Saturation of absorption of the powerful radiation of a system of anharmonic oscillators

Recently, the theory of nonequilibrium systems simulated by a set of anharmonic oscillators has received significant development. The investigation of such kinds of systems is especially important in the study of problems associated with the stimulation of chemical reactions and the development of effective molecular lasers. The systematic analysis of the kinetics of anharmonic oscillators assumes the simultaneous solution of a large number of nonlinear equations describing the population balance of the vibrational levels. Realization of this approach is associated with cumbersome numerical calculations and does not permit obtaining a qualitative picture of the behavior of the system as a function of the different parameters (pressure, temperature, etc.). An approximate analytical theory has been formulated in [1, 2] which permits finding the distribution function over the vibrational states with the effects of anharmonicity taken into account. We will employ the approach developed in these papers to describe a system of anharmonic oscillators under conditions of powerful optical pumping. This problem was discussed in [3], where it was found that such a system changes into a saturation mode in the case of high pumping levels. The existence of this mode is explained by the fact that the maximum rate of energy input into a vibrational degree of freedom is determined by the rate of distribution of this energy over all the vibrational levels, i.e., by the constant of V—V-exchange. For sufficiently large pumpings the approximation of the Boltzmann distribution function adopted in [3] in connection with the calculation of the saturation parameters is too crude. The goal of this paper is to derive in explicit form expressions for the vibrational energy supply, the absorbed power, and so on, under saturation conditions without the use of the approximation indicated above [3].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 10–15, September–October, 1978.     

To the solution of the thermoelasticity problem for conical shells

We consider the stress-strain state of thin conical shells in the case of arbitary distribution of the temperature field over the shell. We obtain equations of the general theory based on the classical Kirchhoff-Love hypotheses alone. But since these equations are very complicated, attempts to construct exact solutions by analytic methods encounter considerable or insurmountable difficulties. Therefore, the present paper deals with boundary value problems posed for simplified differential equations. The total stress-strain state is constructed by “gluing” together the solutions of these equations. Such an approach (the asymptotic synthesis method) turns out to be efficient in studying not only shells of positive and zero curvature [1, 2] and cylindrical shells [3] but also conical shells [4, 5]. Here we illustrate it by an example of an arbitrary temperature field, and the problem is reduced to solving differential equations with polynomial coefficients and with right-hand side containing the Heaviside function, the delta function, and their derivatives.     

NEW THEORY FOR EQUATIONS OF NON-FUGHSIAN TYPE REPRESENTATION THEOREM OF TREE SERIES SOLUTION (Ⅰ)

In the analytic theory of differential equations the exact explicit analytic solution has not been obtained for equations of the non-Fuchsian type (Poincare's problem). The new theory proposed in this paper for the first time affords a general method of finding exact analytic expres-sion for irregular integrals.By discarding the assumption of formal solution of classical theory,our method consists in deriving a cor-respondence relation from the equation itself and providing the analytic structure of irregular integrals naturally by the residue theorem. Irregular integrals are made up of three parts: noncontracted part,represented by ordinary recursion series,all-and semi-contracted part by the so-called tree series. Tree series solutions belong to analytic function of the new kind with recursion series as the special case only.     

Estigation of twisted flows of an ideal gas in nozzles under super- and subcritical conditions

The article discusses weakly twisted flows of an ideal gas with arbitrary thermodynamoc properties in a supersonic nozzle with a central body, under super- and subcritical working conditions. The results of the investigation are generalized for the case of flow with an arbitrary (not necessarily weak) nonuniformity of the stagnation parameters of the flow (the entropy and the total enthalpy) over the cross section of the nozzle. An evaluation is made of the range of applicability of the linear theory with respect to the rate of twisting. The investigation is a generalization of the ideas of Chernyi, developed in [1].     

