Bracket Integrals Research Articles

Navier–Stokes equations with rovibrational state-resolved transport flux closure

In the time since the Navier–Stokes equations were introduced more than two centuries ago, their application to problems involving real gas effects has relied on appropriate closure for the mass, momentum, and energy transport fluxes via the constitutive laws. Determination of the corresponding transport coefficients, most readily obtained through generalized Chapman–Enskog theory, requires knowledge of the intermolecular potentials for rotationally and vibrationally excited molecules. Recent advances in computational chemistry provide extraordinary detail of interactions involving rovibrationally excited molecules, offering a means for transport flux closure with unprecedented accuracy. Here, the bracket integrals for rovibrationally resolved molecular states are developed, and the resulting transport flux closure is presented for the rovibrationally resolved Navier–Stokes equations. The accompanying continuum breakdown parameters are also derived as a rigorous metric to establish the range of applicability of the aforementioned equations in flow conditions approaching the rarefied regime.

Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory Made Amenable to Computer Simulations

In the kinetic theory of dense fluids the many-particle collision bracket integral is given in terms of a classical collision operator defined in the phase space. To find an algorithm to compute the collision bracket integrals, we revisit the eigenvalue problem of the Liouville operator and re-examine the method previously reported [Chem. Phys. 1977, 20, 93]. Then we apply the notion and concept of the eigenfunctions of the Liouville operator and knowledge acquired in the study of the eigenfunctions to cast collision bracket integrals into more convenient and suitable forms for numerical simulations. One of the alternative forms is given in the form of time correlation function. This form, on a further manipulation, assumes a form reminiscent of the ChapmanEnskog collision bracket integrals, but for dense gases and liquids as well as solids. In the dilute gas limit it would give rise precisely to the Chapman-Enskog collision bracket integrals for two-particle collision. The alternative forms obtained are more readily amenable to numerical simulation methods than the collision bracket integrals expressed in terms of a classical collision operator, which requires solution of classical Lippmann-Schwinger integral equations. This way, the aforementioned kinetic theory of dense fluids is made fully accessible by numerical computation/simulation methods, and the transport coefficients thereof are made computationally as accessible as those in the linear response theory.

Application of computer algebra for the calculation of bracket integrals

Computer algebra is used to develop a system for analytical calculation of bracket integrals on a computer. The results produced by this system can be used for calculating kinetic coefficients of nonideal plasma based on the solution of the Boltzmann kinetic equation.

Chapman–Enskog solutions to arbitrary order in Sonine polynomials V: Summational expressions for the viscosity-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 (I), 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 (II–III), we 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 a fourth paper (IV) we derived general summational representations for the diffusion- and thermal conductivity-related bracket integrals and provided compact, explicit expressions for all of these bracket integrals needed to compute the diffusion- and thermal conductivity-related transport coefficients up to order 5 in the Sonine polynomial expansions used. 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 expressions for the needed bracket integrals have not previously been available in the literature beyond orders 2 or 3, and since such 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 viscosity-related Chapman–Enskog solutions for gas mixtures. Specifically, we have obtained such expressions in summational form that are conducive to use in high-order viscosity coefficient computations for arbitrary gas mixtures and have computed and reported explicit expressions for all of the orders up to 5.

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 properties of partially ionized and unmagnetized plasmas

