Boundary conditions on a rigid surface in a two-phase flow

A study is made of the boundary conditions on a rigid surface in a two-component disperse flow. Appropriate boundary conditions are obtained for the kinetic equation and macroscopic equations of a pseudogas of solid particles proposed in [1–3]. The reasons for the occurrence of bubbles in two-phase systems are discussed. On the basis of the similitude parameters of the kinetic equation of the pseudogas, disperse systems are classified generally on the basis of the concentration of solid particles and their diameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 46–51, May–June, 1980.I thank V. P. Myasnikov for suggesting the problem and for a helpful discussion.     

Kinetic theory of disperse systems

A kinetic equation for the motion of solid particles in a liquid or gas is derived on the basis of the Fokker-Planck-Kolmogorov diffusion equation for the N particle distribution function. It is shown that, under appropriate assumptions, Bogolyubov's method can also be applied to equations of diffusion type. The obtained kinetic equation is a generalization of the one proposed earlier in [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 128–132, January–February, 1980.I thank V. P. Myasnikov for suggesting the problem and for helpful discussions.     

A model and kinetic equations for a charged mixture of gases in the presence of phase transitions

A model of a gas mixture is studied in which one of the components can carry electric charge and undergo phase transitions. Under a number of assumptions, Boltzmann kinetic equations are written down and the form of the collision integral determined. Conservation equations for the components of the mixture are found. The conservation equations for a charged mixture of gases in the absence of phase transitions have been discussed earlier [1]. Collision integrals for a reacting gas mixture in the case of chemical reactions of bimolecular type and when the mixture is described by Boltzmann kinetic equations are derived in [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 3, pp. 118–127, May–June, 1980.     

Group-invariant solutions of kinetic equations

In the development of analytic methods of solution of kinetic equations, it is expedient to use group raetliods. The establishment of a symmetry group makes it possible to justify the choice of a definite model of kinetic equation corresponding to the physical formulation of the problem, to solve the Cauchy problem in a number of cases, and to obtain classes of new exact solutions that can be used as standards in the construction of numerical algorithms for solving kinetic equations. Bobylev [1–4] and Krook and Wu [5, 6] used group methods to analyze the spatially homogeneous Boltzmann equation in the case of isotropy with respect to the velocities and Maxwellian molecules. They obtained exact solutions and investigated the asymptotic behavior of the main equation. In the present paper, group methods are used to find and analyze exact solutions of the Bhatnagar-Gross-Krook kinetic equation, which successfully simulates the basic properties of the Boltzmann equation. Conclusions are drawn about the symmetries of the Boltzmann equation. To simplify the calculations, the exposition is presented for the case of the one-dimensional Bhatnagar-Gross-Krook equation with constant effective collision frequency.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 135–140, July–August, 1982.     

The dynamic equations of motion for a two-component system

The motion of solid particles in a fluid flow is represented as a random process with independent increments. The resulting kinetic equation for the particle distribution has the form previously proposed [1]. The solution to this equation provides a system of equations for the hydrodynamics of the assembly of solid particles. These equations differ from ones previously proposed [2, 3] in having additional terms related to relative motion of the components, whose presence is due to anisotropy in the distribution of the normal stresses in the pseudogas.I am indebted to V. G. Levich for valuable discussions and for constant interest in the work.     

Macrokinetic model of the trapping process in two-phase fluid displacement in a porous medium

A microinhomogeneity-averaged model of the kinetics of the trapping process is proposed for a porous medium in which two fluids are mutually displaced. The traps are treated as a new hydrodynamic phase, and the trapping process as a phase transition. Kinetic relations for the average trapping process are obtained. The structure and quantitative values of the kinetic coefficients are obtained for a model of a porous medium in the form of a system of doublets. The dependence of the characteristic time of the process on the degree of inhomogeneity of the medium is investigated. A variant of the macroscopic model of the process of two-phase flow, in which the kinetic relations obtained are used as the closing relations, is proposed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 92–101, May–June, 1995.     

A method for deriving hydrodynamic equations for complicated systems

A general algorithm is proposed for constructing a uniformly valid asymptotic solution of the kinetic equations under conditions when the number of slowly varying macroscopic variables is greater than the number of integral invariants of the collision operator. The case of a chemically reacting gas mixture is considered, and a method for constructing the asymptotic solution for this case is described. The hydrodynamic equations for reacting and relaxing gas mixtures are described in general form and it is noted that consistent allowance for the disequilibrium of the reaction and relaxation processes leads to the appearance in the hydrodynamic equations of a number of additional terms, which describe the dependence of the rates of these processes on the spatial derivatives of the hydrodynamic variables.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 96–105, May–June, 1981.     

