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.     

Development of magnetohydrodynamic boundary layers

Many authors have studied the problem of the development of a hydrodynamic boundary layer when a body is suddenly set in motion. The results obtained are presented most fully in the monographs of H. Schlichting [1] and L. G. Loitsyanskii [2]. In magnetohydrodynamics the development of the boundary layer over the surface of an infinite flat plate for uniform oncoming flow has been closely studied (for example [3, 4]). Below, the problem of the development of a plane magnetohydrodynamic boundary layer is considered in a different formulation. We shall suppose that the distributions of velocity U(x) and enthalpy h(x) are given along the body contour for t=0. At that moment the viscosity and thermal conductivity mechanisms are instantaneously switched on. Viscous and thermal boundary layers begin to grow in a direction normal to the body. The medium in the boundary layer interacts with the magnetic field. This formulation corresponds to the development of a magnetohydrodynamic boundary layer on a body which is set in motion with a jerk, in the case where the rate of establishment of magnetohydrodynamic flow of the inviscid, thermally nonconducting fluid around the body is much less than the rate of development of the boundary layer. Then U(x) and h(x) are found by solving the problem of stationary magnetohydrodynamic flow of an inviscid thermally nonconducting fluid around a body, or simply the hydrodynamic flow if the medium interacts with the field only in the boundary layer.     

Investigation of the properties of a shock wave arising in a supersonic flow of plasma,passing through a transverse magnetic field

In a flow of plasma, set up by an ionizing shock wave and moving through a transverse magnetic field, under definite conditions there arises a gasdynamic shock wave. The appearance of such shock waves has been observed in experimental [1–4] and theoretical [5–7] work, where an investigation was made of the interaction between a plasma and electrical and magnetic fields. The aim of the present work was a determination of the effect of the intensity of the interaction between the plasma and the magnetic field on the velocity of the motion of this shock wave. The investigation was carried out in a magnetohydrogasdynamic unit, described in [8]. The process was recorded by the Töpler method (IAB-451 instrument) through a slit along the axis of the channel, on a film moving in a direction perpendicular to the slit. The calculation of the flow is based on the one-dimensional unsteady-state equations of magnetic gasdynamics. Using a model of the process described in [9], calculations were made for conditions close to those realized experimentally. In addition, a simplified calculation is made of the velocity of the motion of the above shock wave, under the assumption that its front moves at a constant velocity ahead of the region of interaction, while in the region of interaction itself the flow is steady-state.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 86–91, January–February, 1975.     

Interaction between the boundary layer and a supersonic flow when part of the wall is rapidly heated

There have been many theoretical studies of aspects of the unsteady interaction of an exterior inviscid flow with a boundary layer [1–9]. The mathematical flow models obtained in these studies by the method of matched asymptotic expansions describe a wide range of phenomena observed experimentally. These include boundary layer separation near the hinge of a flap, the flow in the neighborhood of the trailing edge of an oscillating airfoil [1–2], and the development and propagation of perturbations in a boundary layer excited by an oscillating wall or some other way [3–5]. The present paper studies the interaction of an unsteady boundary layer with a supersonic flow when a small part of the surface of a body in the flow is rapidly heated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 66–70, January–February, 1984.     

Asymptotic analysis of transonic flows

In this paper we derive the equations of the second and third approximations for the stream function of two-dimensional and axisymmetric potential transonic flow of an inviscid gas and find their particular solutions corresponding to certain transonic flows.A similar study concerning the second approximation of subsonic and supersonic flow was made by Van Dyke [1] and Hayes [2]. The second approximation for the velocity potential of transonic flow has been examined in detail by Hayes [3]. Euvrard [4, 5] has investigated the asymptotic behavior of transonic flow far from a body, while Fal'kovich, Chernov, and Gorskii [6] have studied the flow in a nozzle throat.The transonic asymptotic analysis for the stream function is presented in this paper.     

A qualitative investigation of the equations of quasi-one-dimensional magnetohydrodynamic channel flow

A qualitative investigation of the system of differential equations describing the quasi-one-dimensional flow of an electrically conducting medium at small magnetic Reynolds numbers gives an idea of the different possible flow patterns occuring when the electromagnetic field and channel shape are given in different ways. Such a treatment is essential for the calculation of one-dimensional flows, and also for the solution of variational problems [1].In the literature devoted to this question studies have been made of flow in a one-dimensional electromagnetic field and a channel of constant cross section [2], as well as of the flow when the magnetic field is described by specially given functions of the flow velocity [3]. These cases reduce to the analysis of integral curves in a plane.In the present paper the investigation is carried out for an arbitrary distribution of the electric and magnetic fields and channel shape, which leads to a consideration of the behavior of integral curves in three-dimensional space. The qualitative results are illustrated by examples.     

