Transport coefficients for a partially ionized two-temperature plasma with ions and neutral particles having different masses

The viscosity, heat-conduction, and diffusion coefficients are calculated for a two-temperature three-component plasma composed of ions, neutral particles and electrons when the masses of the ions m_i and neutral particles m_a are different. Similar transport coefficients for m_i=m_a were calculated in [1–3]. The numerical values obtained for the transport coefficients are compared with the values calculated from the formulas of [1–3]. Numerical calculations are carried out for helium with a cesium additive (m_i>m_a) and for krypton with a lithium additive (m_ia).Finally, the author is grateful to V. V. Gogosov, under whose direction this work was carried out.     

Calculation of the parameters of two-phase flow in an axisymmetrical laval nozzle allowing for the coagulation and atomization of the particles

The laws governing changes taking place in the parameters of the heterogeneous flow in a Laval nozzle were studied on the one-dimensional approximation in [1, 2]; a flow containing particles of uniform size was considered in [3–5]. In this paper we shall consider a method of calculating the parameters of a two-phase flow in the sub- and supersonic parts of an axisymmetrical Laval nozzle with due allowance for the coagulation and atomization of the particles, and shall present the results of some corresponding calculations.     

Capture of small drops by large drops in an electric field

In the investigation of the process of the breakdown of emulsions by the coalescence of drops, up to the present time, use has been made of the theory of the coagulation of colloids (for example, [1]). However, there is a considerable difference between colloids and emulsions. Forces of attraction, bringing about the coalescence of two colloidal particles, become predominant at distances much greater than the particle size, so that, in a hydrodynamic sense, it can be assumed that colloidal particles do not interact. On the contrary, the disperse phase in emulsions consists of molecularly smooth spherical drops with a size from a few tenths of a micron or more and, with approach of the drops, forces of hydrodynamic interaction, inhibiting coalescence, become substantial. As a rule, the drops can be regarded as rigid undeformed spheres since their surface is stabilized by surface-active substances. With the approach of such spheres, the layer of liquid between them generates a braking force proportional to the rate of approach of the drops and inversely proportional to the distance between their surfaces. As a result, the approach of drops under the action of a finite force takes place over an infinite time. It follows from this that the process of the coalescence of drops requires the presence of a force of attraction, rising to infinity with approach of the drops, and any theory of coalescence must take simultaneous account of the forces of attraction of the drops and of the hydrodynamic forces.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 61–68, September–October, 1973.     

Calculation of the rate of coalescence of an emulsion in a turbulent flow

The central moment of the theory describing the merging (coalescence) of the drops of an emulsion is determination of the time of the approach of a drop or a number of drops colliding with a given drop in unit time. In the stage immediately preceding the merging of the drops the forces of the hydrodynamic braking of the approaching drops are found to be considerable. The role of these forces has been analyzed earlier for the case of the capture of small drops by large drops in an oncoming flow in the presence of an external electrical field [1] and for the problem of the Brownian coalescence of drops, taking account of the effect of the electric double layer and of surface forces of interaction [2–4]. The present article considers the approach of drops with turbulent diffusion in an electrical field. Of the greatest interest is the sharp slowing of the approach due to the hydrodynamic interaction of the drops, considerably sharper than in the case of molecular diffusion [2]. As a result, the sharp acceleration of the approach and coalescence of drops with the action of an electrical field on an emulsion in a turbulent flow becomes understandable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 47–55, May–June, 1976.The authors are grateful to G. I. Barenblatt, A. I. Leonov, V. I. Loginov, and L. P. Smirnov for their evaluation and criticism of the work.     

Aerosol aspiration through a slot of finite width

Aerosol aspiration has been calculated by Levin for a sink simulating a fairly narrow slot or a thin tube [1].In the present study we determine the aspiration coefficient for aerosol particles for a two-dimensional nozzle of finite width. For strongly inertial aerosol particles, when kk_*, where k is the Stokes number and k_* is its critical value, the approximate equations proposed in [2] are applicable.For kk_*, when there is no inertial deposition of aerosol particles on the nozzle wall, we use the small parameter method. The aspiration coefficient is calculated with accuracy to k². Some numerical data and a comparison with sink flow are presented.     

