Charging of disperse particles in two-phase media with unipolar charge

Charging of disperse particles with good conduction in two-phase media with unipolar charge is considered in the case when the volume concentration of the particles is low. For this, in the framework of electrohydro-dynamics [1, 2], a study is made of the charge of one perfectly conducting liquid particle in a gas (or liquid) with unipolar charge in a fairly strong electric field. The influence of the inertial and electric forces on the motion of the gas is ignored, and the velocities are found by solving the Hadamard—Rybczynski problem. We consider the axisymmetric case when the gas velocity and electric field intensity far from the particle are parallel to a straight line. The analogous problem for a solid spherical particle was solved in [3–6] (in [3], the relative motion of the gas was ignored, while in [4–6] Stokes flow around the particle was considered). The two-dimensional problem of the charge of a solid circular, perfectly conducting cylinder in an irrotational flow of gas with unipolar charge was studied in [7].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 108–115, November–December, 1980.We thank L. I. Sedov and V. V. Gogosov for a helpful discussion of the present work.     

Equations of electrohydrodynamics of weakly ionized aerosols with diffusion charging of dispersed phase particles

A closed model describing the motion of weakly ionized aerosols with allowance for dispersed phase particle charging processes due to ion deposition is constructed within the framework of continuum mechanics [1]. Both the general process of particle charging in a weakly ionized gas and its limiting cases, where the limiting stage of the process is the diffusion of the ions towards the particles or the reactions leading to their deposition on the particles, are investigated. Expressions are obtained for the positive and negative ion flows to a particle in a weakly ionized gas. The basic equations of electrohydro-dynamics of weakly ionized aerosols, in which the dispersed phase particle charging mechanism in question leads to the interphase transfer of elctrical charge, are formulated. Cases where the system of equations of electrohydrodynamics obtained can be simplified by replacing the differential equations of motion and charging of the dispersed phase and, moreover, the positive and negative ion balance equations by algebraic relations such as Ohm's law and Saha's equation are investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 54–60, January–February, 1986.     

Inductive acceleration of an electrically conductive particle in a viscous liquid

The characteristics of the motion of a particle in an electrically conducting liquid with constant crossed electric and magnetic fields present have been investigated in connection with the problem of MHD-separation in many papers (for example, see the bibliography in [1]). The separation of electrically conducting particles contained in a dielectric liquid, which can be accomplished with the help of a variable magnetic field [2], is also of practical interest. The ponderomotive force acting on a spherical conducting particle near a straight conductor through which the discharge current of a capacitor bank is flowing is found in this paper, and the motion of a particle in a viscous liquid under the action of this force is investigated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 31–34, November–December, 1984.     

Charging of perfectly conducting and nonconducting particles in a suspension by collisions with walls

The change in the electric charge of perfectly conducting and nonconducting particles of a suspension as a result of their collision with walls in a flow of the suspension is found. Criteria for the applicability of the obtained expressions to real particles are given. For the density of the electric current in rarefied suspensions consisting of a nonconducting gas and charged particles a condition is obtained that is satisfied for flow of a suspension at impermeable walls.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 80–87, July–August, 1980.I thank L. I. Sedov and V. V. Gogosov for a helpful discussion of the work.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

Calculation of a turbulent two-phase isobaric jet

A numerical calculation is carried out by the finite-difference method based on proposed equations for a turbulent submerged jet containing an admixture of solid particles. The relative longitudinal particle velocity and the influence of particles on the turbulence intensity are taken into account. The calculated results adequately agree with available experimental data. A turbulent two-phase jet is examined in [1] on the basis of the theory for a variable density jet, assuming equal mean velocities for the gas and particles and not considering the influence of particles on the turbulence intensity. Particles are analogously taken into account by a noninertial gas mixture in [2, 3], and a particle Schmidt number of 1.1 is assumed in [4]. A model is proposed in [5] which takes into account the influence of particles on the turbulence intensity of the gas phase. Problems concerning the initial and main sections of a submerged jet were solved in [6] by the integral method on the basis of this model and the assumed equality of the mean velocities of the gas and particles. Turbulent mixing of homogeneous two-phase flows with allowance made for dynamic nonequilibrium of the phases is considered in [7]. However, the neglect of turbulent transfer of particle mass and momentum led to a physically unrealistic solution for the particle concentration in the far field of the mixture. A two-phase jet is considered in the present work on the basis of the theory of a two-velocity continuous medium [8, 9] with allowance made for turbulent transfer of particle mass and momentum. The influence of particles on the turbulence intensity of the gas phase is taken into account with the model of [5].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 57–63, September–October, 1976.The author acknowledges useful comments and discussion.of the work by G. N. Abramovich and participants of his seminar. The author sincerely thanks I. N. Murzinov for scientific supervision of the work.     