Quasiequilibrium model of the kinetics of coupled modes of CO2 molecules

The correct allowance for the influence of anharmonicity in the vibrational spectrum of CO₂ on the level distribution of molecules under nonequilibrium conditions, when the vibrational temperature departs significantly from the gas temperature, has become especially urgent in connection with obtaining generation on a number of long-wavelength transitions of CO₂ molecules [1, 2]. The shifts in the levels of coupled modes (symmetric and deformation) are due mainly to Fermi resonance and can reach a considerable value, comparable with the gas temperature even for low levels. In [3] the main features of the quasisteady level distribution of coupled modes were clarified within the framework of the Treanor model of vibrational kinetics. The influence of the ascending flux of quanta, excited by VV exchange under nonequilibrium conditions, on the vibrational distribution was considered in [4–6]. In the present paper we propose a quasiequilibrium model of CO₂ kinetics, obtained without presuming quasisteadiness of the ascending flux of quanta, and making it possible, in contrast to [3–6] to describe the dynamics of the variation of the distribution of molecules among multiplets as a result of processes of VV exchange and VT relaxation between multiplets, with allowance for possible processes of pumping by outside sources. With a Boltzmann population distribution within the multiplets, having the translational temperature of the gas, the problem of studying relaxation in coupled modes is reduced to the equations for an effective anharmonic oscillator with levels corresponding to the multiplets of CO₂ molecules. In this case the levels of the effective oscillator are degenerate with a multiplicity equal to the number of levels in the corresponding multiplet, and they have an anharmonicity constant dependent on the gas temperature. The population distribution of the effective oscillator can be studied by methods developed for the investigation cf a one-mode anharmonic oscillator. The proposed quasiequilibrium model was used for a numerical calculations of the temporal evolution of the distribution function of CO₂ molecules over the levels of coupled modes under the conditions of an extremely maintained discharge.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 16–22, May–June, 1986.     

Laminar boundary layer on swept-back wings of infinite span at an angle of attack

A study is made of the flow of a compressible gas in a laminar boundary layer on swept-back wings of infinite span in a supersonic gas flow at different angles of attack. The surface is assumed to be either impermeable or that gas is blown or sucked through it. For this flow and an axisymmetric flow an analytic solution to the problem is obtained in the first approximation of an integral method of successive approximation. For large values of the blowing or suction parameters, asymptotic solutions are found for the boundary layer equations. Some results of numerical solution of the problem obtained by the finite-difference method are given for wings of various shapes in a wide range of angles characterizing the amount by which the wings are swept back and also the blowing or suction parameters. A numerical solution is obtained for the equations of the three-dimensional mixing layer formed in the case of strong blowing of gas from the surface of the body. The analytic and numerical solutions are compared and the regions of applicability of the analytic expressions are estimated. On the basis of the solutions obtained in the present paper and studies of other authors a formula is proposed for the calculation of the heat fluxes to a perfectly catalytic surface of swept-back wings in a supersonic flow of dissociated and ionized air at different angles of attack. Flow over swept-back wings at zero angle of attack has been considered earlier (see, for example, [1–4]) in the theory of a laminar boundary layer. In [5], a study was made of flow over swept-back wings at nonzero angle of attack at small and moderate Reynolds numbers in the framework of the theory of a hypersonic viscous shock layer.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 27–39, May–June, 1980.We thank G. A. Tirskii for a helpful discussion of the results.     

Plastic flow for axisymmetric-tension specimens with necking

Plastic deformation of a tension specimen bounded by a curved surface of revolution is considered. Such a configuration may occur, for example, as a result of necking. The specimen material satisfies the Tresca yield condition and the associated flow rule. Approximate solutions for the stress distribution in the neck were examined in [6]. The extension of notched bars was investigated by numerical and graphic methods in [2, 4]. Below, the problem is solved analytically for a sufficiently smooth neck; a small degree of nonuniformity of the plastic properties is taken into account.     

Some dynamical behavior of the Stuart-Landau equation with a periodic excitation

The lock-in periodic solutions of the Stuart-Landau equation with a periodic excitation are studied. Using singularity theory, the bifurcation behavior of these solutions with respect to the excitation amplitude and frequency are investigated in detail, respectively. The results show that the universal unfolding with respect to the excitation amplitude possesses codimension 3. The transition sets in unfolding parameter plane and the bifurcation diagrams are plotted under some conditions. Additionally, it has also been proved that the bifurcation problem with respect to frequence possesses infinite codimension. Therefore the dynamical bifurcation behavior is very complex in this case. Some new dynamical phenomena are presented, which are the supplement of the results obtained by Sun Liang et al.     

