On the theory of stationary velocity of propagation of an exothermic reaction front in a condensed medium

It has been demonstrated experimentally that in the combustion of many explosives and powders in the condensed phase (k-phase) an exothermic chemical reaction occurs. Although the heat release in the k-phase is usually small in comparison with the calorific value, it may play an important role in the multistage reaction in the combustion zone.Analysis of the heat balance of the k-phase reveals that in a number of cases heating of the substance before gasification is primarily due to self-heating. According to the thermocouple measurements made by A. A. Zenin, the heat release in the k-phase during combustion of nitroglycerine N powder is more than 80% of the total quantity of heat in the heated layer of the k-phase (pressure 50 atm). This makes it possible to speak of the propagation of the exothermic reaction front in a condensed medium as the first stage in the combustion of condensed systems. Cases are also known where the propagation of the reaction front is maintained only by self-heating (flameless combustion [1]), and there are cases when such propagation is not accompanied by gasification (combustion of thermites, sometimes the polymerization process). Theoretical investigations of stationary propagation of a reaction front in a condensed medium were made in [2–6]. We note that this problem is also of interest in relation to the study of various nonstationary phenomena associated with the combustion of powders [7–9]. One of the principal theoretical problems is the derivation of a formula for the velocity of propagation of the reaction front in the k-phase. The Zel'dovich-Frank-Kamenetskii method [10] was used in [2–5] in the solution of this problem.This paper is an investigation of the applicability of the Zel'dovich-Frank-Kamenetskii method to the case of propagation of a zero-order reaction front in the k phase. A method is proposed for deriving a formula for the propagation velocity of the front leading in the case of a zero-order reaction to a formula identical to that obtained using the Zel'dovich-Frank-Kamenetskii method, and this method is then used to derive a formula for the propagation velocity of a first-order reaction front in the k-phase. The upper and lower limits of the velocity given by this formula are investigated.     

Theory of flame propagation in a gas and drop mixture

The equations of a reacting multiphase continuous medium [1] are used to investigate the problem of steady-state flame front propagation in a gas mixture with evaporating drops. A simple model for ignition of the liquid drops is proposed which is based on the application of the method of equally accessible surfaces [2] to the heat and mass exchange processes between the microflames surrounding the separate drops, the drops, and the carrying gas medium. The parameter distributions in the macroscopic flame front as well as the dependences of the flame propagation velocity in the gas suspension on a number of parameters governing the process under investigation are represented.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 101–108, July–August, 1973.     

Asymptotic analysis of stationary distribution of the front of a two-stage consecutive exothermic reaction in a condensed medium

An approximate theory of the stationary distribution of the plane front of a two-stage exothermic consecutive chemical reaction in a condensed medium is developed in the article. The method of joined asymptotic expansions is used in constructing the solutions. The ratio of the sum of the activation energies of the reactions to the final adiabatic combustion temperature is a parameter of the expansion. The characteristic limiting states of the stationary distribution of the wave corresponding to different values of the parameters figuring in the problem are shown. Approximate analytical expressions for the wave velocity and distribution of concentrations are obtained for each of the states.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 75–87, January–February, 1973.     

Application of the method of matched asymptotic expansions to the calculation of the stationary thermal propagation of the front of an exothermic reaction in a condensed medium

In this paper we use the method of matched asymptotic expansions to establish a two-term formula for the speed of propagation of the front of an exothermic reaction in a condensed medium whose thermophysical characteristics depend on the concentration of the reacting matter and the temperature. As the parameter of the expansion we use the ratio of the activation temperature to the adiabatic combustion temperature. The results are applied to the case of the combustion of nonvolatile condensed systems. We compare the approximate formula obtained with the results of a numerical integration.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 106–112, September–October, 1972.     

Multizone combustion of condensed systems

In the general case the combustion of condensed systems is of a stage-wise character and the combustion front is multizone [1, 2]. Following the investigation of two-zone models [3–5] it became clear that, during multizone combustion, one of the zones of heat evolution is the controlling zone. The velocity of the front is equal to the velocity of the controlling zone; however, with a change in the parameters of the system, there is the possibility of a transition of the controlling role from one zone to another, as well as of the coalescence and splitting of zones. This paper discusses a generalization of the two-zone problem which makes it possible to go over to the analysis of a complex, multizone front and shows that, for a front with two reactions (in the condensed phase and in the gas) and with dispersion, there are in all three possible arrangements of the zones of heat evolution (two three-zone variants and one two-zone variant). All possible types of dependence of the combustion rate on the depth of the dispersion are found.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 99–105, November–December, 1972.     