This work is a comprehensive and theoretical study of transport phenomena in partially ionized and unmagnetized plasmas by means of kinetic theory. The pros and cons of different models encountered in the literature are presented. A dimensional analysis of the Boltzmann equation deals with the disparity of mass between electrons and heavy particles and yields the epochal relaxation concept. First, electrons and heavy particles exhibit distinct kinetic time scales and may have different translational temperatures. The hydrodynamic velocity is assumed to be identical for both types of species. Second, at the hydrodynamic time scale the energy exchanged between electrons and heavy particles tends to equalize both temperatures. Global and species macroscopic fluid conservation equations are given. New constrained integral equations are derived from a modified Chapman-Enskog perturbative method. Adequate bracket integrals are introduced to treat thermal nonequilibrium. A symmetric mathematical formalism is preferred for physical and numerical standpoints. A Laguerre-Sonine polynomial expansion allows for systems of transport to be derived. Momentum, mass, and energy fluxes are associated to shear viscosity, diffusion coefficients, thermal diffusion coefficients, and thermal conductivities. A Goldstein expansion of the perturbation function provides explicit expressions of the thermal diffusion ratios and measurable thermal conductivities. Thermal diffusion terms already found in the Russian literature ensure the exact mass conservation. A generalized Stefan-Maxwell equation is derived following the method of Kolesnikov and Tirskiy. The bracket integral reduction in terms of transport collision integrals is presented in Appendix for the thermal nonequilibrium case. A simple Eucken correction is proposed to deal with the internal degrees of freedom of atoms and polyatomic molecules, neglecting inelastic collisions. The authors believe that the final expressions are readily usable for practical applications in fluid dynamics.

Transport properties in a two-temperature plasma: theory and application.

An alternate derivation of transport properties in a two-temperature plasma has been performed. Indeed, recent works have shown that the simplified theory of transport properties out of thermal equilibrium introduced by Devoto and then Bonnefoi, very often used in two-temperature modeling, is questionable and particularly does not work when calculating the combined diffusion coefficients of Murphy. Thus, in this paper, transport properties are derived without Bonnefoi's assumptions in a nonreactive two-temperature plasma, assuming chemical equilibrium is achieved. The electron kinetic temperature T(e) is supposed to be different from that of heavy species T(h). Only elastic processes are considered in a collision-dominated plasma. The resolution of Boltzmann's equation, thanks to the Chapman-Enskog method, is used to calculate transport coefficients from sets of linear equations. The solution of these systems allows transport coefficients to be written as linear combinations of collision integrals, which take into account the interaction potential for a collision between two particles. These linear combinations are derived by extending the definition and the calculation of bracket integrals introduced by Chapman et al. to the thermal nonequilibrium case. The obtained results are rigorously the same as those of Hirschfelder et al. at thermal equilibrium. The derivation of diffusion velocity and heat flux shows the contribution of a new gradient, that of the temperature ratio straight theta=T(e)/T(h). An application is presented for a two-temperature argon plasma. First, it is shown that the two-temperature linear combinations of collision integrals are drastically modified with respect to equilibrium. Secondly, the two-temperature simplified theory of transport coefficients of Devoto and Bonnefoi underestimates the electron thermal conductivity with respect to the accurate value at T(e)=20 000 K. Lastly, contrary to the simplified theory of transport coefficients, the diffusion coefficients satisfy the symmetry conditions. An example is given at T(e)=6000 K for different values of straight theta for the diffusion coefficient between electrons and heavy species D(e-Ar) as well as for that between argon atoms and argon ions D(Ar-Ar+).

Kinetic theory of a binary mixture of nearly elastic disks

Constitutive relations for a binary mixture of nearly elastic, circular disks are derived using the Revised Enskog Theory. Viscosity results from hard disk simulations are compared to the Enskog viscosity for a radii ratio of 1.25. The agreement is good up to an area fraction of 0.40. In the Appendix, we provide bracket integrals calculated to determine the perturbation coefficients of the expansion parameter used in the theory.

The polynomial expansion method for boundary value problems of transport in rarefied gases

The principles of rarefied gas dynamics are finding application in many areas of practical interest ranging from aerosol science to micro–fabrication to vacuum and space applications. Many methods have become available for solving the Boltzmann equation which is the complicated integro–differential equation that describes the distribution of the gas molecules and forms the basis for the field of rarefied gas dynamics. Moments methods are among the most popular of these solution techniques due to their relative simplicity and overall versatility. In the half–range moment method of solving boundary value problems based on the Boltzmann equation, difficult bracket integrals are encountered. We discuss here a simplification of such integrals specific to the spherical geometry and applicable for arbitrary values of the Knudsen number. A polynomial expansion in velocity space is used to represent the discontinuous factor in these integrals. The accuracy of the method is verified by a comparison of the analytical results with available numerical results.

