Interdisciplinary Applied Mathematics

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Remark: Note that while the second term on the right-hand side of equation (2.20b) (thermal creep effect) appears to be O(Kn2), it actually corresponds to a first-order expansion (in Kn) of the Boltzmann equation. So both velocity jump and thermal creep are derived from an O(Kn) asymptotic expansion of the Boltzmann equation (Sone, 2002).


To determine fully the momentum and energy transport in microdomains, we need the following nondimensional numbers:


•    Reynolds number: Re =


2


•    Eckert number: Ec = CU^T, and


•    Knudsen number: Kn =


h


However, it is possible to introduce a functional relation for Knudsen number and Eckert number in terms of the Mach number


M


u


Vim


The Knudsen number can be written in terms of the Mach number and Reynolds number as


Kn






while the Eckert number can be written as


Ec=(7-1)^M2,    (2.22)


where AT is a specified temperature difference in the domain, and T0 is the reference temperature used to define the Mach number. Using these relations for Ec and M, the independent parameters of the problem are reduced to three:


Prandtl number Pr, Reynolds number Re, and Knudsen numb er Kn.

2.2.2 The Role of the Accommodation Coefficients


Momentum and energy transfer between the gas molecules and the surface requires specification of interactions between the impinging gas molecules and the surface. A detailed analysis of this is quite complicated and requires complete knowledge of the scattering kernels (see Section 15.4). From the macroscopic viewpoint, it is sufficient to know some average parameters in terms of the so-called momentum and thermal accommodation coefficients in order to describe gas-wall interactions. The thermal accommodation coefficient (aT) is defined by

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