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,


£


о


1


1—1


ui = Vi/(2RTo)1/2,


T = T/To — 1,


«хз


«3»


о


1


i—1


Pij pij /po &ij,


Qi = qi/po(2RTo)1/2,


uwi = Vwi/(2RTo)1/2,


Tw = Tw/To — 1,


^w = pw/po 1,


о


1


i—1


(pw^w + Tw + ^w Tw )

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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