Requirements to the accuracy of the Burnett transport coefficients

Some effects in gases are completely or partially determined by the Burnett terms of the stress tensor and the heat flux vector. Usually, approximate values of the Burnett transport coefficients for a monatomic gas are used to calculate the above-mentioned quantities. “Exact” expressions for these coefficients correct to values less than one percent are obtained for elastic-sphere molecules. The results of calculations of certain such effects are compared with one another using the approximate and exact Burnett transport coefficients for a monatomic gas consisting of elastic-sphere molecules.     

Kinetic and phenomenological theories for the linearized Burnett equations of a molecular gas

The linearized Burnett equations for a molecular gas are obtained from a kinetic theory based on the Boltzmann equation, and from a phenomenological theory based on extended thermodynamics. The constitutive equation for the pressure tensor of a molecular gas has three terms that do not have appeared in the corresponding equation for a monatomic ideal gas. One is the well-known term proportional to divergence of velocity whose coefficient is the volume viscosity. The two others are proportional to Laplacians of the temperature and of the density, and are associated with athermal (or temperature) pressure and with adensity pressure, respectively.     

Burnett simulations of gas flow in microchannels

The Burnett equations with slip boundary conditions are used to model the gas flow in microchannels in transition flow regime. As the Navier-Stokes equations are not appropriate to model the gas flow in this regime, the higher-order Burnett equations are adopted in the present study. In earlier studies, convergent solutions of the Burnett equations of microPoiseuille flow could only be obtained when Knudsen number is less than 0.2. By using a relaxation method on the boundary values, convergent solutions of the Burnett equations can be obtained even when Knudsen number reaches 0.4. The solutions of Burnett equations agree very well with experimental data and direct simulation Monte Carlo (DSMC) results. The pressure distributions and velocity profiles are then discussed in detail.     

Status of the Navier–Stokes Equations in Gas Dynamics. A Review

The existing ideas on the status of the Navier–Stokes equations are changed in taking into account the following facts: generally speaking, the terms of these equations neglected in the boundary layer equations are of the order of certain Burnett terms in the conservation equations; the Navier–Stokes equations cannot be used to describe slow nonisothermal gas flows since in this case it is necessary to take the Burnett temperature stresses into account; and in the transport relations the Burnett terms determine certain effects (for example, the mechanocaloric effect).     

Burnett simulation of flow and heat transfer in micro Couette flow using second-order slip conditions

The conventional Burnett equations with second-order velocity slip and temperature jump conditions were applied to the steady-state micro Couette flow of a Maxwellian monatomic gas. An analytical approach as well as a relaxation method was used to determine the velocity slip and temperature jump at the wall. Convergent solutions to the Burnett equations were obtained on arbitrary fine numerical grids for all Knudsen numbers (Kn) up to the limit of the equations’ validity. The Burnett equations with second-order slip conditions indicate a much better agreement with DSMC data over the first-order slip conditions at high Kn. The convergent Burnett solutions were obtained in orders of magnitude quicker than that with the corresponding DSMC simulation. The augmented Burnett equations were also introduced to model the flow but no obvious improvement in the results was found.     

Modifications of Gasdynamic Equations of Higher Approximations of the Chapman-Enskog Method

The non-Navier-Stokes continuum models proposed earlier on the basis of a modification of the gasdynamic equations of the higher (starting from the Burnett) approximations of the Chapman-Enskog method for shock wave flow are generalized to include the case of three-dimensional flows of a simple (monatomic) gas. The models are tested on the problems of shock wave structure and cylindrical Couette rarefied gas flow.     

Gasdynamic equations of higher approximations of the chapman-enskog method. Review

