Flow of a viscous gas in a shock layer with equilibrium chemical reactions

Steady flow of supersonic air over a sphere is examined, allowing for viscosity, heat conduction, and actual physical and chemical processes. Flow in the shock layer at flight speeds in the range 3 km/sec V10 km/sec (10⁴R10⁶) is investigated, under the assumption of local thermodynamic equilibrium. The flow is described by simplified Navier-Stokes equations, which are solved by a finite difference method. The case of a cooled surface is examined. The distribution of gasdynamic parameters is obtained in different flow regimes. The distribution of heat flux and friction coefficient is investigated as a function of the oncoming-stream parameters and the sphere radius. The shape and position of the shock wave are determined, and the stream lines and sonic lines are constructed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 150–153, July–August, 1970.The authors thank Yu. P. Lun'kin and F. D. Popov for their help in formulating the problem and their constant interest.     

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.     

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     

Radial oscillations of gas bubbles in liquids in the presence of a combustible gas component

Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.1, pp. 103–109, January–February, 1993.     

The influence of gas solubility on the stability of a layer of liquid with bubbles and an immobile filling

The instability of a bubbling layer due to the presence of a vertical gradient in the ascent velocity of the bubbles, causing stratification of the layer with respect to density, is considered in [1]. A similar instability mechanism of a fluidized bed is studied in [2]. The stabilizing influence of electrical and magnetic fields on a bubbling layer is shown in [3]. Consideration is given in [4] to the influence of the conditions of supply of the gas on the stability of a bubbling layer with an immobile filling. The present work deals with the stability of the mechanical equilibrium of a horizontal layer of liquid with an immobile filling through which a gas soluble in the liquid is bubbled. It is shown that there exists a critical solubility of the gas at which the mechanical equilibrium is unstable with respect to monotonie perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 68–74, September–October, 1984.The author would like to thank V. P. Myasnikov and V. V. Dil ' man for their interest in this work, and M. H. Rozenberg for assistance with the programming.     

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.     

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.     

Self-similar problems of convective diffusion in the presence of heterogeneous chemical reactions with allowance for thermal diffusion effects

A number of studies have been made of the problem of the effect of a temperature gradient on mass transfer in a forced viscous fluid flow. The question of allowing for thermal diffusion effects has been examined in connection with flow around bodies [1–4], duct flow [5], and jet flows [6,7]. Recently, in addition to the problem of thermal diffusion separation, the attention of investigators has been claimed by the problem of taking into account the effect of thermal diffusion on mass transfer in a convective flow in the presence of chemical reactions on the flow surfaces [4].     

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.     

Dynamics of a single bubble in a liquid in the presence of chemical reactions and interphase heat and mass exchange

The influence of inert and chemically reactive additives in the form of microdrops on the dynamics of a single bubble filled with an active gas mixture and collapsing under the action of a shock wave is considered. The development of a reaction during formation of the mixture is analyzed for instantaneous and dynamic evaporation of drops with allowance for various phases of their injection t _inj . It is shown that in instantaneous evaporation, an increase in the fraction of gaseous argon in the H₂+O₂ system raises the final temperature of the system under cryogenic conditions, lowers it under ordinary conditions, and causes appreciable oscillations of the values of γ, heat release, and molecular weight. It is noted that there are values of t _inj and D₀ at which the final temperature of the mixture decreases practically to the initial temperature. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 119–127, March–April. 1999.     

Conjugated heat and mass transfer between a reactive solid and a gas in the presence of nonequilibrium chemical reactions

An investigation of a multicomponent boundary layer taking account of nonequilibrium chemical reactions has been made in a number of publications [1–3]; here, the temperature of the solid was assumed to be known or was determined from the condition of the conservation of energy at the interface between the gas and the solid, taking account of the solution of the equation of thermal conductivity in the solid phase. At the same time, heating of the material of a coating is an unavoidable step in any mechanism of thermokinetic decomposition and, in view of this, it is necessary to take account of the lag of the heat-transfer process inside the solid. Therefore, it is necessary to solve the equation of the energy balance in the solid phase simultaneously with the system of the equations of the boundary layer, i.e., the conjugate problem. The present article discusses the problem of flow around a solid in the vicinity of a frontal critical point, taking account of the dependence of the processes taking place in the solid body on the time, in the presence of two heterogeneous and one homogeneous reactions. The distributions of the velocity, the temperature, and the concentrations in the boundary layer are obtained, as well as the mass rate of entrainment of the material at different moments of time. The time of the change between kinetic and diffusion conditions of the course of the heterogeneous chemical reactions (the ignition time) is determined. It is established that, in the presence of a homogeneous chemical reaction, the mass rate of entrainment is less than with a frozen flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 121–128, March–April, 1974.     

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.     

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.