Kinetic models for dilute solutions of dumbbells in non-homogeneous flows revisited

We propose a two-fluid theory to model a dilute polymer solution assuming that it consists of two phases, polymer and solvent, with two distinct macroscopic velocities. The solvent phase velocity is governed by the macroscopic Navier–Stokes equations with the addition of a force term describing the interaction between the two phases. The polymer phase is described on the mesoscopic level using a dumbbell model and its macroscopic velocity is obtained through averaging. We start by writing down the full phase-space distribution function for the dumbbells and then obtain the inertialess limits for the Fokker–Planck equation and for the averaged friction force acting between the phases from a rigorous asymptotic analysis. The resulting equations are relevant to the modelling of strongly non-homogeneous flows, while the standard kinetic model is recovered in the locally homogeneous case.     

Formation and evolution of the liquid-phase particle size spectrum in the presence of nonequilibrium homogeneous condensation

The evolution of the liquid droplet size distribution during nucleation, growth and re-evaporation is considered. A closed system of equations, which consistently takes into account the dependence of the growth rate on particle size and ensures a fairly accurate solution of the basic kinetic equation, is obtained for the four lowest moments of the distribution function. Comparative calculations of the condensation in an expanding volume of vapor for constant and periodic expansion rates are carried out on the basis of the system proposed and, moreover, by directly solving the kinetic equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 108–116, January–February, 1990.     

Nonlinear interactions of langmuir waves in a weakly inhomogeneous plasma

Kinetic equations for the scattering of the waves of the one-dimensional spectrum by plasma particles are obtained for a weakly inhomogeneous plasma. The equation for the evolution of the spectrum of the short waves [k² > (m_e/m_i) D_e ^–2] trapped in the inhomogeneities of the plasma density differs significantly from the kinetic equation for the waves in a homogeneous plasma. The problem of localization on the spectrum of the Langmuir waves in regions near the minima of the plasma density is also considered. A solution of the kinetic equation for the waves, which describes this process, is obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 6–13, November–December, 1972.In conclusion, the author thanks A. S. Kingsep for suggesting the problem and for directing the work.     

Averaging of stochastic evolution equations of transport in porous media

A study is made of the problem of averaging the simplest one-dimensional evolution equations of stochastic transport in a porous medium. A number of exact functional equations corresponding to distributions of the random parameters of a special form is obtained. In some cases, the functional equations can be localized and reduced to differential equations of fairly high order. The first part of the paper (Secs. 1–6) considers the process of transport of a neutral admixture in porous media. The functional approach and technique for decoupling the correlations explained by Klyatskin [4] is used. The second part of the paper studies the process of transport in porous media of two immiscible incompressible fluids in the framework of the Buckley—Leverett model. A linear equation is obtained for the joint probability density of the solution of the stochastic quasilinear transport equation and its derivative. An infinite chain of equations for the moments of the solution is obtained. A scheme of approximate closure is proposed, and the solution of the approximate equations for the mean concentration is compared with the exactly averaged concentration.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 127–136, September–October, 1985.We are grateful to A. I. Shnirel'man for pointing out the possibility of obtaining an averaged equation in the case of a velocity distribution in accordance with a Cauchy law.     

Propagation of modulated nonlinear waves in a relaxing gas-liquid mixture

The propagation of nonstationary weak shock waves in a chemically active medium is essentially dispersive and dissipative. The equations for short-wavelength waves for such media were obtained and investigated in [1–4]. It is of interest to study quasimonochromatic waves with slowly varying amplitude and phase. A general method for obtaining the equations for modulated oscillations in nonlinear dispersive media without dissipation was proposed in [5–8]. In the present paper, for a dispersive, weakly nonlinear and weakly dissipative medium we derive in the three-dimensional formulation equations for waves of short wavelength and a Schrödinger equation, which describes slow modulations of the amplitude and phase of an arbitrary wave. The coefficients of the equations are particularized for the considered gas-liquid mixture. Solutions are obtained for narrow beams in a given defocusing medium as well as linear and nonlinear solutions in the neighborhood of a diffraction beam. A solution near a caustic for quasimonochromatic waves was found in [9].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 133–143, January–February, 1980.     