Heat and mass transfer to a reacting particle in a gas flow for arbitrary temperature dependence of the transport coefficients

A study is made of simultaneous heat and mass transfer to a reacting particle of any shape in a translational (and shear) flow of a viscous heat conducting compressible gas for which the thermal conductivity and diffusion coefficient, and also the specific heat depend on the temperature. The first two terms of the asymptotic expansion with respect to the small Reynolds number are obtained for the mean Sherwood and Nusselt numbers. The case of a power-law temperature dependence of the gas viscosity is considered in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–119, January–February, 1984.     

Internal rotation in the hydrodynamics of weakly conducting dielectric suspensions

Hydrodynamic phenomena in weakly conducting single-phase media due to interphase electric stresses are reviewed in [1]. In the present paper, a model is constructed of a dielectric suspension with body couples due to the field acting on free charges distributed on the surface of the particles of the suspension. Averaging of the microscopic fields yields macroscopic equations for the field and the polarization of the dielectric suspension with allowance for the finite relaxation time of the distribution of the free charge on the phase interface. The developed model is used to consider the occurrence of spontaneous rotation of a dielectric cylinder in a weakly conducting suspension in the presence of an electric field; compared with the case of single-phase media [2], this is characterized by a significant reduction in the threshold intensity of the electric field with increasing concentration of the particles [3]. In the present model of a dielectric suspension, the destabilization of the cylinder is due to the occurrence of rotations of the particles of the suspension due to the interaction between the polarization and the motion of the medium. The relaxation equation for the polarization for the given model is analogous to the corresponding equation for media which can be magnetized [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 86–93, March–April, 1980.     

Diffusion to a particle at large péclet numbers and small Reynolds and Hartmann numbers

A study is made of the influence of a homogeneous magnetic field on the mass transfer for a spherical solid particle and a liquid drop in a flow of a viscous electrically conducting fluid. The previously obtained [1] velocity field of the fluid is used to calculate the concentration distribution in the diffusion boundary layer, the density of the diffusion flux, and the Nusselt number, which characterizes the mass transfer between the particle and the surrounding medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 189–192, January–February, 1980.     

Influence of the viscous boundary layer on the deposition of particles from a gas suspension flowing past a sphere

The flow of incompressible gas containing particles past bodies of simple shapes at moderate and high velocities is investigated in [1–5], in which the flow of the carrier medium is assumed to be irrotational. The estimates made in [3] for the neighborhood of the stagnation point show that it is necessary to take into account the viscous boundary layer in the case of fine particles. In the present paper, the viscous flow of a gas suspension over the front surface of a sphere at Reynolds numbers R = 10³–10⁷ is considered. It is assumed that the carrier gas is incompressible and the particle concentra ion negligibly small. The influence of the boundary layer on the particle trajectories and the deposition of the disperse phase on the surface of the sphere is investigated. It is shown that there is a wide range of flow parameters for the gas suspension in which the influence of the boundary layer is important. The limits of this range are established.Translated from Izvestiya Akademli Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 59–66, January–February, 1982.I thank Yu. P. Savel'ev for a helpful discussion of the work.     

Determination of particle charge and size from the propagation of ultrasound in suspensions

The propagation of small perturbations in raulticomponent disperse media consisting of an uncharged dispersion fluid, positive and negative ions and charged particles or droplets of another fluid is investigated. When weak waves pass through emulsions and suspensions, because of the difference in the velocities of the ions and charged particles a non-uniform distribution of electric potential develops in the medium [1–3]. Expressions relating the amplitude of the electric potential and the amplitude of the fluid velocity in the wave, the particle charge and the parameters characterizing the medium are derived. Relations are obtained for the phase shift between the values of the electric potential and the fluid velocity. It is proposed to use the expressions obtained, which describe the propagation of ultrasound, for the experimental determination of the particle charge and other parameters of the disperse medium, in particular, the particle size.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 122–128, January–February, 1988.     

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.     

Boundary layer on a blunt body in a flow of dusty gas

The problem of interaction of gas-dust flows with solid surfaces arose in connection with the study of the motion of aircraft in a dusty atmosphere [1–2], the motion of a gas suspension in power generators, and in a number of other applications [3]. The presence of a disperse admixture may lead to a significant increase in the heat fluxes [4] and to erosion of the surface [5]. These phenomena are due to the joint influence of several factors — the change in the structure of the carrier-phase boundary layer due to the presence of the particles, collisions of the particles with the surface, roughness of the ablating surface, and so forth. This paper continues an investigation begun earlier [6–7] into the influence of particles on the structure of the dynamical and thermal two-phase boundary layer formed around a blunt body in a flow. The model of the dusty gas [8] has an incompressible carrier phase. The method of matched asymptotic expansions [9] is used to obtain the equations of the two-phase boundary layer. In the frame-work of the refined classification made by Stulov [6], it is shown that the form of the boundary layer equations is different in the presence and absence of inertial precipitation of the particles. The equations are solved numerically in the neighborhood of the stagnation point of the blunt body. The temperature and phase velocity distributions in the boundary layer, and also the friction coefficients and the heat transfer of the carrier phase are found for a wide range of the determining parameters. In the case of an admixture of low-inertia particles that are not precipitated on the body, it is shown that even when the mass concentration of the particles in the undisturbed flow is small their accumulation in the boundary layer can lead to a sharp increase in the thermal fluxes at the stagnation point.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 99–107, September–October, 1985.I thank V. P. Strulov for a discussion.     

