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 °

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