Enskog’s theory

Consider a fluid of molecules Interacting with pair additive, centrally symmetric forces In the presence of an external field and assume that the colllslonal contribution to the equation of motion for the singlet distribution function Is given by Enskog s theory. In a multicomponent fluid, the distribution function fi(r,Vj,t) of a particle of type 1 at position r, with velocity Vj at time t obeys the equation of change (Z)... 

In Figure 10, we present flow velocity predictions of the high density approximation, Equations 32 - 33, 38 and 39, of Davis extension of Enskog s theory to flow In strongly Inhomogeneous fluids (1 L). The velocity profile predicted In this way Is also plotted In Figure 10. The predicted profile, the simulated profile, and the profile predicted from the LADM are quite similar. 

The theory which forms the basis for discussions of the transport phenomena in dense gases is Enskog s kinetic theory for a pure gas made up of rigid spheres (E3, C3, Chapter 16 Hll, 9.3). To date, this theory in one of several modifications is the best theory available for calculating the temperature and density dependence of the transport coefficients. Recently Enskog s theory has been extended to a pure gas made up of nonrigid molecules by Curtiss and Snider (CIO, C14). Enskog s theory has also been extended to binary gas mixtures by Thorne (C3, p. 292). 

An alternate approach has been attempted for describing the transport phenomena in dense gas and liquid systems by means of the methods of nonequilibrium statistical mechanics, as developed by Kirkwood (K7, K8) and by Born and Green (B18, G10). Although considerable progress has been made in the development of a formal theory, the method does not at the present time provide a means for the practical calculation of the transport coefficients. Hence in this section we discuss only the applications based on Enskog s theory. 

However, the perfect-gas assumption does not always hold for micro- and nanoscale gas flows. When the gas density is high or the temperature is low, intermolecular attractions become significant as the effect of denseness changes the characteristics of the gas flow. According to Enskog s theory, the molecular mean free path in a dense gas, X, is given by [12]... 

Since ordinary kinetic theory approximations to Z> , such as Enskog s theory, do not include hydrodynamics effects at all, it is conceivable that, with an appropriate choice of k D could be identified in Eq. (148) as Enskog s approximation Z>e to Z>. The mode-mode part of D would then represent the enhancement of D over D, as lately discussed in the literature, due to hydrodynamic effects. The proper tool for the study of this point is kinetic theory, not mode-mode coupling, but we mention the subject due to its extreme interest. 

In Figure 5.3, values for B calculated in this way (MET-I) for the Lennard-Jones (12-6) potential are shown to have the same general temperature dependence as the experimental data (there are other modifications of Enskog s theory which are not considered here.). In Figure 5.4, an analogous comparison is shown for thermal conductivity. The calculated values (MET-I) underestimate the experimental results when the association correction (see Table 5.1) is omitted from the MET approach but are... 

Basically Enskog s kinetic theory extension consists in the introduction of corrections that account for the fact that for dense gases the molecular diameter is no longer small compared with the average intermolecular distance. 

Enskog s dense gas theory for rigid spheres is also used as basis developing granular flow models. The modifications suggested extending the dense gas kinetic theory to particulate flows are discussed in chap 4. 

C. F. Curtiss communicated to the author that Eq. (49) may be deduced from D. Enskog s treatment of gas diffusion. Further applications to the ordinary kinetic gas theory, and to the electron gas theory, were successfully made by Ljimggren, through calculations of on the basis of molecular kinetics. 

A temperature-dependent expression for gas viscosity of a pure mono-atomic gas is given by Chapman-Enskog s kinetic theory as... 

To derive the Enskog s equation for a dense gas, Enskog [37] did departure from the kinetic theory of dilute mono-atomic gases that is described by the Boltzmann... 

Kremer GM, Rosa E Jr (1988) On Enskog s dense gas theory. I. The method of moments for... 

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

Chapman and Enskog s kinetic theory (Hirschfelder, Curtiss, and Bird, 1954)... 

The connection between the classical and quantum formulations of the transport coefficients has been studied by applying the WKB method to the quantum formulation of the kinetic theory (B16, B17). In this way it was shown that at high temperatures the quantum formulas for the transport coefficients may be written as a power series in Planck s constant h. When the classical limit is taken (h approaches zero), then the classical formulas of Chapman and Enskog are obtained. 

In the formulations developed from the renormalized kinetic theory approach, these self-consistencies were avoided either by using values obtained from computer simulation and experiments or by using some exactly known limiting values for the transport coefficient. For example, in the treatment of Mazenko [5-7], and of Mehaffey and Cukier s [8] the transport coefficients are replaced by their Enskog values. In the theory developed by Sjogren and Sjolander [9], the velocity autocorrelation function is required as an input that was obtained from the computer simulated values. This limits the validity of the theories only to certain regimes and for certain systems where the experimental or computer-simulated results are available. 

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