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.     

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.     

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.     

Vapor Flow with Evaporation–Condensation on Solid Particles

Strongly nonequilibrium vapor (gas) flows in a region filled by solid particles are considered with allowance for particlesize variation due to evaporation–condensation on the particle surface. The study is performed by directly solving the kinetic Boltzmann equation with allowance for the transformation of the distribution function of gas molecules due to their interaction with dust particles.     

Rheological relationships for concentrated polymer solutions

Using a network model for concentrated polymer solutions, an expression is calculated for the stress tensor, defined in terms of the moments of the distribution function and the kinetic equation for these moments. In the limiting case the results obtained coincide with known results for normal Newtonian liquids.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 126–132, March–April, 1976.     

Spherical expansion of vapor during evaporation of a droplet

The problem of the evaporation of a spherical particle is solved by a numerical finnite-difference method for the stationary and nonstationary cases on the basis of the generalized Krook kinetic equation [1]. Evaporation into a vacuum and into a flooded space are considered taking into account the reduction in size and cooling of the droplet. The minimum mass outflow is determined for stationary evaporation into a vacuum at small Knudsen numbers. The results are compared with those of other authors for both the spherical and plane problems. Most previous studies have used different approximations which reduce either to linearizing the problem [2, 3] or to use of the Hertz-Knudsen equation [4]. The inaccurate procedure of matching free molecular and diffusive flows at some distance from the surface of the droplet [5] is completely unsuitable in the absence of a neutral gas. Equations for the rate of growth of a droplet in a slightly supercooled vapor were obtained in [6] from a solution of the ellipsoidal kinetic model by the method of (expansion of) moments.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 97–102, March–April, 1976.     

The mathematical simulation of various condensation regimes in turbulent isobaric jets

Mathematical models are considered and calculations made for flows in turbulent isobaric steam—air jets in the presence of condensation of the water vapor they contain. The models consist of gasdynamic equations for a turbulent jet, equations for a differential two-parameter model of turbulence, thermodynamic relations, and kinetic equations. A study is made of steam—air jets in a regime of condensation in equilibrium, when the flow region is broken down into zones of frozen flow and flow in equilibrium, described by the equations for a turbulent jet with the use of the traditional thermodynamic relations and of the thermodynamic relations for condensation in equilibrium. An analysis is made of the influence of pulsating motion on the kinetic parameters: rate of nucleation, the critical size of the nuclei, and rate of growth of the drops. It is shown that the rate of nucleation, determined from a quasilaminar averaging model, is several orders of magnitude less than the mean value obtained by averaging using the density distribution of the passive admixture concentration probability. A numerical study is made of the heterogeneous condensation in turbulent jets on extraneous particles entering from the nozzle. Kinetic equations are written down for the case when the rate of growth of the drops does not depend on their radius. A study is made of the dynamics of the transition of heterogeneous condensation from disequilibrium to equilibriumTranslated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–67, January–February, 1985.     

Effect of the shape of a blunt nose on the drag coefficient of a body in a hypersonic rarefied gas flow

The effect of the shape of a blunt nose of a body located in a hypersonic rarefied gas flow on the field of flow and on the aerodynamic characteristics is studied in the example of flow round ellipsoids of revolution at a zero angle of attack. The problem of the flow in the transition regime is solved on the basis of numerical analysis of the model kinetic Bhatnagar—Gross—Krook (BGK) equation for a monatomic gas. The good agreement of the results of the numerical calculations with the experimental data in a broad range of Mach numbers has shown [1, 2] that the numerical solution of the model kinetic equations is a reliable and effective means for studying flow problems. In the case when the problem is posed of determining the laws of the purely force interaction of a flow with the body, sufficiently good accuracy is given by the use of the model BGK equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 190–192, March–April, 1985.     

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.     

Self‐Similar Solution of the Antiplane Shear Fracture Problem in a Coupled Formulation (Creep–Damage)

A growing antiplane shear crack in a damaged medium is considered. It is assumed that the crack tip neighbors a region of completely damaged material, in which all stress tensor components and the continuity parameter vanish. The stress–strained state is analyzed and the configuration of the region of completely damaged material is determined. The crack growth rate is estimated for various values of the constants included in the constitutive relations and kinetic equation.     

