Boltzmann-Enskog equation

A similar expansion can be carried out for the inelastic Boltzmann-Enskog equation (Garzo et al, 2007 Jenkins Richman, 1985) to find an approximation for Tp that will be valid when TcoI is much smaller than all other characteristic times of the system. 

The approximation wherein one retains the first term in the memory function expression (105) is of special interest because it leads to a kinetic equation that is closely related to the Boltzmann-Enskog equation in transport theory. In this section we will investigate this particular approximation in some detail not only from the standpoint of further analytical analysis but also from the standpoint of practical calculations. We will see that within certain limitations the approximation results in a reasonably realistic description of dense gases and liquids, and in this sense represents the first step in a systematic microscopic calculation. 

This extended Boltzmann equation is called the Enskog equation. 

The form of the Boltzmann-Enskog collision operator is thus specified out task is to find its generalization. We denote the general collision operator by /l (1, 1 t ), where we have allowed for the possibility that it may be nonlocal in time as well as space. The general kinetic equation may then be written as... 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

From our discussions of the Boltzmann equation and the Enskog equation, we know that a quantity of special interest to us is/(r, v, i), which we now interpret as the ensemble average value of the number of particles in dr d about r, v at time t. Since the F-space is properly formulated in r and p instead of... 

Chapman-Enskog Solution to the Boltzmann Transport Equation... 

A3.1.3.2 THE CHAPMAN-ENSKOG NORMAL SOLUTIONS OF THE BOLTZMANN EQUATION... 

The Burnett Expansion.—The Chapman-Enskog solution of the Boltzmann equation can be most easily developed through an expansion procedure due to Burnett.15 For the distribution function of a system that is close to equilibrium, we may use as a zeroth approximation a local equilibrium distribution function given by the maxwellian form ... 

Chapman-Enskog Solution.—The solution of the Boltzmann equation obtained by Chapman and Enskog involves the assumption... 

Block relaxation, 61 Bogoliubov, N., 322,361 Boltzmann distribution, 471 Boltzmann equation Burnett method of solution, 25 Chapman-Enskog method of solution, 24... 

In principle, one should solve the Boltzmann equation Eq. (65) in order to arrive at explicit expressions for the pressure tensor p and heat flux q, which proves not possible, not even for the simple BGK equation Eq. (11). However, one can arrive at an approximate expression via the Chapman Enskog expansion, in which the distribution function is expanded about the equilibrium distribution function fseq, where the expansion parameter is a measure of the variation of the hydrodynamic fields in time and space. To second order, one arrives at the familiar expression for p and q... 

We begin with a simple, physical derivation of the Boltzmann equation, which is the starting point in obtaining the rigorous transport properties. Following this discussion, the theory of Chapman and Enskog [60,114] is presented. 

Solution of the Boltzmann equation gives the velocity distribution function throughout the gas as it evolves through time, for example, due to velocity, temperature, or concentration gradients. A practical solution to the Boltzmann equation was found by Enskog [114], which is discussed in the next section. This approach is used to calculate rigorous expressions for gas transport coefficients. 

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