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.     

A model kinetic equation for a gas with rotational degrees of freedom

A model kinetic equation for a gas with rotational degrees of freedom is obtained. By averaging of the distribution function over quantities corresponding to the rotational degrees of freedom this equation is reduced to a closed system of two kinetic equations, each of which is analogous to the kinetic equation of a monatomic gas.     

Wave interaction in a nonequilibrium gas flow

By employing the method of multiple time scales, we derive here the transport equations for the primary amplitudes of resonantly interacting high-frequency waves propagating into a non-equilibrium gas flow. Evolutionary behavior of non-resonant wave modes culminating into shocks or no shocks, together with their asymptotic decay behavior, is studied. Effects of non-linearity, which are noticeable over times of order O(ε^-1), are examined, and the model evolution equations for resonantly interacting multi-wave modes are derived.     

Structure of a gasdynamic disturbance in a thermodynamically nonequilibrium medium with a power-law relaxation model

The evolution and the steady-state structure of gasdynamic disturbances in a thermodynamically nonequilibrium gas are investigated both analytically and numerically. It is shown that in a medium with negative viscosity steady-state structures different from those in an equilibrium medium can exist. The conditions of existence of stationary shock waves with a discontinuous front and a smooth increase or decrease in the amplitude behind the front, waves with an oscillatory structure, and a stationary self-wave pulse with a power-law trailing front are determined.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 181–191. Original Russian Text Copyright © 2004 by Makaryan and Molevich.     

Approximating kinetic equation for weakly nonequilibrium states of polyatomic gases

The aim of the present paper is to construct an approximate kinetic equation that, first, takes into account correctly the possibility of excitation of both rotational and vibrational degrees of freedom of the molecules and, second, is valid for any law of intermolecular interaction.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 183–187, March–April, 1982.We thank M, Ya. Alievskio for helpful consultations.     

Construction of solutions of the Boltzmann kinetic equation in a Knudsen layer

We consider the moment equation method for solving the Boltzmann equation in a Knudsen layer; the calculation of one of the moments of the collision integral is presented.     

Basic kinetic equation of a rarefied gas

Starting from the Liouville equation, the basic kinetic equation of a rarefied gas is derived for both spatially homogeneous and spatially nonhomogeneous systems. The relation between the equation obtained and the Boltzmann equation is investigated, together with the nature of the dependence of the solutions of the basic kinetic equation on the number of particles in the system.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 154–160, November–December, 1989.The author is grateful to M. S. Ivanov for numerous stimulating discussions and to D. N. Zubarev, E. G. Kolesnichenko, and V. E. Yanitskii for their help in assessing the results.     

New results on the semidiscrete Boltzmann equation for a binary gas mixture

Summary A plane semidiscrete model of the Boltzmann equation for a binary gas mixture with molecular collisions ruled by the hard-spheres interaction potential is described. After establishing a model, a theorem demostrating the global existence of mild solutions of the initial-value problem is given and the propagation of unidimensional shock waves examined.

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Sommario Si propone un modello semidiscreto piano dell'equazione di Boltzmann per una miscela binaria con collisioni molecolari soggette al potenziale di interazione delle sfere rigide. Costruito il modello, si dà un teorema di esistenza globale di soluzioni generalizzate per il problema di Cauchy, e si analizza la propagazione di onde d'urto unidimensionali.

     

Analytical solution of the poiseuille flow problem using the ellipsoidal-statistical model of the Boltzmann kinetic equation

An analytical solution (in the form of a Neumann series) of the problem of rarefied gas flow in a plane channel with infinite walls in the presence of a pressure gradient (Poiseuille flow) parallel to them is constructed within the framework of the kinetic approach in an isothermal approximation. The ellipsoidal-statistical model of the Boltzmann kinetic equation and the diffuse reflection model are used as the basic equation and the boundary condition, respectively. Using the resulting distribution function, the mass and heat flux densities in the direction of the pressure gradient per unit channel length in the y′ direction are calculated, and profiles of the gas mass velocity and heat flux in the channel are constructed. The results obtained for the continuum and free-molecular flow models are analyzed and compared with similar results obtained by numerical methods.     

