundamental iteration technique has been used to compute eigenvalues and corresponding eigenfunctions to represent field quantities with the help of Matlab software. The numerical results have been presented graphically.

Nonuniform particle distribution and interference between removal mechanisms during unsteady aerosol deposition from a rising spherical bubble

ABSTRACT Most of the practical computer models available today on aerosol pool scrubbing are built on the basis of fundamental assumptions introduced by Fuchs for his model published in 1955 (in Russian). The primary assumptions are that particle concentration in the bubble can be regarded as uniform and that particle removal mechanisms can be regarded as mutually independent. The present paper shows that these assumptions can cause considerable under- or overprediction of removal rates when applied to particles with radii in the submicrometer range. This size range, known to be where removal mechanisms are generally inefficient, is of particular interest in pool scrubbing study. The present paper, while not being the first to point out the problems with Fuchs’s assumptions, provides a comprehensive description on how the assumptions fail in the submictometer range based on comparisons of Fuchs model predictions against Lagrangian tracking simulations. The simulations, performed for particles of 0.1 to 1.0 m in radius, advected by circulating gas flow in a spherical bubble rising in a water pool, show interference between inertial migration and Brownian diffusion of particles through the development of nonuniform particle concentration profile.

The Effect of Area Loading and Punching Shear on the Reinforced Concrete Slab Containing Spherical Plastic Bubble Balls

The reinforced bubble deck slab or BubbleDeck is a unique system that improves the building design and performance of structures. This slab structure is a reinforced concrete structure that contains high-density polyethene (HDPE) hollow spherical plastic bubble balls with less concrete volume compared to a normal reinforced concrete slab. The system can facilitate up to a 50% longer span compared to a conventional reinforced concrete solid slab. But, eliminating the deadweight concrete may affect the actual performance of the slab structure such as its flexural and shear capacity. Thus, this paper investigates the effect of area loading and punching shear loading on the reinforced bubble deck slab in terms of flexural performance. The square slabs with 1200mm by 1200mm for width and length with a thickness of 230mm were designed as a one-way supported slab. A total of 36 HDPE hollow spherical plastic bubble balls with a 180mm diameter were placed in each bubble deck slab specimen. The high yield steel DA6 BRC reinforcement steel bar meshes and Grade-30 concrete were used for the slabs. The experimental results of the flexural performance of the reinforced bubble deck slab that were subjected to the static area and punching shear loadings are presented. The effect of the load applied in the experiments on the slabs such as flexural strength, bending stiffness and load-deflection behaviour were discussed including the crack propagation and crack pattern.

Effect of Area Loading on Flexural Performance of Bubble Deck Slab

Reinforced bubble deck slab is a structural slab that contains high-density polyethene (HDPE) hollow spherical plastic bubble balls forming a slab with less concrete volume compared to the normal reinforced concrete slab. Reducing certain volumes of concrete from 30 to 50% will affect the performance of the slab structure in particular the flexural and shear capacity. Thus, in this research the effect of area loading on the flexural performance of bubble deck slabs is investigated by considering the slabs to be one-way supported slabs. The square deck slabs used were 1200mm by 1200mm for the width and length with a thickness of 230mm. A total of 36 HDPE hollow spherical plastic bubble balls with a 180mm diameter were placed in the bubble deck slab specimens which reduce significantly the structural self-weight. In this paper, the experimental results of the flexural performance of the reinforced bubble deck slab, (BD slab) compared with a conventional reinforced concrete slab, simply supported, subjected to static area loadings, are presented. The effect of the load applied in the experiments on the flexural strength, bending stiffness and load-deflection behaviour of both types of slabs have been discussed including the crack propagation and crack pattern. In general, the conventionally reinforced solid slab, simply supported (SS) has a 60.6% higher resistance against bending deformation than the reinforced bubble deck slab.

Interaction of two bubbles with distortion in an acoustic field

Expression of the secondary Bjerknes force of two bubbles is obtained by considering the distrotion of two bubbles. The secondary Bjerknes forces in different acoustic fields are simulated, and the influence factors are analyzed and discussed. It is shown that the distortion of a bubble has an important influence on the interaction of two bubbles. The strength and even the directions of the secondary Bjerknes force of two bubbles with distortion differ considerably from the predictions of the sherical symmetry theory. The results show that when two bubbles oscillated stably in an acoustic field, the secondary Bjerknes force of two bubbles with distortion is several times more than that of two spherical bubbles in the same condition. The secondary Bjerknes force of two bubble with distortion has more interaction distance than that of two spherical bubbles. The secondary Bjerknes force of two bubbles with distortion depends on the distance of two bubbles, the shape mode of two bubbles, the equilibrium radii of two bubbles and the driving acoustic filed. The nonspherical distortion effects of the secondary Bjerknes has an importance on understanding the structure formation of bubbles and evolution process of bubble group in an acoustic field.

