Grad's Moment Research Articles

The kinetic boundary layer for the linearized Boltzmann equation around an absorbing sphere

We construct approximate solutions of the linearized Boltzmann equation for a gas outside of a completely absorbing sphere, a simple model for a liquid droplet growing in a supersaturated vapor. The solutions are linear combinations of two Chapman-Enskog-type solutions, which carry heat and particle currents, and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path. To construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere, we expand the solution in Burnett functions and truncate the resulting system of equations for the expansion coefficients. For one particular truncation prescription, which generalizes Grad's 13-moment scheme, good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere; the results for small radii extrapolate smoothly toward the known limit of zero radius. We present results for the reaction rate (the particle current arriving at the sphere divided by the density at infinity) and for the density and temperature profiles in the boundary layer. The explicit calculations are carried out for Maxwell molecules, but the method appears to be suitable for more general intermolecular potentials.

The moment method for boundary layer problems in Brownian motion theory

We apply Grad's moment method, with Hermite moments and Marshak-type boundary conditions, to several boundary layer problems for the Klein-Kramers equation, the kinetic equation for noninteracting Brownian particles, and study its convergence properties as the number of moments is increased. The errors in various quantities of physical interest decrease asymptotically as inverse powers of this number; the exponent is roughly three times as large as in an earlier variational method, based on an expansion in the exact boundary layer eigenfunctions. For the case of a fully absorbing wall (the Milne problem) we obtain full agreement with the recent exact solution of Marshall and Watson; the relevant slip coefficient, the Milne length, is reproduced with an accuracy better than 10−6. We also consider partially absorbing walls, with specular or diffuse reflection of nonabsorbed particles. In the latter case we allow for a temperature difference between the wall and the medium in which the particles move. There is noa priori reason why our method should work only for Brownian dynamics; one may hope to extend it to a broad class of linear transport equations. As a first test, we looked at the Milne problem for the BGK equation. In spite of the completely different analytic structure of the boundary layer eigenfunctions, the agreement with the exact solution is almost as good as for the Klein-Kramers equation.

Electron thermal transport in laser-target plasmas on the basis of Grad's 13-moment transport equations

Grad's 13-moment theory is used to model thermal transport in laser-produced plasmas where conditions are such that one may not assume that the plasma is collision-dominated. The equations are presented for a multi-fluid model in slab geometry and are solved numerically for various cases. Comparisons are made with classical theory and with simulations of collisionless plasmas. Results show that, although there are differences between 13-moment and classical predictions, the two theories agree on the general behaviour of the plasma. In particular, 13-moment theory fails to explain the heat-flux inhibition implied by some experimental observations. The theory has also been applied to model thermal transport in plasmas with a population of suprathermal electrons.

Grad's 13-moment system for a dense gas of inelastic spheres

Grad's 13-moment system for a dense gas of inelastic spheres

Perturbation method in the theory of kinetic equations

Including in the definitions of correlation functions singular corrections that describe "self-correlatio ns," it is possible to find a rigorous, almost trivial solution of the complete BBGKY hierarchy for a system of charged particles corresponding to motion of them in a self-consistent field. Study of the averaged small deviations from this motion makes it possible to construct a scheme of successive approximations. In this manner, an expansion is obtained for the single-particle distribution function which is equivalent to a generalization of Grad's moment method to the phase space. In the first order of perturbation theory, an approximate Lenard-Balescu equation that differs from the result of its direct linearization is obtained. The proposed approach makes possible a more consistent approximate treatment of statistical systems.

An exact solution of fokker-planck equation and brownian coagulation in the transition regime

The theory of Brownian coagulation was examined to obtain the rate of coagulation. For this purpose an exact analytical solution of the Fokker-Planck equation was obtained for the boundary conditions of the given problem. It is shown that Grad's moment method, if carried out to infinite order, does not yield the complete solution. An exact expression for the coagulation coefficient was derived which agrees with Brock's limiting value when Knudsen number tends to infinity and with Smoulchowski's continuum limit when Knudsen number approaches zero. The results are in agreement with Fuch's semiempirical formula and the experimental results of Chatterjee, Kerker, and Cooke and Wagner and Kerker.