Some features of the derivation of the gasdynamic equations of higher (Burnett and super-Burnett) approximations of the Chapman-Enskog method are discussed for non-dense monatomic gases in the absence of external forces. The equations are given, and their properties and the results of applying them to various problems of gas dynamics are described. The problems of deriving, justifying, and testing macroscopic models are among the fundamental problems of not only continuum mechanics but physical kinetics in general. In the latter case it is a question of reducing more general molecular-kinetic models to simpler macroscopic ones. Historically, the classical Chapman-Enskog method of solution (as the Knudsen number Kn → 0) of the kinetic Boltzmann-Maxwell equation is of primary importance. Its application yields a chain of Euler and Navier-Stokes equations and equations of higher (Burnett, super-Burnett, etc.) approximations of the method. The publication of this review is a result of the increasing interest in the equations of higher approximations of the Chapman-Enskog method, in particular, in the Burnett equations. Primarily, this is a consequence of advances in the application of the latter equations to the problem of shock wave structure and the possible (by virtue thereof) expansion of the region of applicability of the macroscopic rarefied gas flow models. In addition, using the Burnett approximation, a series of new interesting effects has been established. In particular, it has been shown that in the classical problem of heat transfer between bodies heated to different temperatures it is necessary to take into account not only the temperature creep but also the Burnett temperature stresses. At the same time, a number of fundamental problems concerning the status of these equations has not yet been solved. An extensive literature is devoted to this problem. In the present paper, we shall mainly consider studies carried out since the end of the sixties and concerned with the best developed case of neutral monatomic (structureless) non-dense gas flow in the absence of external forces. Purely mathematical problems are not considered. Preference is given to final or review studies appearing in the most accessible publications. In Section 1 we consider the problem of taking out-of-order terms in the Chapman-Enskog method into account and make a brief comparison with other perturbation methods. Section 2 is devoted to the expressions for the transport properties in the Burnett and super-Burnett approximations. The relations between the Burnett approximation and axiomatic continuum mechanics theory, the thermodynamics of irreversible processes, and nonequilibrium statistical mechanics are discussed in Section 3. New effects due to specific terms of the formulas for the Burnett transport properties as Kn → 0 are considered in Section 4. The basic features of formulations of the boundary value problems for the systems of gasdynamic equations considered are outlined in Section 5. Finally, we describe the results of applying these equations to particular problems: some exact results of the kinetic theory (Section 6), the propagation of sound waves and cylindrical Couette flow (Section 7), shock wave structure and hypersonic hydrodynamics (Section 8). Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–28, July–August, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-01244).     

Modification of the burnett equations and shock wave structure

Questions of the applicability of simplifying modifications of the Burnett equations are studied with reference to the problem of the shock wave structure in a monatomic gas. As distinct from the complete system of Burnett equations, the order of the systems of modified equations is the same as that of the Navier-Stokes equations, and the equations are stable with respect to shortwave perturbations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 164–176, May–June, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-01244).     

Some self-similar solutions of Grad's equations

It is shown that the self-similar solutions of the Navier-Stokes and Burnett equations found earlier by the authors [1–9] can be extended to the case of two-dimensional flows of a weakly rarefied gas described by Grad's equations. Examples are given of numerical realization of self-similar solutions for flow in an expanding planar channel. It is found that there are appreciable differences between the behavior of the self-similar solutions of the Navier-Stokes, Burnett, and Grad equations in the neighborhood of a channel wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 88–94, May–June, 1982.     

Effect of the Intermolecular Potential on the Generalized Newton-Fourier Relations in the Problem of Gas Outflow into a Vacuum

Using the numerical method of direct statistical simulation (Monte-Carlo method), the effect of the intermolecular potential exponent on the dependence of the stress and the heat flux on the stress rate tensor components and the temperature gradient is analyzed for the nonequilibrium, spherically symmetric outflow of a monatomic gas into a vacuum. Analytical approximations of the constitutive relations are presented. The dependence of the macroscopic parameters on the distance from the source is found.     

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.     

Derivation of the equations of slow nonisothermal gas flows

In recent years, some new phenomena have been predicted theoretically on the basis of the Burnett approximation. These include thermal-stress and concentration-stress convection [1–3], and also effects due to the influence of a magnetic field in a multiatomic gas (viscomagnetic heat flux, etc., [4]). It has been shown theoretically (see [5]) that under certain conditions various terms of the Burnett approximation must be taken into account in the expression for barodiffusion. The conclusions relating to a viscomagnetic heat flux have recently been confirmed experimentally [4]. The predicted phenomena follow rigorously from the Burnett equations. However, many hydrodynamicists adopt a sceptical attitude to these equations, which is due partly perhaps to attachment to the classical Navier-Stokes equations, which have served theoreticians without fail for a century and a half. In this connection, we discuss the evolution of ideas relating to the validity of the Burnett approximation. We discuss the minimal assumptions which must be made in order to derive the equations of slow [Reynolds number R = 0(1)], essentially nonisothermal [ ln T = 0(1)] flows of a gas as a continuous medium (Knudsen number K O) in the case when the derivatives of the thermal Burnett stresses in the momentum equation have the same order of magnitude as the Euler and Navier-Stokes terms of this equation [1–3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 77–84, November–December, 1979.We thank G. I. Petrov and L. I. Sedov for discussions that stimulated the above analysis.     