Influence of excitation of internal degrees of freedom of molecules on the process of weak evaporation or condensation

A study is made of the process of weak evaporation (or condensation) with allowance for excitation of vibrational and rotational degrees of freedom of diatomic molecules. The solution to the corresponding Knudsen layer problem is obtained on the basis of a model kinetic equation of the type of the Morse equation [1]. A relation is obtained that establishes the connection between the rate of evaporation (or condensation) and the parameters of the surface and the gas above it. The boundary conditions of slip for the equations of gas dynamics are analyzed. The results are compared with the evaporation or condensation in the case of a monatomic gas. The introduction of accommodation coefficients for an evaporating surface is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6. pp. 98–110, November–December, 1979.     

Effect of Allowance for Small‐Scale Perturbations of the Carrier Phase on Properties of the System of Equations of a Two‐Fluid Flow with Incompressible Phases

The effect of the fluctuating components of kinetic energy and stress tensor of the carrier phase, which were previously obtained by the cell technique, on the properties of the system of equations of a gas–liquid flow with incompressible phases is considered. It is shown that the characteristic properties of this system and also the possibility of modeling the Zuber–Findlay empirical relation are determined by the tensor of fluctuating stresses of the carrier phase.     

Calculation by the Krylov-Bogolyubov method of the velocity profile and fluxes in the nonisothermal flow of a rarefied gas in a cylindrical capillary

The Krylov-Bogolyubov numerical method is used to solve integral transfer equations obtained from the kinetic equation with the BHC (Bhatnagar—Cross—Krook) model of the collision operator. The velocity profiles and the thermal-creep flows and Poiseuille flow are calculated in different modes of flow under conditions of incomplete accommodation of the tangential momentum of molecules at the wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 143–150, November–December, 1978.     

Air-steam mixture flow with condensation on ions and electrokinetic processes

A physical model of air-steam flow with homogeneous condensation, condensation on ions, mass exchange between droplets and surrounding medium, and charge exchange between droplets and ion component is presented. A kinetic equation for the droplet distribution over sizes and charges is used in the model. On the basis of this equation, the moment equations are obtained and various approximate ways of closing them are proposed. The electric self-fields produced by the ion component and the charged dispersed phase are taken into account. Modifications of the equations for the case of turbulent flow are given. A one-dimensional flow model taking into account certain special features of the condensation and electrophysical processes in real flows is realized numerically.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 67–77, January–February, 1996.     

Basic kinetic equation of a rarefied gas

Starting from the Liouville equation, the basic kinetic equation of a rarefied gas is derived for both spatially homogeneous and spatially nonhomogeneous systems. The relation between the equation obtained and the Boltzmann equation is investigated, together with the nature of the dependence of the solutions of the basic kinetic equation on the number of particles in the system.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 154–160, November–December, 1989.The author is grateful to M. S. Ivanov for numerous stimulating discussions and to D. N. Zubarev, E. G. Kolesnichenko, and V. E. Yanitskii for their help in assessing the results.     

Similitude of hypersonic flows of low-density gas over blunt bodies

Laws of similitude of hypersonic flows of monatomic gases have been obtained earlier from asymptotic analysis of the equations as S and confirmed by experimental data and numerical results [1], For diatomic gases, dimensionless numbers have not been deduced by analyzing the equations but by general arguments based on analogy with monatomic gases; they were used to compare experimental and calculated results in [1–3]. In the present paper, dimensionless numbers are derived on the basis of model kinetic equations for a diatomic gas, and limits of their applicability are established. Numerical calculations confirm the exact and approximate laws of similitude and permit a comparison with experimental results. The influence of the laws of viscosity on the drag for a sphere as a function of the Reynolds number Re₀ determined using the viscosity at the stagnation point is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 130–135, March–April, 1981.     

Modeling of particle motion in a turbulent flow with allowance for collisions

On the basis of the kinetic equation for the colliding-particle velocity probability density distribution in a turbulent flow, a model for calculating the dispersed phase motion is constructed for a broad range of variation of the number density and particle size.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 62–78, January–February, 1995.     

Heat transfer between infinite parallel plates in a rarefied gas

The problem of heat transfer between two infinite parallel plates is investigated on the basis of equations obtained by averaging the Boltzmann kinetic equation with respect to the transverse velocity. A numerical solution of the problem is accomplished for a temperature ratio between the plates of T₀/T₁=1/4 and for various Knudsen numbers.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 87–91, January–February, 1972.