Averaged equations and wave jumps describing the propagation of stokes waves with slowly varying parameters

The equations describing the stationary envelope of periodic waves on the surface of a liquid of constant or variable depth are investigated. Methods previously used for investigating the propagation of solitons [1–5] are extended to the case of periodic waves. The equations considered are derived from the cubic Schrödinger equation assuming slow variation of the wave parameters. In using these equations it is sometimes necessary to introduce wave jumps. By analogy with the soliton case a wave jump theory in accordance with which the jumps are interpreted as three-wave resonant interactions is considered. The problems of Mach reflection from a vertical wall and the decay of an arbitrary wave jump are solved. In order to provide a basis for the theory solutions describing the interaction of two waves over a horizontal bottom are investigated. The averaging method [6] is used to derive systems of equations describing the propagation of one or two interacting wave's on the surface of a liquid of constant or variable depth. These systems have steady-state solutions and can be written in divergence form.The author wishes to thank A. G. Kulikovskii and A. A. Barmin for useful discussions.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 113–121, September–October, 1989.     

Unsteady three-dimensional viscous shock layer in nonuniform gas flow in the neighborhood of a stagnation point

The combined influence of unsteady effects and free-stream nonuniformity on the variation of the flow structure near the stagnation line and the mechanical and thermal surface loads is investigated within the framework of the thin viscous shock layer model with reference to the example of the motion of blunt bodies at constant velocity through a plane temperature inhomogeneity. The dependence of the friction and heat transfer coefficients on the Reynolds number, the shape of the body and the parameters of the temperature inhomogeneity is analyzed. A number of properties of the flow are established on the basis of numerical solutions obtained over a broad range of variation of the governing parameters. By comparing the solutions obtained in the exact formulation with the calculations made in the quasisteady approximation the region of applicability of the latter is determined. In a number of cases of the motion of a body at supersonic speed in nonuniform media it is necessary to take into account the effect of the nonstationarity of the problem on the flow parameters. In particular, as the results of experiments [1] show, at Strouhal numbers of the order of unity the unsteady effects are important in the problem of the motion of a body through a temperature inhomogeneity. In a number of studies the nonstationary effect associated with supersonic motion in nonuniform media has already been investigated theoretically. In [2] the Euler equations were used, while in [3–5] the equations of a viscous shock layer were used; moreover, whereas in [3–4] the solution was limited to the neighborhood of the stagnation line, in [5] it was obtained for the entire forward surface of a sphere. The effect of free-stream nonuniformity on the structure of the viscous shock layer in steady flow past axisymmetric bodies was studied in [6, 7] and for certain particular cases of three-dimensional flow in [8–11].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 175–180, May–June, 1990.     

Averaging of stochastic evolution equations of transport in porous media

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

Investigation of two-phase flow in inclined macroinhomogeneous porous media

The results of experiments carried out in order to determine the principal characteristics of the process of displacement of one (nonwetting) fluid from inclined macroinhomogeneous porous media by another (wetting) fluid are presented. Irrespective of whether flow in inclined stratified nonhomogeneous formations or in zonally nonhomogeneous media (with a corresponding well distribution) is investigated, the term oblique stratification is used for describing these processes.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 125–131, November–December, 1992.     

Combination of the homogenization and effective medium methods for flow through porous media in a nonperiodic external perturbation field

Models described by parabolic equations with a rapidly oscillating nonperiodic right side are investigated by means of averaging theory methods. For a nonperiodic perturbation field a combined homogenization and effective medium method is developed. This method makes it possible to obtain the solution of the cell problems in a finite form correct to the second order in the inhomogeneity parameter. The method is applied to problems of single-phase and two-phase flow through porous media. The technique of the method is outlined and explicit solutions of cell problems are constructed.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 108–115, July–August, 1995.     

Flow in the neighborhood of the stagnation line on a blunt body traveling at supersonic speed through a zone with elevated temperature and vibrational energy of the molecules

The unsteady flow in the neighborhood of the stagnation line on a sphere traveling at supersonic speed through a plane layer of diatomic gas with elevated temperature and nonequilibrium excitation of the molecular vibrations is investigated. (The source of the inhomogeneity could be a gas discharge [1].) The problem is solved using the viscous shock layer model which makes it possible to take molecular transport processes into account and analyze the unsteady heat transfer. Such flows were previously calculated in [2] within the framework of the inviscid gas model.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i GazNo. 3, pp. 183–185, May–June, 1990.     