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.     

n the fourth order tensor valued function of the stress in return map algorithm

针对各向同性材料,基于一组相互正交的基张量,建立了一套有 效的相关运算方法. 基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则 使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴. 根据张量函数表示 定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组 基所表示. 推演结果表明：这些张量之间的运算,表现为对应系数矩阵之间的简单 关系. 其中,四阶张量求逆归结为对应的3\times3系数矩阵求逆,它对二阶张量的变换 则表现为该矩阵对3times 1列阵的变换. 最后,对这些变换关系应用于返回映 射算法的迭代格式进行了相关讨论.     

The Burnett equations for a relativistic gas

The Burnett constitutive equations for the dynamic pressure, heat flux and pressure deviator are obtained from the relativistic Boltzmann equation by using fourteen moment equations and an iteration method akin to the Maxwellian iteration procedure of the non-relativistic theory. The non-relativistic and ultra-relativistic limits of all Burnett coefficients are given and it is shown that the non-relativistic limit of the Burnett constitutive equations agree with those obtained from the non-relativistic Boltzmann equation. Received July 21, 2000     

The hydrodynamics of convective stellar envelopes

Large-scale movements in convective stellar envelopes are studied by the methods of hydrodynamics, taking account of the compressibility of the gas, rotation, gravitation, and anisotropic turbulent viscosity. From dimensional analysis and observational data, the order of the nondimensional parameters controlling the motion of gaseous masses in the convective zone can be estimated. Estimates of the dimensionless parameters are used to simplify the equations of motion and develop an approximate method of solution. It is assumed that the motion of the gas in the convective envelope is anisotropic and turbulent and that the Reynolds stress tensor is linear in the rates of deformation, while the coefficients of proportionality in a spherical coordinate system form a diagonal tensor. It is assumed that the boundaries of the convective zone are free of turbulent stresses. Three approximations are found for the stream function for meridional motions, and the angular velocity of differential rotation for the case when the forces of turbulent viscosity are very strong. The components of the tensor of turbulent viscosity are assumed to be known functions of the depth of the convective zone.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 27–38, March–April, 1971.     

Relation between the stress and couple-stress tensors in the microcontinuum theory of elasticity

The stress tensor is expressed in terms of an arbitrary symmetric tensor field of second rank and the couple-stress tensor. The stress and couple-stress tensors are represented by arbitrary tensor fields satisfying the homogeneous equilibrium equations. These tensors are also given in the form of the expressions satisfying the inhomogeneous equilibrium equations used in the microcontinuum theory of elasticity. The stress tensor functions are considered.     

Reaction surface (flame-sheet) model in the theory of combustion of unmixed fuels. Molecular-kinetic solution in the reaction region

In the first part [1] of this investigation, a study was made of the main characteristic features of the flow resulting from the mixture of two flows with components that enter into an irreversible chemical reaction. The main errors in the solution of the corresponding gas-dynamic problem resulting from the use of various assumptions were also analyzed. It was found that the influence of the usually ignored Burnett and super-Burnett terms in the transport properties (i.e., in the stress tensor and in the heat flux and diffusion vectors) can be much stronger in this problem than in ordinary mixing problems. In the present, second part, a molecular-kinetic study is made of the flow in the thin reaction region. For the finding of the external solution of the problem in the mixing layer or when the classical flame-sheet model is used, this region is treated as a surface in the flow field on which all the chemical transformations of the components of the gas mixture take place. Gas-dynamic equations are obtained that describe the main and some of the subsequent approximations in the reaction region; these equations confirm the arguments put forward in the first part concerning the part played by the non-Navier-Stokes terms in the transport properties.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 77–90, March–April, 1979.     

关于返回映射算法中应力的四阶张量值函数

针对各向同性材料,基于一组相互正交的基张量,建立了一套有效的相关运算方法.基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴.根据张量函数表示定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组基所表示.推演结果表明:这些张量之间的运算,表现为对应系数矩阵之间的简单关系.其中,四阶张量求逆归结为对应的3×3系数矩阵求逆,它对二阶张量的变换则表现为该矩阵对3×1列阵的变换.最后,对这些变换关系应用于返回映射算法的迭代格式进行了相关讨论.