Motion of a liquid containing gas bubbles

In this paper a class of models of continuous media is constructed whose free energy depends on the density, the rate of change of density, and the temperature. Conditions at discontinuities in such models are found. A Kogarko model for a mixture of liquid with gas bubbles is obtained. Moreover, the propagation of small disturbances is investigated. In the third and fourth sections exact solutions are found for the problems of nonsteady and steady motion of a mixture of liquid and bubbles in a tube.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 111–116, November–December, 1971.The author thanks L. I. Sedov for guidance and valuable advice.     

Motion of gas bubbles in a nonuniformly heated liquid

The thermocapillary drift of air bubbles in water was studied experimentally. The linear connection between the drift velocity and the temperature gradient predicted by theory was confirmed. The experimental and theoretical results are compared.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 55–57, September–October, 1979.We thank M. T. Sharov for assistance in setting up the experiment.     

Motion of large gas bubbles ascending in a liquid

An equation is derived for the ascent velocity of large gas bubbles in a liquid. This velocity is assumed to be governed by the propagation of a wavelike perturbation caused by the bubble in the liquid.Notation w bubble (or drop) velocity - specific gravity - dynamic viscosity - kinematic viscosity - r bubble (or drop) radius - surface tension - coefficient of friction - g gravitational acceleration - D bubble (or drop) diameter - p pressure - c propagation velocity of the wavelike perturbation - wavelength     

Motion of a piston under the influence of gas pressure in the presence of an initial temperature gradient

A study is made of the motion of a piston without initial velocity under the influence of gas pressure. Under the assumption that the temperature gradient is constant and fairly small, expressions are obtained for the distributions of the gas-dynamic parameters in the disturbed region between the piston and the leading edge of the sound wave propagating through the gas at rest.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 177–180, March–April, 1984.     

Motion of bed-loads under the influence of flow of a liquid

Equations are obtained for the motion of a water-soil mixture in the layer next to the base. The water-soil mixture is modeled by means of a viscous-friable medium, and the acceleration of the mixture is assumed to be small and is not taken into account. The validity of the equations is confirmed by the experimental data for the following characteristics of uniform motion of a flow: for the speed of the start of particles touching on an even bed and with allowance for inclines of a bed, and also for the flow rate of loads on an even bed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 182–185.The author is grateful to A. G. Kulikovskii, V. Ya. Shkadov, and M. É. Églit for discussing the results of the study.     

Dynamics of a liquid with gas bubbles

The hydrodynamics and diffusion of an admixture near an isolated bubble, which simulates the rise of either a chain of identical bubbles or a system of regularly arranged bubbles of the same volume, are analyzed by solving the Navier-Stokes equations numerically. Data are presented for a specific liquid. It is shown that in both cases the maximum flow velocity on the surface of identical bubbles is practically the same, although in the former case the ascent velocity is considerably higher. The stationary admixture diffusion from a bubble also proves to be nearly the same.In relation to the bubbling of a gas through a liquid layer, it is shown that the total admixture diffusion is maximum for regularly arranged bubbles whose diameter is comparable with the liquids capillary constant. Although the flow past the bubble remains continuous, the values of the hydrodynamic parameters are no longer small.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 75–88, May–June, 1996.     

Waves in a liquid mixture with gas bubbles

The one-dimensional nonstationary motion of a mixture of an ideal incompressible liquid with gas bubbles in a tube behind a moving piston is considered. An exact solution is obtained. Shock-wave propagation is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 143–145, July–August, 1976.The author thanks L. I. Sedov for his evaluation of the study and advice.     

Motion of a pair of bubbles in a liquid of low viscosity

Lagrange's equations are used to examine the long-range interaction of bubbles. The Lagrange function equals the kinetic energy of an ideal liquid flowing around a bubble. The generalized external forces include the upthrust and the viscous resistance to flow around each buble. The azimuthal angle is increased by the long-range interaction. The locus for the relative motion is calculated for: 1) the case in ~hieh the relative speed is fairly high, which allows one to neglect the effects of viscosity on the collision time. 2) low relative speed, where the viscous forces determine the motion. Estimates are given for the differential effective cross-section for elastic scattering and the coalescence cross-section.We are indebted to V. G. LevichandV. V. Tolmachev for discussions.     