Region of applicability and the main features of the generalized Chapman-Enskog method

The present paper is made necessary by the publication of the foregoing paper in this issue by Kolesnichenko [1]. It considers the basic propositions of the generalized Chapman-Enskog method and analyzes the arguments put forward by Kolesnichenko [1] and the validity of the method. The position of the results obtained by Kolesnichenko [14–17] is indicated. Nonequilibrium flows of multiatomic gases in which there occur processes of exchange of internal energy of the molecules in collisions between them and chemical reactions (such processes are called inelastic) are encountered frequently in nature and technology. It is therefore naturally of interest to derive gas-dynamic equations for such flows. The methods of the kinetic theory of gases were first used to obtain equations describing the limiting cases of very fast inelastic processes that take place in times of the order of the molecule-molecule collision times (equilibrium case) and very slow inelastic processes that take place over times of the order of the characteristic flow time (relaxation case). In [2–5], an algorithm was proposed for deriving gas-dynamic equations valid for arbitrary ratios of the rates of the elastic and inelastic processes and reducing to the well-known equations for the limiting cases already mentioned. The algorithm is called the generalized Chapman-Enskog method (abbreviated to the generalized method). The development, modification, and analysis of its properties can be found in [4, 6–13]. In [1], Kolesnichenko has questioned the validity of this algorithm.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 126–136, May–June, 1984.We thank V. A. Rykov for helpful and constructive discussions of the work.     

Distribution function of isotropically scattered particles in anisotropic structures

A study is made of the properties of solutions for a one-dimensional single-velocity transport equation for particles in a medium in which the probabilities of the elementary processes depend on the direction, and the scattering indicatrix (phase function) is spherical. It is shown that the linear parameter of the asymptotic exponential reduction in the global particle density when orientational inhomogeneity is allowed for can differ strongly from the value found by averaging the probabilities of the elementary processes over the directions. The role played by the structure of the orientational inhomogeneity is considered. The results are generalized to the case of anisotropy of the scattering.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 46–50, September–October, 1971.I should like to thank N. A. Khizhnyak for his constant interest.     

Dispersion of a filtration flow

In this paper we shall consider the transport of a dynamically neutral impurity in a porous medium containing random inhomogeneities. The original versions of the equations for the mean impurity concentration [1, 2] were based on the hyphothesis that the random motions obeyed the Markov principle, use being made of the diffusion equations of A. N. Kolmogorov. Later [3, 4] the method of perturbations was used to study the complete system of equations for the impurity concentration and random filtration velocity in the case of a constant, nonrandom porosity; after an averaging process this yields a generalized equation for the average concentration. In the limiting cases of small- and large-scale inhomogeneities in the permeability of the medium, the basic integrodifferential equation may be, respectively, reduced to parabolic and hyperbolic equations of the second order. In the present analysis we shall use the perturbation method to study the transport of an impurity by a flow when the filtration velocity of the latter fluctuates around inhomogeneities in the permeability field, the porosity of the medium in which the flow is taking place also constituting a random field, correlating with the field of permeability. We shall derive equations for the average concentration and should formulate the corresponding boundary-value problems for these equations; we shall also calculate the components of the dispersion tensor and shall consider the equilibrium sorption of an impurity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 65–69, July–August, 1976.The author is grateful to A. I. Shnirel'man for useful discussions.     

Calculation of the characteristics of a gas-dynamic laser

The dependence of the radiated power on the characteristics of optical cavities in the case of flow systems has been investigated in a number of papers [1–3], in which it is assumed that population inversion of the laser levels is obtained until entry into the cavity. The operation of a cavity is analyzed in [1] in the geometric-optical approximation with allowance for vibrational relaxation in the gas flow. A simplified system of relaxation equations is solved under steady-state lasing conditions and an expression derived for the laser output power on the assumption of constant temperature, density, and flow speed. The vibrational relaxation processes in the cavity itself are ignored in [2, 3]. It is shown in those studies that the solution has a singularity at the cavity input within the context of the model used. In the present article the performance characteristics of a CO₂-N₂-He gas-dynamic laser with a plane cavity are calculated. A set of equations describing the processes in the cavity is analyzed and solved numerically. Population inversion of the CO₂ laser levels is created by pre-expansion of the given mixture through a flat hyperbolic nozzle. The dependence of the output power on the reflectivities of the mirrors, the cavity length, the pressure, and the composition of the active gas medium is determined.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi FiziM, No. 5, pp. 33–40, September–October, 1972.     

