Transient deformation of an inviscid inclusion in a viscoelastic extensional flow

The transient deformation of a bubble in a viscoelastic extentional flow is analyzed by means of a finite element algorithm for viscoelastic moving boundary problems. Using the Oldroyd-B constitutive model, we find that bubbles in a viscoelastic fluid deform to the same steady-state configurations as bubbles in a Newtonian fluid at equal values of the far-field extensional stresses (corresponding to different stretch rates). Vapor bubbles in a developed extensional flow collapse more readily in the viscoelastic liquid than bubbles in Newtonian fluids because of the large compressive stresses associated with the viscoelastic liquid.     

Diffusion-induced growth of a gas bubble in a viscoelastic fluid

The diffusion-induced growth of a spherical gas bubble surrounded by a thin shell of viscoelastic fluid containing a limited amount of dissolved gas is analyzed. This is representative of a situation when a large number of bubbles grows in close proximity in a viscoelastic medium. The upper-convected Maxwell model is employed to describe the rheology of the fluid. Limited quantities of the dissolved gas available in the liquid shell mandates solution of the convection-diffusion equation, as opposed to using similarity solutions or polynomial profiles to describe the mass transport across the interface. Utilizing the properties of a potential field and a Lagrangian transformation, a new approach is introduced to solve the coupled system of integro-differential equations governing the bubble growth. The results indicate that, at the early stages of the growth, bubbles in a viscoelastic fluid grow faster than in a Newtonian fluid. However, eventually they attain the same steady-state configuration.     

Evolution of finite perturbations in a viscoelastic relaxing liquid with gas bubbles

The propagation of one-dimensional perturbations in a viscoelastic relaxing liquid containing gas bubbles is investigated within the framework of the homogeneous model of the medium when the wavelength of the perturbation is much larger than the distance between the bubbles and the bubble radius. The evolution of stationary and nonstationary waves is investigated analytically and with the use of numerical integration; shock waves are also investigated. The results are compared with the behavior of perturbation waves in a Newtonian liquid with gaseous inclusions. The models of the gas-liquid medium [1, 2] are generalized to the case when the liquid phase is a viscoelastic liquid, for example, a weak aqueous solution of polymers. The propagation of longwave perturbations of finite amplitude in such a mixture is investigated using the technique developed in [3].     

Acoustic Waves in a Liquid with Solid Particles and Gas Bubbles

The mathematical model which determines acoustic wave propagation in a mixture of liquid with gas bubbles and solid particles is proposed. A system of differential equations is written and the dispersion relation is derived. Low- and high-frequency asymptotics of the phase velocity in the mixture considered are found and illustrated. The effect of solid particles and gas bubbles on acoustic wave dispersion and dissipation is indicated. For the mixture of fluid with solid particles considered the speed of sound is compared with available experimental data.     

A numerical study of weak shock wave propagation in a reactive bubbly liquid

A numerical study is presented on the response of a weakly shock compressed liquid column that contains reactive gas bubbles. Both the liquid and gas are considered compressible. Compressibility of the liquid allows calculation of shock and rarefaction waves in the pure liquid as well as in the gas/liquid mixture. A microscopic model for local bubble collapse is coupled with a macroscopic model of wave propagation through the gas/liquid mixture. In the particular cases presented here, the characteristic times of propagation of the shock wave and bubble collapse are of the same order of magnitude. Consequently, the coupling between various phenomena is very strong. The present model based on fundamental principles of two-phase fluid mechanics takes into account the coupling of localized bubble oscillations. This model is composed of a microscopic one in the scale of a bubble size, and a macroscopic one which is based on the mixture theory. The liquid under study is water, and the gas is a reactive mixture of argon, hydrogen and oxygen. Received 18 December 1995 / Accepted 2 June 1996     

Evolution of bubble spectrum in the process of cavitational fragmentation by a shock wave reflected from a free liquid surface

Equations which describe the evolution of the bubble spectrum in the process of cavitational fragmentation by a shock wave reflected from a free liquid surface are formulated. As an example, the effect of artificial saturation of the initial fluid with large bubbles on the dispersity of a liquid-drop gas suspension focused by dispersion is investigated.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 51–58, May–June, 1992.     

The transient buoyancy driven motion of bubbles across a two-dimensional quiescent domain

The transient buoyancy driven motion of two-dimensional bubbles across a domain bounded by two horizontal walls is studied by direct numerical simulations. The bubbles are initially released next to the lower wall and as they rise, they disperse. Eventually all the bubbles collect at the top wall. The goal of the study is to examine how a simple one-dimensional model for the averaged void fraction captures the unsteady bubble motion. By using void fraction dependent velocities, where the exact dependency is obtained from simulations of homogeneous bubbly flows, the overall dispersion of the bubbles is predicted. Significant differences remain, however. We suggest that bubble dispersion by the bubble induced liquid velocity must be included, and by using a simple model for the bubble dispersion we show improved agreement.     

Waves in partially saturated porous media

The propagation of compressional waves in a porous medium is investigated in case the pore liquid contains a small volume fraction of gas. The effect of oscillating gas bubbles is taken into account by introducing a frequency-dependent fluid bulk modulus, which is incorporated in the Biot theory. Using a shock tube technique, new experimental data are obtained for a porous column subjected to a pressure step wave. An oscillatory behaviour is observed, consisting of two distinct frequency bands, which is predicted by the theoretical analysis.     