Breakup of gas bubbles in a liquid by shock waves

The nature of the propagation of shock waves in various media is related to the characteristics of the latter, including their compressibility, thermophysical properties, the presence of multiple phases, etc. The structure of a shock wave varies appreciably as a function of the properties of the medium. The most significant property of a liquid mixture with gas bubbles is the compressibility of the latter under the influence of an externally applied pressure, for example, in a shock wave propagating in the liquid—gas medium. The transfer of momentum and energy between phases and the pressure variation behind the wave depends on the behavior of the gas bubbles behind the shock front.     

Dynamics of shock waves in a liquid containing gas bubbles

Results are presented of a numerical solution of the Korteweg-de Vries-Burgers equation that describes the propagation and establishment process for a stationary structure to a shock wave in a gas-liquid medium. Data are obtained on the time for the establishment of a stationary structure of a shock wave, propagation velocity, and amplitude oscillations in the front of the shock wave. Experiments are discussed on the basis of the results obtained for the study of shock waves in a liquid containing gas bubbles.     

Motion of bubbles and bubble-droplets in an immiscible liquid

A method is presented by which the movement of bubbles is recorded using still photography. Data can be obtained such as path, velocity, indications of surface instabilities, variation of size like growth and collapse.Results are presented for rising small and medium size air bubbles, spherical cap-shape butane bubbles, evaporating butane droplets and condensing butane bubbles in distilled water.

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Bewegungen von Blasen und Blasen-Tropfen in nichtmischbarer Flüssigkeit

Zusammenfassung Es wird eine Methode zur Wiedergabe der Blasenbewegung unter Benutzung der Standphotographie mitgeteilt. Man erhält damit Informationen über den Weg, die Geschwindigkeit, Anzeichen von Instabilitäten der Oberfläche und eine Änderung der Größe wie durch Wachstum und Zusammenbruch.Ergebnisse werden für den Aufstieg kleiner und mittlerer Luftblasen, kugeliger hutförmiger Butanblasen, verdampfender Butantropfen und kondensierender Butanblasen in destilliertem Wasser mitgeteilt.

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Nomenclature ATMP Atmospheric pressure (mm Hg) - f Number of bubbles per second - H Water head above the nozzle tip (mm) - T_n Temperature in the nozzle (°C) - T_s Butane saturation temperature at the nozzle tip (°C) - T_w Water temperature (°C) - U Rise velocity (mm/s) - Z Height above the nozzle tip (mm) - T Temperature difference - t Time between every consecutive shot(s)     

On the possibility of steady evolution of gas bubbles in a liquid

The equations describing the development of inclusions of a foreign phase are obtained, as a rule, under the assumption of a steady concentration field [1–3] or velocity field [4–7]. This is justified only if the process relaxes rapidly to a steady development. However, for the majority of systems of practical importance — the ensemble of pores when powders are sintered or an ensemble of gas bubbles in a liquid — even the actual possibility of their steady evolution is far from obvious. The general development of an ensemble of inclusions is associated with successive solution of the smallest of them with subsequent precipitation of the dissolved particles into the largest inclusions and into the exterior medium. Indeed, in the final stages of the process the particle fluxes are directed toward the exterior medium. It is clear that the complete process is strongly unsteady, since as it develops the directions and rates of molecular transport change. It is of interest to establish the conditions under which one can distinguish quasisteady stages characterized by the presence and persistence for an appreciable time of well-defined particle sources and sinks. Analysis of gas-liquid systems of simple configuration shows that steady regimes of their evolution are possible only if the liquid layer between the bubbles has a quite definite thickness.     

Nonlinear magnetoacoustic waves in a conducting liquid containing gas bubbles

The propagation of long waves in an incompressible conducting liquid saturated with nonconducting gas bubbles is considered on the basis of the equations of magnetohydrodynamics of a homogeneous gas-liquid medium. It is shown that the propagation of weakly nonlinear MHD waves in such a medium is described by the Burgers-Korteweg-de Vries (BKdV) equation. The influence of MHD interaction effects on the parameters of fast and slow weak magnetoacoustic shock waves is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 142–147, March–April, 1991.     

Note on the propagation of disturbances in a liquid containing gas bubbles

Summary In a liquid containing gas bubbles the speed of sound is less than that in each phase separately. Using the equations of motion for a homogeneous liquid containing gas bubbles it is shown that the dominating attenuation of an infinitesimal disturbance is that due to the second viscosity. In the propagation of a finite compressive disturbance an expression for the time required for the disturbance to display shock characteristics is found in terms of the initial disturbance profile and the liquid-gas ratio.