A spatial laminar boundary layer on a streamline in conical external flow of a uniform gas in the absence of sources and sinks

Theoretical study of a three-dimensional laminar boundary layer is a complex problem, but it can be substantially simplified in certain particular cases and even reduced to the solution of ordinary differential equations.One such particular case is the flow of a compressible gas on a streamline in conical external flow. The case is of considerable practical importance because the local heat fluxes may take extremal values on such lines.Such flow, except for the conical case, has been examined [1–4], and an approximate method has been given [1] on the basis of integral relationships and a special form for the approximating functions. A numerical solution has been given [2, 3] for such flow around an infinite cylinder. It was assumed in [1–3] that the Prandtl number and the specific heats were constant, and that the dynamic viscosity was proportional to temperature. Heat transfer has been examined [4] near a cylinder exposed to a flow of dissociated air.Here we give results from numerical solution of a system of ordinary differential equations for the flow of a compressible gas in a laminar boundary layer on streamlines in conical external flow, with or without influx or withdrawal of a homogeneous gas. It is assumed that the gas is perfect and that the dynamic viscosity has a power-law temperature dependence.     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Small vibrations of a sphere in a spherical volume of viscous fluid

Problems of the vibration of bodies in confined viscous fluids have been solved to determine the added masses and damping coefficients of rods [1–3] and floats [4–5]. The solutions of these problems, based on the use of simplifications of the boundary-layer method [4–6], are obtained analytically in general form and are in good agreement with the experimental data. However, in each specific case the possibility of using such solutions for given values of the fluid viscosity and vibration frequency must be justified either experimentally [2, 4, 5] or theoretically as, for example, in [1], where an analytic solution was obtained for concentric cylinders. The present paper offers a general solution of the problem of the small vibrations of a sphere in a spherical volume of fluid valid over a broad range of variation of the dimensionless kinematic viscosity. The limiting cases of this solution for both high and low viscosity are considered. The asymptotic expressions obtained are compared with calculations based on the analytic solution.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 29–34, March–April, 1986.     

The uniqueness of the solution of boundary-value problems of a thermal model of discharge

The paper is devoted to a nonlinear analysis of superheating [1, 2] instability of an electric discharge stabilized by electrodes [3] in the framework of a thermal model [4] where the stability of the discharge relative to the long-wave and short-wave perturbations is proved in a linear approximation. Similar boundary-value problems arise in the theories of chemically and biologically reacting mixtures [5–7], thermal breakdown of dielectrics [8], thermal explosion [9], in the investigation of nonlinear waves in semiconductors and superconductors [10, 11], and in the investigation of Couette flow with variable viscosity [12]. The uniqueness of the one-dimensional steady solutions of the thermal model of discharge and the stability relative to the small spatial perturbations, respectively, for the exponential and step dependence of the electrical conductivity on the temperature are proved in [3, 13]. The uniqueness of the solutions in the one-dimensional case for the same electrode temperature and arbitrary dependences of the electrical and thermal conductivity on the temperature is established in paper [14]. In the present paper, the existence and uniqueness of steady solutions of the thermal model of discharge in a three-dimensional formulation for arbitrary fairly smooth electrical and thermal conductivity functions of the temperature in the case of isothermal isopotential electrodes are proved analytically.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 140–145, January–February, 1986.The author expresses his gratitude to A. G. Kulikovskii and A. A. Barmin for the formulation of the problem and their discussions.     

Thermal explosion during the flow of a viscous liquid

Frank-Kamenetskii has discussed a steady-state formulation of thermal explosions [1]. Bostandzhiyan et al. [2] and Bostandzhiyan and Chernyaeva [3] have shown, for the flow in a cylindrical tube of Newtonian and non-Newtonian liquids having a strong (nonlinear) temperature dependence of the viscosity, that a phenomenon analogous to thermal explosion may occur during the flow of a chemically inert liquid. Bostandzhiyan et al. [4] have also studied Couette flow and the flow between two rotating circular cylinders of a Newtonian liquid having the same temperature dependence for its viscosity. It was shown that, although the heat balance equation reduces to the equations of the steady-state theory of thermal explosion for the corresponding region, hydrodynamic thermal “explosion” was not observed in these cases. This phenomenon was found to be characteristic of only pressurized flows. Below, we study thermal explosions during the Poiseuille flow of a viscous, chemically reactive liquid in an infinite circular cylindrical tube, and during the motion of the liquid between infinite rotating cylinders. The combined effect of chemical and mechanical heat cources are considered. Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, Vol. 9, No. 5, pp. 38–43, 1968     

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.     

