# Interdisciplinary Applied Mathematics

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For a hard sphere molecular gas

74 = 0.635021,    75 =

and for the BKG model

74 = 75 = 1.

dxj2

0.961142,

(15.35)

(15.36)

(15.37)

Remark 1: The last terms of Pij3 and Qi3, i.e.,

73 [d2Ti/dxidxj — (1/3)(d2ri/dx|)Sij]

and

(7з/22uii/dxj,

are not present in the Newton and Fourier laws, respectively. The former is called thermal stress and its effect has been discussed in Section 5.1. The terms before these terms (terms with 74 and 75) are due to the dependence of the viscosity and thermal conductivity on the temperature of the gas.

Boundary Conditions: On the wall boundary the following boundary expansions at    different    orders are    valid.    For    expansions    of    first    order    we

have

uH uwil °    (15.38a)

Ti — Tw1 — 0.    (15.38b)

For terms of second order we have

(ui2    uwi2)tik0 Sij1ni tj + K1 Gifii    (15.39a)

Щ2Щ — 0,    (15.39b)

T2Tw2—diGiini,    (15.39c)

where the slip coefficient k0, K1, and d1 are the same as those in the linear theory.

Remark 2: The equations derived from the first order expansion are the incompressible Navier-Stokes equations with no slip. The next-order equations are the compressible Navier-Stokes equations valid for a slightly compressible fluid, but there is a difference. Specifically, the corresponding Navier-Stokes equations have 73 — 0. This difference is due to the thermal stress in Pij3 in equation (15.34). Following Sone’s analysis and introducing a new variable P3t, we obtain

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