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.     

Transition from regular reflection to mach reflection when a shock wave interacts with a wall in a two-phase gas—liquid medium

A study is made of the transition from regular reflection to Mach reflection when a plane moderately strong or weak shock wave interacts with a wall in a two-phase gas—liquid medium. An equilibrium model that differs from the model of Parkin et al. [1] by the introduction of the adiabatic velocity of sound is used to investigate shock wave reflection in the complete range of gas concentrations. For the reflection of weak shock waves, nonlinear asymptotic expansions [2] are used. In the limiting cases, the results agree with those already known for single-phase media [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 190–192, September–October, 1983.     

Modeling of retrograde condensation in the process of steady radial flow through a porous medium

The problem of the steady axisymmetric two-phase flow of a multicomponent mixture through a porous medium with phase transitions is considered. It is shown that the system of equations for the two-phase multicomponent flow process, together with the equations of phase equilibrium, reduces to a system of two ordinary differential equations for the pressures in the gas and liquid phases. A family of numerical solutions is found under certain assumptions concerning the pressure dependence of the molar fraction of the liquid phase.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 92–97, November–December, 1994.     

Hydroshock theory in a two-phase gas-liquid mixture in a slug flow regime

A study is made of the intensity of a hydroshock in a two-phase gas-liquid mixture in a slug flow regime in the case when a pipeline is shut off by a liquid slug. The intensity is studied as a function of the length of the shut-off section of the liquid slug, the content of gas bubbles in the liquid slugs, and the pipeline shut-off law, and with allowance for the shock-wave character of the process [1, 2]. The calculated data using the shock-wave theory agree well with the experimental data of [3] and, unlike the results of the linear theory of [3], make it possible to determine the intensity of the hydroshock not only in the case of weak waves, but also in the case of waves of moderate intensity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 188–190, September–October, 1985.     

Impact Dynamics of Damaged Shells Interacting with a Two-Phase Liquid

The propagation of shock waves in a system consisting of a deformable medium with damage and a two-phase liquid with gas or vapor bubbles are studied. The nonlinear interaction of the media are modeled taking into account phase transformations in the liquid and the damage kinetics of the deformable medium. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 1, pp. 139–152, January–February, 2006.     

Cell model of nonisothermal gas absorption in gas-liquid bubbly media

A model for combined mass and heat transfer during nonisothermal gas absorption in a two-phase gasliquid bubbly medium with a high gas content and/or large times of gas-liquid contact is suggested. Diffusion and thermal interactions between bubbles is taken into account in the approximation of a cellular model of a bubbly medium whereby a bubbly medium is viewed as a periodic structure consisting of identical spherical cells with periodic boundary conditions at a cell boundary. Distribution of concentration of dissolved gas, temperature distribution in liquid and coefficients of mass and heat transfer during nonisothermal absorption of a soluble pure gas from a bubble by liquid are determined. In the limiting case of absorption without heat release the derived formulas recover the expressions for isothermal absorption.     

Steady two-phase flow in a vertical tube subject to combined free and forced convection

Using the two-velocity, two-temperature model of a continuous medium, the viscousgravitational flow of a mixture of incompressible liquid and solid particles in a vertical round tube is considered. The free-convection equations are written down on the basis of the general equation of motion and the energy equation of a two-phase medium [1, 2]. Using a finite Hankel integral transformation, a solution is constructed for the case of a linear wall-temperature distribution along the tube. The results of some practical calculations of the velocity and temperature fields over the cross section of the tube are presented, together with the dimensionless heat-transfer coefficient expressed as a function of the Rayleigh number and phase concentration. Here it is assumed that the dynamic and thermal-interaction coefficients between the phases correspond to the Stokes mode of flow for each particle, as a result of which the velocity and thermal phase lag is very small [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 132–136, July–August, 1975.     

Solution of some variational problems of two-phase gas dynamics

The problem of constructing an optimal-profile nozzle for a two-phase medium is considered in the one-dimensional approximation. A problem of this type to find an optimal-thrust nozzle was considered by Kraiko, Starkov, and Sternin [1]. In contrast to their study, a more complete model of the two-phase medium is used in the present paper, and the nozzles are optimized with respect to the efficiency, gas velocity, and velocity of the suspended particles. The problem is solved using the formalism of optimal control theory [2, 3]. The change in the vapor concentration and phase transitions are taken into account. A method of numerical solution of the problem is proposed. It has been realized on a computer. The method can be used to solve similar problems for a more complicated model of the two-phase medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 52–58, January–February, 1982.     