A kinetic derivation of extended irreversible thermodynamics

The basic postulates of the extended irreversible thermodynamics are derived from the kinetic model for a dilute monoatomic gas. Using the Grad 13-moment method to solve the full nonlinear Boltzmann equation for molecules conceived as soft spheres we obtain the microscopic expressions for the entropy flux, the entropy production, and the generalized Pfaffian for the extended definition of entropy as required by such a theory. Some of the physical implications of these results are discussed.

Generalized Langevin theory for inhomogeneous fluids: The equations of motion

We use the generalized Langevin approach to study the dynamical correlations in an inhomogeneous system. The equations of motion (formally exact) are obtained for the number density, momentum density, energy density, stress tensor, and heat flux. We evaluate all the relevant sum rules appearing in the frequency matrix exactly in terms of microscopic pair potentials and an external field. We show using functional derivatives how these microscopic sum rules relate to more familiar, though now nonlocal, hydrodynamiclike quantities. The set of equations is closed by a Markov approximation in the equations for stress tensor and heat flux. As a result, these equations become analogous to Grad's 13-moment equations for low-density fluids and constitute a generalization to inhomogeneous fluids of the work of Schofield and Akcasu-Daniels. We also indicate how the resulting general set of equations would simplify for systems in which the inhomogeneity is unidirectional, e.g., a liquid-vapor interface.

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.

Grad's moment equations and the conductivity tensor for a warm magneto-plasma

The role of the various Grad moment equations in deriving the conductivity tensor for a warm magneto-plasma has been discussed.

A relativistic relaxation-time model for the Boltzmann equation

A relaxation-time model is proposed for the collisions term of the relativistic Boltzmann equation for a single-component gas. The resultant equation is solved as a power series in the relaxation time and this solution is used to calculate the transport coefficients for the gas. These coefficients are then compared with those obtained using the relativistic Grad moment method and Marle's version of the relativistic relaxation-time model. The results indicate that the proposed relaxation-time model gives values for the transport coefficients which functionally agree with the values obtained using the relativistic Grad moment method in the asymptotic limits.

Stress tensor for a fully ionised collisionless plasma in time dependent electric field

An expression for the viscous stress tensor for a fully ionized collisionless plasma in the presence of a strong magnetic and varying electric fields is derived. It is done by solving the Vlasov equation by Grad's 13-Moment Approximation. Damping and thermal effects have also been considered.

Transport Equations for a Weakly Ionized Electron Gas

The Grad 13-moment method (using separate zero-order species flow velocities and temperatures) is used to calculate the electron traceless pressure Pe and heat flow qe for a weakly ionized gas in a dc magnetic field. Electron-neutral collisions are assumed to dominate and the “transport approximation” is used to obtain expressions for Pe and qe using arbitrary inverse power interparticle force laws. The results, under most conditions, show terms that are similar to the usual expressions for a simple gas except that the electron viscosity and thermal conductivity involve the electron-neutral collision frequency ven rather than the electron self-collision frequency. In addition, there are terms in qe proportional to any differences in the zero-order electron and neutral flow velocities and temperatures. Finally, it is noted that the electron-neutral interparticle force law has very little effect on the form of the results (although it is important in determining the explicit value for νen).

Some properties of homogeneous flows of rarefied gases

A generalization of the existence conditions for homogeneous flows of a rarefied monatomic gas mixture [2, 3] to the case where external forces are present is presented in [1]. Below we obtain for this case the solution of the Cauchy problem for the Boltzmann equation under “free molecular” (collisionless) conditions, when the collision integrals may be neglected (Knudsen number K ≫ 1). On the basis of this solution we construct a general solution for the equations of the kinetic moments of a Maxwellian monatomic gas mixture in the form of a series in inverse powers of K. Some additional remarks are made concerning the properties of the solutions of the second-order kinetic moment equations, and on the applicability of the Grad 13-moment equations and the Chapman-Enskog method [in particular, for the calculation of slow (“Stokesian”) motions of a gas mixture].

On a class of solutions of Grad's moment equations

On a class of solutions of Grad's moment equations