John D. Ramshaw

Self-Consistent Effective Binary Diffusion in Multicomponent Gas Mixtures

The correct treatment of diffusion in multicomponent gas mixtures requires solution of a linear system of equations for the diffusive mass fluxes relative to the mass-averaged velocity of the mixture. Effective binary diffusion approximations are often used to avoid solving this system. These approximations are generally internally inconsistent in the sense that the approximate diffusion fluxes do not properly sum to zero. The origin of this inconsistency is identified, and a general procedure for removing it is presented. This procedure applies equally to concentration, forced, pressure, and thermal diffusion, either separately or in combination. It is used to obtain a self-consistent effective binary diffusion approximation in which the diffusive mass fluxes properly sum to zero and all four types of diffusion are simultaneously accounted for.

Hydrodynamic Theory of Multicomponent Diffusion and Thermal Diffusion in Multitemperature Gas Mixtures

A phenomenological theory is developed for multicomponent diffusion, including thermal diffusion, in gas mixtures in which the components may have different temperatures. The theory is based on the hydrodynamic approach of Maxwell and Stefan, as extended and elaborated by Furry [1] and Williams [2]. The present development further extends these earlier treatments to multiple temperatures and multicomponent thermal diffusion. The resulting diffusion fluxes obey generalized Stefan-Maxwell relations which include the effects of ordinary, forced, pressure, and thermal diffusion. When thermal diffusion is neglected, these relations have the same form as the usual single-temperature ones, except that mole fractions are replaced by pressure fractions (i.e., ratios of partial pressures to total pressure). The binary ap.d thermal diffusion coefficients are given in terms of collision integrals. Single-temperature systems and binary systems are treated as special cases of the general theory. A self-consistent effective binary diffusion approximation for multitemperature systems is presented.

Existence of the Dielectric Constant in Rigid-Dipole Fluids: The Direct Correlation Function

The question of whether the dielectric constant e exists (is well defined) for a finite fluid system of rigid dipolar molecules is reconsidered and reformulated. It is found that this question can most simply be expressed in terms of the behavior of the position‐ and orientation‐dependent direct correlation function c(r1, ω1; r2, ω2). It is shown that e exists if c satisfies the following two conditions: (a) c∼−φ/ kT for |r1−r2| > σ, where φ is the dipole‐dipole potential and σ is a length which is large microscopically but small macroscopically. (b) c(r1, ω1; r2, ω2) is of the form c8(|r1− r2|)+ F(r1−r2):e(ω1) e(ω2) for |r1−r2| < σ, where e(ω) is the unit vector with orientation ω. An explicit (and new) expression for e in terms of c is automatically obtained; its applicability is ensured if the above conditions are satisfied. These results lend new intuition and insight into the question of the existence of e, and suggest a promising approach for future investigations of this question.

Friction-Weighted Self-Consistent Effective Binary Diffusion Approximation

The self-consistent effective binary diffusion (SCEBD) approximation for multicomponent diffusion in gas mixtures is reconsidered and reformulated. The new formulation is based on the fact that a suitable rearrangement of the Stefan-Maxwell equations pro_vides an exact expression for the complementary mean velocity a; for species i as a weighted average of the velocities of all the other species. The coefficients in a; are normalized friction coefficients which are simply related to the true binary diffusion coefficients. A simple factorized bilinear approximation to the friction coefficients then yields approximate species diffusion fluxes identical in form to those of a previous intuitive treatment [ 4], together with a new relation between the previously ambiguous weighting factors wi and the friction coefficients. This relation places the SCEBD approximation on a firm foundation by providing a rational basis for determining the W;. A simple further approximation based on the known form ofthe friction coefficients for hard spheres yields w; = (const.)pJfo., where P; and M; are respectively the mass density and molecular weight of species i. These weighting factors are shown to produce considerably more accurate diffusion velocities than the conventional choice w; = (const.)pJM;.

Fluid dynamics and energetics in ideal gas mixtures

The generalization of fluid dynamics from pure to multicomponent fluids (fluid mixtures composed of different components or species) requires the introduction of new concepts, some of which are rather subtle and are less widely appreciated than they deserve to be. The purpose of this paper is to provide a simple didactic introduction to some of these concepts based on a detailed analysis of the equations governing the flow of ideal gas mixtures. The treatment is based entirely on a continuum description and makes no explicit use of the kinetic theory of gases. We include a straightforward and physically transparent derivation of the additional heat flux arising from the relative motion of the different species, and show why this flux involves species enthalpies rather than energies. Some of the concepts are reminiscent of those used in turbulence modeling, and these analogies are briefly discussed.