Self-similar thinning of a liquid film separating liquid phases or a liquid and a solid phase

The article discusses the thinning of a thin flat film of liquid between coalescing bubbles or coalescing drops due to motion of the liquid under the action of capillary forces, under drag conditions of the surfaces of the film. A study was also made of the outflow of a liquid from a film separating a bubble (drop) from a solid surface. The problem of the decrease in thickness and subsequent breakaway of thin liquid films is of interest in investigations of the kinetics of coalescence and coagulation and in the theory of the stability of foams and lyophobic colloids in the presence of surfactants or electrolytes, as well as in the theory of heterogeneous boiling. Information on some work and on special characteristics of the statement of the problem may be found in [1].Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp, 39–48, March–April, 1973.     

Destruction of coarse and fine water drops by monopulses of a ruby laser

In addition to the known method of evaporating water drops in the intensive radiation field of a CO₂ laser, papers have recently appeared wherein the destruction of water drops without the conversion of the light energy they absorbed into heat is investigated (surveys [1, 2], for example). Papers devoted to nonthermal methods of destroying a water aerosol, although still few in number, indicate the proposal of three methods of destroying the drops: optical breakdown in water, excitation of mechanical vibrations of the drops, and photochemical destruction of the water molecules [1, 2]. The optical breakdown phenomenon, when intense destructive shocks occur in a water drop subjected to a laser monopulse, has been investigated more fully than the other methods but also clearly insufficiently. Experiments on destroying millimeter- and micron-sized drops by ruby laser monopulses are described in this paper, values of the parameters characterizing this process are determined, and an approximate estimate of the energy and power of the laser pulses required to destroy a water aerosol in a track of definite length is also given.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 26–35, May–June, 1979.     

Population inversion behind a detonation wave propagating in a variable-density medium

The creation of an active medium by means of detonation has been investigated on a number of occasions. It was suggested that one could use the expansion of the detonation products of an acetylene-air mixture in vacuum [1] or the cooling of the detonation products of a mixture of hydrocarbons and air through a nozzle [2, 3]. In [4], the detonation of a solid high explosive was used to produce population inversion in the gas mixture CO₂-N₂-He(H₂O). Stimulated emission from HF molecules was observed in [5] behind the front of an overdriven detonation wave propagating in an F₂-H₂-Ar mixture in a shock tube. Population inversion behind a detonation wave was studied in H₂-F₂-He mixtures in [6–8] and in H₂-Cl₂-He mixtures in [9] with energy release on a plane and on a straight line in a medium with constant density. Similar problems were solved for shock waves propagating in both a homogeneous gaseous medium [7, 10] and in the supersonic part of an expanding nozzle. In the present paper, we study theoretically population inversion behind an overdriven detonation wave propagating in a mixture (fine carbon particles + acetylene + air) which flows through a hypersonic nozzle. The propagation of detonation in media with variable density and initial velocity was considered, for example, in [11, 12]. Analysis of the gas parameters behind a detonation wave propagating in a medium with constant density (for a given fuel) showed that the temperature difference across the detonation front is insufficient to produce population inversion of the vibrational levels of the CO₂ molecule.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 65–71, January–February, 1980.I am grateful to V. P. Korobeinikov for a helpful discussion of the results.     

ON THE STRUCTURE OF SOLUTIONS OF LINEAR PARTIAL DIFFERENTIAL EQUATION sum form i+j≤n aijpiqjφ=0 WITH TWO INDEPENDENT VARIABLES AND CONSTANT COEFFICIENTS

This paper is a continuation of [1]. In this paper, the solutions of the more general linear partial differential equation form i+j≤n a_ijpⁱq^jφ=0 with two independent variables and constant coefficients are discussed in detail. The general solution which can be used in the approximation to the conditions of the definite solution of the practical problems is presented. To illustrate the use of the results obtained in this paper, some practical examples in mechanics are given.     

A two-phase stream containing liquid particles in the sub- and transonic regions of plane nozzles

The results of experimental studies of the influence of the entrance conditions, the particle size, the profiles of the sub- and transonic parts of the nozzle, and the initial concentration on the distribution of the solid discrete phase in the exit cross sections of axisymmetric nozzles were analyzed in [1]. The results of a study of the influence of the profiling of the nozzle and the size of the particles at the nozzle entrance on the formation of the distribution fields of the discrete liquid phase and its size at the cut of a plane nozzle are presented in the present report, which is a continuation of [1]. The experimental data presented permit a deeper understanding of the mechanism of flow of a two-phase medium in a nozzle and are required for an evaluation of efficiency of the calculation methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 167–170, March–April, 1978.     