One limiting scheme for the propagation of a pulsating exothermic reaction front in a condensed medium

In connection with studies on the synthesis of various solid substances in a combustion wave (self-propagating high-temperature synthesis) [1–3] a model is proposed for the unsteady combustion of gasless compositions which describes the relaxation mechanism of the propagation of an exothermic chemical reaction front in a condensed system, forming refractory products. The period of the oscillations in the combustion rate, the movement of the reaction front during one oscillation, and other characteristics of the process are determined. A comparison is made with the results of numerical calculations.     

One-dimensional unsteady-state flow in a rotating grid

The article discusses the problem of determining the dynamic characteristics of rotating and fixed grids in turbine-type machines. The grid is regarded as a linear system with distributed parameters. In problems involved with determining the stability of complex systems and in control problems, the dynamic characteristics of the turbine-type machines must be known. Such characteristics have been studied, for example, in [1]. However, in [1] the discussion is limited to a system with lumped parameters, which is admissible only at small perturbation frequencies. Article [2] discusses the problem of the propagation of vibrations in a flow with a constant mean velocity. In what follows, this problem is solved with a variable velocity of the mean motion. In addition, it is assumed that, with vibrations, there is removal or supply of mechanical energy.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 77–83, September–October, 1971.     

Determnation of elasticity constants for microheterogeneous media

The foundations of the theory of stochastically heterogeneous solids were laid a long time ago by Voigt [1J, who developed a method for determining the macroscopic parameters of polycrystalline materials by averaging the appropriate crystallite parameters with respect to orientations. Lifshits and Rozentzveig [2] showed that it was necessary to consider the correlation properties of the field in computations of macroscopic parameters. They calculated the first corrections for the averaged elastic constants of polycrystallites for the case of cubic and hexagonal crystallites. Assuming a low degree of heterogeneity, these authors used an approximation which corresponds to the Born approximation in the theory of scattering [3]. This method and its modifications were subsequently used by several authors for the computation of macroscopic parameters of polycrystallites [4– 6] and of other microheterogeneous materials [8].Moreover, the assumption of a low degree of heterogeneity of the properties is very restrictive. It precludes use of the method in the case of macroscopically isotropic polycrystallites formed from essentially anisotropic crystallite stochastically glass reinforced plastics, and similar microheterogeneous materials. This rises the problem of developing procedures that could be applied in cases of a high degree of heterogeneity. This problem presents serious analytical difficulties, however. It is sufficient to point out that even computation of the second approximation (i.e., the one following the Born approximation) has not yet been completed. Analogous problems in the classical and quantum theories of scattering are also, as a rule, considered only in the Born approximation. More complicated methods (e.g., Feyman's method) make possible only partial summation of infinite sequences in which the result is obtained. A method analogous to that of a selfconsistent field in quantum mechanics [9,10] is promising; however, this method is approximate and the magnitude of its error has not yet been estimated.The possibility of accurate determination of mascroscopic parameters for certain classes of microheterogeneous media was demostrated in [11], in which a detailed analysis was presented of parameters forming a second order tensor and characterizing the distribution in the medium of a certain scalar value obeying an equation similar to the steady-state heat-conduction equation. Accurate formulas for macroscopic coefficients of thermal conductivity (diffusion) were derived for the case of a strongly anisotropic medium and for that of a medium with a high degree of transverse isotropy. We made a comparison with various approximate methods and evaluated their degree of error. This article describes an accurate method of computing macroscopic elastic constants for polycrystalline media with a high degree of anisotropy; for the case of polycrystals with a cubic structure [12] the error margin and range of application of approximate methods are estimated.     

Some problems in the theory of plane jets of conducting fluid

Using the boundary-layer equations as a basis, the author considers the propagation of plane jets of conducting fluid in a transverse magnetic field (noninductive approximation).The propagation of plane jets of conducting fluid is considered in several studies [1–12]. In the first few studies jet flow in a nonuniform magnetic field is considered; here the field strength distribution along the jet axis was chosen in order to obtain self-similar solutions. The solution to such a problem given a constant conductivity of the medium is given in [1–3] for a free jet and in [4] for a semibounded jet; reference [5] contains a solution to the problem of a free jet allowing for the dependence of conductivity on temperature. References [6–8] attempt an exact solution to the problem of jet propagation in any magnetic field. An approximate solution to problems of this type can be obtained by using the integral method. References [9–10] contain the solution obtained by this method for a free jet propagating in a uniform magnetic field.The last study [10] also gives a comparison of the exact solution obtained in [3] with the solution obtained by the integral method using as an example the propagation of a jet in a nonuniform magnetic field. It is shown that for scale values of the jet velocity and thickness the integral method yields almost-exact values. In this study [10], the propagation of a free jet is considered allowing for conduction anisotropy. The solution to the problem of a free jet within the asymptotic boundary layer is obtained in [1] by applying the expansion method to the small magnetic-interaction parameter. With this method, the problem of a turbulent jet is considered in terms of the Prandtl scheme. The Boussinesq formula for the turbulent-viscosity coefficient is used in [12].This study considers the dynamic and thermal problems involved with a laminar free and semibounded jet within the asymptotic boundary layer, propagating in a magnetic field with any distribution. A system of ordinary differential equations and the integral condition are obtained from the initial partial differential equations. The solution of the derived equations is illustrated by the example of jet propagation in a uniform magnetic field. A similar solution is obtained for a turbulent free jet with the turbulent-exchange coefficient defined by the Prandtl scheme.     

