Invariant submodels of the model of thermal motion of gas in a rarefied space

A model describing the thermal motion of a gas in a rarefied space is investigated. This model can be used in the study of the motion of gas in outer space, and the processes occurring inside the tornado, and the state of the medium behind the shock front of the wave after a very intense explosion. For a given initial pressure distribution, a special choice of mass Lagrange variables leads to a reduced system of differential equations describing this motion, in which the number of independent variables is one less than the original system. This means that there is a stratification of a highly rarefied gas with respect to pressure. Namely, in a strongly rarefied space for each given initial pressure distribution, at each instant of time all gas particles are localized on a two-dimensional surface moving in this space. At each point of this surface, the acceleration vector is collinear with its normal vector. The resulting system admits an infinite Lie transformation group. All significantly various submodels that are invariant with respect to the subgroups of its eight-parameter subgroup generated by the transfer, extension, rotation, and hyperbolic rotation operators (the Lorentz operator) are found. For invariant submodels of rank 1, the basic mechanical characteristics of the gas flow described by them are obtained. Conditions for the existence of these submodels are given. For invariant submodels of rank 2, integral equations describing these submodels are obtained. For some submodels, the problem of describing the gas flow from the initial location of its particles and the distribution of their velocities has been investigated.     

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.     

Temperature jumps at the boundary of a rarefied gas

Received October 19, 1999     

Heat transfer in a very rarefied gas

A one-dimensional problem of heat transfer in a rarefied gas is considered. It relates to the nonmonotonic variation of the heat flux between two plates when the temperature of one of them is reduced. Attention is drawn to a paradox that arises in this problem if the interaction of the molecules of the gas with the surface is described by means of accommodation coefficients.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1. pp. 195–198, January–February, 1980.     

Thermophoresis and photophoresis in a rarefied gas

The present article discusses the problem of the force acting on a spherical particle in a heated rarefied gas (thermophoresis) and the problem of the force acting on such a particle in an isothermal rarefied gas, heated by an external heat flux (photophoresis). Both problems are solved in a linear statement, i.e., under the assumption of the smallness of the temperature gradient of the gas and of the external heat flux, respectively. The rising interest in these problems is due to problems of atmospheric contamination, the physics of clouds, etc.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 178–182, September–October, 1976.     

Exact solutions of the system of the equations of thermal motion of gas in the rarefied space

We study the model describing thermal motion of gas in the rarefied space. This model can be used, in particular, in the study of the state of the medium behind the front of shock wave after very strong blast, in the study of the processes taking place inside of tornado, in the study of the motion of the gas in outer space. For any given initial distribution of the pressure a specific selection of mass Lagrange variables leads to reduction of the system of differential equations describing this motion to the system, for which the number of independent variables is less on the unit. For the obtained system we found all nontrivial conservation laws of the first order. In addition to the classical conservation laws the system has other conservation laws, which generalizes the energy conservation law. We obtained the exact solutions of this system. These solutions describe a variety of different physical processes taking place in the rarefied medium. Using the symmetry properties of the system we got the generating formulas for the receipt of the new solutions using already found earlier solutions of the system.     

Simulation of interaction between a rarefied gas jet and an obstacle by the methods of molecular dynamics

An efficient method of direct numerical simulation is proposed. The steady-state flow field generated by the impingement of a gas jet on a wall is studied as an example. The numerical results obtained during this study are in good agreement with the numerical solutions of other authors.     

Flow of rarefied gas past a sweptback cylinder

A numerical investigation has been made of the hypersonic flow of a rarefied monatomic gas past the windward part of the side surface of an infinite circular cylinder. The calculation was made by direct statistical Monte Carlo modeling for freestream Mach number M_t8=20, ratio of the surface temperature of the body to the stagnation temperature equal to t_tw =T _tw/T _t0 = 0.03, sweep angle 75°, and Reynolds number Re_t0 30.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 146–154, January–February, 1992.     

Asymptotic theory of the Boltzmann system,for a steady flow of a slightly rarefied gas with a finite Mach number: General theory

A steady rarefied gas flow with Mach number of the order of unity around a body or bodies is considered. The general behaviour of the gas for small Knudsen numbers is studied by asymptotic analysis of the boundary-value problem of the Boltzmann equation for a general domain. The effect of gas rarefaction (or Knudsen number) is expressed as a power series of the square root of the Knudsen number of the system. A series of fluid-dynamic type equations and their associated boundary conditions that determine the component functions of the expansion of the density, flow velocity, and temperature of the gas is obtained by the analysis. The equations up to the order of the square root of the Knudsen number do not contain non-Navier–Stokes stress and heat flow, which differs from the claim by Darrozes (in Rarefied Gas Dynamics, Academic Press, New York, 1969). The contributions up to this order, except in the Knudsen layer, are included in the system of the Navier–Stokes equations and the slip boundary conditions consisting of tangential velocity slip due to the shear of flow and temperature jump due to the temperature gradient normal to the boundary.     

Simulation of the rarefied gas flow around a perpendicular disk

The present work contributes to define the domain of validity of two continuum approaches, based on Navier–Stokes (NS) and quasi gas-dynamic (QGD) equations, respectively. Results obtained using each method are compared with those obtained using a direct simulation Monte Carlo (DSMC) method considered as a reference. QGD equations differ from NS ones by the presence of additional dissipative terms. The present paper includes a brief presentation of QGD equations and DSMC procedures used here. The rarefied flow around a perpendicular disk has been considered for a freestream Mach number varying from 2 to 20, a Knudsen number equal to 0.1 and two levels of wall temperature.     

Supersonic rarefied gas flow through a wire grid

The direct simulation Monte Carlo method is used to study a plane-parallel supersonic gas flow through a grid formed by a series of parallel infinite cylinders. Characteristic features of the shock disturbance formation during the interaction of a supersonic flow with a permeable grid and the effect of this disturbance on the flow parameters behind the grid are revealed. The boundaries of the domain of supersonic flow breakup ahead of the grid and the laws of the total momentum loss on the grid are obtained. Kinetic and energetic characteristics of the flow behind the grid are determined.     

Supersonic transverse rarefied gas flow past a strip

The supersonic flow of a monatomic gas consisting of hard spherical particles past a flat strip normal to the flow is investigated using the direct simulation Monte-Carlo (DSMC) method. The calculations are performed over the Knudsen and Mach number ranges 0.015–5 and 1.8–15, respectively. The structure of the compressed layer and the aerodynamic characteristics are systematically studied for the Mach number 5 and various Knudsen numbers. The dependences of the compressed-layer thickness in molecular free paths are found. The nonequilibrium processes in the neighborhood of the strip are described on the basis of the data on the temperature anisotropy with respect to three coordinates.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, 2005, pp. 159–167. Original Russian Text Copyright © 2005 by Maltsev and Rebrov.     

Nonisothermal rarefied gas flow through a narrow slit

The nonisothermal flow of gas through a narrow slit under the influence of small pressure and temperature differences is investigated. The flow field and the mass and heat fluxes are found. It is shown that the heat transfer between the gas and the diaphragm, caused by the pressure difference, leads to a thermal polarization effect. The Onsager reciprocity relation is checked.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 171–175, July–August, 1990.