Evolution of a Small Distortion of the Spherical Shape of a Gas Bubble under Strong Expansion-Compression

The evolution of a small distortion of the spherical shape of a gas bubble which undergoes strong radial expansion-compression upon a single oscillation of the ambient liquid pressure under a harmonic law are analyzed by numerical experiments. It is assumed that the distortions of the spherical bubble shape are axisymmetric and have the form of individual spherical surface harmonics with numbers of 2–5. Bubble-shape oscillations prior to the beginning of expansion are taken into account. Generally, the distortion value during bubble expansion-compression depends on the phase of bubble-shape oscillation at the beginning of the expansion (initial phase). Emphasis is placed on the dependence of the maximum distortions in the initial phase at certain characteristic times of bubble expansion-compression on the amplitude of the external excitation, liquid viscosity, and distortion mode (harmonic number). The parameters of the problem are typical of the stable periodic sonolumiescence of an individual air bubble in water at room temperature. An exception is the liquid pressure oscillation amplitude, which is varied up to values that are five times the static pressure. That large excitation amplitudes are beyond the stability threshold of periodic oscillations of spherical bubbles. Their consideration is of interest from the point of view of increasing the compression ratio of the bubble gas, i. e., increasing the maximum temperature and density achievable in the final compression stage.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 17–28, July–August, 2005.     

Forced oscillations of two gas bubbles in a fluid in the vicinity of bubble contact

For a theoretical derivation of bubble coalescence conditions, nonlinear forced oscillations of two closely spaced spherical bubbles subjected to the action of a periodic external pressure field are considered. The equations, asymptotic with respect to a small distance between the bubble surfaces, are derived to describe the approach of the bubbles under the action of (i) the Bjerknes attraction force averaged over the oscillation period and (ii) the viscous drag. It is shown that due to nonlinear interaction of the viscous drag with the radial and translational oscillations of the bubbles a unidirectional repulsive force is generated, which prevents the approach of the bubbles. The coalescence of the bubbles is possible when the nondimensional parameter combined from the amplitude and frequency of the external pressure field, the bubble radius, and the fluid viscosity is greater than a certain critical value. The obtained coalescence condition is qualitatively confirmed by experiments.     

Liquid vorticity computation in non‐spherical bubble dynamics

The purpose of this work is to compare efficiency of a number of numerical techniques of computation of liquid vorticity from non‐spherical bubble oscillations. The techniques based on the finite‐difference method (FDM), the collocation method (one with differentiating (CMd) the integral boundary condition and another without it (CM)) and the Galerkin method (GM) have been considered. The central‐difference approximations are used in FDM. Sinus functions are chosen as the basis in GM. Problems of decaying a small distortion of the spherical shape of a bubble and dynamics of a bubble under harmonic liquid pressure variation with various parameters are used for comparison. The FDM technique has been found to be most efficient in all the cases. Copyright © 2004 John Wiley & Sons, Ltd.     

Forced oscillations of a gas bubble in a spherical volume of a compressible liquid

A spherically symmetric problem of oscillations of a single gas bubble at the center of a spherical flask filled with a compressible liquid under the action of pressure oscillations on the flask wall is considered. A system of differential-difference equations is obtained that extends the Rayleigh-Plesset equation to the case of a compressible liquid and takes into account the pressure-wave reflection from the bubble and the flask wall. A linear analysis of solutions of this system of equations is performed for the case of harmonic oscillations of the bubble. Nonlinear resonance oscillations and nearly resonance nonharmonic oscillations of the bubble caused by harmonic pressure oscillations on the flask wall are analyzed. Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 111–118, March–April, 1999.     

Forced nonlinear oscillations of a gas bubble in a large spherical flask (resonator) filled with a fluid

Radial oscillations of a gas bubble in a large spherical flask filled with a fluid are considered. We derive an equation of the change of the bubble radius by the known law of pressure variation at the boundary of the liquid volume (the law of motion of the piston) for a period of time during which, repeatedly reflected from the piston, the leading front of the reflected-from-the bubble perturbations reaches the bubble. For further calculations of the change of the bubble radius, recurrent relations which include the wave reflected from the bubble in the previous cycle and its subsequent reflection from the piston are obtained. Under harmonic action of the piston on the fluid-bubble system, a certain periodic regime with a package of bubble oscillations is established. Institute of Mechanics, Ural Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 77–87, September–October, 1998.     