Circular shallow spherical shells with central circular hole under simultaneous actions of arbitrary unsteady temperature field and arbitrary dynamic normal load

In this paper, under assumption that tempeature is linearly distributed along the thickness of the shell, we deal with problems as indicated in the title and obtain general solutions of them which are expressed in analytic form.In the first part, we investigate free vibration of circular shallow spherical shells with circular holes at the center under usual arbitrary boundary conditions. As an example, we calculate fundamental natural frequency of a circular shallow spherical shell whose edge is fixed (m=0). Results we get are expressed in analytic form and check well with E. Reissner’s [1]. Method for calculating frequency equation is recently suggested by Chien Wei-zang and is to be introduced in appendix 3.In the second part, we investigate forced vibration of shells as indicated in the title under arbitrary harmonic temperature field and arbitrary harmonic dynamic normal load.In the third part, we investigate forced vibration of the above mentioned shells with initial conditions under arbitrary unsteady temperature field and arbitrary normal load.In appendix 1 and 2, we discuss how to express displacement boundary conditions with stress function and boundary conditions in the case m=1.     

Three-dimensional laminar boundary layer on a permeable surface in the neighborhood of a symmetry plane

Analytical and numerical methods are used to investigate a three-dimensional laminar boundary layer near symmetry planes of blunt bodies in supersonic gas flows. In the first approximation of an integral method of successive approximation an analytic solution to the problem is obtained that is valid for an impermeable surface, for small values of the blowing parameter, and arbitrary values of the suction parameter. An asymptotic solution is obtained for large values of the blowing or suction parameters in the case when the velocity vector of the blown gas makes an acute angle with the velocity vector of the external flow on the surface of the body. Some results are given of the numerical solution of the problem for bodies of different shapes and a wide range of angles of attack and blowing and suction parameters. The analytic and numerical solutions are compared and the region of applicability of the analytic expressions is estimated. On the basis of the solutions obtained in the present work and that of other authors, a formula is proposed for calculating the heat fluxes to a perfectly catalytic surface at a symmetry plane of blunt bodies in a supersonic flow of dissociated and ionized air at different angles of attack. Flow near symmetry planes on an impermeable surface or for weak blowing was considered earlier in the framework of the theory of a laminar boundary layer in [1–4]. An asymptotic solution to the equations of a three-dimensional boundary layer in the case of strong normal blowing or suction is given in [5, 6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 37–48, September–October, 1980.     

Vibrational relaxation of diatomic molecules in a non-Boltzmann thermostat

The action of resonance IR laser radiation on a molecular gas leads, at high-power absorption intensity, to a breakdown in the equilibrium (Boltzmann) energy distribution in the internal degrees of freedom [1]. Under realistic conditions, molecular gases usually are (due to small amounts of impurities or isotopic components) multicomponent systems. In this case resonance IR laser radiation (or other methods of selective action), disturbing the distribution function of the primary gas, does not interact directly with impurities. The problem thus arises of determining the distribution function of the impurity gas interacting with the nonequilibrium (non-Boltzmann) thermostat. The present paper, devoted to the solution of this problem, treats the distribution function of harmonic oscillators A, consisting of a small amount of impurities in a system of harmonic oscillators B with given nonequilibrium distribution functions of vibrational energy. The behavior of a system in a nonequilibrium thermostat was first considered in [2, 3] where, as well as in [4, 5], it was shown that in a non-Maxwellian thermostat with a small amount of harmonic oscillator impurities, a Boltzmann distribution in harmonic oscillator vibrational energies is established under stationary conditions, with a temperature differing from the gas-kinetic temperature of the thermostat, defined in terms of the mean-square velocity. The behavior of a small amount of impurities (heavy monoatomic particles and harmonic oscillators) in a non-Maxwellian thermostat of a light gas was further investigated in [6–8]. Unlike the papers mentioned, the present one considers the behavior of a small amount of harmonic oscillator impurities in a thermostat with a Maxwellian velocity distribution and with a nonequilibrium (non-Boltzmann) distribution in vibrational energies.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1978.     

