Example Chapman-Enskog reduction of kinetic theory to fluid mechanics

2 Example Chapman-Enskog reduction of kinetic theory to fluid mechanics 

In this illustration we take the level Li to be the level of kinetic theory (see Section 3.1.3), the level L2 the level of fluid mechanics (see Section 3.1.4), and the projection x2 = x2(xi) is given 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) 

The setting for our discussion is the space Mi equipped with the vector field (v.f.), and the mapping pr. 2 Mi — M2. Our problem is to equip the space M2 with a vector field v.f.)2 that is inherited from the vector field v.f.) 1 in Mi- The elements X of Mi are the distribution functions (7) the elements x2 of M2 are the hydrodynamic fields (85). 

We shall organize the Chapman-Enskog reduction into four steps Step 1) Initial suggestion M w C Mi for the slow manifold Msiow C Mi is made. It is a manifold that has a one-to-one relation with the space M2. We can regard it as an imbedding of M2 in Mi. Step 2) The vector field v.f.) 1 is projected on M w (i.e., v.f. jf 1 denoting the vector field v. /.) 1 attached to a point of M, is projected on the tangent plane of M ow a at point). The projected vector field is denoted by the symbol 

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