Numerical investigation of the expansion dynamics and generation of magnetic fields in plasma jets

The processes of interaction between concentrated energy fluxes and solid targets have been investigated in a number of studies. The generation of magnetic fields in erosional plasma formations has been experimentally observed [1–4]. However, the evolution of magnetic fields in plasma jets has not yet been studied in sufficient detail. This paper is devoted to a numerical investigation of unsteady three-dimensional erosion plasma flows and the generation in those flows of magnetic fields resulting from the action of laser radiation on solid targets. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 135–140, November–December, 1986. The authors are grateful to I. V. Nemchinov and B. T. Fedyushin for useful discussion of their results.     

Stabilization of an alternating-current-carrying plasma filament by a quadrupole magnetic field

Dynamic stabilization of straight and toroidal current-carrying plasma jets by a high-frequency quadrupole magnetic field was proposed by Osovets [1]. A more rigorous theoretical analysis of the problem was performed by Levin and Rabinovich [2], who obtained a Routh function for studying the dynamics of the filament in various magnetic fields for a thin filament experiencing longwave serpentine- and construction-type disturbances. In this paper, the method proposed in [2] is applied to the stabilization of a plasma filament in which (as distinct from [1,2]) flows an alternating current while the quadrupole field is either constant, or varies at a high frequency, as in [1,2].The author is indebted to M. L. Levin for useful discussions and advice.     

The Braginskii model of the Rayleigh–Taylor instability. I. Effects of self-generated magnetic fields and thermal conduction in two dimensions

There exists a substantial disagreement between computer simulation results and high-energy density laboratory experiments of the Rayleigh–Taylor instability [1]. Motivated by the observed discrepancies in morphology and growth rates, we attempt to bring simulations and experiments into better agreement by extending the classic purely hydrodynamic model to include self-generation of magnetic fields and anisotropic thermal conduction.We adopt the Braginskii formulation for transport in hot, dense plasma, implement and verify the additional physics modules, and conduct a computational study of a single-mode RTI in two dimensions with various combinations of the newly implemented modules. We analyze physics effects on the RTI mixing and flow morphology, the effects of mutual physics interactions, and the evolution of magnetic fields.We find that magnetic fields reach levels on the order of 11 MG (plasma β ≈ 9.1 × 10^?2) in the absence of thermal conduction. These fields do not affect the growth of the mixed layer but substantially modify its internal structure on smaller scales. In particular, we observe denting of the RT spike tip and generation of additional higher order modes as a result of these fields. Contrary to interpretation presented in earlier work [2], the additional mode is not generated due to modified anisotropic heat transport effects but due to dynamical effect of self-generated magnetic fields. The overall flow morphology in self-magnetized, non-conducting models is qualitatively different from models with a pre-existing uniform field oriented perpendicular to the interface. This puts the usefulness of simple MHD models for interpreting the evolution of self-magnetizing HED systems with zero-field initial conditions into doubt.The main effects of thermal conduction are a reduction of the RT instability growth rate (by about 20% for conditions considered here) and inhibited mixing on small scales. In this case, the maximum self-generated magnetic fields are weaker (approximately 1.7 MG; plasma β ≈ 49). This is due to reduction of temperature and density gradients due to conduction. These self-generated magnetic fields are of very similar strength compared to magnetic fields observed recently in HED laboratory experiments [3].We find that thermal conduction plays the dominant role in the evolution of the model RTI system considered. It smears out small-scale structure and reduces the RTI growth rate. This may account for the relatively featureless RT spikes seen in experiments, but does not explain mass extensions observed in experiments.Resistivity, related heat source terms and the thermo-electric contribution to the heat flow were not included in the present work. We estimate their impact on RTI as modest and not affecting our main conclusions. These effects will be discussed in detail in the next paper in the series.     