Dynamic compactness of coupled thermoelastic waves in continuously nonuniform anisotropic media

In the present work, the dynamic problem of coupled thermoelasticity with the most general type of nonuniformity and anisotropy is analyzed. The hyperbolic nature of the system of equations of coupled thermoelasticity is demonstrated, effects of extinction of separate waves by superposition of elastic and thermoelastic wave fronts are investigated, and the interrelationship of different orders of discontinuity of stresses, displacements, and temperature is determined. The case of the uncoupled problem of thermoelasticity is especially analyzed. Sufficient conditions are obtained for the dynamic density for wave processes in thermoelasticity, previously investigated for boundary value problems of hyperbolic systems of second order differential equations [1], andelastic stress waves [2] are obtained. The generally accepted system of tensor notation for the theory of thermoelasticity is used [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 154–163, May–June, 1981.     

Hydrodynamics of a liquid with deformable and interacting particles

Interest in the hydrodynamics of a liquid with particle rotations and microdeformations has recently intensified [1–9] in connection with the technical applications of different artificially synthesized structured media. A model of a liquid with deformable microstructure was first proposed in [4] and was thermodynamically analyzed in [6], in which a model of a liquid was constructed by means of methods from the thermodynamics of irreversible processes. A model of a macro- and microincompressible liquid with particle rotations and deformations has been proposed [7, 8] based on constitutive equations from [6]. Below we will solve the sphere rotation problem in an infinite liquid given different boundary conditions on the rates of particle rotation and microdeformation within the context of the system of equations presented in [7]. The solution of an analogous problem for a micropolar liquid simulating a suspension with solid particles has been obtained [9] and the solution for a viscous liquid was found by Stokes in [10].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnieheskoi Fiziki, No. 1, pp. 79–87, January–February, 1976.     

A Decomposition Method for Solving Coupled Multi–Species Reactive Transport Problems

Concerns over the problems associated with mixed waste groundwater contamination have created a need for more complex models that can represent reactive contaminant fate and transport in the subsurface. In the literature, partial differential equations describing solute transport in porous media are solved either for a single reactive species in one, two or three dimensions, or for a limited number of reactive species in one dimension. Those solutions are constrained by many simplifying assumptions. Often, it is desirable to simulate transport in two or three dimensions for a more practical system that might have multiple reactive species. This paper presents a decomposition method to solve the partial differential equations of multi–dimensional, multi–species transport problems that are coupled by linear reactions. A matrix method is suggested as a tool for describing the reaction network. In this way, the level of complexity required to solve the multi–species reactive transport problem is significantly reduced.     

Wave propagation in the combustion of a low-concentration aerial suspension

In the general case the convective combustion of aerial suspensions is described by the equations of mechanics of multiphase media [1]. If the volume particle content is neglected and it is assumed that in the initial stage of convective front propagation the particles are stationary, and that during combustion their temperature is constant, then the equations for describing the combustion process reduce to the equations of gas dynamics for a distributed supply of heat and mass [2, 3]. The equations and model constant mass burning rate kinetics are used to solve the plane one-dimensional problem of the combustion of an aerial suspension in part of a region bounded on one side by a fixed wall. A small parameter proportional to the mass concentration and the heat value of the fuel is introduced. The method of matched asymptotic expansions [4] is used to construct a uniformly applicable first approximation. The solution obtained describes the wave propagation in aerial suspension combustion processes. The resulting pattern includes an inclined compression wave propagated with the speed of sound followed by a convective hot reaction product front whose propagation velocity is much less (in conformity with the small parameter introduced) than the speed of sound.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 63–73, March–April, 1986.     