The boundary layer of a compressible electrically conducting gas at the electrode of a magnetohydrodynamic channel in the absence of ionization equilibrium

We consider the laminar boundary layer of a compressible electrically conducting gas formed at the conducting wall of the channel. We assume that the charged particle concentration in the field of the flow is distinct from the equilibrium distribution. We take into account the destruction of the quasi-neutrality of the gas in a narrow layer at the wall. We assume that the Debye length is much greater than the mean free path length of the charged particles. We investigate the case when the emission currents are substantial.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 164–169, May–June, 1971.     

Motion of a two-temperature plasma in the channel of a disc-type magnetohydrodynamic generator

A study was made of the fully developed homogeneous flow of a two-temperature partially ionized plasma in the channel of a disc-type Hall generator. Experiments with a disc-type generator are described in [1, 2]. In a simplified statement, the problem is analogous to that considered in [3]. The present article takes the chemical reactions of ionization and recombination into account. The energy equation for an electron gas is brought down to a differential form which permits clarification of the question of the applicability of the Kerrebrock [4] formula for the difference in the temperatures of the electrons and the heavy particles.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 18–25, November–December, 1970.In conclusion, the author thanks V. V. Gogosov for his interest in the work and for his valuable observations.     

Numerical solution of an inverse problem of nozzle theory for a two-phase gas-particle mixture

In the present paper gas flows with monodisperse and polydisperse particles in plane and axisymmetric nozzles are calculated by the inverse method [1, 2]. The gas velocity distribution is specified on the axis of symmetry of the nozzle, while the gas and particle parameters are specified in the entrance section. As a result of the numerical integration of a system of equations describing a flow of gas with condensate particles in it we determine the gas and particle parameters, the gas streamlines, and the particle trajectories with allowance for the mutual influence of the gas and particles. One of the gas streamlines is taken as the nozzle contour and the limiting trajectories and pure gas zone are found. A difference method is described which makes it possible to calculate the subsonic, transonic, and supersonic flow regions using a single algorithm, its features are noted, and the results of the calculation for monodisperse mixtures with particle diameters 1 and 5 m and fractions by weight 0.3 are given. A comparison is made with the results of calculations by other methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 106–114, July–August, 1986.The authors express their gratitude to N. B. Ponomarev and G. E. Dumnov for their useful discussions and help in carrying out the calculations.     

Plane one-dimensional stationary flow of an ideal charged gas in its own electric field

The plane one-dimensional flow of an incompressible gas consisting of a neutral and a charged component in its own electric field has been investigated by Stuetzer [1]. Stuetzer's results are valid when the electrostatic pressure is small compared with the hydraulic pressure. In the present paper an analogous problem is considered for a compressible gas under the more general assumption that the pressures are comparable. Three cases are analyzed: a) the velocity of the relative motion of the charged and the neutral particles is equal to zero; b) it is nonvanishing but the flow can be assumed to be approximately isentropic; c) a nonisentropic flow, i.e., one cannot ignore irreversible losses due to the relative motion of the charged and neutral particles. In the first two cases, closed solutions are obtained.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 24–32, January–February, 1971.I should like to thank I. V. Bespalov and Yu. M. Trushin for their interest and helpful comments.     

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.     

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.     

Supersonic two-phase flow around a wedge

Supersonic two-phase flow around bodies is encountered in calculating the flow around the last stages of blades of condensing turbines, in studying the motion of airplanes under cloudy conditions, etc. In the latter case, there is, along with erosion of the forward edges of the wing profiles, a change in the wave structure and interference situation in the flow about the airplane, leading to off-design regimes of motion. Supersonic flow of a two-phase mixture around a wedge, without taking account of the influence of the particles on the flow, was investigated in [1–3]. In [4], also in this kind of simplified setting, a study was made of the interaction of particles with the surface of a wedge in which reflection of the particles from the wall was taken into account. Morganthaler [5] made an experimental study of the flow of a mixture of air and aluminum oxide particles around a wedge. In [6] a theoretical study was made of a supersonic two-phase flow around thin flat axially-symmetric bodies. In particular, for the flow around a wedge, closed form solutions were obtained for the form of the shock wave, the gas streamlines and particle paths, and the distribution of all the parameters along the surface of the wedge. On the basis of the equations given in [7] and the method of characteristics, which were developed for flows consisting of a mixture of a gas and heterogeneous particles in nozzles [8,9], we present below a study of a supersonic two-phase flow around a wedge.Moscow. Translated from Izvestiya Akademii Nauk SSSR. Mekhanika Zhidkosti i Gaza, No. 2, pp. 83–88, March–April, 1972.