Soliton solutions of weakly bound states for higher-order Ito equation

Nonlinear Dynamics - By utilizing an ingenious limit method proposed in this paper, the soliton solutions of weakly bound state including the multiple-pole solutions and the degenerate solution of...     

Wall jumps in temperature,partial pressures and populations of multicomponent mixtures of nonequilibrium polyatomic gases

The Maxwell-Loyalka method is used to derive formulas for the jumps in the macroparameters of a multicomponent nonequilibrium gas mixture. It is assumed that within the kinetic (Knudsen) wall layer one group of internal states of the molecules is close to equilibrium at the translational temperature, while another group and homogeneous chemical reactions are excited fairly slowly, so that they may be considered to be frozen and the corresponding inelastic collision integrals in the kinetic equations can be neglected. However, at the wall different groups of internal states of the molecules may be excited and chemical reactions may take place. The final calculation formulas are obtained under a series of simplifying assumptions in accordance with the recommendations made in [1, 2].Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 133–141, March–April, 1993.The author is grateful to N. K. Makashev for important comments.     

Transport properties of nonequilibrium polyatomic gas mixtures

It is shown that the well-known Hirschfelder-Euken correction to the thermal conductivity of a polyatomic gas mixture given by the first approximation in Sonine polynomials can be less than the corresponding exact value (for a Lorentz mixture of light and heavy molecules interacting in accordance with Coulomb's law) by a factor of 3.4. Fairly high accuracy is achieved in the second approximation in Sonine polynomials. Within the framework of the latter, similar corrections to the nonequilibrium heat and diffusion fluxes are found. On the basis of the generalized Chapman-Enskog method a more general case is studied. In this case some of the nonelastic collision integrals is also taken into account in calculating the transport coefficients. The transport coefficients are either represented in terms of the well-known formulas for fast and retarded internal molecular energy exchange or convenient approximate expressions are obtained.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 183–189, March–April, 1995.     

Nonlinear nonequilibrium kinetic model of the Boltzmann equation for a gas with power-law interaction

A model kinetic equation approximating the Boltzmann equation on a wide range of the intensities of nonequilibrium states of gases is derived to describe rarefied gas flows. The kinetic model is based on a distribution function dependent on the absolute velocity of gas particles. Themodel kinetic equation possesses a high computational efficiency and the problem of shock wave structure is solved on its basis. The calculated and experimental data for argon are compared.     

On the ellipsoidal statistical model for polyatomic gases

The aim of this article is to construct a BGK-type model for polyatomic gases which gives in the hydrodynamic limit the proper transport coefficient. Its construction relies upon a systematic procedure: minimizing Boltzmann entropy under suitable moments constraints (Levermore in J Stat Phys 83:1021–1065, 1996; Brull and Schneider in Cont Mech Thermodyn 20(2):63–74, 2008). The obtained model corresponds to the ellipsoidal statistical model introduced in Andries et al. (Eur J Mech B Fluids 19:813–830, 2000). We also study the return to equilibrium of its solutions in the homogeneous case.      

Kinetic equations for vibrational relaxation in a mixture of polyatomic gases

Kinetic equations are derived for the relaxation of the vibrational energy in a mixture of polyatomic gases, which are ones with molecules simulated by harmonic oscillators. The most general case is envisaged, where the energy relaxation occurs not only via vibrational-translational transitions but also via multiquantum vibrational exchange involving an arbitrary number of vibrational modes. The analysis also incorporates the possible degeneracy of each mode when the molecules colliding are the same. An expression is derived that extends previous results [1–6] and that relates the vibrational temperatures in the case of quasiequilibrium. Equations are derived for the vibrational relaxation for the CO₂-N₂ case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 6, pp. 29–37, November–December, 1972.We are indebted to L. A. Shelepin for valuable discussions on the results.     

Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics

We study the dispersion relation for sound in rarefied polyatomic gases (hydrogen, deuterium and hydrogen deuteride gases) basing on the recently developed theory of extended thermodynamics (ET) of dense gases. We compare the relation with those obtained in experiments and by the classical Navier–Stokes Fourier (NSF) theory. The applicable frequency range of the ET theory is proved to be much wider than that of the NSF theory. We evaluate the values of the bulk viscosity and the relaxation times involved in nonequilibrium processes. The relaxation time related to the dynamic pressure has a possibility to become much larger than the other relaxation times related to the shear stress and the heat flux.     

Approximate method for solving the kinetic equation for moderately dense gases near a boundary. Slip of a binary mixture

The problem of the slip of a binary gas mixture of moderate density along a flat surface is considered. A solution is obtained by a modification of Maxwell's method by means of the Enskog-Thorne kinetic equations for dense gases. The velocities of isothermal, thermal, and diffusion slip for rarefied and moderately dense binary mixtures are compared.     