Investigation of the flow and heat transfer of viscous reacting fluids in long tubes

Heat transfer and resistance in the case of laminar flow of inert gases and liquids in a circular tube were considered in [1–4], the justification of the use of boundary-layer type equations for investigating two-dimensional flows in tubes being provided in [4]. The flow of strongly viscous, chemically reacting fluids in an infinite tube has been investigated analytically and numerically in the case of a constant pressure gradient or constant flow rate of the fluid [5–8]. An analytic analysis of the flow of viscous reacting fluids in tubes of finite length was made in [9, 10]. However, by virtue of the averaging of the unknown functions over the volume of the tube in these investigations, the allowance for the finite length of the tube reduced to an analysis of the influence of the time the fluid remains in the tube on the thermal regime of the flow, and the details of the flow and the heat transfer in the initial section of the tube were not taken into account. In [11], the development of chemical reactions in displacement reactors were studied under the condition that a Poiseuille velocity profile is realized and the viscosity does not depend on the temperature or the concentration of the reactant; in [12], a study was made of the regimes of an adiabatic reactor of finite length, and in [13] of the flow regimes of reacting fluids in long tubes in the case of a constant flow rate. The aim of the present paper is to analyze analytically and numerically in the two-dimensional formulation the approach to the regimes of thermal and hydrodynamic stabilization in the case of the flow of viscous inert fluids and details of the flow of strongly viscous reacting fluids.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 17–25, January–February, 1930.     

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.     

Numerical investigation of the hydrodynamic drag of a circular cylinder coated with a thin layer of magnetic fluid

In order to reduce the drag of bodies in a viscous flow it has been proposed to apply to the surface exposed to the flow a layer of magnetic fluid, which can be retained by means of a magnetic field and thus act as a lubricant between the external flow and the body [1, 2]. In [1] the hydrodynamic drag of a current-carrying cylindrical conductor coated with a uniform layer of magnetic fluid was theoretically investigated at small Reynolds numbers. In order to simplify the equations of motion, the Oseen approximation was introduced for the free stream and the Stokes approximation for the magnetic fluid [3]. This approach has led to the finding of an exact analytic solution from which it follows that at Reynolds numbers Re 1 the drag of the cylinder can be considerably reduced if the viscosity of its magnetic-fluid coating is much less than the viscosity of the flow. The main purpose of the present study is to investigate, with reference to the same problem, how the magnetic-fluid coating affects the hydrodynamic drag at Reynolds numbers 1 Re 10²–10³, i.e., under separated flow conditions. In this case the simplifications associated with neglecting the nonlinear inertial terms in the Navier—Stokes equation are inadmissible, so that a solution can be obtained only by numerical methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 11–16, May–June, 1986.     

Nonisothermal steady flow of a viscoplastic liquid between two coaxial rotating cylinders

Rotational viscosimeters are widely used to determine liquid viscosity. The technique for processing the experimental data is based fundamentally on the analytic solution of the problem of isothermal flow of a viscous liquid between two rotating cylinders.If in the course of the experiment the heat released due to the internal friction leads to significant heating, then the processing of the experimental results using the equations obtained on the assumption of isothermocity of the flow may lead to large errors. The dissipative heating may be reduced by reducing the angular velocity of rotation of the cylinder; however extensive reduction of the angular velocity is not desirable, since this leads to an increase of the measurement relative error. In addition, there is the possibility of conducting the experiments with a wide variation of the angular velocities in order to identify the structural-rheological peculiarities of the liquid. In the latter case we must be able to separate the purely thermal effects from the influence of the rheological factors. All these questions are discussed in detail in [1]. The authors of [1] obtained the solution of the problem of nonisothermal flow of a Newtonian fluid between two rotating cylinders and gave a technique for processing the experimental data which takes account of the dissipative heating of the fluid. The present paper pursues the same objective for a visco-plastic fluid.An attempt to solve the problem of nonisothermal flow of a viscoplastic fluid was made by Dzhafarov in [2], where the problem was solved in two versions. In the first version it was considered that the viscosity varies hyperbolically with the temperature and the gap between the cylinders is small in comparison with the radius of the inner cylinder. As a result of the linearization of the equations of motion and heat balance, it turned out that in fact the problem of Couette flow of a viscoplastic fluid was solved rather than the original problem. In this case, naturally, such a characteristic of the flow of a viscoplastic fluid in an annular gap as the possibility of the formation of an elastic zone was not covered. In the second version the problem was solved under the assumption that the viscosity is independent of the temperature and the angular velocity is small.In the present study the problem is solved without the limitations discussed above on the angular velocity, the fluid visoosity, and the gap between the cylinders. In this case we consider two types of temperature boundary conditions: a) constant temperatures are specified on the surfaces of the cylinders, which in the general case may be different; and b) a constant temperature is given on the surface of the outer cylinder and the inner cylinder is thermally insulated.     