Generalized variables in boundary-layer theory

Independent variables are widely used in boundary-layer theory to construct efficient methods of solving problems. The Dorodnitsyn variables in Lees' form [1] are the most common and general. This form combines the transformations proposed by Dorodnitsyn [2], Blasius [3], and Mangler-Stepanov [4, 5]. As is well known, transformation of the boundary-layer equations to Dorodnitsyn variables in Lees' form leads to a generalized single system of equations describing plane and axisymmetric gas flows. An analogous generalization of the Mises [6] and Crocco [7] variables is carried out below.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 166–168, September–October, 1976.     

Nonlinear Disturbances and Weak Discontinuities in a Supersonic Boundary Layer

The processes of wave disturbance propagation in a supersonic boundary layer with self-induced pressure [1–4] are analyzed. The application of a new mathematical apparatus, namely, the theory of characteristics for systems of differential equations with operator coefficients [5–8], makes it possible to obtain generalized characteristics of the discrete and continuous spectra of the governing system of equations. It is shown that the discontinuities in the derivatives of the solution of the boundary layer equations are concentrated on the generalized characteristics. It is established that in the process of flow evolution the amplitude of the weak discontinuity in the derivatives may increase without bound, which indicates the possibility of breaking of nonlinear waves traveling in the boundary layer.     

Rayleigh stability of shear flow in relaxing media

The stability of steady-state flow is considered in a medium with a nonlocal coupling between pressure and density. The equations for perturbations in such a medium are derived in the linear approximation. The results of numerical integration are given for shear motion. The stability of parallel layered flow in an inviscid homogeneous fluid has been studied for a hundred years. The mathematics for investigating an inviscid instability has been developed, and it has been given a physical interpretation. The first important results in flow stability of an incompressible fluid were obtained in the papers of Helmholtz, Rayleigh, and Kelvin [1] in the last century. Heisenberg [2] worked on this problem in the 1920's, and a series of interesting papers by Tollmien [3] appeared subsequently. Apparently one of the first problems in the stability of a compressible fluid was solved by Landau [4]. The first investigations on the boundary-layer stability of an ideal gas were carried out by Lees and Lin [5], and Dunn and Lin [6]. Mention should be made of a series of papers which have appeared quite recently [7–9]. In all the papers mentioned flow stability is investigated in the framework of classical single-phase hydrodynamics. Meanwhile, in recent years, the processes by which perturbations propagate in media with relaxation have been intensively studied [10–12].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 87–93, May–June, 1976.     

Theory of perfect plasticity of composite materials,taking account of volume compressibility

The practical importance and timeliness of the study of the mechanical macroscopic behavior of composite microinhomogeneous media are determined by its ability to give criteria for estimating the limiting load of various structural elements, the flow of multiphase dispersed systems, the deformation of materials made by powder metallurgy methods, etc. The theoretical prediction of the properties of composites is generally most effectively realized when their structural representations are based on the theory of random fields. Application of the strong isotropy hypothesis [1] for the statistical averaging of certain relations of a perfectly plastic body with microstructure permits the determination of its macroscopic yield surface.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 164–167, May–June, 1981.     

Nonisothermal Couette flow of a non-Newtonian fluid under the action of a pressure gradient

Nonisothermal Couette flow has been studied in a number of papers [1–11] for various laws of the temperature dependence of viscosity. In [1] the viscosity of the medium was assumed constant; in [2–5] a hyperbolic law of variation of viscosity with temperature was used; in [6–8] the Reynolds relation was assumed; in [9] the investigation was performed for an arbitrary temperature dependence of viscosity. Flows of media with an exponential temperature dependence of viscosity are characterized by large temperature gradients in the flow. This permits the treatment of the temperature variation in the flow of the fluid as a hydrodynamic thermal explosion [8, 10, 11]. The conditions of the formulation of the problem of the articles mentioned were limited by the possibility of obtaining an analytic solution. In the present article we consider nonisothermal Couette flows of a non-Newtonian fluid under the action of a pressure gradient along the plates. The equations for this case do not have an analytic solution. Methods developed in [12–14] for the qualitative study of differential equations in three-dimensional phase spaces were used in the analysis. The calculations were performed by computer.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 26–30, May–June, 1981.