Interdisciplinary Applied Mathematics

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Ps3 = Ps3 + g7з •    (15.40)

Then, we can incorporate the y3 term in the pressure term. So in the case of small temperature differences, we recover exactly the compressible Navier-Stokes equations by absorbing the thermal stress in the pressure term. This is not,    however,    true    for    large Reynolds    number    or    for    large    temperature

variations, as we will see below.

Remark 3: The above analysis illustrates that the slip boundary condition should be used in conjunction with the compressible Navier-Stokes equations. The combination of slip boundary conditions and the incompressible Navier-Stokes equations, which is often used because of convenience, is theoretically inconsistent (Aoki, 2001).

Remark 4: Unlike the Chapman-Enskog expansion, Sone’s asymptotic theory leads to a set of equations whose degree of differentiation does not increase with the order of approximation. In addition, Sone’s theory proves directly that the velocity distribution function depends on the spatial variables only through the five macroscopic variables and their derivatives and is not used as an assumption as done in the classical theory.

Case with M « 1

This case corresponds to a finite Mach number, which for a small Knudsen number that we study here implies that the Reynolds number can be very large, since Re ж 0(1/ Kn) ^ 1. The method of analysis of Sone is to first obtain the solution [дф/dxi = О(ф)] describing the overall behavior of the gas without limiting the size of ф. To this end, the Hilbert expansion in the Knudsen number is introduced in the form

ф = фо + фк + ••• .    (15.41)

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