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.     

Extended thermodynamics of dense gases

We study extended thermodynamics of dense gases by adopting the system of field equations with a different hierarchy structure to that adopted in the previous works. It is the theory of 14 fields of mass density, velocity, temperature, viscous stress, dynamic pressure, and heat flux. As a result, most of the constitutive equations can be determined explicitly by the caloric and thermal equations of state. It is shown that the rarefied-gas limit of the theory is consistent with the kinetic theory of gases. We also analyze three physically important systems, that is, a gas with the virial equations of state, a hard-sphere system, and a van der Waals fluid, by using the general theory developed in the former part of the present work.     

Light scattering experiments and extended thermodynamics

Light in a gas is scattered on density fluctuations and the spectrum of the scattered light is influenced by the constitutive properties of the gas. The Navier-Stokes-Fourier theory does not always describe the spectrum of the scattered light in gases satisfactorily; it fails for small densities. Extended thermodynamics of many moments however may be used to predict the scattering spectra of dilute gases correctly. In this paper we compare the results of extended thermodynamics with measurements. Received: January 2, 1997     

Extended thermodynamics of viscoelastic materials

Extended thermodynamics is a field theory with the principal objective of determining the fields of deformation, temperature, stress and heat flux. With its hyperbolic field equations it ensures finite speeds for shear waves and thermal waves. The theory has previously been formulated for gases where its tenets could be well motivated by the kinetic theory of gases. The present paper is an attempt to incorporate solids into extended thermodynamics. In particular, it considers linearly viscoelastic solids and provides generalizations of the standard stress-strain relations of classical viscoelasticity.     

Original Article On the frame dependence and form invariance of the transport equations for the Reynolds stress tensor and the turbulent heat flux vector: its consequences on closure models in turbulence modelling

The present paper shows that the transport equations governing second order turbulent closures are form invariant, but remain frame dependent through the emergence of the body force; thus they do not fulfil the principle of material frame indifference as formulated by Truesdell & Noll (1965). However, this frame dependence corresponds to that first discussed by Müller (1972) and today developed in the framework of the new concept of extended thermodynamics. Following this new concept, these relations are consequently incorporated as additional basic balance laws. The results are: 1) in the case of the Reynolds-stress-transport equation, this eliminates the so-called constraints imposed in [15–17, 19] on turbulence models; 2) to ensure the closure of the new set of basic balance laws, closure assumptions can then be considered as proper constitutive equations which must be restricted by the well known constitutive theory principles in extended thermodynamics. Received: April 4, 1996     

Extended thermodynamics, viscoelasticity, and strain of solids

In this paper will be presented a formulation of extended thermodynamics for viscoelastic materials with heat conduction. The application of the Galilean invariance of the balance equations and the principle of entropy lead to the introduction of Lagrange multipliers, which provide constitutive equations for the flows. A condition of hyperbolicity system of equations is achieved by the concavity of the entropy density. The balance equations are linearized.     

Chapman–Enskog solutions to arbitrary order in Sonine polynomials III: Diffusion,thermal diffusion,and thermal conductivity in a binary,rigid-sphere,gas mixture

The Chapman–Enskog solutions of the Boltzmann equation provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper (I), for simple, rigid-sphere gases (i.e. single-component, monatomic gases) we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are free of numerical error and, in a second paper (II), we have extended our initial simple gas work to modeling the viscosity in a binary, rigid-sphere, gas mixture. In this latter paper we reported an extensive set of order 60 results which are believed to constitute the best currently available benchmark viscosity values for binary, rigid-sphere, gas mixtures. It is our purpose in this paper to similarly report the results of our investigation of relatively high-order (order 70), standard, Sonine polynomial expansions for the diffusion- and thermal conductivity-related Chapman–Enskog solutions for binary gas mixtures of rigid-sphere molecules. We note that in this work, as in our previous work, we have retained the full dependence of the solution on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals. For rigid-sphere gases, all of the relevant omega integrals needed for these solutions are analytically evaluated and, thus, results to any desired precision can be obtained. The values of the transport coefficients obtained using Sonine polynomial expansions for the Chapman–Enskog solutions converge and, therefore, the exact diffusion and thermal conductivity solutions to a given degree of convergence can be determined with certainty by expanding to sufficiently high an order. We have used Mathematica® for its versatility in permitting both symbolic and high-precision computations. Our results also establish confidence in the results reported recently by other authors who used direct numerical techniques to solve the relevant Chapman–Enskog equations. While in all of the direct numerical methods more-or-less full calculations need to be carried out with each variation in molecular parameters, our work has utilized explicit, general expressions for the necessary matrix elements that retain the complete parametric dependence of the problem and, thus, only a matrix inversion at the final step is needed as a parameter is varied. This work also indicates how similar results may be obtained for more realistic intermolecular potential models and how other gas-mixture problems may also be addressed with some additional effort.     