Acoustic waves in liquids with polydisperse gas bubbles. Comparison of theory and experiment

A mathematical model describing the propagation of acoustic waves of different geometry in two-fraction mixtures of a liquid with polydisperse gas bubbles of different composition is presented. A system of differential equations for the perturbed motion of the two-phase mixture is formulated and a dispersion relation is obtained. The theory developed is compared with known experimental data, including those for a near-resonance bubble frequency.     

On the structure of shock waves in liquid-bubble mixtures

Asbtract The structure of shock waves in liquids containing gas bubbles is investigated theoretically. The mechanisms taken into account are the steepening of compression waves in the mixture by convection and the effects due to the motion of the bubbles with respect to the surrounding fluid. This relative motion, radial and translational, gives rise to dissipation and to dispersion caused by the inertia of the radial flow associated with an expanding or compressed bubble. For not too thick shocks the dissipation by radial motion around the bubbles dominates over the dissipation by relative translational motion, in mixtures with low gas content. The overall thickness of the shock appears to be determined by the dispersion effect. Dissipation, however, is necessary to permit a steady shock wave. It is shown that, analogous to undular bores, a stationary wave train may exist behind the shock wave.     

Dynamics of soluble gas bubbles

The problem of the mass, thermal and dynamic interaction between a bubble containing a soluble gas and a liquid is considered. It is shown that this problem can be reduced to the problem of the behavior of a vapor bubble with phase transitions investigated in detail in [1–3]. Expressions are obtained for the rate of decay of the radially symmetric oscillations of the bubbles due to the solubility of the gas in the liquid. The effective coefficients of mass transfer between the radially pulsating bubbles and the liquid are determined. A numerical solution is obtained for the problem of the radial motion of a bubble created by a sudden change of pressure in the liquid which, in particular, corresponds to the behavior of the bubbles behind the shock front when a shock wave enters a bubble screen.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 52–59, November–December, 1985.     

Dissipation of the energy of sound waves in small gas bubbles

One of the main factors affecting the dynamics of homogeneous solution type pulse reactors is the formation of gas bubbles on the fission-fragment tracks [1, 2]. The behavior of the reactor depends very considerably on the size (10^?5 cm) and growth rate of these bubbles [2], and it is, accordingly, a very important matter to study these properties. One convenient means of doing this lies in the acoustic method. The behavior of gas bubbles in the field of a sound wave has been studied in a large number of papers and reviews [3, 4]. In this paper we shall see the approximation of a sound wave of small amplitude to consider the dissipation of sound-wave energy in a gas bubble, at the same time allowing for inertia, surface tension, viscosity, heat transfer, and the diffusion of gas through the surface of the bubble.     

Pressure wave propagation in liquid-filled tubes of viscoelastic material

The propagation of small-amplitude waves in a thick-walled long viscoelastic tube of variable cross-section, filled with a viscous incompressible fluid, is considered with account for wave reflection at the tube end in application to arterial pulse wave propagation. A solution is obtained in the form of expansions in a small parameter. The effect of the coefficient of wave reflection at the tube end and the wall material parameters on the fluid volume flow-rate and the tube wall displacement is investigated. It is shown that the volume flow-rate phase spectrum characteristics depend only slightly on the wall properties and can be used in clinical diagnostics for finding the reflection coefficient from pressure and flow-rate records.     

Propagation of pressure waves in two-phase flow

We deal with a pressure wave of finite amplitude propagating in a gas and liquid medium or in the fluid in an elastic tube. We study the effects of pipe elasticity on the propagation velocity of the pressure wave. Pressure waves of finite amplitude progressing in the two-phase flow are treated considering the void fraction change due to pressure rise. The propagation velocity of the two-phase shock wave is also investigated, and the behavior of the reflection of the pressure wave at the rigid wall is analyzed and compared to that in a pure gas or liquid. The results are compared to experimental data of a pressure wave propagating in the two-phase flow in a vertical shock tube.     

Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations

Shock wave structure in a bubbly mixture composed of a cluster of gas bubbles in a quiescent liquid with initial void fractions around 10% inside a 3D rectangular domain excited by a sudden increase in the pressure at one boundary is investigated using the front tracking/finite volume method. The effects of bubble/bubble interactions and bubble deformations are, therefore, investigated for further modeling. The liquid is taken to be incompressible while the bubbles are assumed to be compressible. The gas pressure inside the bubbles is taken uniform and is assumed to vary isothermally. Results obtained for the pressure distribution at different locations along the direction of propagation show the characteristics of one-dimensional unsteady shock propagation evolving towards steady-state. The steady-state shock structures obtained by the present direct numerical simulations, which show a transition from A-type to C-type steady-state shock structures, are compared with those obtained by the classical Rayleigh–Plesset equation and by a modified Rayleigh–Plesset equation accounting for bubble/bubble interactions in the mean-field theory.      

Disengagement of the gas phase in bubble columns

The technique of gas disengagement is popularly used to assess the bubble size distribution in bubble columns. The technique involves the dynamic measurement of dispersion height when the gas supply is stopped. In this paper a mathematical model has been proposed for the process of dynamic gas disengagement. It has been shown that the initial faster disengagement is due to the presence of internal liquid circulation and not due to the presence of very large bubbles. Further, slower disengagement has been attributed to the transition from heterogeneous dispersion to homogeneous dispersion. The new model also explains the effects of superficial gas velocity, column diameter, column height and liquid phase physical properties on the gas disengagement.