Numerical investigation of nonequilibrium flows of mixtures of fusible metal particles and gases in a laval nozzle

There are numerous papers [1–11] on the determination of the parameters of condensed oxide particles which are formed during combustion of metallized fuels. The ambiguity, and sometimes the contradictoriness, of the test results obtained [3–5, 9–11] indicate the difficulties in conducting correct experimental investigations. In this connection, numerical studies using mixtures of calibrated liquid-metal particles and different gases are of practical interest. Different probes can be calibrated by using calibrated two-phase flows, the two-phase flow around models and probes can be studied, as can the interaction between liquid-metal particles and the front of an aerodynamic compression shock, their intrusion in different entraining media, the interaction between fine particles (particle-projectiles) and large size particles (particle-targets), etc. In many cases, the prehistory of the flow and the parameters of the gas mixture with the particles in the area of the nozzle exit section must be known to investigate the above-mentioned phenomena. The parameters of different nonequilibrium flows of mixtures of gallium particles and gases in a Laval nozzle are investigated numerically in this paper; the maximum diameter (upper boundary of the spectrum) of the particles (d_s = 30 ) which are not destroyed in the nozzle under the effect of the aerodynamic forces and are suitable for use in a calibrated two-phase stream is determined. The computations were carried out in a one-dimensional approximation according to [12–14].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 86–91, March–April, 1976.The authors are grateful to V. K. Starkov and U. G. Pirumov for discussing the results of the research and to N. M. Alekseev for aid in constructing the graphs.     

[0, k i]1 m -Factorizations Orthogonal to a Subgraph

Let G be a graph,k₁,…,k_m be positive integers. If the edges of graph G can be decomposed into some edge disjoint. [0,k₁]-factor. F₁,…,[0,k_m]-factor F_m, then we can say F={F₁,…,F_m, is a [0,k_i]_m¹-factorization of G. If H is a subgraph with m edges in graph G and |E(H)∩E(F_i)|=1 for all 1≤i≤m, then we can call that F is orthogonal to H. It is proved that if G is a..[0,k₁+…+k_m-m+1]-graph, H is a subgraph with m edges in G, then graph G has a. [0,k_i]₁^m-factorization orthogonal to H.     

Dissipative particle dynamics simulation of polymer drops in a periodic shear flow

The steady-state and transient shear flow dynamics of polymer drops in a microchannel are investigated using the dissipative particle dynamics (DPD) method. The polymer drop is made up of 10% DPD solvent particles and 90% finite extensible non-linear elastic (FENE) bead spring chains, with each chain consisting of 16 beads. The channel’s upper and lower walls are made up of three layers of DPD particles, respectively, perpendicular to Z-axis, and moving in opposite directions to generate the shear flow field. Periodic boundary conditions are implemented in the X and Y directions. With FENE chains, shear thinning and normal stress difference effects are observed. The “colour” method is employed to model immiscible fluids according to Rothman–Keller method; the χ-parameters in Flory–Huggins-type models are also analysed accordingly. The interfacial tension is computed using the Irving–Kirkwood equation. For polymer drops in a steady-state shear field, the relationship between the deformation parameter (D_def) and the capillary number (Ca) can be delineated into a linear and nonlinear regime, in qualitative agreement with experimental results of Guido et al. [J. Rheol. 42 (2) (1998) 395]. In the present study, Ca<0.22, in the linear regime. As the shear rate increases further, the drop elongates; a sufficiently deformed drop will break up; and a possible coalescence may occur for two neighbouring drops. Dynamical equilibrium between break-up and coalescence results in a steady-state average droplet-size distribution. In a shear reversal flow, an elongated and oriented polymer drop retracts towards a roughly spherical shape, with a decrease in the first normal stress difference. The polymer drop is found to undergo a tumbling mode at high Schmidt numbers. A stress analysis shows that the stress response is different from that of a suspension of solid spheres. An overshoot in the strain is observed for the polymer drop under extension due to the memory of the FENE chains.     