Investigation of the hypersonic viscous shock layer around blunt bodies in a nonuniform flow

Unseparated viscous gas flow past a body is numerically investigated within the framework of the theory of a thin viscous shock layer [13–15]. The equations of the hypersonic viscous shock layer with generalized Rankine-Hugoniot conditions at the shock wave are solved by a finite-difference method [16] over a broad interval of Reynolds numbers and values of the temperature factor and nonuniformity parameters. Calculation results characterizing the effect of free-stream nonuniformity on the velocity and temperature profiles across the shock layer, the friction and heat transfer coefficients and the shock wave standoff distance are presented. The unseparated flow conditions are investigated and the critical values of the nonuniformity parameter a_k [10] at which reverse-circulatory zones develop on the front of the body are obtained as a function of the Reynolds number. The calculations are compared with the asymptotic solutions [10, 12].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 154–159, May–June, 1987.     

Analyticity of the solution of a plane elastoplastic problem

Elastoplastic problems of the theory of ideal plasticity were studied numerically by the method of small parameter in the monographs [1–3], and strengthening elastoplastic problems were studied in the monograph [4], where a rather complete bibliography of studies in this direction is given. The conditions for the continuous dependence of the the stress-strain state characteristics on the boundary conditions and the material inhomogeneity are defined using the implicit function theorem in [5, 6]. In the present paper, in the framework of the perturbation method, we determine the stress state in a cylindrical tube with nearly circular boundaries under the action of pressure. Using the implicit function theorem, we examine the existence and uniqueness of the solution of the problem. We show that its solution can be obtained in the case of any analytic shape of the boundary and any number of material physical-mechanical characteristics depending on finitely many small parameters. The results can be generalized to more complicated models of media. The results obtained in this paper are compared with already known results.     

Effect of laser radiation on the vibrational distribution of CO2

In connection with progress in the field of CO₂ lasers, questions of the vibrational kinetics of molecules of CO₂ have been discussed in many communications. In a majority of cases of practical importance, the distribution of CO₂ is due to processes of vibrational exchange (V-V) on which is based the well-known thermodynamic model [1]. In other cases, the V-V exchange does not determine the vibrational distribution, since the perturbation is small; therefore, it is found sufficient to consider a small number of levels of CO₂ (usually three), whose populations satisfy the linear equations of the balance [2]. There is the possibility of conditions where the vibrations are strongly excited and, at the same time, V-V processes are insignificant (a very small CO₂ impurity in the inert gas, with a high degree of ionization). Then the number of equations becomes large. The present article discusses one such case: the excitation of a steady-state vibrational distribution in a glow discharge by laser radiation, whose solution is rather graphic.     

Stability of a nonuniformly heated fluid in a porous horizontal layer

We investigate the stability of a nonuniformly heated fluid in the gravitational field in a plane horizontal porous layer through which vertical forced motion is effected. A similar system was studied in [1, 2]. In the present paper, the nonuniformity of the permeability of the porous layer with respect to the depth and the dependence of the viscosity of the saturating fluid on the temperature are taken into account in addition. As a result of the application of the linear stability theory, an eigenvalue problem arises, which is solved numerically. A family of curves representing the dependence of the critical modified Rayleigh number Ra _k ^⋆ on the injection parameter (the Péclet number Pe) for different degrees of inhomogeneity of the permeability and the viscosity is obtained. It is found that although Pe=0 corresponds to Ra _k ^⋆ for uniform permeability and viscosity and the stability increases monotonically as Pe increases, the presence of nonuniformity of the permeability or the viscosity leads to the appearance of a stability minimum in the region Pe≈1, while under the simultaneous influence of these two factors, the minimum is shifted into the region Pe≈2. The results of the paper can be used, for example, in the investigation of heat transfer in the case of forced fluid motion in the fissures of a permeable rock mass, when, in the case of pumping through a horizontal fissure, the fluid penetrates vertically across its permeable walls into the stratum. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 3–7, November–December, 1986.     