Structure of weak shock waves in 2-D layered material systems

Shock waves in homogeneous materials in the absence of phase transitions are understood to have a one-wave structure. However, upon loading of a layered heterogeneous material system a two-wave structure is obtained––a leading shock front followed by a complex pattern that varies with time. This dual shock-wave pattern can be attributed to material architecture through which the shock wave propagates, i.e. the impedance (and geometric) mismatch present at various length scales, and nonlinearities arising from material inelasticity and failure.The objective of the present paper is to provide a better understanding of the role of material architecture in determining the structure of weak shock waves in 2-D layered material systems. Normal plate-impact experiments are conducted on 2-D layered material targets to obtain both the precursor decay and the late-time dispersion. The particle velocity at the free surface of the target plate is measured by using a multi-beam VALYN VISAR. In order to understand the effects of layer thickness and the distance of wave propagation on elastic precursor decay and late-time dispersion several different targets with various layer and target thicknesses are employed. Moreover, in order to understand the effects of material inelasticity both elastic–elastic and elastic–viscoelastic bilaminates are utilized.The results of these experiments are interpreted by using asymptotic techniques to analyze propagation of acceleration waves in 2-D layered material systems. The analysis makes use of the Laplace transform and Floquet theory for ODE’s with periodic coefficients [Asymptotic solutions for wave propagation in elastic and viscoelastic bilaminates. In: Developments in Mechanics, Proceedings of the 14th Mid-Eastern Mechanics Conference, vol. 26, no. 8, pp. 399–417]. Both wave-front and late-time solutions for step-pulse loading on layered half-space are compared with the experimental observations. The results of the study indicate that the structure of acceleration waves is strongly influenced by impedance mismatch of the layers constituting the laminates, density of interfaces, distance of wave propagation, and the material inelasticity.     

Construction of approximate solutions of boundary value impact strain problems

The method for constructing approximate solutions of boundary value problems of impact strain dynamics in the form of ray expansions behind the strain discontinuity fronts is generalized to the case of curvilinear and diverging rays. This proposed generalization is illustrated by an example of dynamics of an antiplane motion of an elastic medium. The ray method is one of the methods for constructing approximate solutions of nonstationary boundary value problems of strain dynamics. It was proposed in [1, 2] and then widely used in nonstationary problems of mathematical physics involving surfaces on which the desired function or its derivatives have discontinuities [3–7]. A complete, qualified survey of papers in this direction can be found in [8]. This method is based on the expansion of the solution in a Taylor-type series behind the moving discontinuity surface rather than in a neighborhood of a stationary point. The coefficients of this series are the jumps of the derivatives of the unknown functions, for which, as a consequence of the compatibility conditions, one can obtain ordinary differential equations, i.e., discontinuity damping equations. In the case where the problem with velocity discontinuity surfaces is considered in a nonlinear medium, this method cannot be used directly, because one cannot obtain the damping equation. A modification of this method for the purpose of using it to solve problems of that type was proposed in [9–11], where, as an example, the solutions of several one-dimensional problems were considered. In the present paper, we show how this method can be transferred to the case of multidimensional impact strain problems in which the geometry of the ray is not known in advance and the rays become curvilinear and diverging. By way of example, we consider a simple problem on the antiplane motion of a nonlinearly elastic incompressible medium.     

Shock wave propagation through a model one dimensional heterogeneous medium

We study the problem of impact-induced shock wave propagation through a model one-dimensional heterogeneous medium. This medium is made of a model material with spatially varying parameters such that it is heterogeneous to shock waves but homogeneous to elastic waves. Using the jump conditions and maximal dissipation criteria, we obtain the exact solution to the shock propagation problem. We use it to study how the nature of the heterogeneity changes material response, the structure of the shock front and the dissipation.     