Investigation of supersonic isobaric submerged turbulent jets

Turbulent supersonic submerged air jets have been investigated on the Mach number interval M_a = 1.5–3.4 and on the interval of ratios of the total enthalpies in the external medium and the jet i₀ = 0.01 – 1. Oxyhydrogen jets with oxidizer ratios = 0.3–5 flowing from a nozzle at Mach numbers M_a = 1 and 2.4 have also been investigated. When < 1 the excess hydrogen in the jet burns up on mixing with the air. Special attention has been paid to obtaining experimental data free of the influence on the level of turbulence in the jet of the initial turbulence in the nozzle forechamber, shock waves occurring in the nozzle or in the jet at the nozzle exit, and the external acoustic field. The jet can be divided into two parts: an initial part and a main part. The initial part extends from the nozzle exit from the section x, in which the dimensionless velocity on the jet axis u_m = u_x/u_d = 0.75. Here, u_x is the velocity on the jet axis at distance x from the nozzle exit, and u_a is the nozzle exit velocity. The main part of the jet extends downstream from the section x. For the dimensionless length of the initial part x_m = x/d_a, where d_a is the diameter of the nozzle outlet section, empirical dependences on M_a and i₀ are obtained. It is shown, that in the main part of the jet the parameters on the flow axis — the dimensionless velocity and temperature — vary in inverse proportion to the distance, measured in units of length x, and do not depend on the flow characteristics which determine the length of the initial part of the jet. The angles of expansion of the viscous turbulent mixing layer in the submerged heated or burning jet increase with decrease in i₀ and M_a and are practically independent of the afterburning process.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza. No. 4, pp. 56–62, July–August, 1988.     

Investigation of a Gas Flow with Solid Particles in a Supersonic Nozzle

A two-phase flow with high Reynolds numbers in the subsonic, transonic, and supersonic parts of the nozzle is considered within the framework of the Prandtl model, i.e., the flow is divided into an inviscid core and a thin boundary layer. Mutual influence of the gas and solid particles is taken into account. The Euler equations are solved for the gas in the flow core, and the boundary-layer equations are used in the near-wall region. The particle motion in the inviscid region is described by the Lagrangian approach, and trajectories and temperatures of particle packets are tracked. The behavior of particles in the boundary layer is described by the Euler equations for volume-averaged parameters of particles. The computed particle-velocity distributions are compared with experiments in a plane nozzle. It is noted that particles inserted in the subsonic part of the nozzle are focused at the nozzle centerline, which leads to substantial flow deceleration in the supersonic part of the nozzle. The effect of various boundary conditions for the flow of particles in the inviscid region is considered. For an axisymmetric nozzle, the influence of the contour of the subsonic part of the nozzle, the loading ratio, and the particle diameter on the particle-flow parameters in the inviscid region and in the boundary layer is studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 65–77, November–December, 2005.     

Effect of inelastic collisions on the first-order slip coefficients for a diatomic gas with rotational degrees of freedom

When solving problems of inhomogeneous gas dynamics in the slip regime, it is necessary to know the boundary conditions for the velocity, temperature, heat fluxes, etc., that is, the boundary conditions for the gas macroparameters. In particular, such problems arise in developing the theory of thermophoresis of moderately large aerosol particles [1].The problem of monatomic and molecular (di- and polyatomic) gas slip along a boundary surface is considered in many publications (see, for example, [2–8]). The first-order effects include the isothermal and thermal gas slips characterized by the coefficients C_m and K_TS, respectively.In contrast to a monatomic gas, the molecules of diatomic and polyatomic gases have internal degrees of freedom, which considerably complicates the kinetic equation [9]. The lack of reliable models for the intermolecular interaction potential predetermines the need to construct model kinetic equations [10].In this study, for a diatomic gas whose molecules have rotational degrees of freedom, we propose a model kinetic equation obtained by developing the approach described in [6]. With the use of this model equation, the problem of diatomic gas slip along a plane surface is solved. As a result, for diatomic gases the coefficients C_m and K_TS, which depend on the thermophysical gas parameters and the intensity of inelastic collisions, are obtained.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 176–182. Original Russian Text Copyright © 2004 by Poddoskin.     