Numerical analysis of shock interaction with a spherical bubble

Two-dimensional and three-dimensional computational fluid dynamics studies of a spherical bubble impacted by a supersonic shock wave (Mach 1.25) have been performed to fully understand the complex process involved in shock–bubble interaction (SBI). The unsteady Reynolds-averaged Navier–Stokes computational approach with a coupled level set and volume of fluid method has been employed in the present study. The predicted velocities of refracted wave, transmitted wave, upstream interface, downstream interface, jet, and vortex ring agree very well with the relevant available experimental data. The predicted non-dimensional bubble and vortex velocities are also in much better agreement with the experiment data than values computed from a simple model of shock-induced Rayleigh–Taylor instability (the Richtmyer–Meshkov instability). Comprehensive flow visualization has been presented and analyzed to elucidate the SBI process from the beginning of bubble compression (continuous reflection and refraction of the acoustic wave fronts as well as the location of the incident, refracted and transmitted waves at the bubble compression stage) up to the formation of vortex rings as well as the production and distribution of vorticity. Furthermore, it is demonstrated that turbulence is generated with some small flow structures formed and more intensive mixing, i.e., turbulent mixing of helium with air starts to develop at the later stage of SBI.

Analytical study on the dynamic characteristics of multiple gas-filled spherical bubbles in typical spatial locations

The dynamic characteristics of multiple gas-filled spherical bubbles in three types of typical spatial locations are investigated analytically through a modified Rayleigh–Plesset equation. In the first type, two bubble centers form a one-dimensional straight line; the second type consists of any number of bubbles whose centers form a regular polygon in a two-dimensional plane; and in the third type, the bubble centers form a regular polyhedron in three-dimensional space. We show that physically these cases correspond qualitatively to periodic oscillations. Analytical expressions are derived for the maximum and minimum radii, based on which the oscillation amplitude and period are studied analytically. Parametric analytical solutions are also obtained. The influences of physical parameters on the multibubble motion are determined with the aid of these analytical results. We also study the limiting behavior of the analytical results for multiple bubbles, with the corresponding results for single bubbles being obtained as the distance between bubble centers approaches infinity.

A compressible 3D finite volume approach for the simulation of unsteady viscoelastic cavitating flows

We present a fully compressible, density-based finite volume solver for the simulation of 3D cavitating flows in viscoelastic Maxwell-/Oldroyd-like fluids. The upper-convected Maxwell model, the Oldroyd-B model and the linear and exponential simplified Phan-Thien Tanner models are implemented as viscoelastic constitutive equations in conservative formulation, and we identified the Truesdell rate as appropriate objective time derivative for compressible flows. Cavitation is modeled by a single-fluid homogeneous mixture equilibrium approach considering condensation and evaporation assuming volume averaged mixture quantities. The corresponding simplified quasilinear system is analyzed, and the wave speeds are calculated in order to adapt the employed four step Runge–Kutta explicit time stepping concerning the viscoelastic transport equations. We introduce a novel ghost cell boundary condition for the viscoelastic stress tensor. The approach is tested against (semi-)analytical unsteady and steady-state references and shows very good agreement. 3D simulations of the spherical vapor bubble collapse are performed for all implemented viscoelastic models and show a distinct influence compared to the Newtonian case. For the upper-convected Maxwell fluid a variation of the relaxation time exhibits its perspicuous influence on the dynamics of the collapse.

Effects of helium cavity size and morphology on the strength of pure titanium

High-purity α-titanium was implanted with helium to observe cavity morphology – size, number density, and form (i.e., bubbles vs. voids) – effects on materials strength. Increasing implantation temperatures lead to an Arrhenius-type (i.e., exponential) increase in cavity size, a transition from spherical bubbles to faceted voids, and a marked increase in strength (∼20–50%). Implanted samples deformed uniformly in contrast to nano-compression of unimplanted or alloyed titanium, but pillars with the largest faceted voids developed a type of bulging localized deformation. Affected voids were repeatedly sheared, becoming rough and shrunken. Dislocation-cavity interactions and the role of helium content within cavities are considered.

