Interdisciplinary Applied Mathematics

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(15.23b)


The boundary conditions up to first order are


(ui1 uwi1)ti — k0 Sij0ni ~^j + K1 Gi0ti


(15.24a)


ui1ni — °


(15.24b)


T1 Tw1d1Gi0ni •


(15.24c)

The boundary conditions up to second order are


(ui2 uwi2)ti



k0 Sij1nitj + a1



+ Я4



dSij0


dxr


dGi0


dxj



V -y О, / j I 0’21’ ‘S/’-yi i!> j I ,j | a^KijSjrQnrti nj ti + a5^Gi0ti + a6Kij G j0ti



— Ki



drw i dxi



ti



(15.25a)



Ui2Ui



bi



dSij0 , ,


—-щщпг + b2


dxr



dGi0


dxj



UiUj + 2nGioUi



(15.25b)



T2 — Tw2



diGiini



d4



dSjj о dxr



ninj nr





d5 K,Gi0ni:


where


к = ^(«1 + re2),    Kij = K-liiij + K2mimj,


uwi    uwi0 + uwi1k + *** :    Tw    Tw0 + Tw1k + *** •


The terms k1/L and k2/L are the principal curvatures of the boundary, where k1 or k2 is taken negative when the corresponding center of curvature lies on the side of the gas; the £i and mi are the direction cosines of the principal directions corresponding to k1 and k2, respectively. Also, /2RTQUWi (uWini = 0) and Tq(1 + tw) are the velocity and temperature of the boundary, respectively; k0,Ki,ai,…,a6,b1,b2, and d1id3,d4, and dare constants called slip coefficients, determined by the molecular model (e.g., hard sphere, BKG) and the reflection law on the boundary (e.g., diffuse reflection). The slip coefficients depend on the parameter (U0/kBT0except for a hard sphere molecular gas and the BKG model.

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