Size-dependent coalescence kernel in fertilizer granulation-A comparative study

Granulation is a key process in several industries like pharmaceutical, food, fertilizer, agrochemicals, etc. Population balance modeling has been used extensively for modeling agglomeration in many systems such as crystallization, aerosols, pelletisation, etc. The key parameter is the coalescence kernel, β（ij） which dictates the overall rate of coalescence as well as the effect of granule size on coalescence rate. Adetayo, Litster, Pratsinis, and Ennis （1995） studied fertilizer granulation with a broad size distribution and modeled it with a two-stage kernel. A constant kernel can be applied to those granules which coalesce successfully. The coalescence model gives conditions for two types of coalescence, Type I and II. A twostage kernel, which is necessary to model granule size distribution over a wide size distribution, is applied in the present fluidized bed spray granulation process. The first stage is size-independent and non-inertial regime, and is followed by a size-dependent stage in which collisions between particles are non-random, i.e. inertial regime. The present work is focused on the second stage kernel where the feed particles of volume i and j collide and form final granule ij instead of i ＋j （Adetayo et al., 1995） which gives a wider particle size distribution of granules than proposed earlier.     

Occurrence of periodic regimes in steady supersonic mid flows due to loss of electrical conductivity of medium

A study is made of the features of supersonic magnetohydrodynamic (MHD) flows due to the vanishing of the electrical conductivity of the gas as a result of its cooling. The study is based on the example of the exhausting from an expanding nozzle of gas into which a magnetic field (Re_m 1) perpendicular to the plane of the flow is initially frozen. It is demonstrated analytically on the basis of a qualitative model [1] and by numerical experiment that besides the steady flow there is also a periodic regime in which a layer of heated gas of electric arc type periodically separates from the conducting region in the upper part of the nozzle. A gas-dynamic flow zone with homogeneous magnetic field different from that at the exit from the nozzle forms between this layer and the conducting gas in the initial section. After the layer has left the nozzle, the process is repeated. It is established that the occurrence of such layers is due to the development of overheating instability in the regions with low electrical conductivity, in which the temperature is approximately constant due to the competition of the processes of Joule heating and cooling as a result of expansion. The periodic regimes occur for magnetic fields at the exit from the nozzle both greater and smaller than the initial field when the above-mentioned Isothermal zones exist in the steady flow. The formation of periodic regimes in steady MHD flows in a Laval nozzle when the conductivity of the gas grows from a small quantity at the entrance due to Joule heating has been observed in numerical experiments [2, 3]. It appears that the oscillations which occur here are due to the boundary condition. The occurrence of narrow highly-conductive layers of plasma due to an initial perturbation of the temperature in the nonconducting gas has previously been observed in numerical studies of one-dimensional flows in a pulsed accelerator [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 138–149, July–August, 1985.     

On some new solutions obtainable by means of invariant transformations

With reference to the example of the equations of monoenergetic nonrelativistic beam of particles of like charge, it is shown how new noninvariant solutions can be obtained by means of invariant transformations (§ 1). The conditions under which Lorentz forces can be ignored and the electric field considered a potential field are obtained for nonstationary flows. Solutions that describe the passage through a plane diode of high-frequency current from the emitter in a high-frequency electric field for an arbitrary relationship between the constant component of the collector potential and the amplitude of the ac voltage across it are derived (§2). Multivelocity (the velocity vector is a multivalued function) beams, and also electrostatic beams that can be described by Vlasov's equations are examined (§3).Given a system of differential equations (S) for m 1 unknown functions u^k (k=1,.,m) of n – m 1 independent variables xⁱ (i=1, ., n – m). The set of values (x, u) is considered as the set of coordinates of a point in n-dimensional space E_n. Any solution of this system u=u(x) defines some manifold in E_n. All possible solutions of (SI specify in E_n some set M. Any invariant transformation of system (S) has the property that it does not lead out of M. In a number of cases, this makes it possible to obtain new solutions by means of invariant transformations, no limitations being imposed on the solutions transformed. For a given system (S), all transformations that preserve (Si and form a continuous group, can be obtained by the method developed by L. V. Ovsyannikov [1–3]. Note that new solutions arise only when the principal group G of system (S) allows other than merely elementary transformations: magnifications, rotations, and translations are, as a rule, useless.Below, solutions of the equations of a monoenergetic nonrelativistic beam of particles of like charge are examined as an example [6–8].