Modeling the response of filled elastomers at finite strains by rigid-rod networks

Summary  A constitutive model is developed for the isothermal response of particle-reinforced elastomers at finite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations: flexed and extended. The mechanical energy of a bond in the flexed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress–strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At finite strains, the governing equations are determined by four adjustable parameters which are found by fitting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several filled and unfilled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, filler content, crosslink density and temperature on the adjustable constants. Received 3 January 2001; accepted for publication 12 July 2001     

Modification of the first approximation of the Chapman-Enskog method for a gas mixture

The authors propose a transformation of the equations of the first approximation of the Chapman-Enskog method for a gas mixture with frozen internal degrees of freedom. As a result, the solution for the perturbation of the distribution functions can be written in terms of the diffusion velocities and temperature gradient, and the derivation of the Stefan-Maxwell relations and the heat flux calculations can be much simplified. The transformations are extended to a mixture of polyatomic gases with nonequilibrium excitation of the internal degrees of freedom of the molecules. The modification of the first approximation of the Chapman-Enskog method does not affect the relations used for calculating the viscosity of the mixture. Accordingly, that part of the solution is not considered.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 178–185, July–August, 1992.     

Non-isothermal polymer melt flow in sudden expansions

This paper presents numerical results for laminar, incompressible and non-isothermal polymer melt flow in sudden expansions. The mathematical model includes the mass, momentum and energy conservation laws within the framework of a generalized Newtonian formulation. Two constitutive relations are adopted to describe the non-Newtonian behavior of the flow, namely Cross and Modified Arrhenius Power-Law models. The governing equations are discretized using the finite difference method based on central, second-order accurate formulas for both convective and diffusive terms. The pressure–velocity coupling is treated by solving a Poisson equation for pressure. The results are presented for two commercial polymers and demonstrate that important flow parameters, such as pressure drop and viscosity distribution, are strongly affected by heat transfer features.     

Chapman–Enskog solutions to arbitrary order in Sonine polynomials IV: Summational expressions for the diffusion- and thermal conductivity-related bracket integrals

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper on simple gases, we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are free of numerical error. In two subsequent papers, we have extended our original simple gas work to encompass binary gas mixture computations of the viscosity, thermal conductivity, diffusion, and thermal diffusion coefficients to high-order. In all of this previous work we retained the full dependence of our solutions on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals up to the final point of solution via matrix inversion. The elements of the matrices to be inverted are, in each case, determined by appropriate combinations of bracket integrals which contain, in general form, all of the various dependencies. Since accurate, explicit, general expressions for bracket integrals are not available in the literature beyond order 3, and since such general expressions are necessary for any extensive program of computations of the transport coefficients involving Sonine polynomial expansions to higher orders, we have investigated alternative methods of constructing appropriately general bracket integral expressions that do not rely on the term-by-term, expansion and pattern matching techniques that we developed for our previous work. It is our purpose in this paper to report the results of our efforts to obtain useful, alternative, general expressions for the bracket integrals associated with the diffusion- and thermal conductivity-related Chapman–Enskog solutions for gas mixtures. Specifically, we have obtained such expressions in summational form that are conducive to use in high-order transport coefficient computations for arbitrary gas mixtures and have computed and reported explicit expressions for all of the orders up to 5.     

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.     

Attainment of maximum levels of natural convective heat transfer across major cavities using as a coolant a mixture of two pure gases instead of air

The enhancement of heat transfer in natural convection cavities is a very difficult task because of the intervening low fluid velocities. It is of fundamental and practical interest to explore alternative instruments that are power-independent and exclude surface modifications for the augmentation of heat transfer in these cavities. One feasible way for enhancing heat transfer rates passively in cavities filled with a gas is to stimulate the mechanism by natural convection of heat. The central objective of this paper is to employ a mixture of two pure gases that yields levels of heat transfer increments that are unattainable by each pure gas acting along (or even by air). In general, dimensional analysis insinuates that four transport properties affect natural convection flows: density, isobaric specific heat capacity, dynamic viscosity and thermal conductivity. Simple correlation equations of power form are useful to engineers for a quick estimate of the magnitudes of the space-mean heat transfer coefficient. Detailed computations were made for four different gases: air, pure helium, pure argon, and a mixture of pure helium and pure argon and the relative merits of each of them have been discussed. Five major cavities of relevance in applications of thermal engineering have been analyzed in this work. Received on 6 August 1999     

An extended thermodynamic approach to non-Newtonian fluids and related results in Marangoni instability problem

It is shown that extended irreversible thermodynamics provides a simple and coherent modelling of non-Newtonian fluids. The basic hypothesis underlyig the present formalism is to raise the stress tensor to the status of independent variable. Restrictions are placed on the constitutive equations and the material coefficients by the second law of thermodynamics, stability of equilibrium and the objectivity principle. The general procedure is applied to derive the Reiner-Rivlin and Rivlin-Ericksen second-order fluids. As an illustration, Marangoni convection in a thin horizontal layer of a non-Newtonian fluid submitted to a temperature gardient and placed in a microgravity environment is treated.     