The importance of chemical mechanisms in sonochemical modelling

A state-of-the-art chemical mechanism is introduced to properly describe chemical processes inside a harmonically excited spherical bubble placed in water and saturated with oxygen. The model uses up-to-date Arrhenius-constants, collision efficiency factors and takes into account the pressure-dependency of the reactions. Duplicated reactions are also applied, and the backward reactions rates are calculated via suitable thermodynamic equilibrium conditions. Our proposed reaction mechanism is compared to three other chemical models that are widely applied in sonochemistry and lack most of the aforementioned modelling issues. In the governing equations, only the reaction mechanisms are compared, all other parts of the models are identical. The chemical yields obtained by the different modelling techniques are taken at the maximum expansion of the bubble. A brief parameter study is made with different pressure amplitudes and driving frequencies at two equilibrium bubble sizes. The results show that due to the deficiencies of the former reaction mechanisms employed in the sonochemical literature, several orders of magnitude differences of the chemical yields can be observed. In addition, the trends along a control parameter can also have dissimilar characteristics that might lead to false optimal operating conditions. Consequently, an up-to-date and accurate chemical model is crucial to make qualitatively and quantitatively correct conclusions in sonochemistry.

Nonuniform Collective Dissolution of Bubbles in Regular Pore Networks

Understanding the evolution of solute concentration gradients underpins the prediction of porous media processes limited by mass transfer. Here, we present the development of a mathematical model that describes the dissolution of spherical bubbles in two-dimensional regular pore networks. The model is solved numerically for lattices with up to 169 bubbles by evaluating the role of pore network connectivity, vacant lattice sites and the initial bubble size distribution. In dense lattices, diffusive shielding prolongs the average dissolution time of the lattice, and the strength of the phenomenon depends on the network connectivity. The extension of the final dissolution time relative to the unbounded (bulk) case follows the power-law function, {B^k/ell }, where the constant ell is the inter-bubble spacing, B is the number of bubbles, and the exponent k depends on the network connectivity. The solute concentration field is both the consequence and a factor affecting bubble dissolution or growth. The geometry of the pore network perturbs the inward propagation of the dissolution front and can generate vacant sites within the bubble lattice. This effect is enhanced by increasing the lattice size and decreasing the network connectivity, yielding strongly nonuniform solute concentration fields. Sparse bubble lattices experience decreased collective effects, but they feature a more complex evolution, because the solute concentration field is nonuniform from the outset.

Physics-based modelling and validation of inter-granular helium behaviour in SCIANTIX

In this work, we propose a new mechanistic model for the treatment of helium behaviour at the grain boundaries in oxide nuclear fuel. The model provides a rate-theory description of helium inter-granular behaviour, considering diffusion towards grain edges, trapping in lenticular bubbles, and thermal re-solution. It is paired with a rate-theory description of helium intra-granular behaviour that includes diffusion towards grain boundaries, trapping in spherical bubbles, and thermal re-solution. The proposed model has been implemented in the meso-scale software designed for coupling with fuel performance codes SCIANTIX. It is validated against thermal desorption experiments performed on doped UO2 samples annealed at different temperatures. The overall agreement of the new model with the experimental data is improved, both in terms of integral helium release and of the helium release rate. By considering the contribution of helium at the grain boundaries in the new model, it is possible to represent the kinetics of helium release rate at high temperature. Given the uncertainties involved in the initial conditions for the inter-granular part of the model and the uncertainties associated to some model parameters for which limited lower-length scale information is available, such as the helium diffusivity at the grain boundaries, the results are complemented by a dedicated uncertainty analysis. This assessment demonstrates that the initial conditions, chosen in a reasonable range, have limited impact on the results, and confirms that it is possible to achieve satisfying results using sound values for the uncertain physical parameters.