Method for calculating the dynamics of vapor bubbles when a liquid boils in a centrifugal force field

The specific feature of the study of the dynamics of vapor bubbles during boiling of a liquid in a centrifugal force field is the fact that the velocity of a bubble is much greater than the rate of change of its radius, and its movement occurs in fields of variable pressure and underheating that have to be determined in the solution of the problem. In addition, when investigating processes occurring when liquid helium boils in a centrifugal force field, its thermodynamic parameters may be close to the critical values, and the dependences of the thermophysical properties of the liquid and vapor on the temperature and pressure must be taken into consideration. The equation of state of a substance close to its critical thermodynamic point cannot be an approximation to the equation of state of an ideal gas, as has been suggested in a series of articles. The nonequilibrium nature of the phase transition must be taken into consideration in the case of substances existing at near-critical parameters and substances with a low coefficient of accommodation. A marked deformation of the bubbles, which also has to be taken into account, will be observed in strong centrifugal force fields. Such studies have not appeared in the specialist journals. Equations of the two-temperature and two-velocity hydrodynamics of two-phase media in a one-dimensional form for substances obeying the equation of state for an ideal gas were discussed in [1, 2] with allowance for the dependence of the thermophysical properties on the temperature and pressure. In strong centrifugal force fields the one-dimensional approach is unacceptable and the flow of liquid around a buoyant bubble must be taken into account. A joint examination of the change in the temperature field with time in the vicinity of a vapor bubble with changes in its dimensions and position was made for the first time in [3–8]. The present article is an extension of the latter work and takes the aforementioned factors into account.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 66–71, July–August, 1984.     

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.     

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.     

Radiation of waves by a sphere harmonically oscillating to-and-fro in a viscous medium

The theory of the radiation of sound by a sphere in an ideal medium is presented in detail in [1–3]. The emission of waves by a sphere oscillating to-and-fro in a viscous incompressible liquid is analyzed in [4, 5]. The present paper gives a precise solution to the problem of the radiation of sound by a sphere oscillating in a viscous medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 101–106, September–October, 1970.     

Numerical study of the dynamics of a vapor envelope round a heated solid particle in a liquid

A study is made of the radial motion of a vapor envelope surrounding an isolated spherical particle in an unbounded mass of liquid. It is assumed that the liquid is viscous and incompressible and that the temperature is distributed uniformly in the solid particle. A model of a calorifically perfect gas is used for the vapor phase. The same assumptions are made as in Rayleigh's formulation for the problem of the dynamics of a single bubble: that the process is spherically symmetric and that the pressure P₂ (t) in the vapor phase is homogeneous. The justification for making these assumptions in problems of the dynamics of gas, vapor, and vaporgas bubbles is discussed in [1–5]. In this paper, the collapse of the vapor layer and the boiling of the liquid on the surface of the heated particle are not considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 154–157, July–August, 1985.     

Nonlinear waves in a liquid containing gas bubbles with allowance for phase transition

An equation describing the kinetics of the mass transfer accompanying the process of gas bubble growth in an incompressible liquid is proposed. The equation of state of the two-phase system, whose derivation is based on the mechanism of interphase mass transfer due to the solubility of the gas atoms in the liquid, is obtained. An evolution equation generalizing the usual Burgers-Korteweg-de Vries (BKdV) equation is derived for describing the nonlinear waves in a gas-containing liquid. The solution of this equation is obtained in the form of a solitary wave.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 114–119, May–June, 1989.     

Hydrodynamical problems of first-order phase transitions

The two-phase liquid-vapor system in a state of thermodynamic equilibrium is considered. If a shock wave propagates in this medium, during its passage the material undergoes shock compression and transforms into a new equilibrium state. Not only the initial velocity changes in this case, but so does the quantitative composition of the phases. Due to the complication of the process, analytic results have practically not been available so far. Calculations of parameters behind the shock discontinuity were carried out approximately by using various tables and nomograms, restricted basically to only one two-phase system, H₂O. Thus, condensation jumps were treated in [1–4] in two-phase supersonic flows within the single-velocity model and a low content of the liquid phase in the mixture. Using the assumptions mentioned, the various parameters were found at the front of the shock wave by numerical solution of the conservation equations of mass, momentum, and energy at the discontinuity. The thermodynamic parameters are usually given in tabulated form as a function of pressure or temperature for equilibrium conditions of the phases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 81–87, September–October, 1977.     

Passage of shock waves through an interface of two-phase gas-liquid media

Among the problems connected with the motion of shock waves in two-phase media consisting of a mixture of a liquid and bubbles of gas, investigations of the passage of shock waves through an interface inside of a two-phase system, or at its surface are of special interest. Inside a two-phase system, interfaces between two two-phase media with different volumetric concentrations of the gas Β₁, Β₂ are possible. One of the values of Β, for example, Β₁, can revert to zero. There is then a passage of the wave from a two-phase system into an incompressible liquid, or vice versa. The investigation of both of the above cases, as well as of the transition two-phase medium (Β₁)—two-phase medium (Β₂) with Β₁≠Β₂ is not only of scientific, but also of practical importance. As is well known [1], the density of a two-phase mixture ρ with a small volumetric concentration of gas is calculated using the relationship ρ=(1?β)ρ_l+αρ₁.     

Dynamics of vapor-gas bubbles

The nonlinear problem of thermal, mass, and dynamic interaction between a vapor-gas bubble and a liquid is considered. The results of numerical solution of the problem of radial motion of the bubble caused by a sudden pressure change in the liquid, illustrating the behavior of vapor-gas bubbles in compression and rarefaction waves, are presented. The corresponding problem for single-component gas and vapor bubbles was considered in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 56–61, November–December. 1976.