Nonisothermal flow of a viscous incompressible fluid in a two-dimensional diffuser

The velocity and temperature distributions in a viscous incompressible fluid flow in a two-dimensional diffuser are analyzed. Fully developed flow is considered, i.e., the influence of the entrant section is disregarded. It is assumed that the diffuser walls are maintained at a temperature depending on the polar radius. The dynamic viscosity is considered to be an exponential function of the temperature. The problem is reduced to the solution of a system of ordinary differential equations, which is solved by the method of successive approximations. The convergence of the iterative scheme is proved.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 40–48, July–August, 1973.The author is indebted to L.A. Galin and N. N. Gvozdkov for assistance with the study.     

Nonisothermal flow of plasma in a plane MHD-channel

There is a large number of published papers (see the references given in the review [1]) in which various cases of the steady-state laminar flow of a conducting medium in a plane channel in the presence of a transverse magnetic field are considered. However, it has always been assumed that the transport coefficients were constants independent of the flow parameters such as the temperature. As a result, the dynamic and thermal problems were separable, and me temperature distribution had no effect on the dynamic flow parameters.In a low-temperature dense plasma, the conductivity is a very rapidly increasing function of temperature (it is approximately given by e^–A/T or T^10–13). It is clear that, in this case, it is necessary to take into account the fact that the transport coefficients are not constants, and the dynamic and thermal problems are not separable even for an incompressible fluid. We shall refer to such flows as nonisothermal, and contrast them with flows for which the transport coefficients are constants, and which we shall refer to as isothermal. for the sake of brevity.The importance of effects due to the nonisothermal nature of a flow was demonstrated in [2, 3], Hysteresis effects in friction and heat transfer, which were found in these papers, were discussed qualitatively in [4], Finally, the flow of a fluid with temperature-dependent conductivity in an MHD-channel was considered in [5] where it was. noted that the nonisothermal nature of the flow must be taken into account. In particular, the appearance of nonmonotonic velocity profiles with points of inflection was demonstrated [5], However, the latter paper included a number of conflicting assumptions. For example, when the propagation of heat was considered along the flow, it was assumed that the conductivity varied only across the channel. The temperature dependence of the conductivity used [5] was quite unrealistic, while the temperature dependence of viscosity and thermal conductivity was not taken into account at all.In the present paper we investigate the flow of plasma in a plane MHD-channel in the absence of a longitudinal flow of heat but with allowance for the temperature dependence of the transport coefficients. We shall use a more realistic form of the temperature dependence for the above parameters, and will take viscous energy dissipation into account.The authors are deeply indebted to M. V. Maslennikov and Yu. S. Sigov for valuable advice in connection with the numerical method of solution.     

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.     

Stability of nonplane-parallel three-dimensional jet flows

The problem of the stability of nonplane-parallel flows is one of the most difficult and least studied problems in the theory of hydrodynamic stability [1]. In contrast to the Heisenberg approximation [1], the basic state whose stability is investigated depends on several variables, and the stability problem reduces to the solution of an eigenvalue problem for partial differential equations in which the coefficients depend on several variables [2–7]. In the case of a periodic dependence of these coefficients on the time [2] or the spatial coordinates [3, 4], the analog of Floquet theory for the partial differential equations is constructed. With rare exceptions, the case of a nonperiodic dependence has usually been considered under the assumption of weak nonplane-parallelism, i.e., a fairly small deviation from the plane-parallel case has been assumed and the corresponding asymptotic expansions in the linear [6] and nonlinear [7] stability analyses considered. The present paper considers the case of an arbitrary dependence of the velocity profile of the basic flow on two spatial variables. The deviation from the plane-parallel case is not assumed to be small, and the corresponding eigenvalue problem for the partial differential equations is solved by means of the direct methods of [5], which were introduced for the first time and justified in the theory of hydrodynamic stability by Petrov [8].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–28, May–June, 1987.     

Self-similar movement of a viscous gas in a channel

The results presented in [1] refer primarily to dropping liquids for which the influence exerted by the thermal conditions on the flow is related to the temperature dependence of the viscosity. The self-similar flow of a viscous gas in a channel with a linearly increasing wall temperature is examined in this paper. The influence exerted by the Reynolds and Prandtl numbers on heat exchange and the hydrodynamics of the flow is analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 47–54, May–June, 1976.The author wishes to thank A. F. Seleznev for carrying out the calculations and V. N. Shtern for discussing the paper.     

