# Interdisciplinary Applied Mathematics

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The macroscopic variables ш, ui, t, etc. are also expanded in a similar fashion. The leading term ф0 of the expansion is the local Maxwellian, characterized by the leading terms ш0, ui0, and т0 of the five macroscopic variables, i.e., density,    flow    velocity,    and    temperature.    The    variables    u0,ui0,    and    т0

are governed by the Euler set of equations:

 дсооЩо = 0, (15.42a) dxi dui0 1    dP0 — = 0 (15.42b) i 2    dxi ’ XlQUjO + ~ <5 2 5 (15.42c) uoujO-r^p(ui0 + 2T°) — 0;

where

ш 0 = 1 + Ш0, Т0 = 1 + Т0, P0 = Ш0 + Т0 + Ш0Т0.    (15.43)

The higher-order macroscopic variables штт, and Tm (m > 1) are governed by inhomogeneous linear Euler-type equations.

In addition, still in leading order (Kn0) a correction is required in order to make the solution satisfy the kinetic boundary condition. This introduces boundary layer terms with the по-slip condition. To this end, the solution is expanded in powers of %/Kn (Sone et ah, 2000). Thus, the next-order term is С(а/Kn) rather than 0(Kn). The governing equations at this order are also perturbed Euler equations with boundary layer corrections but with slip boundary conditions representing velocity jump and thermal creep. In addition, at this order a Knudsen layer correction is required. The technical

details of this approach can be found in (Sone et al., 2000), where some earlier erroneous published results are rectified.