Interdisciplinary Applied Mathematics

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Case with Re « 1 and Small AT


The next step in Sone’s theory is to extend the asymptotic theory to the case the Reynolds number takes a finite value. In that case, the Mach number is of the same order of magnitude as the Knudsen number. Correspondingly, the deviation of the velocity distribution function from a uniform equilibrium state at rest is O(Kn). In addition, the temperature variation should be O(Kn)    for    the    theory    to be    valid.    The    solutions    are    obtained


from the steady-state nonlinear Boltzmann equation. Specifically, the nonlinear terms of the (perturbed) velocity distribution function ф cannot be neglected when powers of к are considered, since ф = O(k) by assumption. Sone uses the so-called S expansion to expand ф in the form


ф = фк + ф2к2 + ••• ,    (15.28)


looking for a moderately varying solution whose length scale of variation is on order of the characteristic length L of the system [дф/dxj, = О(ф)]. Here the series starts from the first-order term of к, since ф = О(к), and фт = O(1), in contrast to the previous Grad-Hilbert expansion. The macroscopic variables w, ui, t, etc. are also expanded in к in a similar fashion, i.e.,


h = Ьк + h2h2 + . ..,    (15.29)


where h = w, ui, ….


The governing equations obtained from the expansion at various orders (m) are:


First, the solvability pressure condition:


8P1 _


dxi


The first-order conservation equations are



0.



dui



l



dxi


dun


дт

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