Modeling unsteady vapor condensation by rapid expansion

The process of vapor condensation in a spherically symmetric explosion is studied using a kinetic unsteady nucleation model. For describing the motion of the two-phase system, consisting of vapor and liquid droplets, a one-temperature one-velocity model is used. The size distribution functions of the liquid droplets formed and the gas dynamic characteristics of the vapor at various moments of time are obtained. The results of comparing quasisteady and unsteady models of the vapor condensation kinetics are discussed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 81–86, May–June, 1992.     

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.     

Generation and evolution of cavitation in magma under dynamic unloading

The full system of equations for the problem of rarefaction-wave passage over the magma-melt column in the gravity field is derived with the use of the kinetic theory of phase transformations, and the problem is numerically solved. With allowance for diffusion zones and nucleation frequency as a function of supersaturation, the dependence of the number of cavitation nuclei formed in the course of phase transformations behind the rarefaction-wave front is found. The dynamics of the size distribution of cavitation bubbles along the magma-melt column (1 km) whose viscosity varies dynamically as a function of the concentration of dissolved water is studied.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 71–80, March–April, 2005.     

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.     

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.     

Macroscopic equations of motion of disperse systems

The macroscopic equations of motion of a two-component system consisting of a continuous phase and a large number of solid particles are considered. The generalized kinetic equation of a pseudogas obtained earlier by the author is expressed in a form more convenient for calculations. The Chapman-Enskog method is used to solve the kinetic equation at small Knudsen numbers and dimensionless number characterizing the transfer of momentum between the phases of order unity. Because of the influence of the continuous phase, the stress tensor in the macroscopic conservation equations of the pseudogas is anisotropic. The obtained macroscopic equations of the pseudogas are more general than the ones proposed earlier by Myasnikov, this being due to the anisotropy of the time constants which occur in the operator of the hydrodynamic interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 39–44, March–April, 1980.I thank V. P. Myasnikov for posing the problem and for helpful discussions.     

Motion of a rarefied gas between infinite plane-parallel emitting and absorbing surfaces

The problem of the motion of a rarefied gas between infinite plane-parallel emitting and absorbing surfaces is solved numerically on the basis of the Boltzmann kinetic equation.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 176–178, March–April, 1972.     

Mutual effect of long waves and turbulence on the surface of a liquid

The interaction of a long coherent wave with the turbulence on the surface of a liquid is investigated within the framework of the theory of weak turbulence. A closed system of equations is obtained which consists of the dynamic equation for the coherent wave and equations of kinetic type describing the turbulent subsystem. It is shown that because of the interaction with the turbulent subsystem, coherent waves with wave vectors identical in magnitude but opposite in direction are coupled. The additional attenuation of the coherent wave because of the interaction is estimated; this attenuation may be considerably greater than that caused by molecular viscosity. A change in the spectrum of height correlators of the liquid surface is seen in the presence of a coherent wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 100–109, January–February, 1973.     

Influence of heat transfer on the dynamic response of a spherical gas/vapour bubble

The standard approach to analyse the bubble motion is the well known Rayleigh–Plesset equation. When applying the toolbox of nonlinear dynamical systems to this problem several aspects of physical modelling are usually sacrificed. Particularly in vapour bubbles the heat transfer in the liquid domain has a significant effect on the bubble motion; therefore the nonlinear energy equation coupled with the Rayleigh–Plesset equation must be solved. The main aim of this paper is to find an efficient numerical method to transform the energy equation into an ODE system, which, after coupling with the Rayleigh–Plesset equation can be analysed with the help of bifurcation theory. Due to the strong nonlinearity and violent bubble motions the computational effort can be high, thus it is essential to reduce the size of the problem as much as possible. In the first part of the paper finite difference, Galerkin and spectral collocation methods are examined and compared in terms of efficiency. In the second part free and forced oscillations are analysed with an emphasis on the influence of heat transfer. In the case of forced oscillations the unstable branches of the amplification diagrams are also computed.     

Solution to problem of strong evaporation of a monatomic gas by the Monte Carlo method

The Monte Carlo method has been used to obtain a numerical solution to the problem of strong evaporation of a monatomic gas in which the molecules are modeled by pseudo-Maxwellian and hard spheres. A comparison with the results of other authors is made. The results agree well with the solution of the problem obtained on the basis of the model Bhatnagar—Gross—Krook kinetic equation.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 185–188, January–February, 1984.I should like to thank M. N. Kogan for discussing the results.