Front Regime of Heat and Mass Transfer in a Gas Hydrate Reservoir under the Negative Temperature Conditions

The analytical self-similar solution to the nonlinear problem of the front regime of heatand- mass transfer in a gas hydrate reservoir under the negative temperature conditions is obtained. In the initial state the reservoir is assumed to be saturated with a heterogeneous gas hydrate–ice–gas mixture. In particular cases there may be no ice or/and gas. The ice and gas are formed behind the gas hydrate dissociation front. The calculations are presented for a stable hydrate–gas system. The critical curves are constructed in the well-pressure–reservoir-permeability plane. These curves separate the domains of the front regime and the regime of volume gas hydrate dissociation ahead of the front. The velocity of the gas hydrate dissociation front is investigated as a function of various problem parameters. The characteristic temperature and pressure distributions corresponding to various regimes on the diagram are investigated.     

A spherical wave in a viscoelastic half-space

The propagation of spherical waves in an isotropie elastic medium has been studied sufficiently completely (see, e.g., [1–4]). it is proved [5, 6] that in imperfect solid media, the formation and propagation of waves similar to waves in elastic media are possible. With the use of asymptotic transform inversion methods in [7] a problem of an internal point source in a viscoelastic medium was investigated. The problem of an explosion in rocks in a half-space was considered in [8]. A numerical Laplace transform inversion, proposed by Bellman, is presented in [9] for the study of the action of an explosive pulse on the surface of a spherical cavity in a viscoelastic medium of Voigt type. In the present study we investigate the propagation of a spherical wave formed from the action of a pulsed load on the internal surface of a spherical cavity in a viscoelastic half-space. The potentials of the waves propagating in the medium are constructed in the form of series in special functions. In order to realize viscoelasticity we use a correspondence method [10]. The transform inversion is carried out by means of a representation of the potentials in integral form and subsequent use of asymptotic methods for their calculation. Thus, it becomes possible to investigate the behavior of a medium near the wave fronts. The radial stress is calculated on the surface of the cavity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 139–146, March–April, 1976.     

On radiative heat transfer in a plane layer of an absorbing medium

Recently there has arisen increased interest in the study of radiative heat transfer between geometrically simple systems, both as autonomous problems and as elements of more complex problems.Problems of this kind have been treated by many authors [1–111 who have considered gray, diffusely emitting and absorbing boundaries and gray nonscattering media. In most cases these investigations were restricted either to the derivation of approximate formulas for the net radiative flux, without an exact analysis of the temperature distribution in the layer [5–7], or to numerical computation [1–4], In the latter case, with the exception of [8], which contains a numerical analysis for the case of optical symmetry, no attempt was made to analyze the effect of the optical properties of the boundaries on the temperature field in the layer.These papers can be divided into two groups according to the method of analysis used. The first group includes papers based on the integral equations of radiative transfer, with the corresponding integral analytical methods [1, 2], Similar in nature are [3, 4] which use the slab method, applicable to electrical-analog computation, as well as a recent paper [8] based on probability methods.The second group of papers [5–7] is based on the so-called differential methods. Of particular interest is [7], which develops these methods to an advanced degree. In several papers the problem of radiative transfer is analyzed in conjunction with more complex problems (cf., e.g. [10, 11]).In the present work we shall attempt to carry out an approximate analytical study of problems connected with radiative heat transfer in a plane layer of an absorbing, emitting, nonscattering gray medium with temperature-independent optical properties. The layer is bounded by two parallel, diffusely emitting and diffusely reflecting, isothermal, gray planes.The paper presents the fundamental formulation of the problem, which consists in: (a) the determination of the net heat flux on the basis of given temperature distribution (direct formulation), and (b) the determination of the temperature distribution on the basis of given distribution of the net radiative heat source per unit volume and boundary temperatures (inverse formulation). The analysis is based on integral methods appropriate to the integral equations which represent the net total and hemispherical radiation flux densities [12].The author would like to thank S. S. Kutateladze for his interest in this work.     

Waves from an underground explosion

The problem of the propagation of a spherical detonation wave in water-saturated soil was solved in [1, 2] by using a model of a liquid porous multicomponent medium with bulk viscosity. Experiments show that soils which are not water saturated are solid porous multicomponent media having a viscosity, nonlinear bulk compression limit diagrams, and irreversible deformations. Taking account of these properties, and using the model in [2], we have solved the problem of the propagation of a spherical detonation wave from an underground explosion. The solution was obtained by computer, using the finite difference method [3]. The basic wave parameters were determined at various distances from the site of the explosion. The values obtained are in good agreement with experiment. Models of soils as viscous media which take account of the dependence of deformations on the rate of loading were proposed in [4–7] also. In [8] a model was proposed corresponding to a liquid multicomponent medium with a variable viscosity.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 34–41, May–June, 1984.