Debye–Hückel theory for rigid‐dipole fluids

The dipolar analog of classical linearized Debye–Huckel theory is formulated for a finite fluid system of arbitrary shape composed of rigid polar molecules. In contrast to the ionic case, the dipolar Debye–Huckel (DDH) theory is nonunique due to an inherent arbitrariness in the choice of a local field E*. This nonuniqueness is expressed in terms of a parameter ϑ related to the ellipticity of the spheroidal cavity used to define E*. The theory then leads to an expression for the direct correlation function c (ϑ) as a function of ϑ. Only the short‐range part of c (ϑ) depends upon ϑ; the long‐range part equals −φd/kT for all ϑ, where φd is the bare dipole–dipole potential. This result for c (ϑ) implies the existence of the dielectric constant e for all ϑ and leads to a formula for e (ϑ). The DDH results for c (ϑ) and e (ϑ) are formally identical to the ’’mean‐field’’ results of Ho/ye and Stell (obtained for an infinite system by a γ→0 limiting procedure) in which ϑ represents a ’’core parameter.’’

Simple Approximation for Thermal Diffusion in Ionized Gas Mixtures

A simple approximation for thermal diffusion in gas mixtures was recently proposed [1]. This approximation was based upon relations valid for rigid spheres. It is therefore appropriate for molecules with steep repulsive potentials, but not for ionized species interacting via the Coulomb potential. Here we formulate an analogous approximation for ionized species and free electrons. The resulting thermal diffusion coefficients differ in sign from those for hard molecules.

Self-Consistent Effective Binary Interaction Approximation For Strongly Coupled Multifluid Dynamics

An improved self-consistent effective binary diffusion approximation for multicomponent diffusion was recently described [1]. Here we develop an analogous self-consistent effective binary interaction (SCEBI) approximation for simplifying multifluid dynamical descriptions in which each fluid is strongly coupled to the other fluids by pairwise frictional forces. The net drag force on each fluid is the summation of the drag forces due to each of the other fluids. This summation is approximated by a single term proportional to the velocity of the fluid in question relative to an appropriately weighted average velocity. This approximation permits an explicit numerical solution for the fluid velocities even when the drag terms are evaluated at the advanced time level to avoid explicit stability restrictions on the time step. Introduction and Summary An improved self-consistent effective binary diffusion (SCEBD) approximation for multicomponent diffusion was recently described [1]. This approximation is used to obtain an explicit expression for the diffusion velocities without solving the full StefanMaxwell equations. Our purpose here is to develop an analogous approximation within the context of multifluid dynamical models in which each fluid satisfies its own momentum equation and is strongly coupled to the other fluids by pairwise frictional interaction forces. Such models are used, for example, in plasma physics [2] and multiphase flow [3]. In models of this type, the net drag force on each fluid is obtained by summing up the individual pairwise drag forces due to each of the other fluids, which are presumed to be proportional to the relative velocities of the fluid pairs. We shall approximate this summation by a single term proportional to the velocity of the fluid in question relative to an appropriately weighted average velocity. This approximation will be referred to as the self-consistent effective binary interaction (SCEBI) approximation. It is self-consistent in the sense that it introduces no net force into the total momentum of the multifluid mixture, which would of course be undesirable. 1>This work was performed under the auspices ofthe U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48. J. Non-Equilib. Thermodyn. · 19.98 ·Vol. 23 ·No.2

Dielectric constant in fluids of classical deformable molecules

Classical statistical mechanical description of dielectric fluids is further discussed. Dipole moment correlations of polarizable molecules are discussed.(AIP)

On the reduction of many‐body dielectric theories to the Onsager equation

An approximate theory for the dielectric constant e of a dense polar fluid was derived by Ramshaw, Schaefer, Waugh, and Deutch (RSWD). In the present article, the RSWD theory is generalized and made rigorous by another method of derivation. The result is a rigorous expression for e which differs from the RSWD expression by the presence of a fluctuation term. Both the rigorous expression and the RSWD expression are then specialized to the Onsager model. It is found that the rigorous expression for e reduces to the Onsager equation, but that the RSWD expression does not because the fluctuation term is nonzero (and nonnegligible) for the Onsager model. The well‐known discrepancy between the Onsager equation and the theory of Harris and Alder is found to have the same origin.