Solution of the temperature jump (Knudsen layer flow) and linear heat-transfer problems for two parallel plates in a rarefied gas

The Monte Carlo method [1, 2] is used to solve the linearized Boltzmann equation for the problem of heat transfer between parallel plates with a wall temperature jump (Knudsen layer flow). The linear Couette problem can be separated into two problems: the problem of pure shear and the problem of heat transfer between two parallel plates. The Knudsen layer problem is also linear [3] and, like the Couette problem, can be separated into the velocity slip and temperature jump problems. The problems of pure shear and velocity slip have been examined in [2].The temperature jump problem was examined in [4] for a model Boltzmann equation. For the linearized Boltzmann equation the problems noted above have been solved either by expanding the distribution function in orthogonal polynomials [5–7], which yields satisfactory results for small Knudsen numbers, or by the method of moments, with an approximation for the distribution function selected from physical considerations in the form of polynomials [8–10]. The solution presented below does not require any assumptions on the form of the distribution function.The concrete calculations were made for a molecular model that we call the Maxwell sphere model. It is assumed that the molecules collide like hard elastic spheres whose sections are inversely proportional to the relative velocity of the colliding molecules. A gas of these molecules is close to Maxwellian or to a gas consisting of pseudo-Maxwell molecules [3].     

Calculation of accelerated motion of fluid in a tube

The results of calculation of accelerated flow of a fluid in a tube are compared with known experiments [1] in the laminar regime. The difference method was used to obtain a solution for unsteady axisymmetric flow that becomes steady over the length of the tube; this case was calculated earlier by Gromeka in the form of a series. An expression is derived for the coefficient of friction as a function of the Reynolds number Re and the acceleration of the fluid. The comparison reveals agreement between the results with an error not worse than 37%. However, the calculation gives a coefficient of friction proportional to Re to the power –1.5, whereas the experiment [1] reveals a weaker dependence proportional to Re to the power –1.15.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 158–160, September–October, 1981.     

Turbulent convection in a plane horizontal layer

The problem of convection in an incompressible fluid between two horizontal planes maintained at a constant temperature without friction on the boundaries is considered. The medium is assumed to be turbulent. A theoretical model is constructed using mathematical modeling of the coherent structure in the turbulent flow. This turbulent convection-model has one empirical constant in the relations closing the generalized Reynolds equations. The problem formulated is solved analytically by means of the Stuart-Landau method. The main characteristics of the finite-amplitude ordered convection are obtained and their dependence on the empirical constant is studied.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 49–56, November–December, 1993.     

Investigation of the stability of a heavy fluid film in a hot heat-conducting gas jet on an inclined phase-transition surface

The stability of the flow of a heavy viscous fluid film flowing along the inclined phase-transition surface is examined. In contrast to [1] wherein it was assumed that a constant temperature is maintained on the free surface, it is assumed here that the fluid film is on the boundary with a gas jet which has finite specific heat and heat conduction. In this connection, the stability criteria differ substantially from [1].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 10–18, July–August, 1974.     

Distribution of radiant thermal flux over the surface of a sphere for hypersonic flow of a nonviscous radiating gas past the sphere

In the present study using the Newtonian approximation [1] we obtain an analytical solution to the problem of flow of a steady, uniform, hypersonic, nonviscous, radiating gas past a sphere. The three-dimensional radiative-loss approximation is used. A distribution is found for the gasdynamic parameters in the shock layer, the withdrawal of the shock wave and the radiant thermal flux to the surface of the sphere. The Newtonian approximation was used earlier in [2, 3] to analyze a gas flow with radiation near the critical line. In [2] the radiation field was considered in the differential approximation, with the optical absorption coefficient being assumed constant. In [3] the integrodifferential energy equation with account of radiation was solved numerically for a gray gas. In [4–7] the problem of the flow of a nonviscous, nonheat-conducting gas behind a shock wave with account of radiation was solved numerically. To calculate the radiation field in [4, 7] the three-dimensional radiative-loss approximation was used; in [5, 6] the self-absorption of the gas was taken into account. A comparison of the equations obtained in the present study for radiant flow from radiating air to a sphere with the numerical calculations [4–7] shows them to have satisfactory accuracy.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 44–49, November–December, 1972.In conclusion the author thanks G. A. Tirskii and É. A. Gershbein for discussion and valuable remarks.