# Interdisciplinary Applied Mathematics

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ф = фы (ZiUi > 0)    (15.16)

on the    boundary,    where    фы    may depend    on    ф    (Zini    < 0)    linearly.

We are looking for the asymptotic behavior of ф for small к following the method of analysis by (Sone, 1969; Sone, 1971) and (Sone and Aoki, 1994). In particular, expansions of the Grad-Hilbert form are sought for a moderately varying solution of equation (15.15), whose length scale of variation is on the    order    of    the    characteristic length    L of the    system    [Зф/dx.i    =    О(ф)],

in a power series of к:

ф = ф + фк + фк2 + ••• .    (15.17)

Corresponding to this expansion, the macroscopic variables u, ui, t, etc. are also expanded in к:

h = h + hk + hk2 + ••• ,    (15.18)

where h = u,ui,….

The asymptotic solution obtained by Sone is summarized below: A solvability condition yields the zeroth-order equation for the pressure, i.e.,

dPo

dxi

0,

(15.19)

and the following governing equations for different orders (m) of the expansion:

where

duj.

dxi

dPm+i

dxi д 2тт

7i-

д2г

im

ax

(m = 0, 1, 2,…)

(15.20a)

(15.20b)

(15.20c)

(15.20d)

and 7i is a constant related to the collision integral.

The corresponding stress tensor and temperature gradient vector of the Grad-Hilbert solution are

Pij 0    P0 &ij,    Pij 1    P1 pj + 71 Sij 0,

Pij2 = P2pj + 71Sij1 + 73

Pij3 = P3pj + 71 Sij 2 + 73