Chapman-Enskog expansion

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. 

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... 

The continuity and momentum equations can be obtained for the fluid mixture as a single fluid using Chapman-Enskog expansion procedure in the nearly incompressible limit 43... 

The macroscopic equations for fluid flowing in porous media may be recovered by Taylor expansion and Chapman-Enskog expansion, which become... 

Corrections to the Smoluchowski equation (1.13) to order have been evaluated by Titulaer, using a Chapman-Enskog expansion. 

Alternative estimates of the transport coefficients can be obtained from the rigorous Chapman-Enskog expansion method of mono-atomic gases at low densities (e.g., [24] [25] [12] [61] (p 202) [28]). The transport coefficients deduced from the Chapman-Enskog kinetic theory with the rigid elastic spheres interaction model yield (e.g., [39] sect 8.2 [61], p 202) ... 

Hydrodynamic models are derived from the mesoscale model (e.g. the Boltzmann equation) using a Chapman-Enskog expansion in powers of the Knudsen number (Bardos et al., 1991 Cercignani et al, 1994 Chapman Cowling, 1961 Ferziger Kaper, 1972 Jenkins Mancini, 1989). The basic idea is that the collision term will drive the velocity distribution n towards an equilibrium function eq (i-e. the solution to C( eq) = 0), and thus the deviation from equilibrium can be approximated by n -i- Knui. From the... 

In the limit of small Knudsen number, the Chapman-Enskog expansion (Chapman, 1916 Enksog, 1921) of the elastic Boltzmann equation yields a first-order term for CTp of the form... 

Originated from the lattice gas automata (LGA), the LBM has been widely applied in simulating the rarefied gaseous flow in microchannel. Recently the LBM has also found success in applications to the liquid microflows, particularly the electrokinetic flows. For the continuous liquid, the Navier-Stokes equations can be recovered from the Boltzmann equation by Chapman-Enskog expansion or multiscale analysis, in which the Boltzmann equation is split into different scales for space and time variables. The lattice that an LBM operates is usually designated... 

The transport equations associated with the Chapman-Enskog expansion can be obtained from the general equations change given earlier. From the truncated expansion... 

Other parameters, including the lattice sound speed Cs and weight factor fj, are lattice structure dependent. For example, for a typical D2Q9 (two dimensions and nine lattice velocities see Fig. 1) lattice structure, we have tQ = 4/9, ii 4 = 1/9, f5 8 = 1/36, and = A /3Afi, where Ax is the spatial distance between two nearest lattice nodes. Through the Chapman-Enskog expansion, one can recover the macroscopic continuity and momentum (Navier-Stokes) equations from the above-defined LBM dynamics ... 

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. 

We know from experience with particnlar classes of problems that it is possible to write predictive, deterministic laws for the behavior (predictive over relevant space/time scales that are nseful in engineering practice) observed at the level of concentrations or velocity fields. Knowing the right level of observation at which we can be practically predictive, we attempt to write closed evolution equations for the system at this level. The closures may be based on experiment (e.g., through engineering correlations) or on mathematical modeling and approximation of what happens at more microscopic scales (e.g., the Chapman-Enskog expansion). 

The following mass transfer (species conservation) equation can be derived from lattice Boltzmann equation after Chapman-Enskog expansion [18] (also see Appendix 3). 

The coefficients tu, depend on the gas model and R is the specified gas constant. Since the Burnett equations are obtained by a second-order Chapman-Enskog expansion in Kn, they require second-order slip boundary condition. However, it may be noted that it has been observed that the second-order slip b.c. are inaccurate for Kn > 0.2. The Burnett equation can be used to obtain analytical/numerical solutions for at least a portion of the transition regime for a monoatomic gas. 

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