Optimization of shock-wave action on a spherical bubble in an ideal incompressible liquid

The possibility of controlling the oscillations of a spherical gas bubble in an ideal incompressible liquid is subjected to theoretical analysis. Liquid surface tension forces are not taken into account. The optimization process realizing a maximum of the radius amplitude and a maximum of the gas pressure in the bubble for a given impulsive change of pressure at infinity is considered. A shock-resonance bubble oscillation procedure giving stepwise pressure changes at the extrema of the radius is constructed. This problem is of interest in connection with the investigation of cavitation erosion [1] and processes in biological tissues [2–4]. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 175–178, September–October, 1988.     

Evolution of deviations from the spherical shape of a vapor bubble in supercompression

This paper considers the evolution of small deviations of a cavitation bubble from a spherical shape during its single compression under conditions of experiments on acoustic cavitation of deuterated acetone. Vapor motion in the bubble and the surrounding liquid is defined as a superposition of the spherical component and its non-spherical perturbation. The spherical component is described taking into account the nonstationary heat conductivity of the liquid and vapor and the nonequilibrium nature of the vaporization and condensation on the interface. At the beginning of the compression process, the vapor in the bubble is considered an ideal gas with a nearly uniform pressure. In the simulation of the high-rate compression stage, realistic equations of state are used. The non-spherical component of motion is described taking into account the effect of liquid viscosity, surface tension, vapor density in the bubble, and nonuniformity of its pressure. Estimates are obtained for the amplitude of small perturbations (in the form of harmonics of degree n = 2, 3, ... with the wavelength λ = 2πR/n, where R is the bubble radius) of the spherical shape of the bubble during its compression until reaching extreme values of pressure, density, and temperature. These results are of interest in the study of bubble fusion since the non-sphericity of the bubble prevents its strong compression.     

Potential flow around a deforming bubble in a Venturi

The differential pressure between the entrance and throat of a Venturi will fluctuate if the liquid flowing through the Venturi contains bubbles. This paper reports computations of the pressure fluctuation due to the passage of a single bubble. The liquid is assumed inviscid and its velocity, assumed irrotational, is computed by means of a boundary integral technique. The liquid velocity at the entrance to the Venturi is assumed constant and uniform across the pipe, as is the pressure at the outlet. The bubble is initially far upstream of the Venturi and moves with velocity equal to that of the liquid. Buoyancy is neglected. If the bubble is sufficiently small that interactions with the Venturi walls may be neglected, a simple one-dimensional model for the bubble velocity is in good agreement with the full boundary integral computations. The differential pressure (taken to be positive) decreases when the bubble enters the converging section of the Venturi, and then increases to a value higher than for liquid alone as the bubble passes the pressure measurement position within the throat. The changes can be estimated using the one-dimensional model, if the bubble is small. The bubble is initially spherical (due to surface tension) but is perturbed by the low pressure within the Venturi throat. In the absence of viscosity, the bubble oscillates after leaving the Venturi. The quadrupole oscillations of the bubble are similar in frequency to those of a bubble in unbounded fluid; the frequency of the monopole oscillations is modified by the presence of the pipe walls. Numerical results for the frequency of monopole oscillations of a bubble in a uniform tube of finite length are in good agreement with analytic predictions, as is the computed drift of the oscillating bubble.     

Motion of a gas bubble in fluid under vibration

This paper is concerned with the analysis of motion of a gas bubble in a uniformly oscillating incompressible fluid. A theoretical model explaining the effect of sinking of gas bubbles in the absence of a standing pressure wave is validated experimentally. The conditions under which this effect occurs are determined, and a simple formula is derived for the average velocity of a gas bubble in the fluid.     