Shock wave solution of the Boltzmann kinetic equation in a 13-moment approximation

We consider the classic problem of a one-dimensional steady shock-wave solution of the Boltzmann kinetic equation utilizing a new type of 13-moment approximation proposed by Oguchi (1997). The model, unlike previous ones, expresses the collision term in an explicit function of the molecular velocity. This enables us to examine directly the nature of the singularity of the distribution function to this particular problem caused by the vanishing molecular velocity. We can thus obtain moment integrals directly because of its explicit expression. The principal value is utilized for the moment integral to cope with the singularity, and we can have five relations for five unknown functions to be determined with respect to the coordinate x. These relations can be reduced to a first-order differential equation that is solved to provide the familiar smooth monotonic transition from the upstream supersonic state to the subsonic downstream state. Computed values of shock thickness for various shock Mach numbers agree well with existing results obtained by different methods to the certain Mach number beyond which no solution exists.Received: 17 May 2002, Accepted: 1 May 2003, Published online: 15 August 2003PACS: 51.10. + y     

Study of a new semi-continuous model of the Boltzmann equation

For a semi-continuous model of the Boltzmann equation (1) peculiar solutions are obtained and generally the global existence of solutions of the initial value problem is discussed. The global existence is possible even in some cases for partially negative initial number densities, which are not physical problems, but mathematical ones. It can be shown that in some cases the entropy begins to increase, reaches a maximum and decreases again.     

Boundary-layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation

In this paper, we study the fluid-dynamic limit for the one-dimensional Broadwell model of the nonlinear Boltzmann equation in the presence of boundaries. We consider an analogue of Maxwell's diffusive and reflective boundary conditions. The boundary layers can be classified as either compressive or expansive in terms of the associated characteristic fields. We show that both expansive and compressive boundary layers (before detachment) are nonlinearly stable and that the layer effects are localized so that the fluid dynamic approximation is valid away from the boundary. We also show that the same conclusion holds for short time without the structural conditions on the boundary layers. A rigorous estimate for the distance between the kinetic solution and the fluid-dynamic solution in terms of the mean-free path in theL -norm is obtained provided that the interior fluid flow is smooth. The rate of convergence is optimal.     

Boundary value problems for a model Boltzmann equation with frequency proportional to the molecule velocity

Exact solutions of a model Boltzmann equation with a collision frequency that depends on the molecule velocity and with a BGK (Bhatnagar-Gross-Krook) collision operator are constructed for the problems of weak evaporation and temperature jump in a rarefied vapor above a plane surface. The numerical calculations and a comparison with previous results are given.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 140–153, May–June, 1996.     

Averaging the Boltzmann kinetics equation with respect to transverse velocity for one-dimensional gas flows

By averaging the Boltzmann kinetics equation with respect to the transverse velocities we obtain a system of two integrodifferential equations for two unknown functions that depend on the longitudinal velocity u, time t, and the x coordinate.It is assumed that the particles interact with one another like perfectly elastic spheres. The integrals appearing in the equations are double integrals. The reduction of the number of variables, with the unknown functions and the low multiplicity of the integrals make possible a computer solution of the one-dimensional problems in both the steady and unsteady cases.As an example, the resulting equations are solved numerically for the problem of shock wave structure.     

Analytic solution of the ellipsoidal-statistical model Boltzmann equation

An exact solution of the ellipsoidal-statistical model Boltzmann equation is constructed. The problem of the temperature jump in a rarefied gas is considered by way of illustration. By expanding the distribution function in two orthogonal directions the problem is reduced to the solution of a vector transport equation with polynomial boundary conditions. The Case approach reduces the equation to a characteristic equation for which generalized eigenvectors and eigenvalues are found. A theorem of existence and uniqueness of the solution, represented in the form of an expansion in eigenvectors, is proved. The proof reduces to solving a Riemann-Hilbert vector boundary-value problem with a matrix coefficient whose diagonalizing matrix has branch points in the complex plane. The value of the temperature jump is found from the conditions of solvability of the boundary-value problem.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.2, pp. 151–164, March–April, 1992.     

High-order hybridisable discontinuous Galerkin method for the gas kinetic equation

ABSTRACT

The high-order hybridisable discontinuous Galerkin (HDG) method is used to find steady-state solution of gas kinetic equations on two-dimensional geometry. The velocity distribution function and its traces are approximated in piecewise polynomial space on triangular mesh and mesh skeleton, respectively. By employing a numerical flux derived from the upwind scheme and imposing its continuity on mesh skeleton, the global system for unknown traces is obtained with fewer coupled degrees of freedom, compared to the original DG method. The solutions of model equation for the Poiseuille flow through square channel show the higher order solver is faster than the lower order one. Moreover, the HDG scheme is more efficient than the original DG method when the degree of approximating polynomial is larger than 2. Finally, the developed scheme is extended to solve the Boltzmann equation with full collision operator, which can produce accurate results for shear-driven and thermally induced flows.