Three-Phase Flow Tomography System in Upward-Vertical High-Viscous-Oil/Water/Gas Flow

A new three-phase flow tomography system is presented in this work. The system was developed to be applied in high-viscous-oil/water/gas flow arranged in different flow patterns, with the three fluids present simultaneously in a crossing point. The primary measurement is based on a simpler and reliable digital in-phase/quadrature (IQ) demodulator and a wire-mesh sensor (WMS), but the system is capable of including different types of primary measurements. Dynamic three-phase flow tests were carried out to verify the system performance and compare its results with the ones obtained by quick-closing valves (QCVs) and a high-speed video camera. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">In situ</i> volumetric fractions and flow patterns were compared. The experiments were conducted in a vertical glass pipe of 50-mm i.d. and 12-m height. The observed three-phase flow patterns were water-continuous churn flow with dispersed bubbles and oil drops (WChBoDo), water-continuous slug flow with Taylor bubbles, spherical bubbles, and oil drops (WPiBoDo), and oil-continuous slug flow with Taylor bubbles, spherical bubbles, and water drops (OPiBoDi). Compared with the measurements using QCVs, the average relative errors of the measured volumetric phase fractions were 12% for water, 13% for air, and 22% for oil. The experimental results, including the details of the flow inside the pipe, the phase distributions, and the objective characterization of flow patterns, suggest that the proposed system is a promising option for three-phase flow-related studies and developments. The proposed technique can reach 8930 frames/s with a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$16\times16$ </tex-math></inline-formula> WMS.

Bubble formation behavior in hydrogen supply system using lithium metal

When a lithium-air battery is discharged, lithium ions normally react with oxygen at the positive electrode, but when the positive electrode potential drops significantly, water in the electrolyte reacts with lithium ions to generate hydrogen. In this study, we proposed a new system for high-density energy storage and hydrogen supply using lithium by utilizing the structure of lithium-air batteries and this working principle, and conducted basic experiments. As a result, it was confirmed that a large overvoltage is involved for the hydrogen production reaction compared to the theoretical potential. In addition, the formation behavior of hydrogen bubbles was observed, and it was confirmed that the almost spherical bubbles were generated in this experimental system. In the future, we will investigate the detailed causes of the overvoltage and hydrogen production behavior in the hydrogen production reaction and propose methods to reduce the overvoltage in order to improve the efficiency of hydrogen production.

Size quantification of non-spherical bubbles by ultrasound

Ultrasonic detection is an effective method to quantify bubbles in opaque liquid, and acoustic scattering model is the key in ultrasonic inversion technique. Classical scattering models are usually based on the spherical assumption and &lt;em&gt;ka&lt;/em&gt; is much less than 1. However, these conditions are not always satisfied in practical applications. In this study, a quantitative strategy of ultrasonic inversion is proposed for non-spherical bubbles and &lt;em&gt;ka&lt;/em&gt; deviation assumption. A series solution model for a spherical gas bubble is established without considering the &lt;em&gt;ka&lt;/em&gt; constraint, and it was compared with the classical Medwin (&lt;em&gt;ka&lt;/em&gt;&lt;&lt;1) and Anderson (&lt;em&gt;ka&lt;/em&gt;≈1) models. The difference in their scattering cross sections &lt;em&gt;σ&lt;sub&gt;bs&lt;/sub&gt;&lt;/em&gt; is only at the higher order formants of scattering and so the fitted line can be used to solve the multi-valued problem between &lt;em&gt;σ&lt;sub&gt;bs&lt;/sub&gt;&lt;/em&gt; and &lt;em&gt;ka&lt;/em&gt;. For a non-spherical bubble, &lt;em&gt;σ&lt;sub&gt;bs&lt;/sub&gt;&lt;/em&gt; is determined by the frequency domain backscattering signal and size is characterized by the equivalent radius&lt;em&gt; a&lt;/em&gt;&lt;sup&gt;*&lt;/sup&gt;and the inversion is performed by fitted curve from series solution model. Ultrasonic quantitative results were examined by high-speed photography. Results show that bubble rises in a zigzag pattern and non-spherical bubbles, their scattering cross sections are measured by the frequency domain scattering signal obtained at a position of ultrasonic measurement and the equivalent radius is inverted by the series solution fitting curve. The deviation between the results and the actual results &lt;em&gt;r&lt;/em&gt;&lt;sub&gt;0&lt;/sub&gt; is about 1mm(relative error less than 45%) when 9≤&lt;em&gt;kr&lt;/em&gt;&lt;sub&gt;0&lt;/sub&gt;≤35. This method can be used for acoustic inversion of non-spherical bubbles in a certain range of measurement accuracy.