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.     

Rotational inviscid flow with circulation zones in a channel of variable cross section

The development of the theory of rotational motion of inviscid fluids for the purposes of describing channel flow encounters certain difficulties in connection with the appearance of viscosity effects near the walls. In the potential-rotational model [1], in which the vorticity is nonzero only in a closed circulation zone surrounded by potential flow, it is assumed that the separation and attachment points are known in advance. For example, for flow around a cavity these points coincide with the extreme corner points of the contour. The problem of determining the vorticity in a closed zone for the potential-rotational model has been investigated in a number of studies [2, 3], etc. In the case of an incompressible fluid the vorticity in the circulation zone is constant for two-dimensional flow and proportional to the distance from the axis for axisymmetric flow. The value of the constant is found from the steady-state condition for the adjoining viscous layers. If the channel walls have a smooth profile without corner points, then for determining the boundaries of the circulation zones additional conditions must be used. This study employs another scheme, in which the vorticity is formed outside the region of flow and in a particular problem is specified in the form of a boundary condition. An analytic solution describing the rotational flow of an inviscid fluid in a channel with a slightly varying cross section is obtained. Three types of entrance flow nonuniformity are considered: 1) uniform shear flow, 2) wake-type flow, and 3) potential flow with a narrow wall boundary layer. Streamline patterns with circulation zones are constructed for flows in diffuser channels with the above-mentioned types of entrance nonuniformity. A model of flow separation in a channel with a turbulent boundary layer on the walls is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 31–37, March–April, 1985.In conclusion the author wishes to thank E. Yu. Shal'man, A. N. Kraiko, and A. B. Vatazhin for useful discussions and advice.     

Application of Special Nonsmooth Temporal Transformations to Linear and Nonlinear Systems under Discontinuous and Impulsive Excitation

Linear and nonlinear mechanical systems under periodic impulsive excitation are considered. Solutions of the differential equations of motion are represented in a special form which contains a standard pair of nonsmooth periodic functions and possesses a convenient structure. This form is also suitable in the case of excitation with a periodic series of discontinuities of the first kind (a stepwise excitation). The transformations are illustrated in a series of examples. An explicit form of analytical solutions has been obtained for periodic regimes. In the case of parametric impulsive excitation, it is shown that a nonequidistant distribution of the impulses with dipole-like temporal shifts may significantly effect the qualitative characteristics of the response. For example, the sequence of instability zones loses its different subsequences depending on the parameter of the shifts. It is shown that the method's applicability can be extended for nonperiodic regimes by involving the idea of averaging.     

Self-Switching of Waves in Materials

The problem is formulated on self-switching of acoustic waves in materials when the wavelength does not exceed considerably the representative dimension of the microstructure. Use is made of the microstructural theory of two-component mixtures, which is adapted well to study waves in composites. Truncated and evolutionary equations describing the interaction of two longitudinal plane waves — a strong pumping wave and a weak signal wave — are derived. A procedure of finding the exact solution to the evolutionary equations is described for the case where the wave numbers of the pumping and signal waves are equal. The solution is expressed in terms of the elliptic Jacobi function     

Nonisothermal viscoplastic fluid flow through porous media

The Alishaev model [1] is extended to the case of nonisothermal flow. Neglecting conductive heat transfer, it is shown that for the model in question in the plane of the complex potential not only are the problems linear but the decoupling of the thermal and hydrodynamic problems is also allowed. The latter is reduced to a mixed problem for an analytic function. This makes it possible to use the wellknown methods and results of the theory of limiting equilibrium pillars for isothermal flow [2–5]. It is also established that the solutions of the unsteady problems tend asymptotically to the solutions of the corresponding steady-state problems and can be obtained from the latter by simpler conversion. The effectiveness of the approach proposed is illustrated with reference to the problem of a source-sink system [1–4].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 117–122, July–August, 1990.