Interdisciplinary Applied Mathematics

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For a hard sphere molecular gas, the continuum equations governing the macroscopic variables at their leading-order term are given as follows (Sone, 2002):



dP0    dP1


—- = 0, —-


dxi ’ dxi



0,



(15.44a)



d ujQUji dxj



(15.44b)



UjQUjl



дщ i


dxj



1 дР2    1 d


2 dxi    2 71 dxj



1    1 d


‘-q’



2 » Pq dxj



1    1 d


‘2



1/2



q



дт0дт0 dxi dx



tq



j


д2т0


dxidxj



duii    duji


—1——


dxj    dxi



2 ^


з ~d^7 ij)



Это


k


1 d2 Tq



2



дх: I 6ij



Sin



3 dx



дт0 _ 1 d f1/2 <9 T0



“ 272 ax



q



dxn



(15.44c)


(15.44d)


where


CTq = 1 + Wq, Tq = 1 + Tq, Pq = 1 + Pq = LTq Tq,


71 = 1.270042427, y2 = 1.922284066, = 1.947906335,77 = 0.188106.


In the present expansion, ui1 is the leading term of the flow velocity, since the case with uiQ = 0 is considered.


The boundary conditions are


TQ — тwQ,    ui1



1/2


Tq’ d TQ


Pq ® XJ



( Sij    ni nj ),



(15.45)


for the temperature and velocity, respectively.


Remark: The third and fourth terms on the right-hand side of equation (15.44c) are    due    to thermal    stress.    The    thermal    stress    contribution    is    not

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