Kinetic theory of dense gases

Choh, S., and Uhlenbeck, G., The Kinetic Theory of Dense Gases, University of Michigan, 1958. 

Further development of the kinetic theory of dense gases and liquids. 

For a dense system of hard, smooth, and elastic spherical particles, a transport theorem based on the analogy of the kinetic theory of dense gases [Reif, 1965] may be derived. Define an ensemble average of any property xjr of a particle as... 

The kinetic theory of dense gases began with the work of Enskog, who in 1922, generalized Boltzmann s derivation of the transport equation to apply it to a dense gas of hard spheres. Enskog showed that for dense gases there is a mechanism for the transport of momentum and energy by means of the intermolecular potential, which is not taken into account by the Boltzmann equation at low densities, and he derived expressions for the transport coeffi-... 

Yet even this systematic theory is not free of difficulties, and their resolution remains one of the central problems in the kinetic theory of dense gases. ... 

Problems Connected with the Kinetic Theory of Dense Gases... 

All of the problems discussed above in connection with the Boltzmann equation carry over to the kinetic theory of dense gases. In addition to the obvious analogs of the problems mentioned above, the following problems need to be solved. 

The equations of motion for granular flows have been derived by adopting the kinetic theory of dense gases. This approach involves a statistical-mechanical treatment of transport phenomena rather than the kinematic treatment more commonly employed to derive these relationships for fluids. The motivation for going to the formal approach (i.e., dense gas theory) is that the stress field consists of static, translational, and collisional components and the net effect of these can be better handled by statistical mechanics because of its capability for keeping track of collisional trajectories. However, when the static and collisional contributions are removed, the equations of motion derived from dense gas theory should (and do) reduce to the same form as the continuity and momentum equations derived using the traditional continuum fluid dynamics approach. In fact, the difference between the derivation of the granular flow equations by the kinetic approach described above and the conventional approach via the Navier Stokes equations is that, in the latter, the material properties, such as viscosity, are determined by experiment while in the former the fluid properties are mathematically deduced by statistical mechanics of interparticle collision. 

For the exact evaluation of the two-monomer contribution Bf the Rainwater-Friend theory includes results from the kinetic theory of dense gases (see, for example, Cohen 1969), which means that direct kinetic as well as collisional transfer effects are considered. In order to calculate reliable values for B Rainwater (1984) extended earlier results of Snider Curtiss (1958), Hoffman Curtiss (1965) and Bennett Curtiss (1969) so that the dynamics of the two particles for a more realistic potential (here the Lennard-Jones (12-6) potential) were included. At the same time, the effects of bound states have been excluded from the evaluation of the complicated set of integrals that defines the two-monomer contribution (see Rainwater 1984). 

Hoffman, D. K. Curtiss, C. F. (1965). Kinetic theory of dense gases. V. Evaluation of the second transport virial coefficients. Phys. Fluids, 8,890-895. 

The collisional pressure tensor represents the instantaneous momentum transfer at binary particle collisions, over the distance separating the centers of the two colliding particles. The pressure tensor closure is derived based on an extension of the kinetic theory of dense gases. The collisional pressure tensor is thus the second out of the two pressure tensor components that is calculated by use of the KTGF. 

Variable in simplified PD algorithm solution in SQMOM Enskog s volume correction function in kinetic theory of dense gases (—)... 

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