Dynamics of Oscillating Drops with Thermocapillary Effects

The dynamics of oscillating drops warming up in a hot gas environment is investigated via numerical simulation. The formation of surface and internal flows, due to variation of surface tension with temperature, and their impact on the oscillations are discussed. Both surface and ambient temperature disturbances are considered in terms of spherical harmonics. The effects of various parameters including modes of surface and temperature disturbances on period and amplitude of oscillations, kinetic and surface energies, and temperature field are studied. The most obvious feature of thermocapillary flows is demonstrated by vortices whose number and strength varies with the mode of temperature disturbance. These vortices tend to modify the amplitude of oscillations and enhance the kinetic energy. It is also shown that the decrease of the surface tension with increasing temperatures results in the increase of the period of oscillations while decreasing the surface energy. Due to the presence of thermocapillary flows, at long times, the equilibrium shape of the drop is not spherical and the kinetic energy approaches nonzero asymptotic values. The average temperature shows a nearly linear increase in time while the root mean square temperature, used to indicate the spatial variation, levels off after a fast initial growth. Received 3 January 2000 and accepted 7 July 2000     

The effect of high viscosity on compressible and incompressible Rayleigh–Plesset-type bubble models

Free oscillations of a single spherical gas bubble in glycerol have been examined numerically and experimentally at different ambient temperatures and pressures. The bubble was generated using a Q-switched Nd:YAG laser and the unsteady radius measurement was based on a shadowing technique of a He–Ne laser beam. The measurements were compared to computations obtained from two models, first taking into consideration the liquid compressibility and then assuming an incompressible liquid domain, respectively. In both cases the temperature fields inside and outside the bubble were computed by solving the energy equation in both phases as the thermodynamic processes have great importance to the bubble behavior. For high amplitude oscillations the incompressible model provides poor agreement with the measurements and the modeling of the liquid compressibility becomes necessary. In contrast to the standard method, a practical region of applicability for the incompressible approach was determined as a function of the instantaneous Mach and Reynolds numbers, rather than specifying a simple threshold Mach number.     

Map of regimes of pressure oscillations induced by absorption during gas jet injection through a submerged nozzle

Map of regimes of pressure oscillations induced by absorption during rapid injection of a soluble gas jet through a submerged nozzle into liquid, namely, oscillations during absorption, bubbling, internal chugging and small chugging, is suggested. Boundaries between various pressure oscillations regimes occurring when rapidly soluble gas is absorbed in water are investigated theoretically. It is showed that these boundaries are determined by four equations. It is showed that regime of high-frequency pressure oscillations during absorption occurs due to gas bubble oscillations, and other regimes of oscillations occur due to pressure oscillations in the whole system comprising the header, vent tubes and gas bubble. The conditions for excitation of high-frequency and low-frequency oscillations and boundaries between different regimes of pressure pulsations are determined. In a case when Henri's law for soluble gases is valid the developed model predicts that oscillations during absorption are not excited.     

Evolution of distortions of the spherical shape of a cavitation bubble in acoustic supercompression

The evolution of small perturbations of the spherical shape of a vapor bubble in the process of its single strong expansion and compression in deuterated acetone is studied. In the mathematical model used the motion of vapor and liquid is broken down into the spherical component and its small nonspherical perturbation. The spherical component is described by the fluid dynamics equations with account for time-dependent heat conduction and evaporation and condensation on the liquid-vapor interface using equations of state constructed from experimental data. In describing the nonspherical component the liquid viscosity and the surface tension are taken into account, while the effect of the bubble content is disregarded. Certain simple analytical formulas are presented which describe the bubble radius at the moment of maximum expansion, its variation in the compression stage, and the evolution of the bubble sphericity distortion in compression.     

Effect of weak compressibility of a fluid on bubble-bubble interaction in strong acoustic fields

The effect of weak compressibility of a fluid on the interaction between spherical bubbles in a strong acoustic field is considered. A small parameter ɛ which represents the ratio of the characteristic velocity of radial oscillations of the bubbles to the speed of sound in the fluid is used as a parameter characterizing the fluid compressibility. The equations governing the interaction between two bubbles are derived with an accuracy O(ɛ) in the case in which the ratio of the characteristic velocities of their translational and radial motions is of the order of ɛ. It is shown that neglecting the fluid compressibility effect due to the bubble interaction can lead to either enhancement or attenuation of their radial oscillations following the main compression stage, variation in the oscillation frequency, the bubble approach velocity, and the velocity of the spatial motion of the coupled pair, and the bubble approach and collision rather than their moving away from one another with the formation of a coupled pair.     

