Kinetic theory dense fluids

We can describe irreversibility by using the kinetic theory relationships in maximum entropy formalism, and obtain kinetic equations for both dilute and dense fluids. A derivation of the second law, which states that the entropy production must be positive in any irreversible process, appears within the framework of the kinetic theory. This is known as Boltzmann s H-theorem. Both conservation laws and transport coefficient expressions can be obtained via the generalized maximum entropy approach. Thermodynamic and kinetic approaches can be used to determine the values of transport coefficients in mixtures and in the experimental validation of Onsager s reciprocal relations. 

Ding and Gidaspow [16], for example, derived a two-phase flow model starting with the Boltzmann equation for the distribution function of particles and incorporated fluid-particle interactions into the macroscopic equations. The governing equations were derived using the classical concepts of kinetic theory. However, to determine the constitutive equations they used the ad hoc distribution functions proposed by Savage and Jeffery [65]. The resulting macroscopic equations contain both kinetic - and collisional pressures but only the collisional deviatoric stresses. The model is thus primarily intended for dense particle flows. 

The main advantage associated with the kinetic theory approach for dense suspensions is the appearance of two extra pressure terms in addition to the interstitial fluid phase pressure, one kinetic pressure tensor accounting for the transport phenomena due to the translational particle movement and one collisional pressure tensor accounting for the transport phenomena due to particle collisions. 

Hence it follows that for multicomponent reactive mixtures the problem has to be re-considered deriving the governing equations again from scratch, this time in terms of an extended distribution function for molecules of a particular type. The multicomponent kinetic theory for a dilute gas has been considered by Hirschfelder et al [28]. Tham and Gubbins [82] extended the Enskog theory of dense, rigid sphere fluids to multicomponent systems. 

Tham MK, Gubbins KE (1971) Kinetic theory of multicomponent dense fluid mixtures of rigid spheres. J Chem Phys 55 268-279... 

The kinetic theory of condensed-phase chemical reactions is a direct outgrowth of kinetic theory and mode coupling descriptions of dense, simple fluids, which have been developed primarily in the past 10 years. This work in turn relies on an older body of literature, but we shall, when possible, draw parallels with the more recent interpretations of liquid-state dynamics. At present there are a variety of techniques available for constructing kinetic equations that are useful for describing dense, simple, nonreacting liquids. These range from approaches based on the dynamic hierarchy ... 

These ring collision events are now a familiar part of the kinetic theory description of dynamic processes in simple dense fluids. A brief comparison of the theory for the velocity autocorrelation function with that for the chemically reacting fluid will help motivate our description. Recent developments in the theory of the velocity autocorrelation function have arisen out of an attempt to understand the slow t power law decay observed by Alder and Wainwright in a computer simulation of a dense hard-sphere fluid. This work also showed that the translational motion of a small hard sphere in a fluid of similar hard spheres has a significant collective (hydro-dynamic) component. On the theoretical side, this type of behavior was discussed from the kinetic theory point of view in terms of the ring collision events described above and provided a microscopic basis for the introduction of collective effects. In addition, it was shown that mode... 

The kinetic theory result for the rate kernel given above provides a much more explicit and tractable description of the dynamic processes that contribute to the rate coefficient in a dense fluid. We analyze these processes in Section X. 

The specific examples chosen in this section, to illustrate the dynamics in condensed phases for the variety of system-specific situations outlined above, correspond to long-wavelength and low-frequency phenomena. In such cases, conservation laws and broken symmetry play important roles in the dynamics, and a macroscopic hydrodynamic description is either adequate or is amenable to an appropriate generalization. There are other examples where short-wavelength and/or high-frequency behaviour is evident. If this is the case, one would require a more microscopic description. For fluid systems which are the focus of this section, such descriptions may involve a kinetic theory of dense fluids or generalized hydrodynamics which may be linear or may involve nonlinear mode coupling. Such microscopic descriptions are not considered in this section. 

The physics of motion in a layer adjacent to the solid surface is quite different from the bulk motion described by the Stokes equation. This generates effective slip at a microscopic scale comparable with intermolecular distances. The presence of a slip in dense fluids it is confirmed by molecular dynamics simulations [17, 18] as well as experiment [19]. The two alternatives are shp conditions of hydrodynamic and kinetic type. The version of the slip condition most commonly used in fluid-mechairical theory is a linear relation between the velocity component along the solid surface and the shear stress... 

Chapter 4. Renormalized Kinetic Theory of Dense Fluids Gene F. Mazenko and Sidney Yip... 

The principal advantage of the time correlation function method is that it provides a new set of microscopic functions for a fluid, the time correlation functions, which can be studied directly by experimental observations of the fluidt or by computer-simulated molecular dynamics. The time correlation functions depend even more sensitively on the microscopic properties of the fluid molecules than the transport coefficients, which are expressed as time integrals of the correlation functions. Thus, a further test of kinetic theory has been found it must not only lead to expressions for the transport coefficients for dilute and dense gases that are in agreement with experiment, but also describe the dependence of the time correlation functions on both time and the density of the gas. One of the principal successes of kinetic theory is that it provides a quantitatively correct description of the short- and long-time... 

It has been recognized for some time that a fully microscopic treatment of the dynamics of fluids is feasible at low densities. For dilute gases methods based on density expansion prove to be useful and a Boltzmann-like equation can be used to calculate time correlation functions. " The notion that microscopic or kinetic theory calculations could be formulated for dense fluids is much more recent. The difficulty with dense systems is that there is no small parameter in the problem. Consequently a direct density expansion where one computes the first few corrections to the low-density result is not likely to give useful results. At first sight things are worse than this because the first few density corrections are not only mathematically exhausting, but the... 

The purpose of this chapter is to explain how a tractable microscopic theory of dense fluids can be constructed that takes into account the exact static correlation functions and the effects of correlated dynamical processes. There are two separate stages in the development of such a theory. The first stage is to construct a flexible formalism that is useful for making approximations applicable to dense fluids. In the second stage, which follows after an appropriate collision kernel for a kinetic equation has been derived, one is concerned with solving this kinetic equation to find the time correlation functions of interest. We will consider both stages in our discussions. 

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