Methods of calculating thermal conductivity of binary mixtures involving polyatomic gases

Summary The experimental binary thermal conductivity data of nineteen different gas pairs have been discussed and the competence of the rigorous, approximate and empirical procedures to represent them is investigated and discussed. In addition a new semi-theoretical method is suggested and tested. The suggested semi-theoretical procedure works very satisfactorily and is of good accuracy. It also compares favourably with the other methods. We also suggest a procedure for estimating thermal conductivity values at high temperature. This is an interesting and useful suggestion in view of the great practical need and their meagre availability.     

On the energy transfer equation for weakly interacting waves

The derivation of the transfer equation based on analysis of the equations for spectral semi-invariant and not invoking equations for realization of the random wave field is presented. Uniformly valid asymptotic expansions for the third and the fourth spectral semi-invariant are constructed using the multiple scale method and the matched asymptotic expansion method. This approach makes it possible to investigate the boundary layer in a neighbourhood of the resonant surface where intensive growth in time of the third spectral semi-invariant occurs. This boundary layer defines the form of the transfer equations. An analogous boundary layer for the fourth spectral semiinvariant and its influence on the second and the third spectral semi-invariants are also investigated.     

On the kinetic theory of relativistic gases

Based on the relativistic Boltzmann equation and on the combined method of Chapman-Enskog and Grad, the laws of Navier-Stokes and Fourier are obtained for a single relativistic gas. Besides, it is shown that there exits only one potential function that corresponds to the relativistic Maxwellian potential function. Received: October 30, 1996     

Computer solution of a kinetic equation for electrons

Physical and mathematical approaches are presented for the behavior of a weakly ionized plasma in a thermoelectronic converter. Numerical solutions are obtained by computer methods. The distribution function for the electrons is examined in series form for a Boltzmann kinetic equation subject to boundary conditions; the coefficients of the series are deduced via moment equations. The electric field is incorporated in the quasineutrality approximation. An exampIe envisaging only electron-atom collisions is presented. Consider two unbounded planar electrodes (cathode and anode) heated to different temperatures, between which lies a weakly ionized plasma subject to a potential difference. From the electrodes flow ion and electron fluxes into the plasma, where ionization and recombination can occur. The quantities to be determined are the current, the potential distribution, the temperature, and the charge density. This problem occurs for a cesium converter in the arc mode. If the volume ionization can be neglected, the processes are ctosely described by the diffusion theory [1], but it is desirable to have more detailed information about the distribution function for the electrons when ionization, excitation, and recombination become important. The diffusion theory is then replaced by a Boltzmann kinetic equation, but this greatly increases the computational difficulties. The present approach envisages the use of computers.The method of solution is basically as follows. The electron-distribution function in the kinetic equation is replaced by a series in some complete set of functions of the velocity coordinates. There is a second system of independent functions; these are mukiplied by the two parts of the kinetic equation, whereupon integration over velocity space gives differential equations of first order in the spatial coordinates for the parameters of the series for the distribution function. These are balance equations or equations for the moments with respect to the above system of independent functions (usuaIly these are polynomials in the velocity coordinates).We select from this system a subsystem of functions, which we multiply by the boundary conditions for the kinetic equation and integrate over the region where they are given (i.e., with respect to the velocity of the electrons leaving the electrode). This gives the boundary conditions for the differential equations for the moments. Varieties of this method are to be seen in Grad's [2] and Weitzsch's [3] methods in gas dynamics, or the method of spherical harmonics [4, 8] in neutron physics; see [6] for review. The method of expansion used here differs from Grad's method in that I use functions of the energy and spherical angles in velocity space, whereas Grad used functions of the cartesian coordinates of the velocity. Moreover, the zero-th-approximation function is taken as the isotropic exp(--mu a/2kT) instead of Grad's anisotropic exp[--m(v-v0)/2kT] (m is electron mass, T is temperature, k is Boltzmann's constant, v is particle velocity, and v0 is the mean particle velocity). These differences are introduced for the following reasons. The electrons in a weakly ionized plasma collide frequently with neutral atoms, so there is more rapid relaxation in momentum than in energy [7], and the distribution function differs little from isotropic. On the other hand, a principal purpose here is to examine the inelastic processes of ionization and excitation, and the major feature is the energy distribution of the electrons without reference to the orientation of the momentum vector. Hence we need take only the first two terms in the expansion with respect to the spherical coordinate ~=vx/U (the Pt approximation in the method of spherical harmonics).We also take account of the electric field set up by the space charge.Let d be the distance between cathode and anode, V be the potential differences, r the Oebye radius, n+ and n. the concentrations of ions and electrons, and q the charge on an electron. As in [1], we consider the case where the main change in the electrical potential U occurs near the electrodes in regions of scale r, while the rest of the region obeys the quasineutrality condition n₊ - n_ n₊ (Fig. 1). The size of the space-charge regions is less than the mean free path for any of the bulk processes, so no scattering occurs in these regions, while their presence is allowed for by the additional potential barriers U t (cathode) and U 2-V (anode), Both physical conditions are obeyed for r sufficiently small.We also assume that the potential changes monotonically in the space-charge regions, as in Fig. 1, where 1 is the cathode, 2 is the anode, a are the space-charge regions, and 4 is the quasineutral plasma.I am indebted to G. E. Pikus for direction and assistance, and to L. A. Oganesyan for assistance in programming the problem for the computer,     

The inertial fractal set for weakly damped forced korteweg-de-vries equation

ＴＨＥＩＮＥＲＴＩＡＬＦＲＡＣＴＡＬＳＥＴＦＯＲＷＥＡＫＬＹＤＡＭＰＥＤＦＯＲＣＥＤＫＯＲＴＥＷＥＧ－ＤＥ－ＶＲＩＥＳＥＱＵＡＴＩＯＮＤａｉＺｈｅｎｇ－ｄｅ（戴正德）（ＩｎｓｔｉｌｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ,ＹｕｎｎａｎＰｒｏｖｉ...     

Laser diagnostics of molecular states in nonequilibrium gas flows

Laser diagnostics techniques are developed, and some typical results are presented on the vibrational CO₂ molecule level population measurements under highly nonequilibrium conditions for two types of flows: (i) in a conventional gasdynamic lasers and (ii) in a flow mixing gasdynamic laser. The measuring procedure is based on recording spectral gain coefficient distributions at several resolved rotational transitions of different vibrational bands. The laser optical system with spatial selection of single lines is described. The system allows fast (about 10^–5–10^–4 s) line tuning, thus providing a variable and arbitrary choice of the selected line sequences. The optimum choice of the laser generation spectrum is discussed, and the sources of measuring uncertainties are analyzed. Typical illustrations and results are given and discussed.     

Light scattering and sound propagation in polyatomic gases with classical degrees of freedom

In this work we analyze time-dependent problems like sound propagation and light scattering in dilute polyatomic gases with classical internal degrees of freedom by using a kinetic model of the Boltzmann equation that replaces the collision operator with a single relaxation-time term. Comparison of the theoretical results with available experimental data shows that the model equation can be used to describe the acoustic properties and the light scattering spectrum of dilute polyatomic gases as long as the external oscillation frequency is smaller than the frequency required for the translational and the internal degrees of freedom to come to thermal equilibrium.Received: 12 January 2004, Accepted: 2 February 2004, Published online: 16 April 2004PACS: 51.10. + y; 51.40. + p     

Exact solution of the kinetic equation in the problem of nonisothermal flow in the neighborhood of a weakly curved surface

An analytic method for solving the half-space boundary value problem for the inhomogeneous Boltzmann equation with the collision operator in the form of an elliptico-statistical model (the ES-model of the Boltzmann equation) is proposed for the problem of nonisothermal rarefied gas flow in the neighborhood of a curved surface. An exact analytic expression is derived for the thermal slip of a monatomic gas along the surface of a rigid spherical aerosol particle. A numerical value of the gas-kinetic coefficient which takes into account the effect of the curvature of the surface on the thermal slip coefficient is obtained. A comparison with published data is carried out. Moscow, Arkhangelsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 165–173, March–April, 1998.     

Simulation method in the kinetic theory of gases

We develop a method that is based on use of the simulation equation vf/v=vl ^–1(f ₀–f), where v is the modulus of the molecular velocity andl is the mean free-path length. A number of general properties of the model are clarified and the transition to the limit of a continuous medium is discussed in detail.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 110–115, November–December, 1973.     

On the development of a kinetic model for a gas with allowance for nonequilibrium processes

The problems of developing a kinetic model of a medium (gas and plasma) are considered from the viewpoint of choice of the most important physicochemical processes. For the problem of a direct shock wave propagating in the atmosphere, kinetic models are selected with allowance for the error in specifying reaction-rate constants. The investigation was performed using an automated system that incorporates structured bases of physicochemical data, a generator of kinetic equations, a complex of programs for direct calculation, and program modules for determining, from a set of admissible solutions, the one satisfying specified criteria. Institute of Mechanics, Moscow State University, Moscow 119899. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 36–43, July–August, 1999.