Ordinary and thermal diffusions in polyatomic binary fluid mixtures

General formulas for the ordinary and Soret diffusion coefficients D12 and DT, and the thermal-diffusion ratio kT (or the thermal-diffusion factor α12) of a binary and dilute fluid mixture consisting of nonspherical molecules have been obtained utilizing the classical transport theory recently developed by us [G. S. Singh and B. Kumar, J. Chem. Phys. 104, 5604 (1996)]. The general results involve up to the infinite order of approximations but in the present work we have restricted ourselves up to the second order only. We thus obtain general expressions for the second approximation to both D12 and DT, and the first approximation to kT (or α12) in terms of different square bracket integrals. The velocity as well as angular velocity parts of these integrals for the fluid mixtures of hard biaxial ellipsoids have been analytically performed with the help of the modified Hoffman procedure. The expressions for [D12]2, [DT]2, and [kT]1 have thus ultimately been obtained in terms of five four-dimensional quadratures over the orientational coordinates of like or unlike pairs of colliding ellipsoidal molecules.

Transport properties of a binary gas mixture of molecules with internal energy. I. Shear and volume viscosities

General expressions in terms of the square bracket integrals for the shear and volume viscosities of a classical dilute binary gas mixture of nonspherical molecules have been derived by extending the Taxman classical kinetic theory of transport phenomena within the framework of the Chapman–Enskog second approximation method for a mixture of spherical molecules. The dimensionality of integrals for the binary mixture of rigid spheroidal molecules having C∞v symmetry has been reduced analytically to get ultimately the forms wherein the integrals are to be performed only over the orientational coordinates of two colliding like or unlike spheroids. The general expressions for the nonspherical molecules as well as the final integrals for the spheroidal molecules are shown to be correct. The shear- and volume-viscosity formulas for a classical dilute binary gas mixture of hard sphere molecules have been retrieved as the limiting case of our corresponding expressions for the spheroidal systems.

A method for solving linearized problems of the transfer theory for spherical geometry at arbitrary Knudsen numbers

A method of calculating the bracket integrals containing a discontinuous two-sided Maxwellian distribution function is developed. This method makes it possible to obtain analytic solutions for different linearized systems of moment equations for all Knudsen numbers. The particular problem of heat transfer from a heated sphere is considered. New special functions associated with a specific moment system are introduced. For these functions a table of numerical values is given. Approximate analytic expressions for calculating the bracket integrals are also presented.

Cluster expansion for transport coefficients for dense simple fluids

Cluster expansions are presented for the collision bracket integrals for transport cofficients of a dense simple fluid which appear in the kinetic theory of dense fluid reported previously. The density series calculated with the cluster expansions are given in exponential forms and their leading terms consist of a connected three-particle collision operator. The three-particle contributions are discussed in terms of mass-normalized coordinates which make it possible to put them in forms structurally rather similar to the two-particle collision bracket integrals in the Chapman-Enskog theory. The three particle contributions are thus cast into forms suitable for numerical computation.

Extended irreversible thermodynamics, generalized hydrodynamics, kinetic theory, and rheology

The gist of extended irreversible thermodynamics and generalized hydrodynamics is presented within the context of rheology of complex molecules (e.g., polymers) in this paper. Then, the constitutive equation for stress developed for polyatomic fluids in a previous paper is applied to rheology of polymeric fluids. This constitutive equation is fully consistent with the thermodynamic laws. It is shown that the collision bracket integrals appearing in the constitutive equation can be recast in terms of friction tensors of beads and equilibrium force-force correlation functions if the momentum relaxation is much faster than the configuration relaxation and there exist such relaxation times. The force-force correlation functions reduce to those related to the mean square radius of gyration of the polymer if the Hookean model is taken for forces. By treating the recast collision bracket integrals in the constitutive equation as empirical parameters, we analyze some experimental data on shear rate and elongation rate dependence of polymeric melts and obtain excellent agreement with experiment. We show that the empirical parameters can be related to the zero shear rate viscosity and the ratio of the secondary to the primary normal stress coefficient. Therefore, for the plane Couette flow geometry considered in the paper, the constitutive equation is completely specified by the limiting material functions at zero shear rate and relaxation times.