Oscillations of a gas bubble in a non-Newtonian liquid under the action of an acoustic field

The evolution of the radius of a spherical cavitation bubble in an incompressible non-Newtonian liquid under the action of an external acoustic field is investigated. Non-Newtonian liquids having relaxation properties and also pseudoplastic and dilatant liquids with powerlaw equation of state are studied. The equations for the oscillation of the gas bubble are derived, the stability of its radial oscillation and its spherical form are investigated, and formulas are given for the characteristic frequency of oscillations of the cavitation hollow in a relaxing liquid. The equations are integrated numerically. It is shown that in a relaxing non-Newtonian liquid the viscosity may lead to the instability of the radial oscillations and the spherical form of the bubble. The results obtained here are compared with the behavior of a gas bubble in a Newtonian liquid.     

Parametric Analysis for a Single Collapsing Bubble

This work presents a sensitivity analysis for cavitation processes, studying in detail the effect of various model parameters on the bubble collapse. A complete model (Hauke et al. Phys Rev E 75:1–14, 2007) is used to obtain how different parameters influence the collapse in SBSL experiments, providing some clues on how to enhance the bubble implosion in real systems. The initial bubble radius, the frequency and the amplitude of the pressure wave are the most important parameters determining under which conditions cavitation occurs. The range of bubble sizes inducing strong implosions for different frequencies is computed; the initial radius is the most important parameter characterized the intensity of the cavitation processes. However, other parameters like the gas and liquid conductivity or the liquid viscosity can have an important effect under certain conditions. It is shown that mass transfer processes play an important role in order to correctly predict the trends related with the effect of the liquid temperature, which translates into the bubble dynamics. Moreover, under some particular circumstances, evaporation can be encountered during the bubble collapse; this can be profitably exploited in order to feed reactants when the most extreme conditions inside the bubbles are reached. Thus, this paper aims at providing a global assessment of the effect of the different parameters on the entire cycle of a single cavitating spherical bubble immersed in an ultrasonic field. This work has been partially supported by Ministerio de Ciencia y Tecnologia, under grant number CTM2004-06184-C02-02.     

Acoustic radiation induced by bubble motion in compressible fluid

Based on the theory of compressible fluid, a three-dimension boundary element method is utilized to research the motion of bubble. The far-field noise radiation during the growth and contraction is calculated by the Kirchhoff formula and the Ffowcs Williams-Hawkings （FW-H） formula with a fixed radiation surface being arranged at the near-field of bubble as a new acoustic source. The results show that the amplitude of the sound pressure induced by non-spherical bubble is lower than that of spherical bubble in the contraction phase. The retardance effect is more obvious when the observer is farther away from the bubble. In the anaphase of contraction, the observer with the maximum amplitude of sound pressure moves up with the obvious jet. Larger buoyance parameters will generate lower sound pressure amplitudes in the anaphase, while larger intensive parameters will cause higher sound pressure amplitudes in the whole procedure of bubble motion.     

PRESSURE FLUCTUATIONS IN GAS-SOLIDS FLUIDIZED BEDS

Pressure fluctuation data measured in a series of fluidized beds with diameters of 0.05, 0.1, 0.29, 0.60 and 1.56 m showed that the maximum amplitude or standard deviation increased with increasing the superficial gas velocity and static bed height for relatively shallow beds and became insensitive to the increase in static bed height in relatively deep beds. The amplitude appeared to be less dependent on the measurement location in the dense bed. Predictions based on bubble passage, bubble eruption at the upper bed surface and bed oscillation all failed to explain all observed trends and underestimated the amplitude of pressure fluctuations, suggesting that the global pressure fluctuations in gas-solids bubbling fluidized beds are the superposition of local pressure variations, bed oscillations and pressure waves generated from the bubble formation in the distributor region, bubble coalescence during their rise and bubble eruption at the upper bed surface.     

Spherical bubble collapse in viscoelastic fluids

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius.