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.     

Degeneracy of Chapman-Enskog series for the transport properties in the case of slow steady flows of weakly rarefied gases

The structure of the Chapman-Enskog solution of the Boltzmann equation linearized with respect to the absolute Maxwell equilibrium is studied. Under the assumption of uniqueness and existence of a solution it is shown that in the steady case the series describing the transport phenomena consist of a finite number of terms, and the heat fluxes and diffusion rates are given by the Burnett approximation and the stresses by the super-Burnett approximation, the following terms of the series vanishing. At the same time, the gas-dynamic variables in all approximations in the small Knudsen number K satisfy the conservation equations in the Stokes approximation; the forces and moments acting on bodies placed in a mixture of gases can be calculated from the Navier-Stokes stresses without allowance for their reprocessing in Knudsen layers. A problem is formulated for a simple gas, and the transport properties are analyzed by using the invariance properties of the linearized Boltzmann equation and by means of the algorithm of the Chapman-Enskog method, and then the results are generalized to a mixture of gases, and the question of the forces and moments is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 157–163, July–August, 1988.I thank M. N. Gaidukov and O. G. Fridlender for fruitful discussions.     

Transport coefficients of a dissociating gas

The calculation of the transport coefficients of a dissociating gas involves fundamental difficulties which arise when the internal degrees of freedom of the molecules are taken strictly into account. In practical calculations extensive use is made of the approximation proposed in [1], in the context of which the dependence of the diffusion velocity of the molecule on its internal state is totally neglected. In this case the expressions for the stress tensor and the diffusion velocities coincide with the corresponding expressions for a mixture of structureless particles; in the expression for the heat flux the diffusion transport of internal energy is taken only approximately into account. Here, analytic expressions for the diffusion velocities, heat flux and stress tensor are obtained without introducing simplifying assumptions. The calculation method is based on the results of [2], in which an approximate method of calculating the transport coefficients of a multicomponent mixture of structureless particles was proposed, and [3], in which the transport coefficients of a rotationally excited gas were calculated. The relations obtained are analyzed and compared with the existing results; their accuracy is estimated. A closed system of equations of gas dynamics is presented for a number of cases of practical importance.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 158–165, January–February, 1987.     

On the boundary conditions for the equations of gas dynamics corresponding to the higher approximations in the Chapman-Enskog method of solution of the Boltzmann equation

When the gas-dynamic equations are derived from the solution of the Boltzmann equation by the Chapman-Enskog method, the order of the system of partial differential equations tends to increase with increasing number of the approximation. As a result, it is necessary to have more and more boundary conditions for these equations, which, however, are at present definitely known only for the first two approximations (models of an ideal gas and a viscous gas). A method is proposed for constructing additional boundary conditions; the method is illustrated in a number of examples.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 77–87, May–June, 1979.I should like to thank M. N. Kogan, V. S. Galkin, O. G. Fridlender, and S. A. Regirer for their interest in the work and for valuable discussions.     

Reduced numerical modeling of flows involving liquid-crystalline polymers

Kinetic theory models involving the Fokker–Planck equation can be accurately discretized using a mesh support (Finite Elements, Finite Differences, Finite Volumes, Spectral Techniques, …). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function has only local support and is related to a node that defines the associated degree of freedom. In the technique proposed here, a reduced approximation basis is constructed. The new shape functions have extended support and are defined in the whole domain in an appropriate manner (the most characteristic functions related to the model solution). Thus, the number of degrees of freedom involved in the solution of the Fokker–Planck equation is very significantly reduced. The construction of those new approximation functions is done with an ‘a priori’ approach, which combines a basis reduction (using the Karhunen–Loève decomposition) with a basis enrichment based on the use of some Krylov subspaces. This paper analyzes the application of model reduction to the simulation of non-linear kinetic theory models involving complex behaviors, such as those coming from stability analysis, complex geometries and coupled models. We apply our model reduction approach to the Doi's classical constitutive equation for viscoelasticity of liquid-crystalline polymer.     

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

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

Hypersonic viscous shock layer on sweptback wings of infinite span at different angles of attack

A study is made of the flow of a viscous compressible gas in a hypersonic shock layer on sweptback wings of infinite span with blunt leading edge at different angles of attack. The equations of the hypersonic viscous shock layer with modified Rankine-Hugoniot relations across the shock wave and boundary conditions on the surface of the body that take into account slip and discontinuity of the temperature are solved by a method of successive approximation which yields not only an analytic solution for the first approximations but also an exact numerical solution when the method is implemented on a computer. The analytic solution of the problem is found in the first approximation. Expressions are obtained for the coefficients of friction and heat transfer on the surface of the body, and also for the profiles of the velocities and the temperature across the shock layer. Comparison of the analytic solution with the numerical solution reveals a satisfactory accuracy of the analytic solution for not too large Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 91–102, March–April, 1979.We thank G. A. Tirskii for his interest in the work and valuable discussions.     

Systematization of relaxation gas dynamics equations. A review

The phenomenological theory of relaxation gas dynamics equations is outlined for laminar flows of multicomponent reacting gases in an approximation analogous to the Navier-Stokes approximation. A system of general equations of relaxation gas dynamics including the level kinetics equations for all excited internal degrees of freedom is formulated on the basis of notions of continuum mechanics. A procedure of going over to particular cases characterized by certain relations between the relaxation times is described and examples of the corresponding closed systems of gas dynamics equations including systems containing the balance equations of the level or mode approximations for the vibrational energy levels of molecules of a gas mixture are given. A method of constructing a database of the models of the rate constants of physicochemical processes as coefficients in the source terms of the balance equations is considered.     

Dynamics of an elliptic vortex

The two-dimensional flow of an ideal incompressible fluid is considered. Within an ellipse, the flow has a constant vorticity, while outside the ellipse there is an irrotational flow with circulation. Such motion can be described by Lagrange equations for a dynamical system with an infinite number of degrees of freedom. For a system with four degrees of freedom a new steady solution is obtained and its stability investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 55–60, July–August, 1983.We thank L. I. Sedov for his interest in the work.     

Nonlinear surface waves in media with complex rheology

Weak nonlinear waves in a generalized viscoelastic medium with internal oscillators are considered. The rheological relations contain higher time derivatives of the stresses and strains as well as their tensor products. The method of expansion in a small parameter with the introduction of slow time and a running space coordinate is employed. The first approximation gives wave velocities and relations between the parameters equivalent to the results of an acoustic analysis at elastic wave fronts [1]. The second approximation leads to an evolution equation for the displacement velocity. For this a Fourier-Laplace double integral transformation is used. Reversion to the inverse transforms of the unknown functions leads to an integrodifferential evolution equation, which contains a Hubert transform and is a generalization of the Benjamin-Ono equation of deep water theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 95–103, September–October, 1990.     

Stefan-Maxwell Relations in Nonequilibrium Polyatomic Reacting Gas Mixtures

The Stefan-Maxwell relations in nonequilibrium (with respect to the internal degrees of freedom) flows of multicomponent gas mixtures are obtained in an arbitrary approximation in Sonine polynomials.     

Estimation of the effect of vibrational excitation of diatomic molecules on their diffusional transport and dissociation in a boundary layer

Relations for the diffusion fluxes of vibrationally highly excited diatomic molecules are found by means of the Chapman-Enskog method. On the basis of these relations a quantitative estimate of the changes in the diffusion coefficients under vibrational excitation and the corresponding changes in the macroscopic dissociation rate is obtained under conditions of disequilibrium of the upper vibrational levels of the dissociating molecules caused by the diffusion processes. The diffusion relations obtained are used in deriving the boundary conditions for the equations of level vibrational kinetics. A simplified version of this derivation is presented for noncatalytic and catalytic surfaces.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 169–182, January–February, 1996.     

Structure of the equations of hydrodynamics for a medium consisting of inelastic rough spheres

It is proposed to determine the structure of the equations of hydrodynamics of a medium consisting of identical inelastic rough spheres corresponding to the solution of a kinetic equation of the Enskog type by the Chapman-Enskog method over the entire range of variation of the parameters characterizing the inelasticity and roughness of the particles for which a solution exists. The equations necessary for calculating the coefficients entering into the equations of hydrodynamics are constructed. It is shown that the Euler equations for a gas of inelastic spheres contain source terms which must also be taken into account in the conditions at the shock.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 166–175, March–April, 1990.     

Influence of the macrotransport processes in gases on the change in the hydrodynamic parameters in the presence of a resonance radiation pulse

The effect of a resonance radiation pulse on the gas parameters is investigated. It is assumed that the pulse duration is large enough for the diffusion and heat conduction to be taken into account. It is shown that the diffusion and heat conduction may be significantly influenced by the processes of excitation of the internal degrees of freedom of the molecules.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 120–128, July–August, 1992.     

Determination of optimum nozzle parameters in a gasdynamic laser

In connection with the use of supersonic nozzles to create lasers, the question arises of the optimum parameters of the nozzle and the gas mixture from the aspect of obtaining the greatest population inversion of the energy levels of internal degrees of freedom of molecules of the working gas and the greatest output power of the lasers. A rather complete concept of the kinetic processes taking place during the escape of a relaxing gas mixture containing carbon dioxide through a supersonic nozzle has now been developed on the basis of calculated and experimental data. In [1–4] the problems of optimization of the parameters of a CO₂-N₂-H₂O-He mixture and of the shape of the nozzle were set up and solved in a one-dimensional steady-state formulation. The influence of the two-dimensionality of the stream in an optimum nozzle on the laser characteristics is studied in the present report. The method of through calculation suggested in [5] is used to calculate the two-dimensional flow of a relaxing gas.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 23–26, September–October, 1978.     

Theory of the relaxation pressure in a dissociating gas

The influence of a slow dissociation reaction (the reaction rate is much lower than the rate of energy exchange between the translational and internal degrees of freedom of the gas particles) on the transfer coefficients is studied using the Chapman—Enskog method. It is shown that changes occur only in the diagonal part of the pressure tensor, where an additional coefficient, the so-called relaxation pressure, appears.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 101–107, March–April, 1978.The author thanks A. S. Pleshanov for useful discussions and valuable comments.     

Evolution of quasiplane weakly nonlinear internal waves in a viscous fluid

The method of multiscale asymptotic expansions is used to derive a model equation describing the evolution of internal waves in a viscous stratified fluid with allowance for nonlinearity, dispersion, and diffraction in the diffusion approximation. The approximate analytic solution of the obtained equation in the case of weak nonlinearity is analyzed. The possibility of using the Boussinesq approximation is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 158–162, October–December, 1981.     

Diagnostics of rarefied gases using hot-wire anemometer

Hot-wire anemometry has been quite widely used in recent years to study low-density flows [1–3]. High sensitivity and good spatial resolution of the method make its application effective in flows with large gradients. The technique permits the determination of two thermodynamic quantities, the coefficient of heat transfer h and recovery temperature T_r. Relations coupling h and T_r with flow parameters were obtained in [4]. These relations were generalized in [3] for the case of unequal accommodation of translational and internal energy of molecules on the wire surface and nonequilibrium energy distribution in internal degrees of freedom. In order to obtain a more detailed information on the distribution of gasdynamic parameters, the hot-wire anemometer is, as a rule, used in conjunction with other diagnostic tools like the Pitot tube or electron beam [2]. The present work deals with the application of a twin-wire (perpendicular and parallel to the flow direction) hot-wire anemometer in rarefied flows without the benefit of other techniques. Furthermore, the influence of differences in the accommodation coefficients of translational and internal energies and the degree of rarefaction on the hot-wire anemometer is also investigated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 126–130, September–October, 1986.In conclusion, the authors acknowledge A. K. Rebrov for useful discussions stimulating development of studies with hot-wire anemometry.     

Sound absorption in a shock wave

It was shown in [1–4] that the reflection of a sound wave or its transmission through a shock front should be accompanied by attenuation or intensification of the wave is regarded as a discontinuity. In accordance with current representations [5, 6], a shock wave includes a viscous shock and a lengthy relaxation zone. Equilibrium is established with respect to translational and rotational degrees of freedom in the viscous shock and with respect to internal degrees of freedom in the relaxation zone. The result of the interaction of the shock and sound waves is determined by the relationship between the length of the sound wave and the width of the shock wave.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 90–94, May–June, 1986.