Grad's Method Research Articles

Kinetic theory of droplet evaporation rate from the solution of the Boltzmann equation

The moments method of Grad is applied to the problem of condensation-evaporation of a spherical droplet immersed in its pure vapor. The vapor molecules are assumed to be interacting with the particle as hard spheres. Using the thirteen-moments method and linearization in the solution of the Boltzmann equation, an analytical expression for the rate of condensation-evaporation in the stationary state is derived. Temperature and density profiles of the vapor are obtained as functions of: distance from the center of the droplet, Knudsen number, accommodation coefficient, saturation number density of the vapor, and assumed temperature of the droplet surface. The results are supposed to be applicable in the case when the system is not very far from equilibrium.

Collision Frequencies for Electrical and Heat Conductivity in Dense Cold Plasmas

AbstractFormulas for electrical and heat conductivity are derived from the Boltzmann equation with Grad's method of orthogonalized moments, retaining the coupling between diffusion and heat fluxes. The results are expressed by transport collision frequencies. These are calculated for Coulomb interaction potentials screened with the Debye‐Hückel length. Quantum corrections to the classical results are included.

Analysis of flux-relaxed electron heat waves

It has long been known that a classical description of electron heat conduction results in energy transport speeds which are unphysically large and that the results of model calculations using the classical description are inconsistent with the results of many laser fusion experiments. Some insight into the proper corrective measures is obtained if the plasma kinetic equations are solved via the ‘‘13 moment’’ approximation of Grad, rather than the more usual but incomplete Chapman–Enskog method. The Grad method naturally incorporates time variation of the electron distribution function and leads to a ‘‘flux-relaxation’’ equation for the heat flux, resulting in a hyperbolic temperature equation in which heat flow is restricted to speeds less than a characteristic electron thermal velocity. This paper contains some new insight into the subject of flux relaxation in the form of similarity solutions for the heat flow, as well as numerical solutions for boundary conditions appropriate to laser fusion modeling. Included in the treatment is the effect of hydrodynamics on the flux-relaxed heat flow. The results are that although flux relaxation naturally incorporates classical flux-limiting it does not play a role in the observed ‘‘anomalous’’ flux inhibition.

Thirteen-moment solution of the steady-state fokker—planck equation for Brownian motion in a homogeneous medium occupying the region bounded internally by an absorbing sphere

The steady, centrally symmetric motion of Brownian particles in a homogeneous medium occupying the region bounded internally by an absorbing sphere of radius R is considered. The distribution function of the particles, assumed to obey the generalized fokker—Planck equation, is found (in three successive approximations) by applying Grad's method of moments, which consists in expanding the distribution function in Hermitean tensors and truncating the series after the term of Nth order. Approximations up to and including the so-called 13-moment approximation (corresponding to N = 1, 2,and3) are studied, and a previous 13-moment analysis is criticized; it is stressed that the results furnished by the Nth-order approximation are valid to O ( Λ/ R) N , where Λ is the velocity persistence length.

Comments on the Grad procedure for the Fokker-Planck equation

If the kinetic equation of a macroscopic system is expanded with respect to the velocity in terms of orthogonal functions, e.g., in terms of Hermite functions, one obtains an infinite hierarchy of equations for the expansion coefficients. Grad's method consists in truncating this hierarchy and investigating the remaining finite system. In this paper we set up conditions under which this procedure is rigorously justified in case of the Fokker-Planck equation.

Adiabatic compression of rotating plasmas

The adiabatic compression of plasmas with mass flow is treated via a variational principle based on a few constants of motion. It is then possible to adapt Grad's alternating-dimensions numerical method to this case.

Transient relativistic thermodynamics of a system of particles with variable masses

Abstract This paper is devoted to the study in general relativity of the transient thermodynamical properties of a self-graviting gas, the particles of which may exchange mass during collisions. Such a work is carried out by means of the relativistic version of the kinetic Grad method which Israel and Stewart proposed. The transfer phenomena are investigated and the propagating velocities obtained.

