# Interdisciplinary Applied Mathematics

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dxidxj ’

a2T1    276 a2P1

(15.21a)

dxidxj

71 dxidxj’,

Qio = 0,

Qi

f72pi0,

73 aP1

Qi2 §72pil + X-~T)’ Qi3 ~ |72pi2 +

73 aP2

(15.21b)

271 dxi ’    ‘“bi3    4 ,2~12 271 dxi

Here we have defined the strain tensor and the heat flux as

dTm

Si

ijm

f ^ dUjrn

dx

j

dxi

r*

Gim

dxi

The constants are as follows:

For a hard sphere molecular gas (Ohwada and Sone, 1992):

(15.22a)

71 = 1.270042427,    72 = 1.922284066,

73 = 1.947906335,    76 = 1.419423836.

• For the BKG model,

Remark: The governing equations are the steady-state Stokes and heat conduction equation at various orders with appropriately defined stress tensor and heat flux. In particular, in equation (15.21a) the term proportional to Sjm    corresponds to    viscous    stress    in    the    classical    fluid    dynamics,    the

higher-order term proportional to d2Tm/dxidxj is called thermal stress, and the    term    proportional    to    d2P1/dxidxj    is    the    pressure stress.    In    the

second and higher orders, the heat flux vector Qim depends on pressure gradient as well as on temperature gradient.

Boundary Conditions: We will ignore here the Knudsen correction, which is also derived asymptotically by (Sone, 2002). Instead, we will focus our attention to consistent boundary conditions, which have been presented in Chapter 2 from a different perspective.

The boundary conditions up to zeroth order are

 ui0 uwi0 — ° (15.23a) T0 — Tw0 — ° Скачать в pdf «Interdisciplinary Applied Mathematics» Н. В. Камышова; С.-Петерб. национал. исслед. ун-т информ. технологий, механики и оптики, Ин-т холода и биотехнологийМетки учебные пособия