Exact solutions to the equations of perfect gases through Lie group analysis and substitution principles

In this paper we consider the equations that govern the motion of perfect gases. We explicitly characterize some classes of steady solutions in two and three space dimensions, by introducing invertible point transformations suggested by Lie group analysis; moreover, by using various transformations known as substitution principles, new steady and unsteady solutions are constructed.     

A short history of balance equations for rarefied monatomic gases and plasmas

The main steps in the development and generalizations of balance equations and representations of velocity distributions are listed from Maxwell's first attempt in 1867 until today.     

Exact solutions to the unsteady equations of perfect gases through Lie group analysis and substitution principles

In this paper, we consider the unsteady equations that govern two- and three-dimensional flows of a perfect gas. We explicitly characterize various classes of exact solutions by introducing some invertible transformations suggested by the invariance with respect to Lie groups of point symmetries and using suitable transformations known in literature as substitution principles.     

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.     

Exact solution of the system of equations of the second-order kinetic moments for a two-scale homoenergetic affine monatomic gas flow

The exact solution of the system of equations for the second-order kinetic moments (stresses) is investigated for a flow with two microscales when an unsteady shear flow is superimposed on a one-dimensional unsteady gas flow.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 156–166, May–June, 1995.The work was carried out with financial support from the Russian Foundation for Fundamental Research (project No. 93-013-16407).     

An exact solution of the navier-stokes equations for a chemically reacting mixture of gases

An exact solution of the Navier-Stokes equations is presented for the flow of a viscous heat-conducting chemically reacting mixture of gases in a two-dimensional expanding channel. The conditions for the existence of an exact solution of the source type are: the chemical reactions must be equimolecular in an equilibrium mixture of gases, and they must be second order forward and backward in a nonequilibrium mixture. Numerical results are obtained for the flow of a four-component equilibrium mixture of gases in a two-dimensional nozzle for various Mach numbers (M) and Reynolds numbers (Re).Translated from Zhurnal Prikladnoi Mekhanika i Tekhnicheskoi Fiziki, No. 4, pp. 49–56, July–August, 1973.     

Exact Thomas precession related solutions of the inertial navigation equations

The aim of the present paper is to give a survey of those exact solutions of relativistic inertial navigation equations which are directly or indirectly related to Thomas precession. Various cases of uniform circular motion of an object in gravity-free space close to motion with inertial or orbital attitude are considered.     

Exact geodesic precession related solutions of the inertial navigation equations

Various cases of uniform circular motion of an object subjected to gravitational acceleration are considered. Special attention is paid to the exact geodesic precession related solution of the relativistic inertial navigation equations.     

Exact solutions of the boundary layer equations for power-law fluids

Exact analytic solutions of the equations of the hydrodynamic boundary layer are obtained for pseudoplastic fluids with exponents n=1/5, 1/4, 1/2, 3/5, 5/7 flowing longitudinally over a flat plate.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 39–42, September–October, 1989.     

Exact solutions of linearized equations of convection of a weakly compressible fluid

A mathematical model of fluid convection under microgravity conditions is considered. The equation of state is used in a form that allows considering the fluid as a weakly compressible medium. Based on the previously proposed mathematical model of convection of a weakly compressible fluid, unsteady convective motion in a vertical band, with a heat flux periodic in time set on the solid boundaries of this band, is considered. This model of convection allows one to study the problem with the boundary thermal model oscillating in an antiphase rather than in-phase mode, while the latter was required for the model of microconvection of an isothermally incompressible fluid. Exact solutions for velocity components and temperature are derived, and the trajectories of fluid particles are constructed. For comparison, the trajectories predicted by the classical Oberbeck-Boussinesq model of convection and by the microconvection model are presented.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 2, pp. 52–63, March–April, 2005.     

Exact solutions of some general nonlinear wave equations in elasticity

A similarity analysis of a nonlinear wave equation in elasticity is studied; in particular, one with anharmonic corrections. The symmetry transformation give rise to exact solutions via the method of invariants. In some cases, graphical figure of the solutions are presented. Furthermore, we consider some cases wherein the velocities of the longitudinal and transversal plane waves are variable. Finally, a brief discussion on how a symmetry analysis on a perturbation of the elasticity equation can be pursued.     

A model and kinetic equations for a charged mixture of gases in the presence of phase transitions

A model of a gas mixture is studied in which one of the components can carry electric charge and undergo phase transitions. Under a number of assumptions, Boltzmann kinetic equations are written down and the form of the collision integral determined. Conservation equations for the components of the mixture are found. The conservation equations for a charged mixture of gases in the absence of phase transitions have been discussed earlier [1]. Collision integrals for a reacting gas mixture in the case of chemical reactions of bimolecular type and when the mixture is described by Boltzmann kinetic equations are derived in [2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 3, pp. 118–127, May–June, 1980.     

Exact solutions of the generalized Navier– Stokes equations for benchmarking

The generalized Navier– Stokes equations for incompressible viscous flows through isotropic granular porous medium are studied. Some analytical classic solutions of the Navier– Stokes equations are generalized to the case of the considered equations. Obtained solutions of generalized equations reduce to classic ones as porosity effect disappears. Average velocity of generalized solutions is calculated and evaluated in two limiting regimes of flow. In the shallow conduit, the generalized flow rate approximates the free (without porous medium) flow rate and in the case of removed boundaries this approaches Darcy's law. The use of the derived exact solutions for benchmarking purposes is described. Copyright © 2002 John Wiley & Sons, Ltd.     

Exact solutions of incompressible Couette flow with porous walls for slightly rarefied gases

In this work, the transient incompressible Couette flow and steady-state temperature profiles between two porous parallel plates for slightly rarefied gases are solved exactly. The first-order approximation of slip velocity at the boundaries is used in the formulation. The solution is also applicable for Couette flow in micro-channels under certain circumstances. The influences of mass transfer and a nondimensional slip parameter on slip velocities are discussed. It is also found that the transient slip velocities at the walls are greatly different from the steady-state velocity slips. The influences of velocity slip and temperature slip parameters on the temperature distribution and heat transfer at the walls are analyzed and discussed. It is shown that the slip parameters can greatly change the temperature profiles and heat transfer characteristics at the walls.