Bregman's balancing method

In this paper we observe that most of the independently discovered balancing methods, used in transportation planning and in other fields, are in fact special cases of a method of Bregman. Examples include the usual Kruithof or Furness method, the Evans-Kirby three dimensional balancing procedure, the Murchland multiproportional balancing procedure, the Osborne or Grad method for preconditioning matrices, the Jefferson-Scott procedure for gravity models with inequality constraints, and the method considered by Macgill for partially constrained gravity models. The convergence of all of these methods follows from Bregman's general result.

ChemInform Abstract: EFFECT OF BROWNIAN DIFFUSION ON CHEMICAL KINETICS

A new method of theoretical prediction of the kinetic rate constants of fast chemical reactions in solutions is presented. It takes into account the effect of finite diffusive displacements of the reacting molecules. The approach is based on the solution of the steady-state Fokker–Planck equation by the moments method of Grad developed in the theory of coagulation of aerosol particles. A comparison of the predicted rate constants with the experimental data provided by Schuh and Fischer for the self-reaction of tert-butyl radicals in n-alkanes shows a good correspondence.

Effect of brownian diffusion on chemical kinetics

AbstractA new method of theoretical prediction of the kinetic rate constants of fast chemical reactions in solutions is presented. It takes into account the effect of finite diffusive displacements of the reacting molecules. The approach is based on the solution of the steady‐state Fokker–Planck equation by the moments method of Grad developed in the theory of coagulation of aerosol particles. A comparison of the predicted rate constants with the experimental data provided by Schuh and Fischer for the self‐reaction of tert‐butyl radicals in n‐alkanes shows a good correspondence.

Condensation rate of trace vapor on Knudsen aerosols from the solution of the Boltzmann equation

The condensation rate of trace vapor on a stationary, spherical aerosol particle was studied theoretically. Under the assumption that flow of the vapor toward the droplet does not disturb the Maxwellian distribution of molecules of the multicomponent bath gas, the stationary Boltzmann equation was solved analytically by the method of Grad. Heat effects associated with the phase transition were neglected. The expressions obtained for the condensation rate and density profile are functions of the Knudsen number, the accommodation coefficient, and the ratios of masses of vapor and bath gas molecules. The limiting behavior of the solutions for very large and very small Knudsen numbers was studied as well.

Transport equations for multispecies plasmas based on individual bi-Maxwellian distributions

The authors have derived a closed system of transport equations for an anisotropic plasma of arbitrary degree of ionisation. The system is based on an anisotropic bi-Maxwellian species distribution function, and therefore, should provide a better description of flow conditions characterised by large temperature anisotropies. The method used to derive the transport equations is an extension of Grad's method and corresponds to a 16-moment approximation for the species distribution function. The relevant collision terms were calculated for an arbitrary inverse-power interaction potential and for a resonant charge exchange interaction between an ion and its parent neutral. In the collisionless limit, the system of equations reduces to the collisionless transport equations which include the effects of collisionless 'viscosity' and heat flow.

On transient relativistic thermodynamics and kinetic theory. Il

This paper, the sequel to Stewart (1977), discusses the relation between transient relativistic thermodynamics and kinetic theory in the formulation of Israel (1972, 1976) which differs (mathematically) from that of the previous paper. The difference is that the moment expansions are truncated in different ways. However, identical results for the characteristic velocities of thermal transients are obtained, and this lends credence to both approaches. In this approximation at least, the Grad method of moments is insensitive to the truncation procedure.

On the relativistic Boltzmann equation

Using harmonic analysis techniques, we construct orthogonal polynomials on Minkowski’s hyperboloid in order to interpret in the framework of the theory of group representation the tensorial polynomials introduced by Marle for the resolution of the relativistic Boltzmann equation by the Grad method. Calculations are performed in an adapted coordinate system by using zonal spherical functions on the hyperboloid of Minkowski, which as a matter of fact are Jacobi polynomials. Moreover, the homogeneous components of the Maxwell distribution function give an interpretation of Juttner’s formulas and, consequently, of the thermodynamic quantities of the fluid.

