Enskog

The Chapman-Enskog solution of the Boltzmaim equation [112] leads to the following expressions for the transport coefficients. The viscosity of a pure, monatomic gas can be written as... 

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

When ions move under equilibrium conditions in a gas and an external electric field, the energy gained from the electric field E between collisions is lost to the gas upon collision so that the ions move with a constant drift speed v = KE. The mobility K of ions of charge e in a gas of density N is given in tenns of the collision integral by the Chapman-Enskog fomuila [2]... 

Chapman-Enskog (Bird et al.) and Wilke-Lee The inherent assumptions of these equations are quite restrictive (i.e., low density. 

Binary Mixtures—Low Pressure—Polar Components The Brokaw correlation was based on the Chapman-Enskog equation, but 0 g and were evaluated with a modified Stockmayer potential for polar molecules. Hence, slightly different symbols are used. That potential model reduces to the Lennard-Jones 6-12 potential for interactions between nonpolar molecules. As a result, the method should yield accurate predictions for polar as well as nonpolar gas mixtures. Brokaw presented data for 9 relatively polar pairs along with the prediction. The agreement was good an average absolute error of 6.4 percent, considering the complexity of some of... 

Riazi-Whitson They presented a generahzed correlation in terms of viscosity and molar density that was apphcable to both gases and liqmds. The average absolute deviation for gases was only about 8 percent, while for liquids it was 15 percent. Their expression relies on the Chapman-Enskog correlation [Eq. (5-194)] for the low-pressure diffusivity and the Stiel-Thodos correlation for low-pressure viscosity ... 

Chapman and Enskog (see Chapman and Cowling, 1951) made a semi-empirical study of tire physical properties of gases using the Lennard-Jones... 

Assuming that A << /o and that /o varies appreciably only over distances x L, it is easy to show that A//o —XjL, where A is the mean free path length i.e. /o is a good approximation if the characteristic wavelengths of p, T and u are all much greater than the mean free path. The exact solution / can then be expanded in powers of the factor X/L. This systematic expansion is called the CAia.pma.n-Enskog expansion, and is the subject of the next section. 

Chapman-Enskog Expansion As we have seen above, the momentum flux density tensor depends on the one-particle distribution function /g, which is itself a solution of the discrete Boltzman s equation (9.80). As in the continuous case, finding the full solution is in general an intractable problem. Nonetheless, we can still obtain a useful approximation through a perturbative Chapman-Enskog expansion. 

Enskog expansion for fp and this definition for f may be written in the following... 

Bather than using the Chapman-Enskog procedure directly, we shall employ the technique of Burnett,12 which involves an expansion of the distribution function in a set of orthogonal polynomials in particle-velocity space. 

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... 

Chapman-Enskog solution, 35 coefficicent equations, 28 derivation from Liouville s equation, 41... 

The correction of mean free path, hi by the nanoscale effect function results in a smaller mean free path, or a smaller Knudsen number in other word. As a matter of fact, a similar effect is able to be achieved even if we use the conventional definition of mean free path, / = irSn, and the Chapmann-Enskog viscosity equation, /r = (5/16)... 

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)... 

The Chapman-Enskog method has been used to solve for steady state tracer diffusion (. ). According to the method the singlet distribution function for the diffusing species 1, present In a trace amount n nj, 1 1) In an otherwise equilibrium fluid. Is approximated by... 

The Chapman-Enskog theory of flow In a one-component fluid yields the following approximation to the momentum balance equation (Jil). 

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. 

Two predictions of the LADM for the effective viscosity are shown In Table II. The first was made by using the Enskog hard-sphere theory for the calculation of the local viscosities. It agrees qualitatively with the simulation result In that It predicts a large decrease of the effective viscosity as a result of the density structure. For the second prediction the local... 

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