Investigation of the supersonic flow around a pointed elliptical cone at an angle of attack

The picture of ideal gas flow around cones at zero and low angles of attack has been well studied by using approximate methods [1], and results for high angles of attack have been obtained mainly numerically [2–7]. At high angles of attack it is sensible to examine inviscid flow only up to some generator on the downwind side of the cone at which boundary-layer separation occurs. Hence, the domain where the flow can be considered inviscid yields the main contribution to the magnitude of the aerodynamic forces and the heat fluxes [5, 9]. A picture of the supersonic flow around a pointed elliptical cone is obtained in this paper by the numerical solution of the gasdynamics equations. The whole flow domain is computed at low angles of attack while the solution at high angles is obtained in a domain bounded by some surface of three-dimensional type [10]. The dependence of the flow parameters on the angle of attack is studied when the shock is attached to the cone apex. In contrast to a circular cone, at low angles of attack two spreading lines occur on the surface of an elliptical cone, to which the maximum pressure corresponds. As the angle of attack increases, these lines come together and merge at a certain time. At high angles of attack the flow picture is analogous to a circular cone with a pressure maximum in the plane of symmetry.     

Effect of viscosity on the lift-drag ratio of a thin blunt wing at hypersonic rates of flow round it

The effect of viscosity on the carrying properties of hypersonic aircraft appears at great flight altitudes, where an important factor is the interaction of the laminar boundary layer with inviscid flow. In the present study the method of bands is used to make an approximate calculation of this effect for a regime of weak viscous interaction [1]. The results of [2] are used for conditions of inviscid flow round a body. The local coefficient of friction and coefficients of the additional pressure induced by the boundary layer are determined from the data for a plate of infinite width [3]. Simple relationships are obtained which make it possible to estimate the effect of viscosity on the magnitude of the maximum lift-drag ratio and the value of the angle of attack corresponding to it. The results are given of an experimental study of hypersonic flow round a plane triangular wing in a broad range of Reynolds numbers, and these confirm the relationships obtained and indicate the region in which they are applicable.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 149–152, November–December, 1988.     

Self-similar magnetogasdynamic flows accompanied by detonation and combustion waves

Magnetogasdynamic (MGD) flows with detonation waves and combustion fronts have attracted more and more attention in recent years. Intensive heat supply assures such a significant increase in the temperature and pressure behind the heat liberation fronts that the gaseous combustion products become conductive so that the flow map in the electric and magnetic fields can vary substantially as compared with ordinary gasdynamics. In the case of finite gas conductivity, when the magnetic Reynolds numbers R_m are low, the asymptotic laws of detonation wave propagation which either go over into the Chapman-Jouguet (CJ) mode (in a number of cases at a finite distance from the initiation source) or remain overcompressed, have been studied [1]. Stationary flow modes behind detonation waves have been investigated in [2] and the problem of the detonation wave originating at the closed end of the tube emerging in the stationary mode in crossed homogeneous magnetic and electric fields has been examined. Results are presented in this paper of an investigation of one-dimensional self-similar flows caused by piston motion in a hot gas mixture in which a detonation wave or combustion front is propagated. The motion is realized in external electric and magnetic fields which exert a substantial effect on the flow of the conductive combustion products. Domains of application of the governing parameters in which the various flow modes are realized are found by using a qualitative and numerical analysis. The results obtained are used to solve problems about the hypersonic gas flow around a thin wedge in an axial magnetic field.     

Application of unsteady mixing equations to some aerodynamic problems

Tangential discontinuities [1] are introduced in solving several transient and steady-state problems of gas dynamics. These discontinuities are unstable [2] as a result of the effects of viscosity and thermal conductivity. Therefore it is advisable to replace the tangential discontinuity by a mixing region and account for its interaction with the inviscid flows, establishing on the boundaries of this region the conditions of vanishing friction stress and equality of the velocity and temperature components to the corresponding velocity and temperature components of the inviscid flows. This formulation improves the accuracy of the solution of such problems by posing them as problems with irregular reflection and intersection of shock waves [1].The consideration of the interaction of unsteady turbulent mixing regions with the inviscid flow also permits the formulation of several problems in which the effects of viscosity lead to complete rearrangement of the flow pattern (the lambda-configuration) with the interaction of the reflected shock wave with the boundary layer in the shock tube [3,4], the formation of zones of developed separation ahead of obstacles, etc.).In this connection, §1 presents an analysis of the self-similar solutions of the unsteady turbulent mixing equations (a corresponding analysis of the laminar mixing equations which coincide with the boundary layer equations is presented in [1]). It is shown that these self-similar solutions describe, along with the several problems noted above, the problems of the formation of steady jets and mixing zones in the base wake.As an example, §2 presents, within the framework of the proposed schematization, an approximate solution of the problem of the interaction of a shock wave reflected from a semi-infinite wall with the boundary layer on a horizontal plate behind the incident shock wave. The results obtained are applied to the analysis of reflection in a shock tube. Computational results are presented which are in qualitative agreement with experiment [3, 4].     

