Chapman—Enskog method

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

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... 

Reduction of kinetic theory to fluid mechanics is historically the first example of a successful reduction of a mesoscopic dynamical theory to a more macroscopic dynamical theory. The method (called the Chapman-Enskog method) that was invented by Chapman and Enskog for this particular reduction remains still a principal inspiration for all other types of reduction (see, e.g.,. Gorban and Karlin, 2003, 2005), Yablonskii et al., 1991). In this example we briefly recall the geometrical viewpoint of the Chapman-Enskog method. We shall also illustrate the point (IV)... 

In the present contribution, we propose mathematical description for the chemical kinetics in gas flows on the basis of the Chapman-Enskog method, generalized for strongly non-equilibrium reacting gas mixtures. 

In the frame of the method proposed in Kustova Nagnibeda (1998) Nagnibeda Kustova (2009) for the solution of Eqs. (2), the distribution functions are expanded in a power series of the small parameter e. The peculiarity of the modified Chapman-Enskog method is that the distribution functions and macroscopic parameters are determined by the collision invariants of the most frequent collisions. Under condition (1), the set of collision invariants contains the invariants of any collision (momentum and total energy) and the additional invariants of rapid processes. In our case, these additional invariants are any variables indepiendent of the velocity and internal energy and depending arbitrary on chemical species c because chemical reactions are supposed to be frozen in rapid processes This set of collision invariants provides the following set of macroscopic parameters for a closed flow description number densities of species Tic r,t) (c = 1,..., L), gas velocity v(r, f) and temperature T(r,f). 

It is obvious that the system of governing equations in the multi-tempierature approach is considerably simpler than the corresponding system in the state-to-state approach, since it contains much fewer equations. In the zero-order approximation of the Chapman-Enskog method, the system of governing equations takes the form typical for inviscid non-conductive flows. In this case equations (85), (86) read ... 

Alexeev, B., Chikhaoui, A. Grushin, I. (1994). Application of the generalized Chapman-Enskog method to the transpxjrt-coefficients calculation in a reacting gas mixture, Phys. Beview E 49 2809. 

Equations (6.85) and (6.86) refiect the local equilibrium nature of the solution to Eq. (6.83). It is, therefore, seen that the so-called Chapman-Enskog method of the solution is based on an expansion about local equilibrium conditions. Pitfalls of this approach have been previously noted. Writing, without loss of generality. 

We have at several points noted similarities to the Chapman-Enskog method in kinetic theory. This resemblance actually extends quite deep, and in order to demonstrate it in detail we obtain the normal solution to the Boltzmann equation in a form which will facilitate comparisons. For simplicity we consider linear processes only and therefore write the Boltzmann distribution function /5(q,c,<) as... 

The Chapman-Enskog method may be applied directly to Eq. (223) to obtain/g as an expansion in powers of the operator V, and in this manner one can verify that the solution obtained is in fact the usual normal solution, in the linear approximation. On the other hand, the closed form (223) may be useful for rapidly varying processes to which the Chapman-Enskog expansion is not applica.ble. 

One of the solution approaches of the Boltzmann equation is Chapman Enskog method. Here, the velocity distribution function is expanded into a perturbation series with the Knudsen number being the small parameter. 

For the first-order solution of the Boltzmann equation using Chapman-Enskog method, we get the constitutive laws of the N-S equation as... 

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