Interdisciplinary Applied Mathematics

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/ H(vn)R(v^ v; x,t)dv = 1,

and also the scattering kernel satisfies a reciprocity condition; see (Sone, 2002; Cercignani, 1988).

Different types of scattering kernels express different gas-surface interactions and define the accommodation coefficient introduced earlier in Section 2.2.2. Assuming, as before, that av molecules are reflected diffusely and (1 — av) are reflected specularly, then the popular Maxwell’s scattering kernel, used exclusively before the 1960s, has the form

R(vr —► v; x) = (1 — av)6(vr — v + 2nvn) H—exp(—/3^n2), (15.10)

where f3w    involves    the    wall    temperature    Tw.    For    av    = 1    we    obtain    solely

diffuse scattering, which physically means that we have perfect accommodation, in the sense that the molecules “forget their past” and reemerge after the wall collision with a Maxwellian distribution function.

Other scattering kernels have been proposed by many authors, but the most popular one is the Cercignani—Lampis model (Cercignani and Lampis, 1971). It was obtained by other methods using Brownian motion and through an analogy with electromagnetic scattering. It introduces two accommodation coefficients: the tangential accommodation coefficient, 0at < 2, and the normal accommodation coefficient, 0 < an < 1. It has the form

R(v’ ^ v; x) =

2an°t(2 — at)в

x exp    —ft.

2 уП + (1— an)(v’n)2    „2 (vt (1at)vt)2

в

w

n

at (2at)

„Т i 02 2a/1<rtvnv’n

Xio I Pvj

n

where vn and vt are the normal and tangential components of the molecular velocities, and I0 is the zeroth-order modified Bessel function of the first kind.

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