Instability of a layer op lightly ionized plasma in crossed electric and magnetic fields

In a lightly ionized plasma, charged-particle drift due to collisions with neutral atoms occurs at different velocities: $$\begin{array}{*{20}c} {v_{Ea} = \mp \frac{{b_a E}}{{1 + (\omega _a \tau _a )^2 }},v_{ \bot a} = \frac{{b_a E(\omega _a \tau _a )}}{{1 + (\omega _a \tau _a )^2 }}} \\ {\left( {b_a = \frac{{|e|\tau _a }}{{m_a }},\omega _a = \frac{{|e|\tau _a }}{{m_a }}} \right),} \\ \end{array} $$ where b_a is the mobility of particles of the type a;ω_a is the Larmor frequency; the upper sign refers to electrons and the lower sign to ions. A difference in the charged-particle drift velocities can cause instability of an inhomogeneous lightly ionized plasma. Let us consider the following example. Assume that in the initial state of the plasma there is a concentration gradient along the x-axis, that the external electric field is directed along the x-axis, and that the magnetic field coincides with the z-axis. In this system, under the influence of a Lorentz force the charged particles will move in a direction opposite to the y-axis. Since electrons have a higher velocity than ions, an electric field is induced in this direction. This electric field, together with the magnetic field, causes particle drift in the negative direction of the x-axis. Consequently, if the concentration gradient in the initial state is directed opposite to the x-axis this state cannot be stable. Instability of this kind has been examined by Simon [1]. On the basis of studies by Kadomtsev and Nedospasov [2], as well as by Rosenbluth and Longmire [3], Simon developed a theory of instability of a lightly ionized plasma in crossed fields with an inhomogeneous density distribution in the direction of the external electric field. Somewhat later, Simon's theory was developed [4]. In devices with inhomogeneous plasma flow in which the plasma (conducting) layers alternate with nonconducting layers, the external electric field and concentration are normal to one another. We shall bear this case in mind below and shall examine the instability of a lightly ionized plasma in crossed fields when the concentration inhomogeneity is in a direction perpendicular to the external electric field.     

Flow of a conductive gas in a circular tube in a strong axiosymmetric magnetic field

The flow of a conductive gas along a channel in an external axiosymmetric magnetic field with a finite value of the magnetogasodynamic parameter N is examined. Numerical flow calculations are performed for a circular tube in such a field. Gas dynamic parameter fields, total pressure losses, and electric current intensities with the presence of transsonic zones and highly compressed regions are determined. Through comparison of the results obtained with linear theory data, the range of applicability of the latter is determined. Of the works dedicated to study of flow in external magnetic fields with N1, we should take note of [1], in which the process of entry of the gas into a transverse magnetic field was examined; [2], which studied one-dimensional transient motion with shock waves; and [3], where mixed flow in a Laval nozzle with an axiosymmetric homogeneous magnetic field was studied. Flow in a circular tube was examined in [4]; but the analysis performed by the characteristic method permitted calculation of only the initial supersonic flow zone. Motion in circular tubes in the presence of an axiosymmetric, magnetic field was studied in the linear formulation in [4, 5].Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 145–155, September–October, 1972.     

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.     

Method of mean-mass parameters for a three-dimensional boundary layer in a flow with vorticity

When a gas flows with hypersonic velocity over a slender blunt body, the bow shock induces large entropy gradients and vorticity near the wall in the disturbed flow region (in the high-entropy layer) [1]. The boundary layer on the body develops in an essentially inhomogeneous inviscid flow, so that it is necessary to take into account the difference between the values of the gas parameters on the outer edge of the boundary layer and their values on the wall in the inviscid flow. This vortex interaction is usually accompanied by a growth in the frictional stress and heat flux at the wall [2, 3]. In three-dimensional flows in which the spreading of the gas on the windward sections of the body causes the high-entropy layer to become narrower, the vortex interaction can be expected to be particularly important. The first investigations in this direction [4–6] studied the attachment lines of a three-dimensional boundary layer. The method proposed in the present paper for calculating the heat transfer generalizes the approach realized in [5] for the attachment lines and makes it possible to take into account this effect on the complete surface of a blunt body for three-dimensional laminar, transition, or turbulent flow regime in the boundary layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 80–87, January–February, 1981.     