A renormalized kinetic theory of dilute molecular gases: Chattering

The phenomenon of molecular chattering (a collision between two rigid ovaloids involving two or more impulsive hits) is investigated with emphasis on the importance of these collisions on kinetic theory calculations for dilute molecular systems. To facilitate this analysis, the pseudo-Liouville formalism, commonly employed in the kinetic theory of rigid spheres, is generalized to a form appropriate to rigid ovaloids. From this formalism, we derive a chattering expansion of the bracket integrals, the terms of which successively incorporate higher order chattering sequences. Finally, the results of scattering trajectories between a rigid sphere and a rigid ellipsoid are reported. It is demonstrated that the neglect of chattering sequences leads to large errors when calculating the bracket integrals for realistic model parameters; the errors ranging from ∼1% to as high as ∼200%.

On CC, CS, and IOS generalized phenomenological cross sections for atom–diatom mixtures

Close coupled expressions for phenomenological cross sections which describe transport properties of atom–diatom mixtures are obtained in the total-J coupling scheme and are related to the bracket integrals of kinetic theory. Coupled states and infinite order sudden expressions for the generalized phenomenological cross sections using initial, final, and average l-labeling are also given. Particular care is taken to use a phase convention for the CS and IOS approximations which is consistent with the Arthurs–Dalgarno formalism and which gives the correct behavior of degeneracy averaged differential cross sections.

Convergent kinetic equations in the theory of transport properties in multicomponent ionized gas mixtures

The classical unified theory of relaxation phenomena in hot fully ionized gases is applied to multicomponent gas mixtures composed of electrons and several types of ions. The theoretical expressions containing exact numerical factors in the arguments of Coulomb logarithms are developed for the bracket integrals needed in the computations of the third approximations to the thermal and the electrical conductivities, and the diffusion coefficients of multicomponent gas mixtures at intermediate and full ionization within the framework of the Chapman–Enskog–Burnett method.

Kinetic theory of reacting systems

Abstract In this paper transport processes of reacting systems are investigated, based on the Boltzmann equations. The Boltzmann equations are solved by means of Grad's moment method to thirteen moments and some formal results are obtained for transport properties. It is shown that the rate coefficients are quadratic functions of hydrodynamic fluxes and are in the form where are the scalar moments associated with the reaction and q, J, Π are heat flux, material flux and traceless symmetric stress tensor. k(0)i is the usual local equilibrium formula for reaction rate constant. Iterative solutions for the equations of change for , q, J and Π are obtained from which transport coefficients are calculated for the reacting system. It is shown that the solutions, when specialized to nonreacting mixtures, lead to results for the transport coefficients which are exactly in agreement with the Chapman-Enskog theory results. The modifications of the transport coefficients due to reactions are obtained from the iterative solutions and the bracket integrals necessary for their calculations are explicitly given in an appendix.

Heat transfer and temperature jump in a polyatomic gas

A variational principle based on the integro-differential form of the linearised Wang Chang-Uhlenbeck equation, with general boundary conditions, is used to evaluate the heat conducted through a polyatomic gas between parallel plates. The result is an accurate, closed form expression for the heat transfer, valid for all degrees of rarefaction, rational in the inverse Knudsen number, and parametrised by the thermal accommodation coefficients, the heat capacity of the internal modes, full range moments of the Chapman-Enskog solution and half range bracket integrals of the free molecular solution. The temperature jump coefficient is obtained from the high density expansion of the heat flux and is dependent on the thermal accommodation coefficients, the internal heat capacity, and moments of the Chapman-Enskog solution. In the limit of vanishing internal specific heat, both the heat transfer and the temperature jump reduce to results previously given for the monatomic gas.

The higher order Chapman-Cowling bracket integrals for the viscosity coefficient of multicomponent gas mixtures

The rigorous kinetic theory bracket integrals for the Chapman-Cowling third order approximation to the viscosity coefficient of multicomponent gas mixtures are reported. These expressions will be found useful in interpreting the highly accurate viscosity data on multicomponent gas mixtures now possible. Specially for mixtures involving molecules quite different in nature the contribution is comparable with experimental accuracy.