# Interdisciplinary Applied Mathematics

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M ж Kn -Re.

We first introduce the nondimensional variables, which are chosen in such a way that they express a perturbation from a Maxwellian distribution with Vi = 0.    Let    L,    to,    T0,    and    po    be    the    reference length,    time, temperature,

and pressure, and

Po = Po(RTo) 1,    (15.11)

where the reference state is the Maxwellian distribution with vi = 0, p = p0, and T = T0:

/o = (2тгДТ0)3/2 6XP (~2RT0) ‘    ^1512^

Then, the nondimensional variables are defined as follows:

 xi Xi/L, t = t/to, Ci = £i/(2RTo )1/2, Ф = f/fo — 1, £ ui = Vi/(2RTo)1/2, о 1 1—1 T = T/To — 1, «хз Pij pij /po &ij, «3» о 1 i—1 Qi = qi/po(2RTo)1/2, uwi = Vwi/(2RTo)1/2, Tw = Tw/To — 1, ^w = pw/po 1, о (pw — ^w + Tw + ^w Tw ) 1 i—1

We also redefine the Knudsen number to be consistent with Sone’s notation as follows:

(15.14)

/2RTq _ л/7г Ло РоАсЬ 2 L

Case with Re ^ 1

Following (Sone, 2002), we first analyze the small Reynolds number regime, for which we have

Re ^ Kn ^ 1,

and thus small deviations from equilibrium, i.e., M ^ 1. The steady-state Boltzmann equation in nondimensional abstract form is

= ^ЦФ),    (15.15)

OXi к

where the right-hand side denotes the collision operator. The boundary condition expressed also in abstract form is

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