Supersonic flow around the trailing edge of a thin profile

The method of mergeable asymptotic expansions has recently been used effectively in investigations devoted to the study of boundary layer interaction with an external inviscid flow at high subcritical Reynolds numbers Re. The asymptotic analysis permits obtaining a limit pattern of the flow around a solid as Re þ, and determining the similarity and quantitative regularity laws which are in good agreement with experimental results. Thus by using the method of mergeable asymptotic expansions it is shown in [1–4] that near sites with high local curvature of the body contour and flow separation and attachment points, an interaction domain appears that has a small length on the order of Re^-3/8. In this flow domain, which has a three-layer structure, the pressure distribution in a first approximation already depends on the change in boundary-layer displacement thickness, while the induced pressure gradient, in turn, influences the flow in the boundary layer. An analogous situation occurs in the neighborhood of the trailing edge of a flat plate where an interaction domain also appears [5, 6]. The flow in the neighborhood of the trailing edge of a flat plate around which a supersonic viscous gas flows was examined in [7]. Numerical results in this paper show that the friction stress on the plate surface remains positive everywhere in the interaction domain, and grows on approaching the trailing edge. The supersonic flow around the trailing edge of a flat plate at a small angle of attack was investigated in [8, 9], Supersonic flow of a viscous gas in the neighborhood of the trailing edge of a flat plate at zero angle of attack is examined in [10], but with different velocity values in the inviscid part of the flow on the upper and lower sides of the plate. The more general problem of the flow around the trailing edge of a profile with small relative thickness is investigated in this paper.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 36–42, May–June, 1981.     

Linear approximation for the flow of a conducting gas with a high magnetic reynolds number through a channel

In this paper, we discuss the flow of a nonviscous and non-heat-conducting gas through a channel of variable cross section under the influence of a transverse magnetic field. For high magnetic Reynolds numbers, the flow is shown to consist of a core and current layers at the electrodes and at the fixed channel walls. The distributions of currents and other parameters in the core and in the current layers are found analytically, in a linear approximation. The Joule dissipation in the current layers may be more intense than that in the core. The longitudinal currents and Joule dissipation increase with increasing Hall parameter in the electrode layers. Zhigulev [1] has shown that magnetic boundary layers may form in the flow of a conducting gas when there is a high magnetic Reynolds number (R_m«1). He illustrated this situation by the shielding of a plasma flow from the magnetic fields produced near a plate which is electrically isolated from the plasma and through which a current is flowing. In an incompressible fluid, the layer thickness is proportional to R_m ^?1/2. Morozov and Shubin [2] have offered a linear-approximation treatment of the structure of the electromagnetic near-electrode layers which arise during the flow of a nonviscous plasma with a high R_m and a small “exchange” parameter ξ≈H/R_m, for flow transverse to a magnetic field and near a corrugated wall. They pointed out the possible formation of “dissipationless” near-electrode layers with thicknesses on the order of the Debye or electron Larmor radii, and a “dissipative” layer whose thickness increases along the length of the electrodes and is proportional to (R_mc_B ²/c_T ²)^?1/2, where c_B and c_T are the magnetic and thermal sound velocities. Morozov and Shubin studied the properties of dissipationless and dissipative electromagnetic layers at segmented accelerator electrodes through which a current is passing, for an arbitrary “exchange” parameter, in [2] and [3], respectively. The exchange parameter ξ was found in [4]. Such layers should also exist at solid electrodes and at the nonconducting walls of an accelerator channel. Study of the two-dimensional flow in a channel is significantly simplified when such layers are present.     

End effects in magnetohydrodynamic channels at finite magnetic reynolds numbers

The effects of the magnetic Reynolds number have been examined via the distribution of the magnetic fields induced by the motion of a medium in a rectangular channel with conducting walls in the presence of an inhomogeneous magnetic field; the effects of wall conductivity and geometry of the external field are also examined as regards the distribution of the induced currents, the Joule loss, and the electric and magnetic fields over the cross section. The problem has previously been considered for a channel with insulating walls [1].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 19–27, May–June, 1971.We are indebted to A. B. Vatazhin for his interest.