Brownian coagulation in the transition regime

The rate of Brownian coagulation of aerosols is studied theoretically. The particle distribution function is assumed to be governed by the Fokker-Planck equation, which is solved analytically by the 13-moment method of Grad. An expression for the coagulation parameter β( a i , a j ) is obtained from the solution over all ranges of particle radii a i and a j . The coagulation rates predicted by the solution agree closely with those predicted by the empirical expression of Fuchs as well as with those recently obtained experimentally by Chatterjee, Kerker, and Cooke.

Variational principles for relativistic transport coefficients

Variational expressions for the transport coefficients of a one-component, relativistic gas are derived from the linearized relativistic Boltzmann equation for both quantum and classical gases. These expressions depend on functions χ of the energy of the particles comprising the gas in such a way that: a) if χ differs from a solution of the linearized Boltzmann equation by ε, then the value of the variational expression calculated with this χ differs from the true value of the corresponding transport coefficient by ε 2; and b) the value of the variational expression is always less than this true value. It is shown that values of the transport coefficients obtained by expanding χ in a particular set of orthogonal polynomials and keeping only the first nontrivial term in the expansion are equivalent to those obtained using the Grad method of moments. It follows therefore that values obtained using this later method represent lower bounds on the true values. We also show that one can obtain simple, closed-form expressions for the various transport coefficients corresponding to an arbitrary number of terms in an expansion of the trial function χ in the above-mentioned set of orthogonal polynomials. Finally we point out that all of our results can be carried over to the nonrelativistic case by taking the limit c → ∞.

Solution of the Boltzmann equation at the upstream and downstream singular points in a shock wave

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocityu, rather than the spatial co-ordinatex, is used as the independent variable, and an equation for ∂f/∂uat a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.

La méthode de Grad des 20 moments et les invariants CGL. Application au vent solaire

The experimental data of the quiet solar wind appear to be the result of perturbations over the simple expansion of the solar plasma. The distribution function obtained by the 20 moments Grad method, but simplified by the parallel flux has been proposed by Whang to interpret the experimental data of Hundhausen. This distribution has been used to close the moments system of a simple expansion which gives the perpendicular and parallel contribution to the parallel heat flux. The terms in the development which immediately follow the CGL invariants are obtained. The numerical value of the heat flux agrees with Whang's and Hundhausen's, and the corrections to the invariants are in agreement with the kinetic results of Hollweg and HD of Fridman if the closure of the moments system is made by the highly supersonic flux condition.

Independent Investigations Concerning Bulk Viscosity in Relativistic Homogeneous Isotropic Cosmologies

view Abstract Citations (44) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Independent Investigations Concerning Bulk Viscosity in Relativistic Homogeneous Isotropic Cosmologies Nightingale, J. D. Abstract Formulae are found (in those cosmologies that involve an isotropic and homogeneous expansion) for the perturbed Weltradius, energy density, and age, when bulk viscosity is considered nonnegligible. The coefficient of bulk viscosity is found via the relativistic Boltzmann equation, following Anderson's work on the Grad method of moments. The specific form of the coefficient is given, and it is found that it does not depend upon the Hubble constant. It is also estimated that this particular coefficient (different from Weinberg's) could not be responsible for the entropy associated with the background radiation. Some of the results overlap and verify (by different methods) some of the work done by Weinberg, Treciokas and Ellis, and Neugebauer and Strobel. Subject heading: cosmology Publication: The Astrophysical Journal Pub Date: October 1973 DOI: 10.1086/152400 Bibcode: 1973ApJ...185..105N full text sources ADS |

Flow of a relaxed gas in the vicinity of a solid surface

Behavior of a polyatomic relaxed gas in the vicinity of a solid surface was studied. The case in which the size of the relaxation zone exceeds considerably the mean free path between the elastic collisions (suppressed exchange of translational and internal energies) was considered. A smooth validity of the asymptotic expansion for the distribution function with defined assumptions was indicated. A solution for the zero-approximation equation and boundary conditions for the surface flow, based on this solution and generalized model for a diffusive gas reflection from the surface were derived. The latter problem was partially studied in [1] by the Grad method and in [2] in terms of an analysis of the temperature jump.