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.     

Modulation equations for a mixture of gas bubbles in an incompressible liquid

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 86–92, March–April, 1989.     

Study of mass transfer in a system of gas bubbles and a liquid

The dissolution of a system of CO₂ bubbles in water and aqueous solutions of dextrose has been studied experimentally. The interfacial surface area is calculated, and the mass-transfer coefficient is determined. A relation is found among, the bubble diameter, the physical properties of the liquid, and the rate at which the gas is bubbled through the liquid. A criterial equation is proposed for.     

Collapse of vapor bubbles in a liquid

The effect of nonequilibrium phase transitions on the vibrations of a vapor bubble in a liquid caused by a suddenly applied pressure drop is considered. This problem is of interest in the study of mixed liquid and vapor flows with a discrete vapor phase. Results are presented of a numerical solution of this problem in the form of dimensionless radius-time curves for various values of the parameter which characterizes the kinetics of the phase transitions. The case of equilibrium phase transitions has been considered in [1, 2]. The thermal and dynamic interactions of a gaseous bubble with the surrounding fluid are the subject of [3, 4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 50–54, November–December, 1973.The author thanks R. I. Nigmatulin for advice and interest in this work and V. Sh. Shagapov for useful discussions.     

Allowance for nonstationary heat and mass transfer in the propagation of small perturbations in a liquid with bubbles

The propagation of small perturbations in a liquid with vapor-gas bubbles is studied. Heat and mass transfer between the phases is taken into account on the basis of the exact equations of heat conduction and diffusion. The aim of the present investigation is to make more precise the results of an earlier paper [1] of the author and verify the applicability of using fixed coefficients of heat and mass transfer for nonstationary heat and mass transfer between a pulsating bubble and the liquid.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 157–162, July–August, 1979.     

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.     

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.     

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.     

Unsteady waves in a liquid containing vapor bubbles

Unsteady wave processes in vapor-liquid media containing bubbles are investigated taking into account the unsteady interphase heat and mass transfer. A single velocity model of the medium with two pressures is used for this, which takes into account the radial inertia of the liquid with a change in volume of the medium and the temperature distribution in it [1]. The system of original differential equations of the model is converted into a form suitable for carrying out numerical integration. The basic principles governing the evolution of unsteady waves are studied. The determining influence of the interphase heat and mass transfer on the wave behavior is demonstrated. It is found that the time and distance at which the waves reach a steady configuration in a vapor-liquid bubble medium are considerably less than the correponding characteristics in a gas-liquid medium. The results of the calculation are compared with experimental data. The propagation of acoustic disturbances in a liquid with vapor bubbles was studied theoretically in [2]. The evolution of waves of small but finite amplitude propagating in one direction in a bubbling vapor-liquid medium is investigated in [3, 4] on the basis of the generalization of the Burgers-Korteweg-de Vries equation obtained by the authors. An experimental investigation of shock waves in such a medium is reported in [5, 6], and the structure of steady shock waves is discussed [7].Translated from Izvestiya Akademii Nauk SSSR, Hekhanika Zhidkosti i Gaza, No. 5, pp. 117–125, September–October, 1984.     

Numerical method for the system of reaction-diffusion equations with a small parameter

This paper deals with the numerical method for the system of reaction-diffusionequations with a small parameter.It is difficult to solve the problems of this kindnumerically because of the boundary layer efect.Besed on singular perturbed theory andGreens’function,we have established the difference scheme that is suited for the solution tothe problems.It e introduce an idea of feasilbe equidistant degree a here.And this provesthat if a≥2.the scheme converges in l,m norm with speed O.h \t)uniformly.     

Mass transfer during gas absorption in a vertical gas-liquid slug flow with small bubbles in liquid plugs

A model is developed for the analysis of mass transfer during isothermal absorption in a vertical gas-liquid slug flow at large Reynolds numbers with liquid plugs containing small bubbles. Simple formulas for mass flux from the N-th unit cell of gas-liquid slug flow and for total mass flux from N unit cells are derived. In the limiting case the derived formulas for mass transfer during gas absorption in a slug flow with liquid plugs containing small bubbles recover the derived expressions for mass transfer in slug flow without small bubbles in the liquid plugs. Using the developed model recommendations concerning the design of the absorber operating in a slug flow regime are suggested. Received on 28 July 1997