Dispersion relation for sound in rarefied polyatomic gases based on extended thermodynamics

We study the dispersion relation for sound in rarefied polyatomic gases (hydrogen, deuterium and hydrogen deuteride gases) basing on the recently developed theory of extended thermodynamics (ET) of dense gases. We compare the relation with those obtained in experiments and by the classical Navier–Stokes Fourier (NSF) theory. The applicable frequency range of the ET theory is proved to be much wider than that of the NSF theory. We evaluate the values of the bulk viscosity and the relaxation times involved in nonequilibrium processes. The relaxation time related to the dynamic pressure has a possibility to become much larger than the other relaxation times related to the shear stress and the heat flux.     

On a model for a non-newtonian fluid with cylindrical symmetry: Wave features and similarity solutions

In the framework of extended thermodynamics, we consider a model describing a compressible and isotropic non-Newtonian fluid, neglecting heat effects. For simplicity we consider the governing equations in the particular case of cylindrical symmetry and look for the propagation of weak discontinuities, comparing the results with those for classical fluids. Moreover, under suitable conditions, the system under investigation proves to be invariant with respect to the dilatation group of transformations so that it is possible to characterize similarity solutions. Some constitutive relations are characterized for the phenomenological parameters.     

Consequences of the second law for a turbulent fluid flow

Second Law statements in thermomechanics applicable to turbulent fluid flow, in which the internal energy in a macroscopic field theory includes contributions both from molecular vibrations and from turbulent fluctuations, are discussed. In the absence of turbulence, these statements naturally reduce to the known and accepted Second Law statements for a nonturbulent medium. The usual version of the Second Law statements — which deny the existence of perpetual motion and place restrictions on the constitutive equations —is extended here in the presence of turbulence; and an additional statement is introduced associated with the tendency of turbulent fluctuations to decay in the absence of external work or the addition of thermal heat. The mathematical representations of various Second Law statements are then used to derive several restrictions on the response variables of the macroscopic turbulence theory. Examples of such variables include the rates of production and dissipation of turbulent fluctuations, the rate of thermal entropy production, internal energy (involving constitutive coefficients which may be taken to be the thermal and turbulent specific heats), turbulent viscosity coefficients and other response functions which control the degree of flow anisotropy in the medium. These Second Law restrictions are then applied to a recent theory of macroscopic turbulent flow by the present authors in which fairly general constitutive equations are presented for the dependent variables of the theory. It is found that not only is the range of values of several constitutive coefficients limited by these Second Law restrictions, but the presence of a number of terms in the constitutive equations is entirely denied.     

Phase Transitions in Gas- and Liquid-Saturated Porous Solids

Phase transitions in porous media consisting of a porous solid filled with liquid and gas constituents can occur, for example, due to freezing and drying processes. Although these phenomena are of certain relevance in soil mechanics and material sciences, a general thermo-dynamical theory is still awaited. Based on recent findings in the porous media theory, this paper is concerned with the development of thermodynamic restrictions for the constitutive relations of an elastic, incompressible porous solid, filled with an incompressible liquid and a compressible gas. The investigations show that mass conversions are related to the differences of the chemical potentials and energy transitions to the differences of temperatures. Thus, they confirm well-known results in classical thermodynamics of gases.     

Chapman–Enskog solutions to arbitrary order in Sonine polynomials II: Viscosity in a binary,rigid-sphere,gas mixture

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper (I), for simple, rigid-sphere gases (i.e. single-component, monatomic gases) we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are error free. It is our purpose in this paper to report the results of our investigation of relatively high-order, standard, Sonine polynomial expansions for the viscosity-related Chapman–Enskog solutions for binary gas mixtures of rigid-sphere molecules. We note that in this work we have retained the full dependence of the solution on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals. For rigid-sphere gases, all of the relevant omega integrals needed for these solutions are analytically evaluated and, thus, results to any desired precision can be obtained. The values of viscosity obtained using Sonine polynomial expansions for the Chapman–Enskog solutions converge monotonically from below and, therefore, the exact viscosity solution to a given degree of convergence can be determined with certainty by expanding to sufficiently high an order. We have used Mathematica® for its versatility in permitting both symbolic and high precision computations. Our results also establish confidence in the results reported recently by other authors who used direct numerical techniques to solve the relevant Chapman–Enskog equations. While in all of the direct numerical methods more-or-less full calculations need to be carried out with each variation in molecular parameters, our work utilizes explicit, general expressions for the necessary matrix elements that retain the complete parametric dependence of the problem and, thus, only a matrix inversion at the final step is needed as a parameter is varied. This work also indicates how similar results may be obtained for more realistic intermolecular potential models and how other gas-mixture problems may also be addressed with some additional effort.